In this post, I share my notes from when I was reading the book Investments by Bodie ~2014. Please refer to the latest edition of the book for updated information.

  • Investments

    • Real Assets vs Financial Assets real assets and financial assets and how they contribute to the material wealth of a country. Real assets, such as land, buildings, machines, and knowledge, determine a country’s productive capacity and generate net income. On the other hand, financial assets, such as stocks and bonds, are merely securities that allow people to hold claims on real assets or income.

      Individuals can choose to invest their wealth in financial assets by buying securities. Companies then use this money to purchase real assets and share the income with the investors. This means that the returns for investors come from the income generated by real assets.

      There are three main types of financial assets: fixed income, equity, and derivatives. Fixed income securities, such as corporate bonds, promise a fixed stream of income, while equity, or common stock, represents a share in a corporation and its real assets. Derivatives, such as options or futures contracts, derive their value from the price of other assets, such as bonds or stocks. These securities are used to hedge risks or transfer them to other parties.

      In addition to these financial assets, corporations regularly engage in currency transfers and investors can directly invest in real assets, such as commodities traded on exchanges. Firms also use commodities and derivatives to manage their exposure to business risks.

    • Financial Markets and the Economy

      Financial markets play a crucial role in allocating capital in market economies through stock prices. The collective assessment of a firm’s current performance and future prospects reflected in stock prices can directly impact a firm’s ability to raise capital and encourage investment. Stock prices can also allow individuals to transfer purchasing power from their youth to older days by storing wealth in financial assets.

      Diverse financial instruments in financial markets also allow investors with different risk tolerance to bear risk. For instance, a firm like Ford can raise funds through both stocks and bonds, with more risk-tolerant investors choosing to buy stocks and less risk-tolerant ones choosing bonds. This allows the inherent risk of the investment to be borne by those most willing to bear it.

      The separation of ownership and management in large corporations also provides stability to the market. Ownership is distributed among a large group of individuals who elect a board of directors to manage the firm. However, there can be conflicts of interest between management and owners known as “agency problems.” To mitigate these issues, firms may tie executive compensation to the success of the company, have a board of directors that can force out underperforming management, and be subject to close monitoring by security analysts and institutional investors. Unhappy shareholders can also launch a proxy contest to elect a new board, or the firm may be at risk of takeover by other companies.

      Transparency in the market is also crucial for informed decision-making by investors. The Sarbanes-Oxley Act of 2002 aimed to tighten rules of corporate governance and prevent misleading information. The act requires corporations to have more independent directors, tighter accounting standards, and increased oversight of public companies. These measures aim to promote ethical and responsible corporate behavior.

    • The Investment Process

      The process of investing involves building a portfolio, a collection of investment assets, which can be updated over time by selling existing securities and using the proceeds to purchase new securities. Investors make two key decisions in constructing their portfolios: asset allocation and security selection.

      Asset allocation involves deciding the proportion of one’s portfolio to allocate to different asset classes, such as stocks, bonds, or safe assets like bank accounts or money market securities. This decision is a crucial factor in determining the overall risk and return of the portfolio.

      Security selection, on the other hand, involves choosing specific securities within each asset class to hold in the portfolio. This requires security analysis to determine the value of the individual securities.

      There are two main approaches to portfolio construction: top-down and bottom-up. The top-down approach starts with the asset allocation decision and then moves on to security selection. The bottom-up approach focuses more on security selection and less on asset allocation, constructing the portfolio from securities that seem attractively priced.

    • Markets Are Competitive

      Financial markets are constantly monitored by intelligent and well-funded analysts, meaning that there are no easy wins or “free lunches”. This idea has two important implications:

      The Risk-Return Trade-Off: Investments come with a degree of risk and investors want to earn the highest returns they can. However, if an investment offers higher returns without taking on extra risk, many investors will jump on the opportunity, driving up the price and reducing the expected return. This means that there is a trade-off between risk and return in the securities market. Higher-risk assets offer higher expected returns, while lower-risk assets have lower expected returns.

      Efficient Markets: The no-free-lunch proposition also implies that it’s rare to find bargains in the security markets. The financial markets process information about securities quickly and efficiently, meaning that security prices usually reflect all the information available to investors about their value. This is known as the efficient market hypothesis. If the markets are indeed efficient, it might be better to follow a passive investment strategy rather than trying to actively identify mispriced securities.

    • Financial Market Players

      Financial markets are made up of three major players: firms, households, and governments. Firms need capital to fund their investments in plant and equipment and they raise this capital by issuing securities. Households, on the other hand, are the net suppliers of capital as they purchase these securities issued by firms. Governments can play both the role of borrowers and lenders, depending on their tax revenue and expenditures.

      However, firms and governments do not sell their securities directly to the public. Instead, they hire financial intermediaries, such as banks, investment companies, insurance companies, and credit unions, to act as the go-between. Financial intermediaries issue their own securities to raise funds to purchase the securities of other corporations. They offer several advantages, including pooling the resources of many small investors, achieving significant diversification, and building expertise through the volume of business they do.

      Investment bankers also play a crucial role in the financial market as they specialize in services for businesses, such as issuing securities, advising on prices and interest rates, and handling the marketing of securities in the primary market. Meanwhile, start-up companies rely on venture capital and private equity investments to fund their operations. These investments are usually made by dedicated venture capital funds, wealthy individuals known as angel investors, or institutions like pension funds.

  • Asset Classes and Financial Instruments

    • Money Market

      The money market is a subsector of the fixed income market that includes highly marketable short-term debt securities. Many of these securities are traded in large denominations, making them unavailable to individual investors. To access these securities, investors can turn to mutual funds, which pool the resources of many individuals to purchase a diverse range of money market securities on their behalf.

      • T-bill

        One of the most marketable money market instruments is the Treasury bill (T-bill), which is issued by the U.S government. T-bills are sold at a discount from their maturity value and provide a payment equal to the face value of the bill at maturity. They are issued with maturities of 4, 13, 26, and 52 weeks and can be purchased at auction or in the secondary market. T-bills are highly liquid, have little risk, and are usually denominated in $100, $10,000, or even $100,000. The income earned from T-bills is exempt from state and local taxes.

        T-bills are typically quoted using the bank-discount method, which calculates the yield as a fraction of the face value and assumes a 360-day year. The bid price is the price at which a dealer is willing to purchase a T-bill, while the ask price is the price at which the T-bill is offered for sale. The difference between the bid and ask price is the bid-ask spread, which is the source of the dealer’s profit. The bond equivalent yield calculates the yield based on a 365-day year and is found by dividing the face value by the ask price and multiplying by 365/156.

      • Ceritficate of Deposti

        A Certificate of Deposit (CD) is a fixed-term deposit with a bank. The investor cannot access the funds until the end of the specified term, at which point the bank pays both the interest and the principal value to the depositor. CDs are typically issued in denominations of $100,000 or more and are negotiable, meaning that the investor can sell the certificate before its maturity if necessary.

        Short-term CDs are highly marketable and are easily sold, while longer-term CDs with a maturity of three months or more become less marketable. The Federal Deposit Insurance Corporation considers CDs to be bank deposits and insures them up to $250,000.

      • Commercial Paper

        Companies that are well established often issue their own short-term debt notes, known as commercial papers, as an alternative to borrowing from banks. To ensure that they have the necessary funds to pay off the commercial paper at maturity, they may secure backing from a bank or a line of credit. The maturities of commercial papers typically range up to 270 days, with most maturities being less than 1 or 2 months and issued in multiples of $100,000. The Securities and Exchange Commission requires registration for maturities longer than 270 days, which is almost never done.

        Individuals can invest in commercial papers through money market mutual funds. Because the firm’s performance can be monitored and predicted over a short term, such as 1 month, commercial papers are considered safe assets, often issued by nonfinancial firms. However, financial firms such as banks may issue asset-backed commercial papers to raise funds for investing in other assets, such as subprime mortgages. These assets were used as collateral for the commercial paper, which led to difficulties starting in 2007 when subprime mortgagors began defaulting and the banks were unable to issue new commercial paper to refinance their positions as the old paper matured.

      • Banker’s Acceptance

        Banker’s Acceptance is a type of financial instrument in which a bank customer orders the bank to pay a specific amount at a future date, usually 6 months. The bank’s endorsement of the order signifies its responsibility for the ultimate payment to the holder of the acceptance. These instruments are often used in foreign trade to ensure payment when the creditworthiness of the counterparty is unknown. Upon endorsement, the Banker’s Acceptance can be traded in the secondary market. They are considered safe assets as they are backed by the credit of the bank rather than the borrower’s. Banker’s Acceptances are sold at a discount from the face value and are mostly used in international trade.

      • Eurodollars

        Eurodollars are U.S. dollar-denominated deposits held in foreign branches of American banks. These banks are able to evade regulations imposed by the Federal Reserve by operating outside of its jurisdiction. Typically, Eurodollars are large deposits with short maturities of less than 6 months. Similar to domestic certificates of deposit, Eurodollar CDs are issued by a non-U.S. branch of a bank, usually located in London. Due to their lower liquidity and higher risk, Eurodollar CDs offer higher yields compared to domestic CDs. While Eurodollar bonds are also an option, they are excluded from the money market due to their longer maturities.

      • Repos and Reverses

        Dealers in government securities often utilize repurchase agreements, also referred to as repos or RPs, as a short-term borrowing solution, typically for just one night. In a repo transaction, the dealer sells government securities to an investor at an agreed upon price, with a commitment to buy back the securities the next day at a slightly higher price, which represents the overnight interest. This operates similarly to a one-day loan from the investor, with the government securities serving as collateral.

        For longer borrowing needs, a term repo operates in a similar manner, with the term of the loan lasting from 30 days or more. Repos are considered very secure as they are backed by government securities.

        A reverse repo, on the other hand, is the opposite of a repo. In this transaction, the dealer buys securities from an investor, with an agreement to sell them back at a higher price in the future.

      • Federal Funds

        The Federal Reserve System, also known as “the Fed,” is a network of banks that work together to ensure the stability and reliability of the US financial system. As part of this system, each member bank is required to maintain a minimum balance in a reserve account with the Fed. These funds in the bank’s reserve account are called “Federal funds” or “fed funds.”

        At any given time, some banks may have more funds in their reserve accounts than required, while others may have a shortage. Big banks in New York tend to have a shortage of fed funds. In the Federal Funds market, banks with excess funds lend to those with a shortage. These loans are overnight transactions and are made at a rate of interest called the federal funds rate.

        The fed funds was initially created as a way for banks to transfer balances to meet their reserve requirements. However, today, the market has evolved and many large banks use federal funds as a component of their sources of income. The fed funds rate is the rate of interest for short-term loans among institutions. Although most investors cannot participate in this market, the fed funds rate is still of great interest as it is seen as a key indicator of monetary policy.

      • Broker’s Call

        Individuals who purchase stocks using margin borrow part of the funds from their broker. The broker may then borrow these funds from a bank, agreeing to repay the loan on demand if the bank requests it. The interest rate on these loans is typically 1% higher than the rate on short-term Treasury bills (T-bills).

      • The LIBOR Market:

        The London Interbank Offered Rate (LIBOR) is a benchmark interest rate at which large banks in London are willing to lend money to one another. It is widely used as a reference rate in the money market and is tied to multiple currencies, including the US dollar. With trillions of dollars and various derivative assets tied to it, the LIBOR is a crucial reference rate in the financial industry.

      • Yiels on Money Market Instruments

        Most money market instruments are considered low-risk, but it’s important to note that they are not completely risk-free. While these securities typically offer yields higher than default-free T-Bills, investors usually prioritize liquidity, leading them to opt for lower yield options such as T-Bills. As shown in Figure 1, there is typically a gap between CD yields and T-Bills, and this gap widens during times of financial crisis. Similarly, the difference between LIBOR rates and T-Bills also increases during periods of financial stress.

  • The Bond Market

    The bond market is a market for long-term debt instruments, as opposed to the short-term debt instruments found in the money market. The bond market includes Treasury notes, Treasury bonds, Corporate bonds, Municipal bonds, Mortgage securities, and federal agency debts. These debt instruments are often referred to as fixed-income capital market securities because they generally offer a fixed stream of income. However, in some cases, the formulas that determine the income stream may result in a flow of income that is far from fixed. As a result, it is more appropriate to refer to these securities as debt instruments or bonds.

    • Treasury Notes and Bonds

      The U.S. government borrows money through the sale of Treasury Notes and Treasury Bonds. T-Notes have maturities of up to 10 years, while T-Bonds have maturities ranging from 1 to 30 years. They are usually quoted in denominations of $100 or $1000, with the latter being more common. Both T-Notes and T-Bonds make semiannual coupon payments, representing the interest earned on the investment.

      An example of a T-Bill entry can be found in financial publications such as the Wall Street Journal, where its maturity date, coupon rate, bid price, asked price, change in price, and yield to maturity are displayed. The yield to maturity is calculated by determining the semiannual yield and then doubling it, instead of compounding it for two half-year periods. This results in an annual percentage rate (APR) rather than an effective annual yield.

      It’s important to note that T-Bills are considered low-risk investments, but are not risk-free. The prices are quoted as a percentage of par-value, usually $1000, and can fluctuate based on market conditions.

    • Inflation-Protected Treasury Bonds

      Many governments, including the U.S., offer bonds known as Inflation-Protected Treasury Bonds or TIPS as a way for citizens to hedge against inflation. These bonds are linked to the Consumer Price Index and their principal amount is adjusted accordingly. The yield on TIPS bonds represents real, inflation-adjusted rates and is a great starting point for building a low-risk investment portfolio.

    • Federal Agency Debt

      Government agencies often issue their own securities to finance their operations, particularly in sectors that may not receive adequate credit through traditional private sources. Some of the major mortgage-related agencies include the Federal Home Loan Bank (FHLB), Federal National Mortgage Association (FNMA or Fannie Mae), Government National Mortgage Association (GNMA or Ginnie Mae), and the Federal Home Loan Mortgage Corporation (FHLMC or Freddie Mac). The FHLB, for example, raises funds by issuing securities and lending the money to savings and loan institutions, which then lend it to individuals for home mortgages. Although the debt of these federal agencies is not explicitly insured by the government, it was traditionally assumed that the government would assist an agency in the event of default. This assumption was validated in September 2008 when Fannie Mae and Freddie Mac experienced severe financial difficulties.

    • International Bonds

      The international capital market, largely centered in London, provides opportunities for firms to borrow and for investors to purchase bonds from foreign issuers. A Eurobond is a bond denominated in a currency other than that of the country in which it is issued. For example, a bond denominated in dollars and sold in Britain would be considered a Eurobond. On the other hand, Yankee bonds are bonds issued by non-U.S. companies but denominated in dollars and sold in the U.S., while Samurai bonds are bonds issued by non-Japanese companies but denominated in yen and sold in Japan.

    • Municipal Bonds

      Municipal bonds, also known as “munis,” are debt securities issued by state or local governments. They offer interest income that is exempt from federal and, in most cases, state and local taxes, making them attractive to many investors. There are two types of municipal bonds: General Obligation bonds, which are backed by the full faith and credit of the issuer, and Revenue Bonds, which are backed by the revenue from a specific project or agency.

      Short-term Tax Anticipation Notes and long-term bonds are both available in the municipal bond market, with maturities ranging from a few months to 30 years. The tax-exempt status of municipal bonds is the main advantage that attracts investors, but to determine if they are a better investment option than taxable bonds, it’s important to consider an investor’s tax bracket and the equivalent taxable yield. The tax bracket at which an investor is indifferent between taxable and tax-exempt bonds can also be calculated by determining the yield ratio.

      In general, the higher an investor’s tax bracket, the more attractive tax-exempt municipal bonds become. However, it’s important to consider the risk profile of the bond and the issuer’s creditworthiness before making any investment decisions.

      The key feature of municipal bonds is their tax exemption status. Since investors are not paying any form of federal or local taxes, they are willing to accept lower rates on these securities.

      How to compare between taxable and tax-exempt bond: let \(t\) denote the investor’s combined federal plus state tax bracket and \(r\) denote the total before tax rate of return available on taxable bond, then \(r(1-t)\) is the after-tax rate available on those securities. If this value exceeds the rate on municipal bond rate \(r_m\), then the investor does better holding the taxable bond. Another way to determine te interest rate on taxable bond that would be nevessary to provide an after-tax return equal to that of municipals. This value would be equal to \(r = r_m/(1-t)\). This is called equivalent taxable yield. Note that in order to find \(t\), a simple way is to add federal plus local tax rate together as \(t = t_{federal} + t_{state}\) . A more precise approach would recognize that the state taxes are deductible at the federal level. You owe federal taxes only on income net of state taxes. Therefore for every dollar of income, your after-tax proceeds would be \((1- t) (1-t_{federal})\times(1-t_{state})\). This way you could calculate combined federal and local taxes.

      Note that the more your tax bracket is the more you are interested in holding municipal bonds that are tax-exempt. Because for every tax-exempt rate you need to get more yield for taxable yield.

      Another, factor that can be computed is the tax bracket at which investors are indifferent between taxable and tax-exempt bonds. This bracket could be found as \(t = 1 - \frac{r_m}{r}\). The yield ratio, \(r_m/r\) is a key determinant of the attractiveness of municipal bonds. The higher the ratio, the lower the cutoff tax bracket, and then more individuals will prefer to hold municipal bonds.

    • Corporate Bonds

      Corporate bonds are a type of debt securities issued by private companies to raise funds from the public. They work similarly to Treasury bonds, paying out semi-annual coupons and returning the face value to the bondholder at maturity. However, they are riskier than Treasury bonds due to the creditworthiness of the issuing company.

      There are several types of corporate bonds, including secured bonds, which have specific collateral backing them in case of bankruptcy, unsecured bonds or debentures, which have no collateral, and subordinated debentures, which have a lower priority claim to the firm’s assets in the event of bankruptcy.

      Additionally, some corporate bonds come with options attached. Callable bonds give the company the option to repurchase the bond from the holder at a predetermined price. Convertible bonds allow the bondholder to convert each bond into a specific number of shares of stock.

    • Mortgages and Mortgage-Backed Securities

      Investing in mortgage-backed securities, which represent ownership or obligation claims in a pool of mortgages, has become a common aspect of the fixed-income market. Traditionally, these securities, also known as pass-throughs, were made up of conforming mortgages that met strict underwriting guidelines set by organizations such as Fannie Mae or Freddie Mac. However, in the lead up to the financial crisis, a significant amount of subprime mortgages were bundled and sold as mortgage-backed securities. These loans, made to financially weaker borrowers, were encouraged by the government to increase housing affordability for low-income households. However, they ended up causing huge losses, not just for banks, hedge funds and other investors, but also for Fannie Mae and Freddie Mac, which incurred billions of dollars in losses on the subprime mortgage pools they purchased.

  • Equity Securities

    • Common stocks

      Common stocks, also known as equity securities or equities, represent ownership in a corporation. When you own a share of common stock, you are entitled to a vote in corporate governance matters and a share of the company’s financial benefits. Some corporations issue two classes of common stock, one with voting rights and one without. The one with restricted voting rights may sell at a lower price.

      The corporation is governed by a board of directors elected by shareholders. The board meets several times a year to select managers who run the day-to-day operations of the company and ensure that it acts in the best interests of shareholders. Shareholders who cannot attend the annual meeting can vote by proxy, giving another party the power to vote on their behalf. Management typically solicits proxies from shareholders and typically receives a large majority of proxy votes.

      There are several ways to make sure that management is following the goal of shareholders want, otherwise agency problems arises. There are several mechanisms that alleviate the agency problem, such (i) compensation schemes that link the success of the manager to that of the firm; (ii) oversight by the board of directors as well as outsiders such as security analysts, creditors, or large institutional investors; (ii) the threat of a proxy contest in which unhappy shareholders attempt to replace the current management team; or (iii) the threat of a takeover by another firm

      The common stock of most large corporations can be bought and sold freely on one or more stock exchanges. A corporation whose stock is not publicly traded is said to be closely held. In most of closely held corporations, the owners of the firm also take an active role in its management. Therefore take overs are generally not a concern.

      • Characteristics of Common Stock

        The two most important characteristics of common stock as an investment are (i) residual claim and limited liability.

        Residual claim means that stock holders are the last ones in line among all those who have a claim on the assets and income of the corporation. In liquidation of the firm’s assets the shareholders have a claim to what is left after all other claims, such as tax authorities, employees, suppliers, bondholders, and other credits that have been paid. For a firm not in liquidation, shareholders have claim to the part of operating income left over after interest and taxes havebeen paid. Management can pay either pay this residual as cash dividends to the shareholders or reinvest in the business to increase the value of shares.

        Limited liability means that the most shareholders can loose in the event of failure of the corporation is their original investment. Unlike owners of unincorporated businesses, whose creditors can lay claim to the personal assets of the owner (like house, car, or furniture), corporate shareholders may at worst have worthless stock. They are not personally liable for the firm’s obligation.

      • Stock Market Listings

        The New York Stock Exchange (NYSE) is one of the platforms where investors can buy and sell stocks. A sample listing of a stock, such as General Electric (GE), might look like this:

          - The closing price of GE stock is 19.72 dollars.
  - The net change from the previous trading day is +0.13 dollars.
  - On this day, 45.3 million shares of GE were traded.
  - The 52-week high and low prices of the stock are 21.00 dollars and 14.02 dollars, respectively.
  - The last quarterly dividend payment was 0.17 dollars per share (0.68 dollars divided by 4).
  - The annual dividend yield is 3.45%, calculated as 0.68 dollars divided by 19.62 dollars.
  - The price-earnings ratio is 16.01, which is the ratio of the current stock price to last year's earnings per share.
  - The stock has increased by 10.11% since the beginning of the year.

        It’s important to note that the dividend yield is part of the return on a stock investment. If a company pays a lower dividend, it’s expected to offer greater prospects for capital gains. The price-earnings ratio is a metric that tells us how much investors must pay for each dollar of earnings generated by the firm. A lower P/E ratio is generally seen as better.

    • Preferred Stock Preferred stock is a hybrid investment that combines features of both equity and debt. Like a bond, it offers a fixed, annual income payment and does not grant the holder voting rights in the management of the company. However, the company is not obligated to pay the preferred dividends and they are usually cumulative, meaning any unpaid dividends must be paid before common stock dividends are paid. On the other hand, the firm has a contractual obligation to make interest payments on its debt and failure to do so may result in bankruptcy proceedings.

      The tax treatment of preferred stock is also different than bonds, as the payments are considered dividends instead of interest and are not tax-deductible for the company. However, corporations may exclude 70% of dividends received from domestic corporations when computing their taxable income, making preferred stock a desirable fixed-income investment for some corporations. For individual investors who cannot use the 70% tax exclusion, preferred stock yields may not be as attractive as other available assets.

      Preferred stock can also be callable by the issuing company or convertible into common stock at a specified ratio. Adjustable-rate preferred stock links its dividend to current market interest rates, similar to adjustable-rate bonds.

    • Depository Receipts

      American Depository Receipts (ADRs) are a convenient way for US investors to own shares in foreign companies. These certificates, traded in US markets, represent ownership in a foreign company’s shares. Each ADR can correspond to a fraction of a foreign share, one share, or multiple shares. ADRs were introduced to simplify the process of complying with US security registration requirements for foreign firms, making it easier for US investors to invest in international companies.

  • Stock and Bond Market Indexes

    Stock Market Indexes are a measure of the overall performance of the stock market. The most well-known index is the Dow Jones Industrial Average, which tracks the performance of 30 large corporations. Additionally, there are foreign market indexes such as the Nikkei Average in Tokyo and the Financial Times Indexes in London, providing a glimpse into the performance of stock markets around the world.

    • Dow Jones Industrial Average

      Aka DJIA is the average of 30 “blue-chip” corporations that is computed since 1896. DJIA covered only 20 stocks untill 1928 and then chaged to 30 stocks.

      Originally, the DJIA was calculated as the average price of the stocks included in the index! Just adding the prices of these 30 stocks and dividing by 30. The percentage change in DJIA would then be the percentage change in the average price of the 30 shares. This means that the percentage change in DJIA measures the return (excluding dividends) on a portfolio that invests one share in each of the 30 stocks in the index. The value of such portfolio is the sum of the 30 prices (one share of each stock). Since the percentage change in the average of the 30 prices is the same as the percentage change in the sum, the index and the portfolio would have the same percentage change!

      Since DJIA corresponds to a portfolio that holds one share of each component stock, the investment is proportional to the company’s share price! Therefore it is a price-weghted average.

      The DJIA is now (Apr 2017) at a level of 20,000 and it is supposed to be average of only 30 stocks! Actually DJIA no longer equals to the average price of the 30 stocks, since the averaging procedure is adjusted whenever a stock splits or pay a stock dividend of more than 10%, or /when the company in the list is replaced by another firm/. When these events occur, the divisor used to compute the average is adjusted so to leave the index unaffected.

      Company Exchange Symbol Industry Date Added
      3M NYSE MMM Conglomerate 1976-08-09
      American Express NYSE AXP Consumer finance 1982-08-30
      Apple NASDAQ AAPL Consumer electronics 2015-03-19
      Boeing NYSE BA Aerospace and defense 1987-03-12
      Caterpillar NYSE CAT Construction and mining equipment 1991-05-06
      Chevron NYSE CVX Oil & gas 2008-02-19
      Cisco Systems NASDAQ CSCO Computer networking 2009-06-08
      Coca-Cola NYSE KO Beverages 1987-03-12
      DuPont NYSE DD Chemical industry 1935-11-20
      ExxonMobil NYSE XOM Oil & gas 1928-10-01
      General Electric NYSE GE Conglomerate 1907-11-07
      Goldman Sachs NYSE GS Banking, Financial services 2013-09-20
      The Home Depot NYSE HD Home improvement retailer 1999-11-01
      IBM NYSE IBM Computers and technology 1979-06-29
      Intel NASDAQ INTC Semiconductors 1999-11-01
      Johnson & Johnson NYSE JNJ Pharmaceuticals 1997-03-17
      JPMorgan Chase NYSE JPM Banking 1991-05-06
      McDonald’s NYSE MCD Fast food 1985-10-30
      Merck NYSE MRK Pharmaceuticals 1979-06-29
      Microsoft NASDAQ MSFT Software 1999-11-01
      Nike NYSE NKE Apparel 2013-09-20
      Pfizer NYSE PFE Pharmaceuticals 2004-04-08
      Procter & Gamble NYSE PG Consumer goods 1932-05-26
      Travelers NYSE TRV Insurance 2009-06-08
      UnitedHealth Group NYSE UNH Managed health care 2012-09-24
      United Technologies NYSE UTX Conglomerate 1939-03-14
      Verizon NYSE VZ Telecommunication 2004-04-08
      Visa NYSE V Consumer banking 2013-09-20
      Wal-Mart NYSE WMT Retail 1997-03-17
      Walt Disney NYSE DIS Broadcasting and entertainment 1991-05-06

      In order to better understand how DJIA works, lets suppose two stocks, ABC and XYS shwon in the following table. If we are introducing the DJIA for this table, the inital index and the final indexes are Initial Index Value \(= (25+100)/2 = 62.5\) and Final Index Value \(= (30+90) /2 = 60\). So at the end, we have Percentage Change \(= -2.5/62.5= -4\)%. Note that this does not represent the total market change, since the total market chages as Percentage Change in total market \(= \frac{690-600}{600} = +15\%\). This is because DJIA is price averaged, instead of the contribution that the stock is making to the market due to number of it shares in the market.

      Stock Initial Price Final Price Shares Initial Value of Outstanding Stock Final Value of Outstanding Stock
      ABC $25 $30 20 m $500 m $600 m
      XYZ $100 $90 1 m $100 m $90 m
      Sum       $600 m $690 m

      Now, suppose that the company XYZ splits it shares to two. The new values are shown in the following table. DJIA, will find a new divisor \(d\), such that the DJIA does not change afterward, as follows \begin{equation} \frac{\text{Price of ABS} + \text{Price of XYZ}}{d} = 62.5 \end{equation} Note that \(62.5\) is the initial DJIA that we had. The new devisor is then obtained as \(d = 1.2\). With this new divisor the final DJIA would be \((30 + 45)/1.2 = 62.5\), the same value at the start of the year. This would mean that the rate of return for such DJIA portfolio is zero. Note that still the whole market with these new values still have \(15\%\) return, if the portfolio was weigthed on the number of shares.

      Stock Initial Price Final Price Shares Initial Value of Outstanding Stock Final Value of Outstanding Stock
      ABC $25 $30 20 m $500 m $600 m
      XYZ $50 $45 2 m $100 m $90 m
      Sum       $600 m $690 m

      The same story happens when we replace a firm. The denominator is updated to leave the average unchanged. By 2017, the divisor of the DJIA has fallen to a value of \(0.146\). Since, DJIA is the average of small number of firms, representing the broad market, they change the composition every so often to reflect the changes in economy.

    • Standard & Poor’s Indexes* or S&P 500

      The S&P 500 is a market-value-weighted index of 500 firms, which means that the weight of a firm in the index is proportional to its market value, calculated as the product of its stock price and number of shares outstanding. The index is computed daily by adding up the total market value of the 500 firms and comparing it to the total market value on the previous day of trading. An investment in the index is equivalent to holding a portfolio of all 500 firms in proportion to their market value, excluding cash dividends.

      Most modern indexes use a modified version of market-value weights, known as free float market value, which only considers the value of shares that are freely available for investors to trade. This distinction is particularly important in Japan and Europe where a larger fraction of shares are held by non-tradable portfolios.

      Both market-value-weighted and price-weighted indexes are straightforward to follow as investment strategies. A market-value-weighted index perfectly tracks the capital gain of a portfolio comprised of shares in each component firm in proportion to its market value. A price-weighted index tracks the returns of a portfolio comprised of an equal number of shares of each firm.

      Investors can easily invest in market indexes by purchasing mutual funds or exchange-traded funds (ETFs). Index funds yield a return equal to the index, making them a low-cost investment strategy for equity investors. ETFs are portfolios of shares that can be bought and sold as a single unit, and there are a variety of ETFs available, from broad global market indexes to narrow industry indexes.

      In addition to the S&P 500, Standard and Poor’s also publishes other stock indexes including the 400-stock Industrial Index, the 20-stock Transportation Index, the 40-stock Utility Index, and the 40-stock Financial Index.

    • Other U.S. Market-Value Indexes

      The New York Stock Exchange (NYSE) publishes a composite index of all stocks listed on the NYSE, which is market-value weighted. This composite index includes subindexes for industrial, utility, transportation, and financial stocks. These indexes provide a broader view of the stock market than the S&P 500.

      The National Association of Securities Dealers (NASD) publishes an index of more than 3,000 firms traded on the NASDAQ market.

      The Wilshire 5000 index is considered the ultimate US equity index, as it tracks the market value of essentially all actively traded stocks in the US. It actually includes more than 5000 stocks. The performance of these indexes can be found in The Wall Street Journal.

    • Equally Weighted Indexes

      Market performance can be measured using an equally-weighted average of the returns of each stock in an index. This averaging technique gives equal weight to each stock’s return, which corresponds to an implicit investment strategy of putting equal dollar amounts into each stock. This is different from both price weighting, which requires an equal number of shares of each stock, and market-value weighting, which requires investments in proportion to a stock’s outstanding value.

    • Foreign and International Stock Market Indexes

      The growth of financial markets worldwide has led to the creation of indexes to track their performance. Some examples of these indexes include the Nikkei in Japan, the FTSE in the UK, the DAX in Germany, the Hang Seng in Hong Kong, and the TSX in Canada. Morgan Stanley Capital International (MSCI) is a leader in the creation of international indexes and calculates over 50 country indexes and several regional indexes.

    • Bond Market Indicators

      Just as stock market indexes provide insight into the performance of the overall stock market, bond market indicators measure the performance of various categories of bonds. The three most well-known groups of bond market indexes are those of Merrill Lynch, Barclays (previously the Lehman Brothers index), and Salomon Smith Barney (now part of Citigroup).

      However, there is a major challenge with bond market indexes: accurately computing the true rates of return on many bonds. This is because the infrequency of bond trades makes it difficult to obtain reliable, up-to-date prices. As a result, some prices must be estimated using bond valuation models, which may not reflect the true market values.

    • Derivative Markets

      The derivative market consists of two main instruments: futures and options. These financial instruments provide payouts that are based on the values of underlying assets, such as commodity prices, bond and stock prices, or market index values. Because the value of these instruments depends on the value of other assets, they are referred to as derivative assets.

      • Options

        A call option gives its holder the right to purchase an asset, such as stock, at a specified exercise price, also known as the strike price, before or on a specified expiration date. For example, a July call option on IBM stock with an exercise price of $180 allows the holder to buy IBM stock for $180 at any time until the expiration date in July. Each option contract covers the purchase of 100 shares, but quotations are made on a per-share basis. The holder of the call option does not have to exercise it, and it will only be profitable to do so if the market value of the asset is higher than the exercise price. When the market price exceeds the exercise price, the option holder can “call away” the asset for the exercise price and receive a payoff equal to the difference between the stock price and the exercise price. If not exercised before the expiration date, the option simply expires and becomes worthless. Call options are bullish investment vehicles that provide greater profits when stock prices increase.

        In contrast, a put option gives its holder the right to sell an asset for a specified exercise price before or on a specified expiration date. For example, a July put option on IBM with an exercise price of $180 allows the holder to sell IBM stock for $180 to the put writer at any time before the expiration date in July, even if the market price of IBM is lower than $180. Profits on put options increase when the asset decreases in value, unlike call options that benefit from an increase in the asset’s value. The put option is exercised only if its holder can deliver an asset worth less than the exercise price in return for the exercise price.

        It is important to note that the price of a call option decreases as the exercise price increases. For instance, a call option for May 5, 2017 with an exercise price of $144 is only $3.13. This makes sense because the right to purchase a share at a higher price is less valuable.

        On the other hand, the price of a put option increases with the exercise price. For example, the right to sell APPL shares for $130 is $0.31, while the right to sell shares for $143 is $3.07.

        Another factor that affects option prices is the expiration date. Option prices increase as the expiration date approaches. For instance, a call option for $130 a share is worth $17.75 if the expiration date is in November, compared to $13.10 if the expiration date is in May.

      • Future Contracts

        A futures contract requires delivery of an asset (or its cash value) on a specified delivery or maturity date, for an agreed-upon price, called the futures price, to be paid at contract maturity.

        The trader who holds the long position commits to purchasing the asset on the delivery date, while the trader who takes the short position commits to delivering the asset at contract maturity.

        Table 8 shows some futures contracts for crude Brent oil listed on the Chicago Mercantile Exchange. Each contract calls for delivery of 1,000 barrels of oil. The table lists the prices for contracts expiring on various dates, with the first row being the nearest or front contract with a maturity date of June 2017. For example, the most recent price was $54.89 per barrel, an increase of $0.3 from the previous day’s close. The table also shows the opening price, as well as the high and low prices for the trading day. The volume column shows the number of contracts traded that day, but there is no column for “Open Interest” which represents the number of outstanding contracts.

        Month Last Change Prior Settle Open High Low Volume
        JUN 2017 55.19 0.3 54.89 54.75 56.07 54.57 33006
        JUL 2017 55.55 0.38 55.17 55.08 56.23 54.86 12892
        AUG 2017 55.64 0.29 55.35 55.37 56.46 55.15 7571
        SEP 2017 55.65 0.2 55.45 56.2 56.27 55.45 6660

        The trader holding the long position in a futures contract profits from price increases. For example, if at contract maturity, the price of oil is $55.50 per barrel and the trader entered the contract at a future price of $55.19 per barrel on June 2017, they would pay the agreed-upon price of $55.19 for each barrel of Brent oil, which would then cost $55.50 at contract maturity. In this case, the profit for the long position would be $2100 (1000 x ($55.50 - $55.19)). Conversely, the short position trader must deliver 1,000 barrels at the previously agreed-upon futures price, and their loss would equal the long position trader’s profit.

        The key difference between a call option and a long position in a futures contract is the obligation to purchase the asset. While a futures contract obliges the long position trader to purchase the asset at the futures price, a call option conveys the right to purchase the asset at the exercise price. A call option holder has a better position than the holder of a long position in a futures contract with a futures price equal to the option’s exercise price, but this advantage comes at a cost. Call options must be purchased, whereas futures contracts are entered into without cost. The purchase price of an option, called the premium, represents the compensation the call option holder must pay for the ability to exercise the option only when it is profitable to do so. Similarly, the difference between a put option and a short futures position is the right, as opposed to the obligation, to sell an asset at an agreed-upon price.

  • How Securities are Traded

    • How Firms Issue Securities

      Firms need capital to finance their investments, which they can raise by either borrowing money or selling shares. Investment bankers are hired to handle the sale of these securities in the primary market, where newly issued securities are sold to the public. After the initial sale, investors can trade their shares of existing securities in the secondary market.

      Publicly listed firms’ shares can be continuously traded on well-known stock markets such as the New York Stock Exchange (NYSE) or NASDAQ Stock Market. Any investor can purchase shares for their own portfolio from these markets. Companies traded on these markets are referred to as publicly traded, publicly owned, or simply public companies. In contrast, private corporations have shares that are held by a small group of managers and investors.

    • Privately Held Firms

      A privately held company is owned by a relatively small group of shareholders, who are not required to publicly disclose financial statements and other information as frequently as publicly traded companies. This allows the company to pursue long-term goals without the pressure of quarterly earnings announcements.

      However, private companies are limited to having up to 499 shareholders, which restricts their ability to raise large amounts of capital. To overcome this restriction, middlemen may form partnerships to buy shares in private companies, which counts as only one investor even though many individuals may participate.

      When a private company needs to raise funds, it sells shares directly to a small number of institutional or wealthy investors in a private placement. The SEC’s Rule 144A allows these companies to make these placements without having to prepare extensive and costly registration statements. Although attractive, shares of privately held companies do not trade in secondary markets such as the NYSE or NASDAQ Stock Exchange, reducing their liquidity and presumably lowering their prices.

      Recently, some firms have set up computer networks to enable private company stockholders to trade among themselves. However, unlike public stock exchanges regulated by the SEC, these networks require little disclosure of financial information and provide limited oversight of the market operations.

    • Publicly Traded Companies

      When a private firm wants to raise capital from a wide range of investors, it may choose to go public by selling its securities to the general public and allowing them to freely trade shares in the established securities market. The first sale of shares to the public is called an Initial Public Offering (IPO). Later, the firm may issue more shares through a seasoned equity offering.

      Investment bankers are responsible for marketing public offerings of both stocks and bonds and are often referred to as underwriters. More than one investment banker may be involved in marketing the security, with a lead firm forming a syndicate of other investment bankers to share the responsibility of the stock issue.

      The investment bankers advise the firm on the terms of the sale and help prepare a preliminary registration statement that is filed with the Securities and Exchange Commission (SEC) to describe the issue and the prospects of the company. When the statement is in final form and accepted by the SEC, it is called a prospectus, and the price at which the securities will be offered to the public is announced.

      In a typical underwriting arrangement, the investment bankers purchase the securities from the issuing company and then resell them to the public. The issuing firm sells the securities to the underwriting syndicate for the public offering price, less a spread that serves as compensation for the underwriters. In addition to the spread, investment bankers may also receive shares of common stock or other securities from the firm.

    • Shelf Registration

      In 1982, the SEC approved Rule 415, which enables firms to register securities and gradually sell them to the public over a period of 2 years after the initial registration. As the securities are registered, they can be sold quickly with limited additional documentation. These securities are referred to as “on the shelf” and ready for issuance, which gave rise to the term “shelf registration.

    • Initial Public Offering

      Investment bankers play a crucial role in the process of issuing new securities to the public. They coordinate road shows to generate interest among potential investors and gather information about the demand and prospects of the security. The underwriters use the information gathered from these road shows, along with feedback from institutional investors, to determine the offering price and number of shares to be offered.

      Investors communicate their interest in purchasing shares of the IPO to the investment bankers through a process called book building. The allocation of shares to investors is partially based on the strength of their expressed interest in the offering. This creates an incentive for investors to truthfully reveal their interest, as underpriced IPOs often result in significant price jumps on the first day of trading.

      However, while IPOs often offer attractive first-day returns, they have been poor long-term investments on average. According to research by Ritter, a portfolio of equal amounts of each U.S. IPO between 1980 and 2009, held for three years, would have underperformed the broad U.S. stock market by 19.8%. The 2011 IPO of Groupon is an example of underpricing, while the IPO of Facebook is an example of overpricing.

  • How Securities are Traded?

    • Types of Markets

      • There are four types of markets: 1) Direct search markets 2) Brokered Markets 3) Dealer Markets 4) Auction Markets

      • Direct Search Markets: are the least organized markets. Buyers and sellers seek each other out directly. An example of a transaction in such markets is the sale of a used TV on craigslist where the seller advertises for buyers in local area. These markets are characterized by sporadic participation (meaning scattered), low-priced and non-standard goods. Firms find it difficult to profit in a such environment.

      • Brokered Markets: are the next level of organization. Brokers offer search serives to buyers and sellers, in markets where trading a good is active. An example, is real state market, where buyers make it worthwhile for participants to pay broker to help them conduct the search. Another example, is the primary market where new issues of securities are offered to the public. Investment bankers act as brokers as they seek investors to purchase securities directly from the issuing corporation.

      • Dealers Market: When trading activity increases in a particular type of asset, then dealer markets arise. Dealers specialized in various assets, buy these assets for their own, and later sell them for a profit from their inventory. The spread between dealer’s buy (or “bid”) and sell (or “ask”) prices are a source of profit They save traders the search costst, because one can easily look up their prices. An example is the “second hand car dealers”. Also most bonds trade in over-the-counter dealer markets.

      • Auction Markets: The most integrated market is an auction market, where traders converge at on eplace (either “physically” or “electronically”) to buy and sell asset. An example of such markets is New York Stock Exchange (NYSE). An advantage of auction markets to dealer markets is that one need not search across dealers to find the best price for a good. If all participants converge, they can arrive at mutually agreeble prices and save the bid-ask spread.

    • Types of Orders

      There are different types of trades an investor is eager to execute in these markets. Broadly speaking we have two types of orders: (i) Market orders and (ii) Price Contingetn Orders.

      • Market Orders

        Market orders are buy or sell orders that are to be executed immediately. For example, an investor might call his broker and ask for the market price of APPL. The broker will reort back that the best bid price is 140$ (the price to buy shares) and the best ask price is 141$ (the price to buy shares). The bid-ask spread in this case is 1$.

        There are several complications here: First, the posted price quote actually represent commitments to trade up to a secific number of shares. If the market order is more than this number of shares, multiple may be filled at multiple prices. For example, if the investor wants to buy 1,500 shares and the ask price is good for 1,000 shares, it may be necessary to pay a slightly higher price for the last 500 shares. The depth of the markets for shares of stock, shows the total number of shares offered for trading at each bid and ask price. Depth is considered a component of liquitidty. Second, another trader may beat the investor to the quote, this means that the investor’s order would then be executed at a worse price. Third and finally, the best price quote might change before the order arrives, causing execution at a price different from the one at the moment of order.

      • Price Contingent Orders

        Investors may also place orders specifying prices at which they are willing to buy or sell a security.

        A limit-buy order instructs the broker to buy some number of shares if the stock price may be obtained at or below a stipulated price.

        Conversely, limit-sell order instructs the broker to sell if and when the stock price rises above a specified limit.

        limit-order book is a collection of limit-orders to be executed. The best orders are at the top of the list. Offers to buy at the highest price, and offers to sell at the lowest price. The buy and sell orders at the top of the list are called inside quotes. The order sizes for the inside quotes can be fairly small. Therefore, investors interested in larger trades face an effective spread greater than the nominal one because they cannot execute their entire trade at the inside quote.

        Stop-orders are similar to limit orders in that the trade is not to be executed unless the stock price hits a limit. For Stop-loss orders, the stock is to be sold if its price falls below a stipulated price. As the name suggests, this order is executed to stop furthe loss accumulation. Similarly Stop-buy orders specify a stock should be bought when the its rice rises above a limit.

        These stop orders often accompany short sales (sales of securities when you don’t own but have borrowed from your broker) and are used to limit potential losses from short position.

        All these price contingent orders all together

          Price Below the limit Price above the limit
        Buy limit-Buy Order Stop-buy order
        Sell Stop-loss order limit-sell order

    • Trading Mechanisms

      An investor who wants to buy or sell stocks, will ask the broker. The broker charges a commission fee for arranging the trade on client’s behalf. Brokers have several options, to execute the trade.

      There are three trading systems used in the US: * over the counter dealer markets * electronic communication networks * specialist markets

      NASDAQ and NYSE are the best well known markets. They use a variety of trading procedures.

      • Dealers Market

        Roughly 35,000 securities trade on over-the-counter or OTC markets. Thousands of brokers register with SEC as security dealers. Dealers quote prices they are willing to buy or sell. Then they contact a dealer listing an attractive quote and execute the trade.

        Before 1971, OTC quotions were recorded manually and published daily on so called pink-sheets. In 1971, the National Association of Securities Dealers introduced Automatic Quotions System, or NASDAQ, to link brokers an dealers in a computer network, to displayrevise quotes on the network. Dealers used this system to post their bidask pric. The difference between bid and ask price, the bid-ask spread, is the source of dealer’s profit. Dealers examine prices online and then contact the delaer with the best quote and execute a trade.

        The NASDAQ as was originally desinged, was more of a price quotion system to post bid/ask prices. Brokers in the search of best trading opportunity, would contact the dealer and directly negotioate over the phone. NASDAQ Stock Market has now been added to NASDAQ and it allows for electronic execution of trades.

      • Electronic Communivation Network (ECNs)

        ECN allows participants to post market limit orders over the computer networks. The limit order book is available for all of the participants. NYSE Arca is one of the leading ECNs. Orders are crossed, when they match on the limit-order book, without intervention of a broker. Therefore, ECNs are true trading systems, not merely price quotion systems.

        ECNs have several benefits: (i) Direct and automatic crossing without using a broker-dealer system (ii) Modest cost trades typically less than a penny per share, since they are automatic, (ii) elimination of bid-ask spread, since trades are crossed automatically, (iv) Fast speed at which trades are done, and (v) anonymity in the trades

      • Specialist Markets

        This has been largely replaced by electronic communication networks. A decade ago, they were dominant form of market organization for rading stocks. In thi system, exchanges such as NYSE assign representativesfor managing the trade in each security to a specialist. Brokers wishing to buy/sell trades would contact the specialist on board of exchange. The specialist would manage all of the order book.The highest bid, and the lowest ask price would win the trade.

  • The Rise of Electronic Trading

    The NASDAQ and NYSE stock markets have undergone significant changes in the past few decades, transitioning from over-the-counter dealer markets and specialist markets to primarily electronic markets. Advances in technology and new regulations have brought about these changes. Regulations allowed brokers to compete for business, breaking their hold on information and reducing price increments. The integration of markets and the availability of technology to rapidly compare prices across markets has also reduced the cost of trade execution.

    In 1975, the fixed commission on the NYSE was eliminated and the Securities and Exchange Act was amended to create the National Market System, which aimed to partially centralize trading across exchanges and enhance competition. The implementation of a centralized reporting of transactions and price quotation system provided traders with a broader view of market opportunities.

    In 1994, a scandal at the NASDAQ involving dealers colluding to maintain wide bid-ask spreads led to the SEC instituting new order-handling rules. Published dealer quotes were now required to reflect customer limit orders and NASDAQ agreed to integrate quotes from electronic communications networks (ECNs) into its public display. This allowed ECNs to compete for trades and led to the SEC adopting Alternative Trading System Regulations, which gave ECNs the right to register as stock exchanges.

    The SEC has continued to reduce the minimum tick size, which has resulted in a reduction of bid-ask spreads. In the 1990s, exchanges around the world began adopting electronic trading systems and the NASDAQ Stock Market was established as a separate entity. In 2006, the NYSE acquired the electronic Archipelago Exchange and renamed it the NYSE Arca. In 2007, the SEC fully implemented Regulation NMS, which required exchanges to honor quotes from other exchanges when they could be executed automatically.

    Today, trading is almost exclusively electronic for stocks, with bonds still being traded in traditional dealer markets.

  • U.S. Markets

    The NYSE and NASDAQ Stock Market are the two largest US stock markets. But other ECNs (namely BATS, NYSE Arca, Direct Edge,…) have steadily increased their market share.

    • NASDAQ

      The NASDAQ Stock Market lists 3,000 firms and its trading platform has improved over the years. Currently, the NASDAQ Market Center integrates NASDAQ’s previous electronic markets into a single system.

      In 2008, NASDAQ merged with OMX, a Swedish-Finnish company that controls seven Nordic and Baltic stock exchanges, to form NASDAQ OMX Group. This group manages not only the NASDAQ Stock Market, but also several stock markets in Europe, as well as options and futures exchanges in the US.

      NASDAQ has three levels of subscribers:

      • Level 3: The highest level of registered market makers. These firms create a market in securities, maintain inventories of securities, and post bid and ask prices at which they are willing to buy or sell shares. They can continuously enter and change bid-ask quotes and have the fastest execution of trades. They profit from the spread between bid and ask prices.

      • Level 2: Receive all bid and ask quotes but cannot enter their own quotes. They can see which market makers are offering the best bid and ask prices. They are usually brokerage firms that execute trades for their clients but do not actively deal in stocks for their own account.

      • Level 1: Receive only inside quotes, i.e., the best bid and ask prices, but do not see the number of shares being offered. They are usually investors who are not actively buying or selling, but want information on current prices.

    • NYSE

      The NYSE (New York Stock Exchange) is the largest stock exchange in the US, measured by the value of stocks listed on the exchange. On a daily basis, the NYSE has a trading volume of approximately one billion shares.

      In 2006, the NYSE merged with Archipelago Exchange to form NYSE Group, a publicly-held company. In 2007, NYSE Group merged with Euronet to form NYSE Euronext. In 2008, NYSE Euronext acquired the American Stock Exchange, which was then renamed as NYSE Amex to focus on smaller firms. NYSE Arca is the firm’s electronic communication network, where the majority of exchange-traded funds are traded. In 2012, NYSE Euronext was acquired by International Exchange (ICE), a company primarily focused on energy-future trading. ICE plans to retain the NYSE Euronext name and the iconic trading floor on Wall Street.

      The NYSE has many trading specialists who rely heavily on human involvement in trade execution. The exchange began its transition to electronic trading in 1976 with the introduction of DOT (Designated Order Turnaround) and later the SuperDOT system, which routes orders directly to specialists. In 2000, the NYSE launched Direct+, which allowed for automatic cross of smaller trades (up to 1,099 shares) without human interaction. In 2004, Direct+ began eliminating the transaction size limit. The transition to electronic trading accelerated in 2006 with the introduction of the NYSE Hybrid Market, which allowed brokers to send orders for either immediate electronic execution or to a specialist for price improvement. The Hybrid System allowed the NYSE to remain as a fast market for purposes of Regulation NMS while still offering the benefits of human intervention for more complex trades. Note that NYSE Arca is fully electronic.

    • ECNs

      Over time, automated markets have gained a larger market share. Today, some of the largest ECNs include BATS, Direct Edge, and NYSE Arca. Brokers affiliated with an ECN have computer access and can enter orders into the limit order book. When orders are received, if a matching trade is found, the trade is immediately executed.

      Initially, ECNs were only open to traders using the same system. However, after the implementation of Regulation NMS, ECNs started to list limit orders on other networks.

      These cross-market links have paved the way for more popular strategies, such as high-frequency trading, which aim to profit from even small price differences across the market. Speed is critical in these strategies, and ECNs compete based on the speed they can offer. Latency refers to the time it takes to accept, process, and execute a trading order. For example, BATS advertises latency times of around 200 milliseconds.

  • New Trading Strategies

    Electronic trading has introduced new opportunities for various trading strategies and tools. One such strategy is algorithmic trading, where the trading decisions are handed over to computer programs. Another type of algorithmic trading is high frequency trading, which involves the use of computer programs to initiate trades in a matter of milliseconds, much faster than a human could process. This type of trading brings a large amount of liquidity to the market, but it can also lead to a rapid withdrawal of liquidity, as seen in the flash crash of 2010. Dark pools are another aspect of electronic trading, where the trading takes place in an anonymous manner, but it can have an impact on market liquidity.

    • Algorithmic Trading

      Algorithmic Trading” involves using computer programs to make trading decisions. It is estimated that more than half of all equities traded in the US are initiated by algorithms. These strategies were made possible by the decimalization of the minimum tick size.

      There are different types of algorithmic trading, including exploiting short-term trends, pair trading to take advantage of temporary disruptions in the normal price relationship between stocks, and exploiting discrepancies between stock prices and stock index future contract prices.

      Some algorithms aim to profit from the bid-ask spread by quickly buying a stock at the bid price and selling it at the ask price before the price changes.

    • High Speed Frequency Trading

      Algorithmic trading strategies often require extremely fast trade initiation and execution, and high-frequency trading is a subset of algorithmic trading that uses computer programs to make rapid decisions. These trades compete for opportunities that offer small profits, but they can add up to significant amounts.

      Some algorithmic strategies profit from the bid-ask spread, while others rely on cross-market arbitrage, taking advantage of even tiny price differences across markets. The firms that are quickest to identify and execute these trades win the profits. Today, trade execution times for high-frequency trades are measured in milliseconds or even microseconds, and firms are locating their trading centers near electronic exchange centers for faster access.

      The location of trading centers has become crucial in this high-speed competition. A trade order originated in Chicago, for example, takes 5 milliseconds to reach New York. During this time, another firm located in New York could win the trade. ECNs today claim latency periods of 1 millisecond.

    • Dark Pools

      Many big traders prefer to remain anonymous when buying or selling large amounts of stocks as the public knowing about their transactions can affect the stock prices.

      Traditionally, large trades (known as “Blocks” with more than 10,000 shares) were handled by “block houses,” which are brokerage firms that specialize in matching buyers and sellers of large blocks. These brokerages execute trades privately, avoiding price movements against their clients.

      However, block trades have now been replaced by “Dark Pools,” which are trading systems where participants can buy or sell large blocks of securities without revealing their identities or the details of the trades. This contributes to the fragmentation of the markets, as fewer orders are visible on consolidated limit order books, potentially leading to an unfair public price that does not reflect all available information about the security demand.

      Another approach to dealing with large trades is to divide them into smaller trades, which can be executed on electronic markets, thereby hiding the fact that a large amount of shares have been bought or sold. This has led to a rapid decline in the average trade size, which is now around 300 shares.

    • Bond Trading

      In 2006, the NYSE gained regulatory approval to list debt issues from any NYSE-listed company on its bond-trading system. This made it easier for bonds to be listed and resulted in the expansion of NYSE’s electronic bond-trading platform, now known as NYSE Bonds, which is now the largest centralized bond market in the US.

      It’s important to note that the majority of bond trading still takes place in the over-the-counter (OTC) market among bond dealers, even for bonds listed on the NYSE. Major players in the bond market include Merrill Lynch (now part of Bank of America), Salomon Smith Barney (a division of Citigroup), and Goldman Sachs.

      However, the bond market also has liquidity risks, which can make it difficult to sell holdings quickly if necessary.

  • Flash Crash of May 2010

    On May 6, 2010, the Dow Jones Industrial average experienced a rapid drop in value. At 2:42 PM New York time, the market was down by approximately 300 points due to concerns about the European debt crisis. However, within the next five minutes, the Dow plummeted an additional 600 points. After 20 minutes, it regained most of those losses. This sudden drop caused a number of stocks, such as Accenture, to plummet, while others, such as Apple and Hewlett-Packard, experienced huge spikes.

    A subsequent report by the SEC revealed that many algorithmic trading programs were the cause of the market’s sudden drop. As these programs withdrew, liquidity evaporated and buyers disappeared. Trading was temporarily suspended, and when it resumed, buyers took advantage of the market and prices bounced back quickly. The NYSE and NASDAQ cancelled trades executed more than 60% away from the opening price of the day.

    In response to this event, the SEC approved a rule to halt trading for five minutes in stocks that experience a rise or fall of more than 10% in a five-minute period. This is intended to prevent algorithmic trading from rapidly moving share prices before human traders have a chance to determine if the changes are due to fundamental information.

  • Globalization of Stock Markets

    NYSE-Euronext is the largest equity market as measured by total market value of listed firms.

    The securities market is facing pressure to form international alliances or mergers due to the rise of electronic trading. Traders view stock markets as computer networks that connect them to other traders, allowing them to trade a wider range of securities from around the world.

    To stay competitive, it is crucial to have efficient and cost-effective mechanisms for executing and clearing trades.

    • In the US:

      • The NYSE merged with Archipelago ECN in 2006 and acquired the American Stock Exchange in 2008.
      • NASDAQ acquired Instinet in 2005 and the Boston Stock Exchange in 2007.
      • In the derivative market, the Chicago Mercantile Exchange acquired the Chicago Board of Trade in 2007 and the New York Mercantile Exchange in 2008.

    • In Europe:

      • Euronext was formed through the merger of the Paris, Brussels, Lisbon, and Amsterdam exchanges, and later purchased Liffe, a derivatives exchange based in London.
      • The London Stock Exchange merged with Borsa Italiana, which operates the Milan exchange, in 2007.
The biggest stock markets in the world bby domestic market capitalization in 2012

  • Intercontinental:

    • The NYSE Group and Euronext merged in 2007.
    • In 2011, Deutsche Borse and NYSE Euronext announced their intention to merge, but the proposed merger fell through in early 2012 after European Union antitrust regulators recommended blocking the combination.
    • The NYSE and the Tokyo Stock Exchange have announced plans to link their networks, giving their customers access to both markets.
    • NASDAQ Stock Market merged with OMX, which operates Nordic and Baltic stock exchanges, in 2007 to form NASDAQ OMX.
    • In 2008, Eurex acquired International Securities Exchange (ISE) to form a major options exchange.

  • Trading Costs

    When trading securities, there is always a fee involved, which is paid to the broker. There are two types of brokers: full-service and discount brokers.

    • Full-service brokers offer a range of services and are often referred to as account executives or financial consultants. In addition to executing orders, holding securities for safekeeping, offering margin loans, and facilitating short sales, they also provide investment information and advice. These brokers have a research team that prepares analysis and forecasts of general economics, industry, and company conditions, and makes specific buy or sell recommendations. Some customers establish discretionary accounts, which allow the broker to make buy and sell decisions on their behalf. However, this requires a high level of trust since the broker could execute trades solely for the purpose of commission.

    • Discount brokers, on the other hand, offer basic services such as buying and selling securities, holding them for safekeeping, offering margin loans, and facilitating short sales. They do not provide any additional information or advice, only price quotes. Many banks, thrift institutions, and mutual fund management companies now offer these services. Some discount brokers, such as Schwab, E*Trade, or TD Ameritrade, now offer commissions below $10.

    In addition to the explicit trading fee, which is the broker’s commission, there is also an implicit fee in the form of the dealer’s bid-ask spread. Some brokers may offer no commission, but instead collect their fee entirely in the form of the bid-ask spread.

  • Buying on Margin

    Investors have the option to finance their securities purchases through a source of debt financing known as broker’s call loans or buying on margin. This means that the investor borrows part of the purchase price from a broker and only contributes a portion of the total cost, known as the margin. The broker then borrows the remaining amount from banks at a call money rate and charges their clients this rate, along with a service charge, for the loan.

    All securities purchased on margin must be kept with the brokerage firm and serve as collateral for the loan. The Federal Reserve System regulates the extent to which stock purchases can be financed through margin loans, with the current requirement being that the investor must pay 50% of the purchase price in cash.

    • Example on Margin Call

      The concept of percentage margin refers to the relationship between the equity value of an investment account and the market value of the securities it holds. It is calculated as the ratio of the net-worth of the account to the market value of the securities.

      For example, if an investor pays $6,000 towards the purchase of $10,000 worth of stock (100 shares at $100 each), and borrows the remaining $4,000 from the broker, the initial percentage margin would be 60% ($6,000/$10,000).

      However, if the stock price drops to $70 per share, the total value of the stock would be $7,000. With the loan from the broker still at $4,000, the investor’s equity would be $3,000 ($7,000 - $4,000), and the percentage margin would drop to 43% ($3,000/$7,000).

      To avoid the situation where the stock value falls below the loan from the broker, resulting in a negative equity, brokers set a maintenance margin. This requires the investor to add new cash or securities to the account if the percentage margin falls below a certain level. If the investor does not act, the broker may sell securities from the account to restore the margin to an acceptable rate.

      For instance, if the maintenance margin is 30%, the stock price can fall to $57.13 before the margin would fall below the required level.

    • Why to buy in Margin

      Investors often buy securities on margin to invest more than their own money allows. This gives them the potential for greater upside, but also exposes them to greater downside risk.

      For example, an investor who buys 100 shares of stock A at $100 each (for a total cost of $10,000) would expect a 30% return if the price increases 30%. However, if the investor takes a margin call with 9% interest per year, they can buy 200 shares for $20,000. If the price of each share increases 30% to $130, the investor would make a profit of $15,100 ($26,000 worth of stock - $10,900 in principal and interest). This results in a 51% return on the initial $10,000 investment, compared to the original 30% return.

      However, if the price of each share drops 30% to $70, the 200 shares would be worth $14,000 and the investor would be left with only $3,100 after paying off the $10,900 in principal and interest, resulting in a 69% loss instead of the original 30% drop.

    • Short Sales

      Short sales are the opposite of buying stocks. In a short sale, an investor borrows a stock from a broker and sells it with the expectation that the stock price will decrease. The investor then must buy the same stock to replace the borrowed one and make a profit if successful. The short seller must also pay any dividends paid during the short sale to the lender of the security. The profits from a short sale must be kept with the broker and the short seller must also provide margin to cover any potential losses if the stock price increases during the short sale. Short sellers must also be mindful of margin calls, just like investors who buy stocks on margin. If the stock price increases, the margin in their account will decrease and if it drops to the maintenance level, they will receive a margin call.

      The short sale process begins when the short seller borrows a stock from their brokerage firm, which holds a variety of securities for other investors. The short seller then sells the stock, hoping that the price will fall, allowing them to buy the same stock at a lower price to replace the borrowed one. If the brokerage cannot locate new shares to replace the ones sold, the short seller will need to purchase shares in the market to repay the loan. The exchange rules require that the profits from a short sale must be kept with the broker and cannot be invested. However, large investors may receive some income from the profits held by the broker. The short seller must also provide margin to cover any potential losses if the stock price increases during the short sale.

    • Example

      Suppose you are pessimistic about a stock with 100$ price per share (you think the price will drop). You order short-sell on 1,000 stocks. Then the brokerage will borrow 1,000 stocks from another investor account or borrows the stock from another brokerage. Suppose, they ask for margin call of 0.5, this means that they require you to have at least 50,000$ in your account that can serve as margin on short-sell. Let’s say you have 50,000$ on T-Bills.

      First of all, your initial percentage of margin (the ratio of the equity in the account, 50,000$, to the current value of the shares you have borrowed and eventually must return, 10,000$) is as

      \[\text{Margin} = \frac{\text{Equity}}{\text{Value of stock owed}} = \frac{50,000}{100,000} = 0.5\]

      Now, suppose the price of the stock drops to 70$ a share, this way, your profit would be 30,000$, since you need to buy 1,000 shares of stocks with 70$ a share to replace.

      Suppose, the broker has a maintenance margin of 30\% on short sales. This means that equity in your account must be at least 30\% of the value of your short position at all times. Then, you might ask, how much can the price increase before a margin call. Suppose \(P\), denotes the price of stock per share. We have

      \[\frac{150,000-1000P}{1000P} = 0.3 \text{ so } P=115.38\$\]

      which means that \(P = 115.38\) per share, you would receive a margin call. This is the whole reason we have stop-buy orders. You can put a stop-buy order at 110$ a share, to limit your losses to 10$ per share, instead of loosing 15$ per share and receiving a margin call order.

      During the financial crisis, the SEC puts restriction on short-sells. For instance in 2008, the SEC voted to restrict short sales in stocks that decline by at least 10\% on a given day.

  • Regulations of Securities Markets

    The United States has a number of laws that regulate the trading of securities. The two main pieces of legislation are the Securities Act of 1933 and the Securities Act of 1934.

    The Securities Act of 1933 requires companies to provide complete and accurate information about new securities they issue and to register these securities with the SEC. The SEC’s approval of the registration only means that all relevant information has been disclosed, not that the security is a good investment. Investors must make their own evaluation of the security’s value.

    The Securities Act of 1934 established the Securities and Exchange Commission (SEC) to enforce the provisions of the 1933 Act. It also requires companies to periodically disclose relevant financial information and gives the SEC the power to regulate securities exchanges, OTC trading, brokers, and dealers. The SEC works with other regulatory agencies, such as the Commodity Futures Trading Commission (CFTC) and the Federal Reserve, to regulate the securities market. The Federal Reserve is responsible for setting margin requirements for stocks and stock options and regulates bank lending to security market participants.

    The Securities Investor Protection Act of 1970 established the Securities Investor Protection Corporation (SIPC) to protect investors in the event of a failed brokerage firm. The SIPC ensures that securities held in street name by a failed brokerage firm will be returned to the investors up to a limit of $500,000 per investor. The SIPC is funded by insurance premiums levied on its member brokerage firms.

    In addition to federal regulations, state laws, known as blue sky laws, also regulate securities trading. These laws aim to provide investors with a clearer view of investment prospects. Many states have adopted portions of the Uniform Securities Act of 1956, which helped to unify state laws.

    The Financial Stability Oversight Council (FSOC) was established in 2008 as part of the Dodd-Frank Wall Street Reform and Consumer Protection Act to monitor the stability of the US financial system and identify risks posed by large, interconnected banks. The FSOC is comprised of the chairpersons of the major US regulatory agencies, and serves as a coordinating body for key financial regulators.

    Finally, two other agencies also protect investors. The Federal Deposit Insurance Corporation provides depositors with federal protection in the event of bank failures. The SIPC ensures that investors will receive their securities in the event of a failed brokerage firm.

    • Self-Regulation

      In addition to government regulation, the securities market has self-regulatory organizations. One of the most important is the Financial Industry Regulatory Authority (FINRA), which is the largest non-government regulator of all securities firms in the United States. FINRA was formed in 2007 through the consolidation of the National Association of Securities Dealers (NASD) and the self-regulatory arm of the New York Stock Exchange. Its main mission is to protect investors and maintain market integrity. It examines securities firms, creates and enforces rules regarding trading practices, and provides a dispute resolution forum for investors and registered firms.

      Another important self-regulatory organization is the CFA Institute, which has established standards of professional conduct for members with the Chartered Financial Analyst (CFA) designation. These standards govern the behavior of CFAs in the securities market.

    • The Sarbanes-Oxley Act

      The stock market scandals of 2000-2002 were largely focused on three unethical practices: (1) unfair allocation of shares in initial public offerings (IPOs), (2) biased securities research, and (3) misleading recommendations to the public.

      In response to these problems, the Sarbanes-Oxley Act was passed by the US Congress in 2002, implementing several key reforms in the financial industry. The Act established the Public Company Accounting Oversight Board to oversee the auditing of public companies, and set rules for independent financial experts to serve on the audit committees of firms’ boards of directors. Additionally, CEOs and CFOs were required to personally certify the accuracy of their firms’ financial reports, and faced penalties if the reports turned out to be misleading. The Act also limited auditors’ ability to provide certain services to clients in order to prevent potential conflicts of interest. Finally, the Board of Directors must be comprised of independent directors and hold regular meetings without the presence of company management.

      However, in recent years, there have been pushbacks against the Sarbanes-Oxley Act. Some believe that the compliance costs imposed by the law are too high, particularly for smaller firms, and that this heavy-handed regulation is giving foreign countries an advantage over the US in attracting securities listings.

    • Insider Trading

      Insider trading refers to the illegal practice of using confidential information to profit from securities transactions. This information, held by officers, directors, or major stockholders of a company, is not yet known to the public. The definition of who is considered an “insider” can be ambiguous, as it is clear that top executives are insiders, but the status of other individuals, such as major suppliers, may be less clear. The dividing line between legal private information and illegal inside information can be vague.

      The SEC requires insiders to report all transactions in the company’s stock and publishes a monthly summary of these transactions to inform the public. However, despite regulations, evidence suggests that insiders do exploit their knowledge for personal gain. This is evidenced by well-publicized convictions of individuals involved in insider trading schemes, evidence of information leaking to some traders before public announcements, and the returns earned on trades by insiders. A study by Jaffee found that stocks had an abnormal return of about 5% over the 8 months following insider purchases, and tended to perform poorly when insiders sold more than they bought.

  • Mutual Funds and Other Investment Companies

    Investors typically work with intermediaries when investing their funds. The most common intermediaries are:

    • Mutual Funds or open-end investment companies (the most important one)
    • Unit investment trusts
    • hedge funds
    • close-end funds

  • Investment Companies

    Investment companies are financial intermediaries that pool money from individual investors and invest those funds in a variety of securities and assets. The concept behind investment companies is to bring together the resources of many small investors to achieve the benefits of large-scale investing.

    These companies provide several important services, including:

    • Record Keeping and Administration: Investment companies issue periodic reports, keep track of investments, dividends, and capital gains distributions, and may automatically reinvest dividends and interest income for shareholders.
    • Diversification and Divisibility: By pooling their money, investment companies allow investors to own fractional shares of a variety of securities, acting as large investors even if an individual shareholder could not.
    • Professional Management: Investment companies employ security analysts and portfolio managers to attempt to achieve superior investment results for their investors.
    • Lower Transaction Costs: By trading large blocks of securities, investment companies can save on brokerage fees and commissions.

    Since investment companies pool assets of individual investors, they also need to divide those assets among investors. Investors buy shares in investment companies and ownership is proportional to the number of shares purchased. The value of each share is called net asset value or NAV. Net asset value equals assets minus liabilities expressed on a per-share basis.

    \[\text{Net Asset Value } = \frac{\text{Market value of assets minus liabilities}}{\text{Shares outstanding}}\]

    For instance, suppose a mutual fund that manages protfolio of securities worth of 120$ million. The company owes 4$ million to investment advisors, and another 1$ million for rent, wages due, and miscellaneous expenses. The company has 5 million shares outstanding, so the net asset value is

    \[\text{Net Asset Value } = \frac{120-4-1}{5} = 23\$ \text{ per share}\]

    • Types of Investment Companies

      The Investment Companies Act of 1940 classifies the investment companies as either unit investment trusts or managed investment companies.

      The portfolio of unit investment companies is fixed and thus are called “unmanaged”. In contrast, managed companies are so named because securities in their investment portfolios continually are bought and sold, hence “managed”. Managed companies are futhur classified as either closed-end or open-end companies. Open-end companies are what we commonly call mutual funds.

  • Unit Investment Trusts

    Unit Investment Trusts (UITs) are investment vehicles that pool money from individual investors to purchase a fixed portfolio of securities. Unlike mutual funds, the composition of the portfolio remains unchanged for the life of the fund.

    UITs are sponsored by brokerage firms or other financial institutions, who purchase the underlying securities and deposit them into a trust. They then sell shares, called redeemable trust certificates, to investors. All income and payments of principal from the portfolio are distributed to the shareholders by the trustees, typically a bank or trust company.

    UITs are considered unmanaged, as there is no active management of the portfolio once it is established. They typically invest in one particular type of asset, such as municipal or corporate bonds, which provides investors with a pool of a specific asset class to invest in.

    The sponsors of UITs earn their profit by selling shares at a premium over the cost of acquiring the underlying assets. Investors who wish to sell their holdings can either receive payment from the trustee or have their shares sold to a new investor.

    However, UITs have steadily lost market share to mutual funds, declining from $105 billion in 1990 to $60 billion in 2012.

  • Managed Investment Companies Actively managed investment companies come in two types: closed-end and open-end. Both types have a board of directors elected by shareholders who hire a management company to manage the portfolio for an annual fee, typically ranging from 0.2% to 1.5% of assets. The management company is often the firm that organized the fund, for example Fidelity Management and Research Corporation sponsors Fidelity mutual funds and manages them, assessing a management fee on each fund. In some cases, a mutual fund may hire an outside portfolio manager, such as Wellington Management for Vanguard.

    Open-end funds are always ready to issue or redeem shares at their net asset value, though both purchases and redemptions may come with sales charges. When an investor wants to cash out, they can sell their shares back to the fund at net asset value. Most mutual funds are open-end.

    Closed-end funds, on the other hand, do not issue or redeem shares. Investors who want to cash out must sell their shares to other investors. Shares of closed-end funds are traded on organized exchanges and can be purchased through brokers, like common stocks. Their price can vary from the net asset value. In 2013, closed-end funds held $265 billion in assets.

    Closed-end funds and open-end funds are two types of actively managed investment companies. The difference between the two is in how shares are bought and sold. A closed-end fund has a fixed number of shares that are traded on an organized exchange and can be purchased through a broker, just like a common stock. The price of these shares can vary from the net asset value (NAV) of the fund. When a closed-end fund is initially issued, the price may be above the NAV, but it often falls to a discount after some time.

    On the other hand, open-end funds have no fixed number of shares, and investors buy and sell shares directly from the investment company at NAV. The number of outstanding shares changes daily. Open-end funds do not trade on organized exchanges, but they may carry a load, which is a sales charge.

    When considering a list of closed-end funds, the first column shows the name and ticker symbol of the fund, followed by the most recent NAV and the closing share price. The premium or discount is the percentage difference between the price and NAV. The last column shows the 52-week return, which is the percentage change in the share price plus dividend income.

    It is important to note that closed-end funds usually sell at a discount to NAV, while the price of open-end funds cannot fall below NAV, as these funds are ready to redeem shares at NAV.

  • Other Investment Organization

    Some intermediaries are not formally organized or regulated as investment companies, nevertheless they have similar functions. Three of the most important ones are: Commingled Funds, Real Estate Investment Trusts, and Hedge Funds.

    • Commingled Funds

      Commingled funds are investment partnerships in which investors pool their funds and the management firm, such as a bank or insurance company, manages the funds for a fee. They are similar to open-end mutual funds and offer units instead of shares, which are bought and sold at net asset value. These funds are often created for trust or retirement accounts that are too small to be managed separately but still larger than the portfolios of most individual investors. A bank or insurance company may offer a variety of commingled funds, including money market funds, bond funds, and stock funds.

    • Real Estate Investment Trust (REITs)

      A Real Estate Investment Trust (REIT) is a type of investment company that invests in real estate or loans secured by real estate. It operates similarly to a closed-end mutual fund, where capital is raised through the issuance of shares and borrowing from banks, issuing bonds or mortgages. Most REITs are highly leveraged, with a typical debt ratio of 70%.

      There are two main types of REITs:

      • Equity Trusts, which invest directly in real estate
      • Mortgage Trusts, which primarily invest in mortgage and construction loans.

    • Hedge Funds

      Hedge funds are investment vehicles that allow private investors to pool their assets and have them managed by a fund manager. Unlike mutual funds, hedge funds are structured as private partnerships and subject to minimal regulation by the SEC. They are usually only open to wealthy individuals or institutions and often require investors to agree to a lock-up period, where the investment cannot be withdrawn for a set amount of time.

      The lack of regulation allows hedge fund managers to pursue investment strategies that are not available to mutual fund managers, such as heavy use of derivatives, short sales, and leverage. Hedge funds can invest in a wide range of assets, with some focusing on derivatives, distressed firms, currency speculation, convertible bonds, emerging markets, merger arbitrage, and more. They are empowered to adjust their investment strategies as investment opportunities change.

      Hedge funds have experienced tremendous growth over the years, increasing from $50 billion in 1990 to $2 trillion in 2012.

    • Mutual Funds

      Mutual funds are widely referred to as open-end investment companies. They are the leading type of investment companies, with a market share of over 90%. In 2013, they managed over $13 trillion in assets in the US and an additional $13.5 trillion in non-US assets.

      • Investment Policies

        Each mutual fund has a specific investment policy outlined in its prospectus. For example, money market funds invest in short-term low-risk money market instruments, while bond funds invest in fixed-income securities. Some funds may be more narrowly defined, such as a bond fund that invests primarily in Treasury bonds or mortgage-backed securities.

        Management companies manage a family or “complex” of funds and collect a management fee for operating them. By managing a collection of funds under one umbrella, they make it easy for investors to allocate assets across market sectors or switch assets across funds while still benefiting from centralized record-keeping. Well-known management companies include Fidelity, Barclays, and T. Rowe Price. In 2013, there were nearly 8,000 mutual funds in the US with over 700 fund complexes.

        Some west known management companies are Fidelity, Barclays, and * T Rowe Price*. Each offer an array of open-end mutual funds with different investment policies. In 2013m thre were nearly 8,000 mutual funds in US which offered more than 700 fund compleses.

    • Funds Categories

      By investment policies funds are commonly classified into one or more of the following groups:

      • Money Market Funds: These funds invest in short-term, low-risk money market securities such as commercial paper, repurchase agreements, or certificates of deposit. They usually offer check-writing features and have a fixed NAV of $1 per share, without any tax implications associated with redemption of shares.

      • Equity Funds: These funds primarily invest in stocks and may also hold fixed-income or other types of securities at the discretion of the portfolio manager. Equity funds typically hold 4-5% of total assets in money market securities for liquidity. They can be further classified by their emphasis on capital appreciation versus current income and the level of risk they assume.

      • Sector Funds: These are a type of equity funds that concentrate on a specific industry, such as biotechnology, utilities, energy, or telecommunications.

      • Bond Funds: These funds specialize in fixed-income securities, including corporate bonds, Treasury bonds, mortgage-backed securities, or municipal bonds. Some funds also specialize by maturity or credit risk of the issuer.

      • International Funds: These funds have an international focus and can be global, international, regional, or emerging market funds.

      • Balanced Funds: These funds are designed to be candidates for an individual’s entire investment portfolio and hold both equities and fixed-income securities in stable proportions. Some balanced funds are also funds of funds, investing in shares of other mutual funds.

      • Asset Allocation and Flexible Funds: These funds hold both stocks and bonds, but the proportions allocated to each market may vary in accordance with the portfolio manager’s forecast of their relative performance.

      • Index Funds: These funds match the performance of a broad market index by buying shares in securities included in the index in proportion to their representation. Index funds can be tied to non-equity markets as well, such as bonds or real estate.

      In 2012, 15% of equity funds were indexed. With nearly 8,000 mutual funds in the US offering over 700 fund complexes, there are many options for investors to choose from to match their investment goals.

  • How Funds are sold?

    Mutual funds can be sold either directly by the underwriter or through brokers acting on behalf of the underwriter.

    Funds that are sold directly are marketed through the mail, various offices of the fund, over the phone, or over the internet. Investors can purchase shares by contacting the fund directly.

    Around half of the mutual funds are sold through a sales force. Brokers or financial advisors receive a commission for selling shares to investors. This commission is ultimately paid by the investor. People who rely on their broker’s advice to choose mutual funds should be aware that brokers may have a conflict of interest when it comes to fund selection. This is because some fund companies practice “revenue sharing,” where they pay the brokerage firm for being favored in investment recommendations.

    Many mutual funds are also sold through “financial supermarkets,” which sell shares in funds from multiple complexes. Instead of charging customers a sales commission, the broker splits the management fees with the mutual fund company. This results in a unified record-keeping system for all funds purchased from the supermarket, even if the funds are offered by different complexes. However, these supermarkets lead to higher expenses ratios because mutual funds pass along the costs of participating in these programs in the form of higher management fees.

  • Costs of Investing in Mutual Funds

    Fee Structure:

    When investing in a mutual fund, it’s important to consider factors like investment policy, past performance, and management fees and expenses. The mutual fund information can be found in the Morningstar’s Mutual Fund Sourcebook, which is widely available at libraries and universities.

    It’s essential for investors to be aware of the different types of fees involved in mutual funds:

    • Operating Expenses: These expenses include administrative costs and advisory fees paid to the investment managers. They are expressed as a percentage of total assets under management and can range from 0.2% to 2%. These expenses are periodically deducted from the fund’s assets, so shareholders pay for them through reduced portfolio value. The average expense ratio for equity funds in the US in 2011 was 1.43%, but larger funds have lower ratios, bringing the weighted average down to 0.79%. Actively managed funds tend to have higher expense ratios than indexed funds (0.93% versus 0.14%).

    • Front-End Load: A front-end load is a sales charge or commission paid when you purchase shares. This charge is usually paid to brokers who sell the funds and can be up to 8.5%, but in practice, it’s usually no more than 6%. Low-load funds have loads of up to 3%, while no-load funds have no front-end sales charges.

    • Back-End Load: A back-end load is a redemption or exit fee incurred when you sell your shares. Funds that impose back-end loads usually start at 5% or 6% and decrease by 1% each year that the fund is invested. These fees, also known as contingent deferred sales charges, decrease over time, for example, starting at 6% and reducing to 4% at the beginning of the third year.

    • 12b-1 Charges: The SEC allows the managers of 12b-1 funds to use fund assets to pay for distribution costs such as advertising, promotional literature, and commissions paid to brokers who sell the fund to investors. These 12b-1 charges, along with front-end loads, are used to pay brokers. Like front-end charges, 12b-1 charges are deducted from the fund’s assets and limited to 1% of an annual fund’s average net assets per year.

    Many funds ofer “classes” that represent ownership in the same portfolio of securities, but with different combinations of fees. For instance look at the following table

      Class A Class B Class C
    Front-end load 0-4.5\% 0 0
    Back-end load 0 0-1\% 0
    12b-1 Fees 0.25\% 1.0\% 0
    Expense Ratio 0.70\% 0.70\% 0.70\%

    When investing in mutual funds, each investor must consider the cost and decide which combination of fees is best for them. Directly purchasing no-load funds from the mutual fund group is often the most cost-effective option, but many investors choose to pay for financial advice and the accompanying commission to their adviser.

    For investors who purchase funds through a broker, the choice between a front-end load and an annual 12b-1 fee will depend on their expected time horizon. Front-end loads are paid only once, while 12b-1 fees are paid annually. For a long-term investment, a one-time front-end load may be more favorable.

  • Fees and Mutual Fund Returns

    The return on investment in a mutual fund is calculated as the change in the net asset value (NAV) plus any income distributions, such as dividends or capital gains, expressed as a percentage of the NAV at the beginning of the investment period.

    \[\text{Rate of Return} = \frac{\text{NAV}_1 - \text{NAV}_0 + \text{Income and capital gain distributions}}{\text{NAV}_0}\]
    • Example: if a fund has an initial NAV of $20 and makes an income distribution of $0.15 and a capital gain distribution of $0.05, and ends with an NAV of $20.10, the monthly rate of return would be calculated as:
    \[\text{Rate of return} = \frac{20.10 - 20.00 + 0.15 + 0.05}{20.00} = 0.015 \text{ or } 1.5\%\]

    It’s important to note that the rate of return is also impacted by the fund’s expenses and 12b-1 fees, which are periodically deducted from the portfolio. The net return is the gross return on the underlying portfolio minus the total expense ratio.

  • Taxation of Mutual Funds Income

    Mutual funds in the US have “pass-through status” under the tax code. This means that taxes are only paid by the investors in the fund, not the fund itself. The income is treated as passed through to the investors as long as the fund meets certain requirements, such as distributing almost all its income to shareholders. The short-term capital gains, long-term capital gains, and dividends are passed through to investors as though they earned the income directly.

    However, this pass-through status has a disadvantage for individual investors as they have no control over the timing of the sale of securities from the fund’s portfolio, reducing their ability to manage their tax liabilities. A fund with a high portfolio turnover rate can be particularly tax-inefficient as a high turnover rate means that capital gains or losses are constantly being realized, leaving the investor unable to time their realization for tax management.

    Index funds, on the other hand, have low turnover rates, making them tax-efficient and economical in terms of trading costs.

  • Exchange-Traded Funds

    Exchange-Traded Funds (ETFs) are a type of investment product that operates like a mutual fund but can be traded like a stock. They were first introduced in 1993 and the first ETF, “SPDR” (Standard & Poor’s Depository Receipt), tracks the S&P 500 Index.

    Unlike mutual funds, which can only be traded at the end of the day after the net asset value (NAV) is calculated, ETFs can be bought and sold throughout the day, just like a stock. This gives investors greater flexibility and control over their investments.

    ETFs are also available for various indexes such as the Dow Jones Industrial Average (DIA) or the NASDAQ 100 Index (QQQ). There are many different ETFs and sponsors to choose from, and they are listed in the following image for easy reference.

    As of 2012, over 1,100 US ETFs had over $1 trillion invested in them. The majority of these ETFs track broad indexes, but there are also ETFs that focus on specific industries and commodities. The fastest growing sector of ETFs is commodities, with gold and silver ETFs being particularly popular. The growth of ETFs can be seen in the rapid increase of investment in this market, as shown in following figure.

    Barclays Global Investors was a market leader in the Exchange-Traded Fund (ETF) market, offering the popular product named “iShares”. In 2009, Barclays merged with Blackrock and iShares has since operated under the Blackrock name. Blackrock is a major sponsor of equity index funds and industry sector funds, both in the US and globally. To learn more, visit iShares.com.

    Exotic variations of ETFs, such as exchange-traded notes (ETNs) or exchange-traded vehicles (ETVs), are known as synthetic ETFs. These are debt securities that pay off based on the performance of an index, often measuring the performance of an illiquid and thinly traded asset class. Instead of directly investing in these assets, the ETF gains exposure through a “total return swap” with an investment bank. The bank agrees to pay the ETF the return on the index in exchange for a fixed fee. However, this approach becomes controversial during financial stress, as the ETF may be exposed to risk and the bank may be unable to fulfill its obligation, leaving investors without the promised return.

    ETFs have several advantages compared to mutual funds, including:

    • Continuous trading: ETFs can be bought and sold continuously, while mutual funds have a single net asset value quoted at the end of each day.
    • Trading options: ETFs can be sold short or bought on margin, similar to other stocks, while this is not possible with mutual funds.
    • Tax advantages: When many mutual fund investors redeem their shares, the fund may need to sell securities to meet these redemptions, which can result in capital gains taxes passed on to remaining shareholders. On the other hand, if small investors wish to redeem their ETF shares, they can simply sell them to other traders. Large investors can also exchange their ETF shares for shares in the underlying portfolio, avoiding any tax implications.

    ETFs have some disadvantages compared to mutual funds, including:

    • Commission fees: Unlike mutual funds, which can be purchased for free from no-load funds, ETFs must be purchased from brokers for a fee.
    • Price fluctuations: The prices of ETFs can deviate from their net asset value (NAV) as they are traded as securities. Although this difference is usually small and only happens for a short period, it can become unpredictable during market stress. This can easily offset the cost advantage that ETFs offer.

    In times of market distress, it can be difficult to accurately measure the NAV of ETFs, especially those that track less liquid assets. Additionally, some ETFs may only have a small number of dealers supporting them. If these dealers exit the market, the price of the ETF can fluctuate wildly.

ETF sponsors and products
Growth of US ETFs overtime

  • Information on Mutual Funds

    There are three main resources for finding information about mutual funds:

    • The first resource is the mutual fund’s prospectus. The SEC requires mutual funds to have a prospectus that includes a concise “Statement of Investment Objectives” and a detailed discussion of investment policies and risks, information about the investment adviser and portfolio manager, and a fee table that covers the costs associated with purchasing shares, front-end and back-end loads, and annual operating expenses such as management fees and 12b-1 fees.

    • The second resource is the Statement of Additional Information (SAI), also known as Part B of the prospectus. It includes a list of the securities in the portfolio at the end of the fiscal year, audited financial statements, a list of the directors and officers of the fund and their personal investments in the fund, and data on brokerage commissions paid by the fund. Unlike the prospectus, the SAI is not automatically provided to investors and must be requested specifically.

    • The third resource is the fund’s Annual Report, which includes portfolio composition and financial statements and a discussion of factors that influenced the fund’s performance over the last reporting period.

    Several publications offer comprehensive information about mutual funds. One such source is Morningstar’s Mutual Fund Sourcebook, which can be found at www.morningstar.com. Another good source is Yahoo’s finance site at finance.yahoo.com/funds. The Investment Company Institute (www.ici.org), the national association of mutual funds, closed-end funds, and unit investment trusts, also publishes an annual Directory of Mutual Funds that includes information on fees and contact information for the funds.

  • Risk, Return, and Historical Record

    • Level of Interest Rates

      The level of interest rates is a crucial factor in investment decisions and determining the future value of investments. However, forecasting interest rates is a challenging task in macroeconomics.

      There are three main factors that determine the level of interest rates:

      • The supply of funds from households as savers
      • The demand for funds from businesses for investment in real assets or capital information
      • The government’s net demand for funds, which can be affected by the actions of the Federal Reserve Bank.

    • Real and Nominal Interest Rates

      The interest rate is the promised rate of return on an investment expressed in a specific unit of account (such as US dollars, yen, euros, etc.) over a specified time period (such as a month, year, 20 years, etc.).

      It is important to note that an interest rate that is considered risk-free in one unit of account and time period may not be risk-free for another unit or period. For example, an interest rate that is considered safe in US dollars may be risky when evaluated in terms of purchasing power due to inflation uncertainty.

      The real return on an investment depends not only on the promised interest rate but also on the purchasing power of the money received. The Consumer Price Index (CPI) measures the purchasing power of money by averaging the prices of goods and services in the consumption basket of an average urban family of four. If the rate of inflation is 6%, it means that the purchasing power of money is reduced by 6% each year.

      For example, if you invest $1000 with an interest rate of 10% (default-free), at the end of the year you would receive $1100. However, if the rate of inflation is 6%, your real return on the investment would actually be 4%.

      Nominal interest rate, \(nr\), is the growth rate of the investment and Real interest rate, \(rr\), is the growth rate of purchasing power. Assume \(i\) denotes the inflation rate, then

      \[rr \approx rn -i\]

      In words, the real rate of interest is the nominal rate reduced by the loss of purchasing power. the exact relation is in fact given by

      \[1 + rr = \frac{1+rn}{1+i}\]

      This basically says that \((1+rr)(1+i)\) is the real rate with considering the inflation and \(1+rn\) is the money you earn. working out the mass, we obtian

      \[rr = \frac{rn-i}{1+i} \approx rn-i -i (rn-i ) + \frac{1}{2}i^2(rn-i)+ ...\]

      where we have expanded the fraction, assuming \(i \ll 1\).

    The conventional certificate of deposit provides a guaranteed nominal interest rate. However, to determine the expected real rate of investment, one has to adjust for the expected inflation rate. The inflation rate is published by the Bureau of Labor Statistics (BLS). However, the future real rate is uncertain and one must rely on expectations. This means that the future inflation rate is uncertain and therefore the real rate of return is also uncertain, even if the nominal rate of return is risk-free.

  • The Equilibrium Real Rate of Interest

    The nominal interest rate is determined by four factors: supply, demand, government actions, and expected inflation rate. The real rate of interest, which is the rate that takes into account the purchasing power of money, is determined by the first three factors: supply, demand, and government actions. The real interest rate is the foundation of the nominal interest rate, which is the real rate plus the expected rate of inflation.

    The supply and demand of funds determine the real rate of interest and can be represented by a supply and demand curve. The supply curve shows that as the amount of money invested increases, the interest rate will also increase. The demand curve shows that as the demand for funds from businesses increases, the interest rate will decrease. The intersection of these two curves represents the equilibrium real interest rate.

    Note that the future rate of inflation is unknown and is considered a risky factor, which in turn makes the real rate of return uncertain.

    The real interest rate is determined by the supply of funds from savers, the demand for funds from businesses, and government actions. The government and central bank (Federal Reserve) can influence these factors through fiscal and monetary policies, causing the real interest rate to change. For example, an increase in the government’s budget deficit will shift the demand curve to the right, causing the real interest rate to rise. The Fed can counteract this rise by implementing an expansionary monetary policy, which will shift the supply curve to the right. Therefore, while the real interest rate is primarily determined by the propensity of households to save and the expected profitability of investments in physical capital, it can also be influenced by government fiscal and monetary policies.

Determination of the Equilibrium real rate of interest

  • The Equilibrium Nominal Rate of Interest

    We know that nominal rate of interest is the real rate of interest plux inflation.

    Irving Fisher (1930) argues that the nominal rate ought to increase one-to-one with the Expected inflation. The fisher equation is as

    \[rn = rr + E(i)\]

    meaning that when the real rates are stable, changes in the nominal rates ought to predict the changes in the inflation rates. This claim has got mixed results empirically and is under the debate. Nominal rates can be viewed as the sum of the required real rate on nominally risk-free assets plus a noisy forecast of inflation.

  • Tax and the Real Rate of Interest

    Tax liabilities are calculated based on nominal income and the tax rate is determined by the investor’s tax bracket. In 1986, the Tax Reform Act addressed the issue of “bracket creep,” where taxpayers move into higher tax brackets due to the growth of nominal income due to inflation.

    However, index-linked tax brackets do not address the effect of inflation on the taxation of savings. With a tax rate of \(t\) and a nominal interest rate of \(rn\), the after-tax interest is \(rn (1-t)\). The real after-tax rate is approximately the after-tax nominal rate minus the inflation rate, calculated as \((rr + i)(1-t) - i = rr(1-t) -it\). This means that as inflation increases, the after-tax real rate decreases.

    For example, suppose you have an investment with yield of \(12\%\), with inflation of \(8\%\). So you real rate is approximately \(4\%\). In an inflation-protected tax system, after taxes a real return of \(4\% (1-0.3) = 2.8\%\). But tax code does not recognizes the first 8\% of your return is just compensation for inflation–not real income–and hence your after tax return is reduced by \(8\% \times 0.3 = 2.4\%\), so that your after-tax real interest rate, at \(4 \%\) is almost wiped out.

  • Rates of Return for Different Holding Periods

    Consider a safe investment in US Treasury securities with zero-coupon and several different maturities. T-bills are sold at a discount from par value and provide their entire return from the difference between the purchase price and the ultimate return payment. Give the price \(p(T)\), for a Treasury bond of \(100\$\) par value and maturity of \(T\) years, the risk free return available for a horizon of \(T\) years is as

    \[r_f (T) = \frac{100}{p(T)} -1\]

    For \(T=1\), this equation provides the risk free for an investment with a horizon of 1 year.

    Obviously, longer horizons, provide greater total returns. The question then arises that how should we compare investment returns with different maturities? We typically express all investment returns as an effective annual rate (EAR), defined as the percentage increase in funds invested over a 1-year horizon.

    For a 1-year investment, the EAR equals the total return \(r_f(1)\), and the gross return \((1 + \text{EAR} )\) s the terminal value of \(1\$\) investment. Now suppose for a 6-month bill, with \(2.71\%\) return, over two semi-annual periods, we obtain \(1 +\text{EAR} = (1.0271)^2 = 1.0549\) implying that \(\text{EAR} = 5.49\%\). Now, suppose for a 25-year bind, the return is \(329.18\%\). The \(\text{EAR}\), is then obtained as

    \[( 1 + \text{EAR})^{25} = 1 +3.2918\] \[1 + \text{EAR} = 4.2918^{1/25} = 1.0600\]

    meaning that the value of return over a year is \(\text{EAR} = 6\%\). Generally when the total return is \(r_f(T)\), over a holding with period length of \(T\), we obtain \(\text{EAR}\), as

    \[1+ \text{EAR} = [1+r_f(T)]^{1/T}\]

  • Annual Percentage Rates

    For shor-term investments \(T<1\), annulaized return rates oftern are reported using a simple, rather than compound interes, called Annual Percentage Rates or APR. For example, the APR corresponding to a monthly rate such as that charged on a credit card is reported as \(12\) times the monthly rate. This means that, if there are \(n\) compounding periods in a year, and the per-period rate is \(r_f(T)\), then \(\text{APR} = n \times r_f(T)\). Conversely, you can find the per-period rate from the \(\text{APR}\) as \(r_f(T) = T \times \text{APR}\).

    • Example: For a 6-month rate of \(2.71 \%\), discussed above, the \(\text{APR}\), is \(2\times 2.71 = 5.42 \%\). Compare this with the compound rate of \(5.49 \%\).

    Note that for a short-term investment of length \(T\), we have \(n= 1/T\) compounding periods. To generalize the relation between \(\text{APR}\) and \(\text{EAR}\) for a short-term investment of length \(T\) we have

    \[1 + \text{EAR} = [ 1+ r_f(T)]^n = [ 1+ r_f(T)]^{1/T} = [ 1+ T\times \text{APR}]^{1/T}\]

    or equivalently,

    \[\text{APR} = \frac{(1+\text{EAR})^T-1}{T}\]

  • Continuous Compounding

    It is evident that the difference between \(\text{APR}\) and \(\text{EAR}\), grows with frequncy of compounding. This raises the question, how far will the two rates diverge as the compounding frequency continuous to grow.

    First lets write a python code, that compares these two values. We basically, fix the \(\text{EAR} = 5\%\), and then find the \(\text{APR}\) that we get for different investment period. We change the investment period from 1-day to 1-year. For the second plot, we fix the value of \(\text{EAR}\) at \(5\%\) and then see what values we get for the \(\text{APR}\). Note that, as is shown in [[fig-6-0][this figure]], the lne for \(\text{APR}\) always is lower than \(\text{EAR}\).

    Let’s see if we can find a mathematical formulae for that. As \(T\) approaches zero, we effectively are reaching the continuous compounding (CC), and the relation between \(\text{EAR}\) to an annual percentage rate \(\text{APR} = r_{cc}\) (since it is a continuous compound), is given aby an exponential as

    \[1 + \text{EAR} = \lim_{T\rightarrow 0} (1 + T\times r_{cc})^{1/T} = \exp (r_{cc})\]

    This means that the continuous annual percentage rate \(r_{cc}\) is as

    \[r_{cc} = \log ( 1+ \text{EAR})\]

    where \(\log()\) is the natural logarithmic. So the maximum difference would be what we just calculated. For example, if the \(\text{EAR} = 5\%\), then \(r_{cc} = 4.88\%\) for the continuous limit, which can be seen in the left of next figure.

    One good application of \(r_{cc}\), is in calculating the total return for any period \(T\), this is simply \(\exp (T\times r_{cc} )\), since

    \[1 + \text{EAR}= \exp(r_{cc})\] \[(1+\text{EAR})^T = \exp(T\times r_{cc})\] 


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import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
import numpy as np
fig=plt.figure(figsize=(15,4));
# T varies between a day (T = 1/365) up to a year (T=1).
T = np.arange(0.0027, 1.0, 0.001);
TT = [0.0027, 0.0833, 0.25, 0.50, 1.0];
Tlabel = ['1 day', '1 mon', '1 quar', '6 mon', '1 yr ' ];
# Calculating APR for a fixed EAR = 5%
ear = 0.05;
apr = (np.power((1+ear),T)-1)/T;
plt.subplot(1,2,1)
plt.plot(T,apr*100, 'r--')
plt.plot(T, np.ones(np.size(T))*ear*100, 'b--')
plt.ylabel('APR and EAR (%)')
plt.title('EAR is fixed at %s' %ear)
plt.xlabel('T varies between a day to a year')
plt.xticks(TT, Tlabel, rotation='vertical')
plt.subplots_adjust(bottom=0.5)
plt.grid()
# Calculating EAR based on a fix APR = 5%
apr = 0.05;
ear = np.power((1+apr*T),1/T) -1 ;
plt.subplot(1,2,2)
EAR, = plt.plot(T,ear*100, 'b--', label='EAR')
APR, = plt.plot(T, np.ones(np.size(T))*apr*100, 'r--',label='APR')
plt.title('APR is fixed at %s' %apr)
plt.xlabel('T varies between a day to a year')
plt.xticks(TT, Tlabel, rotation='vertical')
plt.subplots_adjust(bottom=0.5)
plt.grid()
plt.legend(handles=[APR, EAR]);
# Saving and showing the result
plt.savefig('img/apr-for-fixed-ear.png',bbox_inches='tight')
#'img/apr-for-fixed-ear.png' # return this to org-mode
The figure displays the difference between the Annual Percentage Rate (APR) and the Effective Annual Rate (EAR) for T, for investment periods ranging from 1 day to 1 year. The APR is represented in red and the EAR is represented in blue.

  • Bills and Inflation 1926-2012

    Financial time series often begin in July 1926 when Center for Research in Security Prices at University of Chicago started a widely used accurate return database.

    Looking at the next figure, we see the average annual rates for the various series. The average interest rate over the most recent half (i.e. 1969-2012), is \(5.25\%\), which is much higher than in the earlier half \(1.79\%\). The reason is inflation, which is the main driver of T-bill rates. Inflation, over the recent half has been \(4.26 \%\), compared to the first half of \(1.74 \%\). The average real rate, taking out the inflation from the nominal rate, for the recent half is still higher than the first half. Compare \(0.10\%\) with \(0.95\%\) for the recent half. Note that the real and nominal half are related as

    \[1 + r(\text{real}) = \frac{1 + r(\text{nominal})}{1 + i(\text{inflation})}\]
Statistics for T-bill rates, inflation rates, and real rates, First half is 1926-1965 and recent half is 1969-2012.

  • History lesson:

    A moderate rate of inflation can reduce the nominal gains from low-risk investments, such as T-bills, to a significant extent. In both halves of the sample, the real return was less than one-fifth of the nominal return.

    The Wealth Index measures the cumulative gain from an investment over a certain period. It assumes a starting investment of $1 and compounds the investment value each year by adding the gross annual rate of return to 1. The Wealth Index at the end of the investment period shows the total increase in wealth per dollar invested.

    The previous figure illustrates the wealth index of a $1 investment in T-bills at the start of 1970. The nominal wealth index shows an impressive growth of $9.20 by 2012, but the inflation-adjusted real wealth index is only $1.20. Similarly, a $1 investment in T-bills from 1926 to 2012 would have reached $20.25 (as shown in the inset), but the inflation-adjusted real value would only be $1.55.

    Next figure shows that the standard deviation (SD) shows a lower SD of inflation in recent half \(2.82\%\), than in the earlier period \(4.66\%\). This is contributed to a lower standard deviation of the real rate in the recent half of teh sample, \(2.44 \%\), compared with the earlier half \(4.98 \%\). Notice that the SD of the nominal rate actually was higher in the recent period \(3.02\%\) than the earlier period \(1.56\%\), indicating that the lower variation in realized real returns must be attributed to T-bill rates more closely tracking inflation in that period.

    Investors presumably focus on the real returns they can earn on their investments. For them to realize acceptable real rate, they must earn a higher nominal rate when the inflation is excpected to be higher. Therefore, the nominal T-bill rate at the begining of a period should reflect anticipation of the inflation over that period. When the real rate is stable and realized inflation matches initial expections, the correclation between inflation and nominal T-bill rates will be close to perfect 1.0, while the correlation between inflation and the realized real rate will be close to zero.

    At the other extreme, if investors are ignoring or could not project the inflation at all, the correlation between inflation and nominal T-bill rates would be zero, and the correlation between inflation and real rates would be -1.0, since the real rate would then fall one-for-one with any increase in inflation.

    Looking the figure after, we see that the accuracy of investors predicting the inflation has increased. The correlation between inflation and nominal T-bill rate has increased from \(-0.03\) to \(0.48\) (perfectly it would be \(1.0\) ), and the correlation between inflation and real T-bill rate has approached the perfect value of \(0.0\) (it has changed from \(-0.98\) to \(-0.67\)).

Nominal and real wealth indexes for investment in T-bil 1970-2012 (inset figure is for 1926-2012)
Correlations between inflation & real T-bill rate and nominal T-bill rates in two period

  • Risk and Risk Premiums

    • Holding Period Returns

      Assume you invest in a stock-index fund, where shares currently sells for \(100\\)$ per share. With an investment horizon of 1-year, the realized rate of return depends on (i) the price per share at year’s end and (ii) the cash dividendsyou will collect over the year.

      In this example, if the price of the share at year’s end is \(110\\)$ and cash dividends over the year amount to \(4\\)$, the realized return or holding period return (HPR), is defined as

      \[\text{HPR} = \frac{\text{Ending price of a share} - \text{Beginning price of a share} + \text{Cash dividends}}{\text{Beginning price of a share}}\]

      Here we have

      \[\text{HPR} = \frac{110 -100 + 4 }{100 } = 0.14 \text{ or } 14\%\]

      HPR treats the dividend as paid at the end of the holdin period. When dividends are received earlier, the HPR inores investment income between the receipt of the payment and the end of the holding period. The percent return from dividends is called the dividend yield, and so dividend yield plus the rate of capital gains equals HPR.

    • Expected Return and Standard Deviation

      There is uncertainty in the share’s price plus its devidend after 1-year. We can quantify the belief about the state of the market and the stock-index funds in terms of probabilities. Suppose, each state has a HPR of \(r_i\) with probability of \(p_i\). The expected return is as

      \[E(r) = \sum_i p_i r_i\]

      The standard deviation of the rate of return \(\sigma\) is a measure of risk. It is defined as the square root of the variance, which in turn is the expected value of the squared deviations from the expected return. The higher volatility in outcomes, the higher will be the average value of these squared deviations. Therefore the variance and standard deviation, provide a measure of the uncertainty of outcomes. The variance is defined as

      \[\sigma^2 = \sum_i p_i (r_i - E(r))^2\]

      What troubles the potential investors in the index fund is the downside risk of a crash or poor market, not the upside potential of a good or excellent market. As long as the probability distribution is more or less symmetric about the mean, \(\sigma\) is a reasonable measure of risk.

    • Excess Returns and Risk Premiums

      The main question for investors, is that: how much, if anything, should i invest in an index fund? We should look at the amount of expected reward offered for the risk involved in investing money in stocks!

      The reward is measured as the difference between the expected HPR on the index stock and the risk free rate, that is the rate you can earn by leaving money in risk-free assets such as T-bills, money market funds, or the bank. This difference is called risk premium on common stocks.

      For example if the risk-free rate is \(4\%\) and the expected index fund return is \(10\%\), then the risk premium on stocks is \(6\%\).

      The difference in any particular period between the actual rate of return on a risky asset and the actual risk-free rate is called excess return. Therefore the risk premium is the expected value of the excess return.

      The degree to which investors are willing to commit funds to stocks depends on risk aversion. Investors are risk averse in the sense that, if the risk premium were zero, they would not invest any money in stocks. In theory, there must always be a positive risk premium on stocks in order to induce risk-averse investors to hold the existin supply of stocks instead of placin all their money in risk-free assets.

  • Time Series Analysis of Past Rates of Return

    In forward looking scenario analysis, we determine a set of scenarios and associated investment rates of return, assign probabilities to each, and conclude by computing the risk premium (reward) and standard deviation (risk).

    In historical looking, we have only dates and associated HPRs. We must infer from this limited data the probability distribution from which these returns might have been drawn or, at least, expected return and standard deviation.

  • Expected Returns and the Arithmetic Average

    When we use historical data, we treat each observation as an equally likely “scenario”. This would result in arithmitic average of rates of return as

    \[E(r) = \sum_{i=1}^n p_i r_i = \frac{1}{n}\sum_{i=1}^n r_i\]

    If the time series of historical returns fairly represents the true underlying probability distribution, then the arithmetic average return from a historical period provides a forecast of the investmen’t expected future of HPR.

  • The Geometric (Time-Weighted) Average Return

    An intuitive measure of performance over the sample period is the (fixed) annual HPR that would compound over the period to the same terminal value as obtained from the sequence of actual returns in the time series.

    \[\text{Terminal Value} = (1 + r_1) \times (1+r_2) \times \ldots \times (1+r_5) = (1+g)^n\] \[g = \text{Terminal Value}^{1/n}\]

    where \(1+ g\) is the geometric average of the gross returns \(1+r\) from the time series and \(g\) is the annual HPR that would replicate the final value of our investment.

    \(g\) is known as the time-weighted average return to emphasize that each past return receives an equal weight in the process of averaging.

    The larger the swings in rates of return, the larger the discrepency between *the arithmetic averages and the geometric averages, that is, between the compound rate earned over the sample period and the average of the annual returns.

    If returns come from a normal distribution, the expected difference is exactly half the variance of the distribution, that is

    \[E[\text{Geometric average}] = E[\text{Arithmetic average}] - \frac{1}{2} \sigma^2\]

  • Variance and Standard Deviation

    In risk, we are interested in the likelihood of deviations from the expected return. Since, in practice we cannot directly observe expectations, so we estimate the variance by averaging the squared deviations from our estimate of the expected return, therefore:

    \[\text{Variance} = \sigma^2 = \sum p_i ( r_i - E(r))^2\]

    Using the historical data with \(n\) observation, we could estimate the variance as

    \[\hat{\sigma}^2 = \frac{1}{n} \sum_i (r_i - \bar{r})^2\]

    where $\hat{\sigma}$ replaces $\sigma$ to denote that it is an estimate. The variance estimate here, is biased downward. The reason is that we have taken deviations from the sample arithmetic average, \(\bar{r}\), instead of the unknown, true expected value \(E(r)\), and so have introduced an estimation error. Its effect on the estimated variance is sometimes called a degrees of freedom bias. We can eliminate the bias by multiplying the arithmetic average of squared deviations by the factor of \(n/(n-1)\). The variance and standard deviation then becomes

    \[\hat{\sigma}^2 = \frac{1}{n-1} \sum_i (r_i -\bar{r})^2\]

    For a large sample, \(n\) becomes large and, \(n/(n-1)\) approaches \(1\), and the adjustment for degrees of freedom becomes trivially small.

  • Mean and Standard Deviation Estimates from Higher Frequency Observations

    Does increasing the number of observations lead to more accurate estimates? The answer may surprise you: the frequency of observations has no impact on the accuracy of mean estimates. Instead, it is the duration of the sample time series (not the number of observations) that improves accuracy.

    For example, the average annual return estimated by dividing the total 10-year return by 10 is as accurate as the average of 120 monthly returns multiplied by 12. The average monthly return should align with the average of the 10-year return. If the probability distribution of returns remains unchanged, a longer sample, such as 100 years, provides a more accurate estimate of the mean return than a 10-year sample.

    Therefore, the rule of thumb is to use the longest sample that still comes from the same return distribution. However, in practice, old data may not be as informative and could limit the accuracy of mean return estimates.

    In contrast to the mean, the accuracy of estimates of the standard deviation and higher moments (computed using deviations from the average) can be improved by increasing the number of observations. Therefore, the accuracy of estimates of the SD and higher moments of the distribution can be improved by using more frequent observations.

    When monthly returns are uncorrelated from one month to another month, monthly variances simply add up. Thus, if the variance is the same every month, we annualize by \(\sigma_A^2 = 12 \sigma_M^2\). In general, the \(T\)-month variance is \(T\) times the \(1\)-month variance. Consequently the standard deviation grows at the rate of \(\sqrt{T}\), i.e. \(\sigma_A = \sqrt{12} \sigma_M\). *The mean and variance grow in direct proportion to time, SD grows at the rate of square root of time.

  • The Reward-to-Volatility (Sharpe) Ratio

    Finally, it is worth noting that investors presumably are interested in the expected excess return they can earn by replacing T-bills with a risky portfolio, as well as the risk they would thereby incur. While the T-bill rate is not constant over the entire period, we still know with certainty what nominal rate we will earn if we purchase a bill and hold it to maturity. Other investments typically entail accepting some risk in return for the prospect of earning more than the safe T-bill rate. Investors price risky assets so that the risk premium will be commensurate with the risk of that expected excess return, and hence it’s best to measure risk by the standard deviation of excess, not total, returns.

    The importance of the trade-off between reward (the risk premium) and risk ( measured by standard deviation or SD), suggests that we measure the attraction of a portfolio by the ratio of risk premium to SD of excess returns.

    \[\text{Sharpe Ratio} = \frac{\text{Risk Premium}}{\text{SD of Excess Return}}\]

    The reward-to-volatility measure (proposed by William Sharpe and hence called Sharpe Ratio or SR) is widely used to evaluate the performance of investment managers.

    Notice that the Sharpe Ratio, devides the risk premium (grows directly with time) by the standard deviation (grows proportional to square root of time). Therefore Sharpe Ratio (SR) grows proportional to square root of time, meaning that if we want to annualize SR from monthly value, we multiply that by \(\sqrt{12}\) or \(SR_A = SR_M \sqrt{12}\).

  • The Normal Distribution

    The bell-shaped normal distribution appears naturally in many applications. In fact, many variables that are the end result of multiple random influences will exhibit a normal distribution.

    If return expectations implicit in asset prices are rational, actual rates of return should be normally distributed around these expectations.

    Next figure shows a normal distribution profile. Decreasing \(\sigma\) results in tight figure around the average. The two parameters controlling this distributin are the average and the standard deviation.

The normal distribution

  • Deviations from Normality and Risk Measures

    to be cont’d