Financial Economics summary, Dispense di Economia Finanziaria

Scarica Financial Economics summary e più Dispense in PDF di Economia Finanziaria solo su Docsity! 1 Chapter 1 The Investment Environment 1.1 Real Assets versus Financial Assets The material wealth of a society is ultimately determined by the productive capacity of its economy, that is, the goods and services its members can create. This capacity is a function of the real assets of the economy: the land, buildings, machines, and knowledge that can be used to produce goods and services. In contrast to real assets are financial assets such as stocks and bonds. Such securities are no more than sheets of paper or, more likely, computer entries, and they do not contribute directly to the productive capacity of the economy. Instead, these assets are the means by which individuals in well-developed economies hold their claims on real assets. Financial assets are claims to the income generated by real assets (or claims on income from the government). While real assets generate net income to the economy, financial assets simply define the allocation of income or wealth among investors. 1.2 Financial Assets It is common to distinguish among three broad types of financial assets: 1. Fixed-income or debt securities: they promise either a fixed stream of income or a stream of income determined by a specified formula. They come in a tremendous variety of maturities and payment provisions. At one extreme, the money market refers to debt securities that are short term, highly marketable, and generally of very low risk, for example, U.S. Treasury bills or bank certificates of deposit (CDs). In contrast, the fixed-income capital market includes long-term securities such as Treasury bonds, as well as bonds issued by federal agencies, state and local municipalities, and corporations. These bonds range from very safe in terms of default risk (e.g., Treasury securities) to relatively risky (e.g., high-yield or “junk” bonds). 2. Equity: unlike debt securities, common stock, or equity, in a firm represents an ownership share in the corporation. Equity holders are not promised any particular payment. They receive any dividends the firm may pay and have prorated ownership in the real assets of the firm. The performance of equity investments is tied directly to the success of the firm and its real assets. For this reason, equity investments tend to be riskier than investments in debt securities . 3. Derivative securities: derivative securities such as options and futures contracts provide payoffs that are determined by the prices of other assets such as bond or stock prices. One use of derivatives, perhaps the primary use, is to hedge risks or transfer them to other parties. Derivatives also can be used to take highly speculative positions. 2 1.3 Financial Markets and the Economy The Informational Role of Financial Markets Stock prices reflect investors’ collective assessment of a firm’s current performance and future prospects. When the market is more optimistic about the firm, its share price will rise. That higher price makes it easier for the firm to raise capital and therefore encourages investment. In this manner, stock prices play a major role in the allocation of capital in market economies, directing capital to the firms and applications with the greatest perceived potential. No one knows with certainty which ventures will succeed and which will fail. It is therefore unreasonable to expect that markets will never make mistakes. The stock market encourages allocation of capital to those firms that appear at the time to have the best prospects. Many smart, well-trained, and well-paid professionals analyze the prospects of firms whose shares trade on the stock market. Stock prices reflect their collective judgment. Consumption Timing How can you shift your purchasing power from high-earnings to low-earnings periods of life? One way is to “store” your wealth in financial assets. In high-earnings periods, you can invest your savings in financial assets such as stocks and bonds. In low-earnings periods, you can sell these assets to provide funds for your consumption needs. Financial markets allow individuals to separate decisions concerning current consumption from constraints that otherwise would be imposed by current earnings. Because the bonds promise to provide a fixed payment, the stockholders bear most of the business risk but reap potentially higher rewards. Thus, capital markets allow the risk that is inherent to all investments to be borne by the investors most willing to bear that risk. This allocation of risk also benefits the firms that need to raise capital to finance their investments. When investors are able to select security types with the risk-return characteristics that best suit their preferences, each security can be sold for the best possible price. This facilitates the process of building the economy’s stock of real assets. 1.4 The Investment Process An investor’s portfolio is simply his collection of investment assets. Once the portfolio is established, it is updated or “rebalanced” by selling existing securities and using the proceeds to buy new securities, by investing additional funds to increase the overall size of the portfolio, or by selling securities to decrease the size of the portfolio. Investors make two types of decisions in constructing their portfolios. • Asset allocation decision: the choice among these broad asset classes • Security selection decision: the choice of which particular securities to hold within each asset class. “Top-down” portfolio construction starts with asset allocation. For example, an individual who currently holds all of his money in a bank account would first decide what proportion of the overall portfolio ought to be moved into stocks, bonds, and so on. Security analysis involves the valuation of particular securities that might be included in the portfolio. In contrast to top-down portfolio management is the “bottom-up” strategy. In this process, the portfolio is 5 Chapter 3 How Securities Are Traded 3.1 How Firms Issue Securities Investment bankers are generally hired to manage the sale of these securities in what is called a primary market for newly issued securities. Once these securities are issued, however, investors might well wish to trade them among themselves. For example, you may decide to raise cash by selling some of your shares in Apple to another investor. This transaction would have no impact on the total outstanding number of Apple shares. Trades in existing securities take place in the so-called secondary market. Privately Held Firms A privately held company is owned by a relatively small number of shareholders. Privately held firms have fewer obligations to release financial statements and other information to the public. When private firms wish to raise funds, they sell shares directly to institutional or wealthy investors in a private placement. However, unlike the public stock markets regulated by the SEC, these networks require little disclosure of financial information and provide correspondingly little oversight of the operations of the market. Publicly Traded Companies When a private firm decides that it wishes to raise capital from a wide range of investors, it may decide to go public. This means that it will sell its securities to the general public and allow those investors to freely trade those shares in established securities markets. This first issue of shares to the general public is called the firm’s initial public offering, or IPO. Later, the firm may go back to the public and issue additional shares. A seasoned equity offering is the sale of additional shares in firms that already are publicly traded. When the statement is in final form and accepted by the SEC, it is called the prospectus. At this point, the price at which the securities will be offered to the public is announced. Firm commitment: the issuing firm sells the securities to the underwriting syndicate for the public offering price less a spread that serves as compensation to the underwriters. Investment bankers manage the issuance of new securities to the public. Once the SEC has commented on the registration statement and a preliminary prospectus has been distributed to interested investors, the investment bankers organize road shows in which they travel around the country to publicize the imminent offering. These road shows serve two purposes. First, they generate interest among potential investors and provide information about the offering. Second, they provide information to the issuing firm and its underwriters about the price at which they will be able to market the securities. 6 3.2 How Securities Are Traded Types of Markets We can differentiate four types of markets: direct search markets, brokered markets, dealer markets, and auction markets. • Direct Search Markets: a direct search market is the least organized market. Buyers and sellers must seek each other out directly. Such markets are characterized by sporadic participation and nonstandard goods. Firms would find it difficult to profit by specializing in such an environment. • Brokered Markets: in markets where trading in a good is active, brokers find it profitable to offer search services to buyers and sellers. Brokers in particular markets develop specialized knowledge on valuing assets traded in that market. • Dealer Markets: when trading activity in a particular type of asset increases, dealer markets arise. Dealers specialize in various assets, purchase these assets for their own accounts, and later sell them for a profit from their inventory. The spreads between dealers’ buy (or “bid”) prices and sell (or “ask”) prices are a source of profit. • Auction Markets: the most integrated market is an auction market, in which all traders converge at one place (either physically or “electronically”) to buy or sell an asset. The New York Stock Exchange (NYSE) is an example of an auction market. An advantage of auction markets over 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 agreeable prices and save the bid–ask spread. Types of Orders There are two types of orders: market orders and orders contingent on price. Market Orders They are buy or sell orders that are to be executed immediately at current market prices. For example, our investor might call her broker and ask for the market price of Facebook. The broker might report back that the best bid price is $118.34 and the best ask price is $118.36, meaning that the investor would need to pay $118.36 to purchase a share, and could receive $118.34 a share if she wished to sell some of her own holdings of Facebook. The bid–ask spread in this case is $.02. So an order to buy 100 shares “at market” would result in purchase at $118.36, and an order to “sell at market” would be executed at $118.34. This simple scenario is subject to a few potential complications. First, the posted price quotes actually represent commitments to trade up to a specified number of shares. If the market order is for more than this number of shares, the order may be filled at multiple prices. For example, if the ask price is good for orders up to 300 shares and the investor wishes to purchase 500 shares, it may be necessary to pay a slightly higher price for the last 200 shares. Second, another trader may beat our investor to the quote, meaning that her order would then be executed at a worse price. Finally, the best price quote may change before her order arrives, again causing execution at a price different from the one quoted at the moment of the order. Price-Contingent Orders Investors also may place orders specifying prices at which they are willing to buy or sell a security. A limit buy order may instruct the broker to buy some number of shares if and when Facebook may be obtained at or below a stipulated price. A limit sell order instructs the broker to sell if and when the stock price rises above a specified limit. 7 A collection of limit orders waiting to be executed is called a limit order book. Trading Mechanisms An investor who wishes to buy or sell shares will place an order with a brokerage firm. The broker charges a commission for arranging the trade on the client’s behalf. Brokers have several avenues by which they can execute that trade, that is, find a buyer or seller and arrange for the shares to be exchanged. There are three trading systems employed in the United States: 1. Over-the-counter dealer markets 2. Electronic Communication Networks (ECNs): allow participants to post market and limit orders over computer networks. The limit-order book is available to all participants. 3. Specialist markets 3.7 Trading Costs Besides carrying out the basic services of executing orders, holding securities for safekeeping, extending margin loans, and facilitating short sales, brokers routinely provide information and advice relating to investment alternatives. Full-service brokers usually depend on a research staff that prepares analyses and forecasts of general economic as well as industry and company conditions and often makes specific buy or sell recommendations. Discount brokers, on the other hand, provide “no-frills” services. They buy and sell securities, hold them for safekeeping, offer margin loans, facilitate short sales, and that is all. The only information they provide about the securities they handle is price quotations. Besides the broker’s commission the trading costs include also: - dealer’s bid-ask spread - price concession of the investor to trade for huge quantities 3.8 Buying on Margin When purchasing securities, investors have easy access to a source of debt financing called broker’s call loans. The act of taking advantage of broker’s call loans is called buying on margin. Purchasing stocks on margin means the investor borrows part of the purchase price of the stock from a broker. The margin in the account is the portion of the purchase price contributed by the investor; the remainder is borrowed from the broker. If the stock value in Example 3.1 were to fall below $4,000, owners’ equity would become negative, meaning the value of the stock is no longer sufficient collateral to cover the loan from the broker. To guard against this possibility, the broker sets a maintenance margin. If the percentage margin falls below the maintenance level, the broker will issue a margin call, which requires the investor to add new cash or securities to the margin account. If the investor does not act, the broker may sell securities from the account to pay off enough of the loan to restore the percentage margin to an acceptable level. 10 Chapter 5 Risk, Return, and the Historical Record 5.2 Comparing Rates of Return for Different Holding Periods Zero-coupon bonds are sold at a discount from par value and provide their entire return from the difference between the purchase price and the ultimate repayment of par value. Given the price, P(T), of a Treasury bond with $100 par value and maturity of T years, we calculate the total risk-free return available for a horizon of T years as the percentage increase in the value of the investment. How should we compare returns on investments with differing horizons? This requires that we express each total return as a rate of return for a common period. 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. In general, we can relate EAR to the total return, rf(T), over a holding period of length T by using the following equation: Annual Percentage Rates Annualized rates on short-term investments (by convention, T < 1 year) often are reported using simple rather than compound interest. These are called annual percentage rates, or APRs. More generally, if there are n compounding periods in a year, and the per-period rate is rf (T ), then the APR = n × rf (T). Conversely, you can find the per-period rate from the APR as rf (T ) = APR/n, or equivalently, as T × APR. The relationship among the compounding period, the EAR, and the APR is: 11 Continuous Compounding The difference between APR and EAR grows with the frequency of compounding. What is the limit of [1 + T × APR]1/ T, as T gets ever smaller? As T approaches zero, we effectively approach continuous compounding (CC), and the relation of EAR to the annual percentage rate, denoted by rcc for the continuously compounded case, is given by the exponential function: To find rcc from the effective annual rate: While continuous compounding may at first seem to be a mathematical nuisance, working with such rates can sometimes simplify calculations of expected return and risk. For example, given a continuously compounded rate, the total return for any period T, rcc(T ), is simply exp (T × rcc ). In other words, the total return scales up in direct proportion to the time period, T. This is far simpler than working with the exponents that arise using discrete period compounding. 5.4 Risk and Risk Premiums The fund currently sells for $100 per share. With an investment horizon of 1 year, the realized rate of return on your investment will depend on (a) the price per share at year’s end and (b) the cash dividends you will collect over the year. The realized return, called the holding-period return, or HPR is defined as: This definition of the HPR treats the dividend as paid at the end of the holding period. When dividends are received earlier, the HPR should account for reinvestment 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 considerable uncertainty about the price of a share plus dividend income one year from now, however, so you cannot be sure about your eventual HPR. We will characterize probability distributions of rates of return by their expected or mean return, E(r), and standard deviation, σ. The expected rate of return is a probability-weighted average of the rates of return in each scenario. Calling p(s) the probability of each scenario and r(s) the HPR in each scenario, where scenarios are labeled or “indexed” by s, we write the expected return as: The variance of the rate of return (σ2 ) is a measure of volatility. It measures the dispersion of possible outcomes around the expected value. The higher the dispersion of outcomes, the higher will be the average value of these squared deviations. 12 Therefore, variance is a natural measure of uncertainty. Excess Returns and Risk Premiums Excess return = actual HPR – risk-free rate Risk premium = expected HPR – risk-free rate The risk premium is the expected value of the excess return, and the standard deviation of the excess return is a measure of its risk. The degree to which investors are willing to commit funds to stocks depends on their risk aversion. Investors are risk averse in the sense that, if the risk premium were zero, they would not invest any money in stocks. 5.5 Time Series Analysis of Past Rates of Return Time Series versus Scenario Analysis In a forward-looking scenario analysis, we determine a set of relevant scenarios and associated investment rates of return, assign probabilities to each, and conclude by computing the risk premium (reward) and standard deviation (risk) of the proposed investment. When we use historical data, we treat each observation as an equally likely “scenario.” So if there are n observations, we substitute equal probabilities of 1/n for each p (s). The expected return, E(r), is then estimated by the arithmetic average of the sample rates of return: The arithmetic average provides an unbiased estimate of the expected future return. But what does the time series tell us about the actual performance of a portfolio over the past sample period? Column F in Spreadsheet 5.2 shows the investor’s “wealth index” from investing $1 in an S&P 500 index fund at the beginning of the first year. Wealth in each year increases by the “gross return,” that is, by the multiple (1 + HPR), shown in column E. The wealth index is the cumulative value of $1 invested at the beginning of the sample period. The value of the wealth index at the end of the fifth year, $1.0275, is the terminal value of the 15 Chapter 6 Capital Allocation to Risky Assets 6.1 Risk and Risk Aversion Risk, Speculation, and Gambling • Speculation: assumption of considerable investment risk to obtain commensurate gain. Considerable risk: risk sufficient to affect the decision. Commensurate gain: positive risk premium, that is, an expected return greater than the risk-free alternative. • Gamble: assumption of risk for enjoyment of the risk itself To turn a gamble into a speculative venture requires an adequate risk premium to compensate risk-averse investors for the risks they bear. Hence, risk aversion and speculation are consistent. Notice that a risky investment with a risk premium of zero, sometimes called a fair game, amounts to a gamble because there is no expected gain to compensate for the risk entailed. A risk-averse investor will reject gambles, but not necessarily speculative positions. Risk Aversion and Utility Values Investors who are risk averse reject investment portfolios that are fair games or worse. Risk-averse investors consider only risk-free or speculative prospects with positive risk premiums. Intuitively, a portfolio is more attractive when its expected return is higher and its risk is lower. But when risk increases along with return, the most attractive portfolio is not obvious. We will assume that each investor can assign a welfare, or utility, score to competing portfolios on the basis of the expected return and risk of those portfolios. where U is the utility value and A is an index of the investor’s risk aversion. The factor of ½ is just a scaling convention. To use this equation, rates of return must be expressed as decimals rather than percentages. We can interpret the utility score of risky portfolios as a certainty equivalent rate of return. The certainty equivalent is the rate that a risk-free investment would need to offer to provide the same utility score as the risky portfolio. In other words, it is the rate that, if earned with certainty, would provide a utility score equal to that of the portfolio in question. The certainty equivalent rate of return is a natural way to compare the utility values of competing portfolios. A portfolio can be desirable only if its certainty equivalent return exceeds that of the risk-free alternative. If the risk premium is zero or negative to begin with, any downward adjustment to utility only makes the portfolio look worse. Its certainty equivalent rate will be below that of the risk-free alternative for any risk- averse investor. In contrast to risk-averse investors, risk-neutral investors (with A = 0) judge risky prospects solely by their expected rates of return. The level of risk is irrelevant to the risk-neutral investor, meaning that there is no penalty for risk. For this investor, a portfolio’s certainty equivalent rate is simply its expected rate of return. A risk lover (for whom A < 0) is happy to engage in fair games and gambles; this investor adjusts the expected return upward to take into account the “fun” of confronting the prospect’s risk. Risk lovers will always take a fair game because their upward adjustment of utility for risk gives the fair game a certainty equivalent that exceeds the alternative of the risk-free investment. 16 Portfolio P, which has expected return E(rP) and standard deviation σP, is preferred by risk-averse investors to any portfolio in quadrant IV because its expected return is equal to or greater than any portfolio in that quadrant and its standard deviation is equal to or smaller than any portfolio in that quadrant. Conversely, any portfolio in quadrant I dominates portfolio P because its expected return is equal to or greater than P’s and its standard deviation is equal to or less than P’s This is the mean-standard deviation, or equivalently, the mean-variance (M-V) criterion. It can be stated as follows: portfolio A dominates B if and at least one inequality is strict. What can be said about portfolios in quadrants II and III? Their desirability, compared with P, depends on the degree of the investor’s risk aversion. Suppose an investor identifies all portfolios that are equally attractive as portfolio P. Starting at P, an increase in standard deviation lowers utility; it must be compensated for by an increase in expected return. Thus, point Q in Figure 6.2 is equally desirable to this investor as P. Investors will be equally attracted to portfolios with high risk and high expected returns as to portfolios with lower risk but lower expected returns. These equally preferred portfolios will lie in the mean–standard deviation plane on a curve called the indifference curve, which connects all portfolio points with the same utility value. 17 6.2 Capital Allocation across Risky and Risk-Free Portfolios We start our discussion of the risk–return trade-off available to investors by examining the most basic asset allocation choice: how much of the portfolio should be placed in risk-free money market securities versus other risky asset classes. We denote the investor’s portfolio of risky assets as P and the risk-free asset as F. We assume for the sake of illustration that the risky component of the investor’s overall portfolio comprises two mutual funds, one invested in stocks and the other invested in long-term bonds. For now, we take the composition of the risky portfolio as given and focus only on the allocation between it and risk-free securities. For example, assume that the total market value of an initial portfolio is $300,000, of which $90,000 is invested in the Ready Asset money market fund, a risk-free asset for practical purposes. The remaining $210,000 is invested in risky securities—$113,400 in equities (E) and $96,600 in long-term bonds (B). The equities and bond holdings comprise “the” risky portfolio, 54% in E and 46% in B: The weight of the risky portfolio, P, in the complete portfolio, including risk-free and risky investments, is denoted by y: The weights of each asset class in the complete portfolio are, therefore, as follows: Therefore, rather than thinking of our risky holdings as E and B separately, it is better to view our holdings as if they were in a single fund that holds equities and bonds in fixed proportions. In this sense, we may treat the risky fund as a single risky asset, that asset being a particular bundle of securities. 6.3 The Risk-Free Asset By virtue of its power to tax and control the money supply, only the government can issue default-free bonds. Even the default-free guarantee by itself is not sufficient to make the bonds risk-free in real terms. The only risk-free asset in real terms would be a perfectly price-indexed bond. It is common practice to view Treasury bills as “the” risk-free asset. Their short-term nature makes their prices insensitive to interest rate fluctuations. In practice, most investors use a broad range of money market instruments as a risk-free asset. All the money market instruments are virtually free of interest rate risk because of their short maturities and are fairly safe in terms of default or credit risk. 20 Chapter 7 Optimal Risky Portfolios 7.1 Diversification and Portfolio Risk If you hold a portfolio of only one stock of one firm, you have two sources of uncertainty: • Risk that comes from conditions in the general economy, such as the business cycle, inflation, interest rates, and exchange rates • Firm-specific influences, such as its success in research and development and personnel changes Now consider a naïve diversification strategy, in which you include additional securities in your portfolio. The two effects are offsetting and stabilize portfolio return. If we diversify into many securities, we continue to spread out our exposure to firm-specific factors, and portfolio volatility should continue to fall. Ultimately, however, even with a large number of stocks, we cannot avoid risk altogether because virtually all securities are affected by the common macroeconomic factors. For example, if all stocks are affected by the business cycle, we cannot avoid exposure to business cycle risk no matter how many stocks we hold. When all risk is firm-specific, as in Figure 7.1, Panel A, diversification can reduce risk to arbitrarily low levels. The reason is that with all risk sources independent, the exposure to any particular source of risk is reduced to a negligible level. Risk reduction by spreading exposure across many independent risk sources is sometimes called the insurance principle. When common sources of risk affect all firms, however, even extensive diversification cannot eliminate risk. In Figure 7.1, Panel B, portfolio standard deviation falls as the number of securities increases, but it cannot be reduced to zero. The risk that remains even after extensive diversification is called market risk, risk that is attributable to market-wide risk sources. Such risk is also called systematic risk, or non-diversifiable risk. In contrast, the risk that can be eliminated by diversification is called unique risk, firm-specific risk, nonsystematic risk, or diversifiable risk. 21 7.2 Portfolios of Two Risky Assets Efficient diversification: construct risky portfolios to provide the lowest possible risk for any given level of expected return. Think about a two-asset risky portfolio as an asset allocation decision, and so we consider two mutual funds, a bond portfolio specializing in long-term debt securities, denoted D, and a stock fund that specializes in equity securities, E. Table 7.1 lists the parameters describing the rate-of-return distribution of these funds. The rate of return on this portfolio, rp, will be: The variance of the two-asset portfolio is: Therefore, another way to write the variance of the portfolio is: In words, the variance of the portfolio is a weighted sum of covariances, and each weight is the product of the portfolio proportions of the pair of assets in the covariance term. Notice that the covariance can be computed from the correlation coefficient, ρDE, as: Therefore, Other things equal, portfolio variance is higher when ρDE is higher. If there is perfect positive correlation, ρDE = 1: Therefore, the standard deviation of the portfolio with perfect positive correlation is just the weighted average of the component standard deviations. Portfolios of less than perfectly correlated assets always offer some degree of diversification benefit. The lower the correlation between the assets, the greater the gain in efficiency. If there is perfect negative correlation, ρDE = -1: 22 The solution to this equation is: These weights drive the standard deviation of the portfolio to zero. What happens when wD > 1 and wE < 0? In this case, portfolio strategy would call for selling the equity fund short and investing the proceeds of the short sale in the debt fund. The reverse happens when wD < 0 and wE > 1. This strategy calls for selling the bond fund short and using the proceeds to finance additional purchases of the equity fund. Look first at the solid curve for ρDE = .30. The graph shows that as the portfolio weight in the equity fund increases from zero to 1, portfolio standard deviation first falls with the initial diversification from bonds into stocks, but then rises again as the portfolio becomes heavily concentrated in stocks, and again is undiversified. This pattern will generally hold as long as the correlation coefficient between the funds is not too high. The solid colored line in Figure 7.4 plots the portfolio standard deviation when ρ = .30 as a function of the investment proportions. It passes through the two undiversified portfolios of wD = 1 and wE = 1. Note that the minimum-variance portfolio has a standard deviation smaller than that of either of the individual component assets. This illustrates the effect of diversification. To find the weight of D in the minimum-variance portfolio: The solid dark straight line connecting the undiversified portfolios of all bonds or all stocks, wD = 1 or wE = 1, shows portfolio standard deviation with perfect positive correlation, ρ = 1. In this case there is no advantage from diversification, and the portfolio standard deviation is the simple weighted average of the component asset standard deviations. The dashed colored curve depicts portfolio risk for the case of uncorrelated assets, ρ = 0. With lower correlation between the two assets, diversification is more effective and portfolio risk is lower. 25 In the case of two risky assets, the solution for the weights of the optimal risky portfolio, P, is: Steps to be followed to arrive at the complete portfolio: 1. Specify the return characteristics of all securities (expected returns, variances, covariances). 2. Establish the risky portfolio (asset allocation): a. Calculate the optimal risky portfolio b. Calculate the properties of portfolio P 3. Allocate funds between the risky portfolio and the risk-free asset (capital allocation): a. Calculate the fraction of the complete portfolio allocated to portfolio P (the risky portfolio) and to T- bills (the risk-free asset). b. Calculate the share of the complete portfolio invested in each asset and in T-bills. 7.4 The Markowitz Portfolio Optimization Model Security Selection We can generalize the portfolio construction problem to the case of many risky securities and a risk-free asset. As in the two risky assets example, the problem has three parts. The first step is to determine the risk–return opportunities available to the investor. These are summarized by the minimum-variance frontier of risky assets. This frontier is a graph of the lowest possible variance that can be attained for a given portfolio expected return. Diversification allows us to construct portfolios with higher expected returns and lower standard deviations. The part of the frontier that lies above the global minimum-variance portfolio, therefore, is called the efficient frontier of risky assets. For any portfolio on the lower portion of the minimum-variance frontier, there is a portfolio with the same standard deviation and a greater expected return positioned directly above it. Hence the bottom part of the minimum-variance frontier is inefficient. The second part of the optimization plan involves the risk-free asset. As before, we search for the capital allocation line with the highest Sharpe ratio as shown in Figure 7.11. The CAL generated by the optimal portfolio, P, is the one tangent to the efficient frontier. Finally, in the last part of the problem, the individual investor chooses the appropriate mix between the optimal risky portfolio P and T-bills. 26 Once the estimates for expected returns, variances and covariances are compiled, the expected return and variance of any risky portfolio with weights in each security, wi , can be calculated from the bordered covariance matrix or, equivalently: The model of Markovitz is precisely step one of portfolio management: the identification of the efficient set of portfolios, or the efficient frontier of risky assets. The principal idea behind the frontier set of risky portfolios is that, for any risk level, we are interested only in that portfolio with the highest expected return. Alternatively, the frontier is the set of portfolios that minimizes the variance for any target expected return. The estimates generated by the security analysts were transformed into a set of expected rates of return and a covariance matrix. We call this group of estimates the input list. This input list is then fed into the optimization program. 27 Clients’ possible constraints: - short positions - socially responsible investing: not invest in industries or countries considered ethically or politically undesirable Capital Allocation and the Separation Property Figure 7.13 shows the efficient frontier plus three CALs representing various portfolios from the efficient set. As before, we ratchet up the CAL by selecting different portfolios until we reach portfolio P, which is the tangency point of a line from F to the efficient frontier. Portfolio P maximizes the Sharpe ratio, the slope of the CAL from F to portfolios on the efficient frontier. At this point our portfolio manager is done. Portfolio P is the optimal risky portfolio for the manager’s clients. The program maximizes the Sharpe ratio with no constraint on expected return or variance at all. The most striking conclusion is that a portfolio manager will offer the same risky portfolio, P, to all clients regardless of their degree of risk aversion. The client’s risk aversion comes into play only in capital allocation, the selection of the desired point along the CAL. Thus the only difference between clients’ choices is that the more risk-averse client will invest more in the risk-free asset and less in the optimal risky portfolio than will a less risk-averse client. This result is called a separation property; it tells us that the portfolio choice problem may be separated into two independent tasks. 1. Determination of the optimal risky portfolio: purely technical. The best risky portfolio is the same for all clients, regardless of risk aversion. 2. Capital allocation: depends on personal preference. Here the client is the decision maker. As we have seen, optimal risky portfolios for different clients also may vary because of portfolio constraints such as dividend-yield requirements, tax considerations, or other client preferences. Nevertheless, this analysis suggests that a limited number of portfolios may be sufficient to serve the demands of a wide range of investors. This is the theoretical basis of the mutual fund industry. 30 Chapter 8 Index Models 8.1 A Single-Factor Security Market The Input List of the Markowitz Model The success of a portfolio selection rule depends on the quality of the input list, that is, the estimates of expected security returns and the covariance matrix. In the long run, efficient portfolios will beat portfolios with less reliable input lists and consequently inferior reward-to-risk trade-offs. The problem with the Markowitz model is that to analyze 50 stocks we need 1325 estimates and to analyze 3000 stocks we need 4.5 million estimates. (n expected returns + n variances + (n^2 – n)/2 covariances) Another difficulty in applying the Markowitz model to portfolio optimization is that errors in the assessment or estimation of correlation coefficients can lead to nonsensical results. This can happen because some sets of correlation coefficients are mutually inconsistent. Introducing a model that simplifies the way we describe the sources of security risk allows us to use a smaller, necessarily consistent, and, just as important, more easily interpreted set of estimates of risk parameters and risk premiums. The simplification emerges because positive covariances among security returns arise from common economic forces that affect the fortunes of most firms. Some examples of common economic factors are business cycles, interest rates, and the cost of natural resources. “Shocks” (i.e., unexpected changes) to these macroeconomic variables cause, simultaneously, correlated shocks in the rates of return on stocks across the entire market. By decomposing uncertainty into these systemwide versus firm-specific sources, we vastly simplify the problem of estimating covariance and correlation. Systematic versus Firm-Specific Risk We focus on risk by separating the actual rate of return on any security, i, into the sum of its previously expected value plus an unanticipated surprise: We therefore will decompose the sources of return uncertainty into uncertainty about the economy as a whole, which is captured by: • a systematic market factor that we will call m • uncertainty about the firm in particular, captured by a firm-specific random variable that we will call ei. The market factor, m, measures unanticipated developments in the macroeconomy. It has a mean of zero with standard deviation of σm. In contrast, ei measures only the firm-specific surprise. As a surprise, it too has zero expected value. Notice that m has no subscript because the same common factor affects all securities. Most important is the fact that m and ei are assumed to be uncorrelated. If we denote the sensitivity coefficient for firm i by the Greek letter beta, βi, we can write the return on each stock in any period as the sum of three terms: the originally expected return, the impact of the common macroeconomic surprise (which depends on the firm’s sensitivity to that surprise), and the impact of firm- specific surprises. 31 Equation 8.2 is the algebraic expression of this single-factor model: There are two uncorrelated random terms on the right-hand side of Equation 8.2, so the total variance of ri is Because the index model assumes that firm-specific surprises are mutually uncorrelated, the only source of covariance between any pair of securities is their common dependence on the market return. Therefore, the covariance between two firms’ returns depends on the sensitivity of each to the market, as measured by their betas: 8.2 The Single-Index Model Because the systematic factor affects the rate of return on all stocks, the rate of return on a broad market index can plausibly proxy for that common factor. This approach leads to an equation similar to the single- factor model, which is called a single-index model because it uses the market index to stand in for the common factor. The Regression Equation of the Single-Index Model To describe the typical relation between the return on Ford and the return on the market index, we fit a straight line through this scatter diagram. There is a positive relation between Ford’s return and the market’s. This is evidence for the importance of broad market conditions on the performance of Ford’s stock. The slope of the line reflects the sensitivity of Ford’s return to market conditions: A steeper line would imply that Ford’s rate of return is more responsive to the market return. The scatter of points around the line is evidence that firm-specific events also have a significant impact on Ford’s return. We will denote the market index by M, with excess return of RM = rM − rf, and standard deviation of σM. More generally, for any stock i, denote the pair of excess returns in month t by Ri (t) and RM(t).3 Then the index model can be written as the following regression equation: Alpha is the security’s expected excess return when the market excess return is zero. Beta is the amount by which the security return tends to increase or decrease for every 1% increase or decrease in the return on the index, and therefore measures the security’s sensitivity to the market index. ei is the zero-mean, firm-specific surprise in the security return in month t, also called the residual. The greater the residuals (positive or negative), the wider is the scatter of returns around the straight line in Figure 8.1. 32 The Expected Return–Beta Relationship Because E(ei) = 0, if we take the expected value of both sides of Equation 8.5, we obtain the expected return– beta relationship of the single-index model: The market risk premium is multiplied by the relative sensitivity, or beta, of the individual security. This makes intuitive sense because securities with high betas have a magnified sensitivity to market risk and will therefore enjoy a greater risk premium as compensation for this risk. We call this the systematic risk premium because it derives from the risk premium that characterizes the market index, which in turn proxies for the condition of the full economy or economic system. Alpha is a nonmarket premium. For example, α may be large if you think a security is underpriced and therefore offers an attractive expected return. Risk and Covariance in the Single-Index Model Remember that one of the problems with the Markowitz model is the large number of parameter estimates required to implement it. Now we will see that the index model vastly reduces the number of parameters that must be estimated. In particular, we saw from Equation 8.4 that the covariance between any pair of stocks is determined by their common exposure to market risk; this insight yields great simplification in estimating an otherwise overwhelming set of covariance pairs. Using Equation 8.5, we can derive the following elements of the input list for portfolio optimization from the parameters of the index model: Equations 8.6 and 8.7 imply that the set of parameter estimates needed for the single-index model consists of only αi, βi, and σ(ei) for each individual security, plus the risk premium and variance of the market index. Remember that the market index has a beta of 1. 35 The Explanatory Power of Ford’s SCL The R-square (.394) tells us that variation in the excess returns of the market index explains about 39.4% of the variation in the Ford series. The adjusted R-square (which is slightly smaller) corrects for an upward bias in R-square that arises because we use the estimated values of two parameters. The standard error of the regression is the standard deviation of the residual, e. High standard errors imply greater impact (positive and negative) of firm-specific events from one month to the next. The intercept (−.0098 = −.98% per month) is the estimate of Ford’s alpha for the sample period it is statistically insignificant. This can be seen from the three statistics next to the estimated coefficient. The first is the standard error of the estimate (.0077). This is a measure of the imprecision of the estimate. If the standard error is large, the range of likely estimation error is correspondingly large. Here, the standard error is nearly as large as the estimated alpha coefficient. The t-statistic equals the number of standard errors by which our estimate exceeds zero, and therefore can be used to assess the likelihood that the true but unobserved value might actually be zero rather than the estimate we derived from the data. Large t-statistics imply low probabilities that the true value is zero. 36 8.5 Portfolio Construction Using the Single-Index Model - Treynor-Black model Alpha and Security Analysis The single-index model creates a framework that separates these two quite different sources of return variation and makes it easier to ensure consistency across analysts. We can lay down a hierarchy of the preparation of the input list using the framework of the single-index model. 1. Macroeconomic analysis is used to estimate the risk premium and risk of the market index. 2. Statistical analysis is used to estimate the beta coefficients of all securities and their residual variances, σ2(ei). 3. The portfolio manager uses the estimates for the market-index risk premium and the beta coefficient of a security to establish the expected return of that security absent any contribution from security analysis. The purely market-driven expected return is conditional on information common to all securities, not on information gleaned from security analysis of particular firms. 4. Security-specific expected return forecasts (specifically, security alphas) are derived from various security-valuation models (such as those discussed in Part Five). Thus, the alpha value distills the incremental risk premium attributable to private information developed from security analysis. The end result of security analysis is the list of alpha values. Alpha is more than just one of the components of expected return. It is the key variable that tells us whether a security is a good or a bad buy: • A positive-alpha security is a bargain and therefore should be overweighted in the overall portfolio compared to the passive alternative of using the market-index portfolio as the risky vehicle. • A negative-alpha security is overpriced and, other things equal, its portfolio weight should be reduced. In more extreme cases, the desired portfolio weight might even be negative, that is, a short position (if permitted) would be desirable. The Optimal Risky Portfolio in the Single-Index Model The single-index model allows us to solve for the optimal risky portfolio directly and to gain insight into the nature of the solution. First, we confirm that we easily can set up the optimization process to chart the efficient frontier in this framework along the lines of the Markowitz model. With the estimates of the beta and alpha coefficients, plus the risk premium of the index portfolio, we can generate the n + 1 expected returns using Equation 8.6. With the estimates of the beta coefficients and residual variances, together with the variance of the index portfolio, we can construct the covariance matrix using Equation 8.7. Given the column of risk premiums and the covariance matrix. The alpha, beta, and residual variance of an equally weighted portfolio are the simple averages of those parameters across component securities. This result is not limited to equally weighted portfolios. It applies to any portfolio, where we need only replace “simple average” with “weighted average,” using the portfolio weights. Specifically, 37 The objective is to select portfolio weights, w1, ... , wn + 1, to maximize the Sharpe ratio of the portfolio. With this set of weights, the expected return, standard deviation, and Sharpe ratio of the portfolio are The optimal risky portfolio turns out to be a combination of two component portfolios: 1. An active portfolio, denoted by A, comprised of the n analyzed securities 2. The market index portfolio, the (n + 1)th asset we include to aid in diversification, which we call the passive portfolio and denoted by M. Assume first that the active portfolio has a beta of 1. In that case, the optimal weight in the active portfolio would be proportional to the ratio αA/σ2(eA). The analogous ratio for the index portfolio is E(RM)/σ2M, and hence the initial position in the active portfolio is: For any level of σA2, the correlation between the active and passive portfolios is greater when the beta of the active portfolio is higher. This implies less diversification benefit from the passive portfolio and a lower position in it. The precise modification for the position in the active portfolio is: 40 Expected Returns on Individual Securities The CAPM is built on the insight that the appropriate risk premium on an asset will be determined by its contribution to the risk of investors’ overall portfolios. Portfolio risk is what matters to investors and is what governs the risk premiums they demand. To calculate the variance of the market portfolio, we use the bordered covariance matrix with the market portfolio weights. We highlight GE in this depiction of the n stocks in the market portfolio so that we can measure the contribution of GE to the risk of the market portfolio. The contribution of one stock to portfolio variance therefore can be expressed as the sum of all the covariance terms in the column corresponding to the stock, where each covariance is first multiplied by both the stock’s weight from its row and the weight from its column. 41 If the covariance between GE and the rest of the market is negative, then GE makes a “negative contribution” to portfolio risk: By providing excess returns that move inversely with the rest of the market, GE stabilizes the return on the overall portfolio. If the covariance is positive, GE makes a positive contribution to overall portfolio risk because its returns reinforce swings in the rest of the portfolio. Therefore, the reward-to-risk ratio for investments in GE can be expressed as: The reward-to risk ratio for investment in the market portfolio is: The ratio in Equation 9.5 is often called the market price of risk because it quantifies the extra return that investors demand to bear portfolio risk. Notice that for components of the efficient portfolio, such as shares of GE, we measure risk as the contribution to portfolio variance. A basic principle of equilibrium is that all investments should offer the same reward-to-risk ratio. If the ratio were better for one investment than another, investors would rearrange their portfolios, tilting toward the alternative with the better trade-off and shying away from the other. Such activity would impart pressure on security prices until the ratios were equalized. Therefore we conclude that the reward-to-risk ratios of GE and the market portfolio should be equal: To determine the fair risk premium of GE stock, we rearrange Equation 9.6 slightly to obtain: The ratio Cov (RGE, RM)/σ2M measures the contribution of GE stock to the variance of the market portfolio as a fraction of the total variance of the market portfolio. The ratio is called beta and is denoted by β. Using this measure, we can restate Equation 9.7 as: This expected return–beta (or mean-beta) relationship is the most familiar expression of the CAPM to practitioners. The expected return–beta relationship tells us that the total expected rate of return is the sum of the risk-free rate plus a risk premium. The size of the risk premium is the product of a “benchmark risk premium” and the relative risk of the particular asset as measured by its beta. The CAPM predicts that systematic risk should “be priced,” meaning that it commands a risk premium, but firm-specific risk should not be priced by the market. 42 If the expected return–beta relationship holds for each individual asset, it must hold for any combination or weighted average of assets. Suppose that some portfolio P has weight wk for stock k, where k takes on values 1, . . ., n. Writing out the CAPM Equation 9.8 for each stock, and multiplying each equation by the weight of the stock in the portfolio, we obtain these equations, one for each stock: Summing each column shows that the CAPM holds for the overall portfolio because E(rP) = ∑kwkE(rk) is the expected return on the portfolio and βP= ∑kwkβk is the portfolio beta. Incidentally, this result has to be true for the market portfolio itself, Indeed, this is a tautology because βM = 1, as we can verify by noting that: If the market beta is 1, and the market is a portfolio of all assets in the economy, the weighted-average beta of all assets must be 1. Hence betas greater than 1 are considered aggressive in that investment in high-beta stocks entails above-average sensitivity to market swings. Betas below 1 can be described as defensive. Security prices, in other words, already reflect public information about a firm’s prospects; therefore only the risk of the company (as measured by beta in the context of the CAPM) should affect expected returns. The Security Market Line Risk-averse mean-variance investors measure the risk of the optimal risky portfolio by its variance. Hence, we would expect the risk premium on individual assets to depend on the contribution of the asset to the risk of the portfolio. The beta of a stock measures its contribution to the variance of the market portfolio and therefore the required risk premium is a function of beta. The CAPM confirms this intuition, stating further that the security’s risk premium is directly proportional to both the beta and the risk premium of the market portfolio; that is, the risk premium equals β[E(rM) − rf]. The expected return–beta relationship can be portrayed graphically as the security market line (SML) in Figure 9.2. It is useful to compare the security market line to the capital market line. The CML graphs the risk premiums of efficient portfolios as a function of portfolio standard deviation. This is appropriate because standard deviation is a valid measure of risk for efficiently diversified portfolios that are candidates for an investor’s overall portfolio. The SML, in contrast, graphs individual asset risk premiums as a function of asset risk, that is the asset’s beta. The security market line provides a benchmark for the evaluation of investment performance and the required rate of return necessary to compensate investors for risk as well as the time value of money. All securities must lie on the SML in market equilibrium. 45 Convertible Bonds Convertible bonds give bondholders an option to exchange each bond for a specified number of shares of common stock of the firm. The conversion ratio is the number of shares for which each bond may be exchanged. The market conversion value is the current value of the shares for which the bonds may be exchanged. The conversion premium is the excess of the bond’s value over its conversion value. Convertible bondholders benefit from price appreciation of the company’s stock. Again, this benefit comes at a price: Convertible bonds offer lower coupon rates and stated or promised yields to maturity than do nonconvertible bonds. Puttable Bonds While the callable bond gives the issuer the option to extend or retire the bond at the call date, the extendable or put bond gives this option to the bondholder. If the bond’s coupon rate exceeds current market yields, for instance, the bondholder will choose to extend the bond’s life. Floating-Rate Bonds Floating-rate bonds make interest payments that are tied to some measure of current market rates. For example, the rate might be adjusted annually to the current T-bill rate plus 2%. If the 1-year T-bill rate at the adjustment date is 4%, the bond’s coupon rate over the next year would then be 6%. This arrangement means that the bond always pays approximately current market rates. Preferred Stock Although preferred stock strictly speaking is considered to be equity, it often is included in the fixed-income universe. This is because, like bonds, preferred stock promises to pay a specified stream of dividends. However, unlike bonds, the failure to pay the promised dividend does not result in corporate bankruptcy. Instead, the dividends owed simply cumulate, and the common stockholders may not receive any dividends until the preferred stockholders have been paid in full. In the event of bankruptcy, preferred stockholders’ claims to the firm’s assets have lower priority than those of bondholders but higher priority than those of common stockholders. Unlike interest payments on bonds, dividends on preferred stock are not considered tax-deductible expenses to the firm. This reduces their attractiveness as a source of capital to issuing firms. International Bonds International bonds are commonly divided into two categories, foreign bonds and Eurobonds. Foreign bonds are issued by a borrower from a country other than the one in which the bond is sold. The bond is denominated in the currency of the country in which it is marketed. For example, if a German firm sells a dollar-denominated bond in the United States, the bond is considered a foreign bond. Eurobonds are denominated in one currency, usually that of the issuer, but sold in other national markets. For example, the Eurodollar market refers to dollar-denominated bonds sold outside the United States (not just in Europe), although London is the largest market for Eurodollar bonds. Because the Eurodollar market falls outside U.S. jurisdiction, these bonds are not regulated by U.S. federal agencies. 46 Inverse Floaters These are similar to the floating-rate bonds we described earlier, except that the coupon rate on these bonds falls when the general level of interest rates rises. Investors in these bonds suffer doubly when rates rise. Of course, investors in these bonds benefit doubly when rates fall. Catastrophe Bonds These bonds are a way to transfer “catastrophe risk” from the firm to the capital markets. Investors in these bonds receive compensation for taking on the risk in the form of higher coupon rates. But in the event of a catastrophe, the bondholders will give up all or part of their investments. Indexed Bonds Indexed bonds make payments that are tied to a general price index or the price of a particular commodity. For example, Mexico has issued bonds with payments that depend on the price of oil. Some bonds are indexed to the general price level. Therefore, the cash flows paid by the bond are fixed in real terms. When the bond matures, the investor receives a final coupon payment of $42.44 plus the (price-level-indexed) repayment of principal, $1,061.11. The nominal rate of return on the bond in the first year is: The real rate of return is precisely the 4% real yield on the bond: Bond Pricing Because a bond’s coupon and principal repayments all occur months or years in the future, the price an investor would be willing to pay for a claim to those payments depends on the value of dollars to be received in the future compared to dollars in hand today. This “present value” calculation depends in turn on market interest rates. To value a security, we discount its expected cash flows by the appropriate discount rate. The cash flows from a bond consist of coupon payments until the maturity date plus the final payment of par value. Therefore, Bond value = Present value of coupons + Present value of par value. If we call the maturity date T and call the interest rate r, the bond value can be written as: As a general rule, keeping all other factors the same, the longer the maturity of the bond, the greater the sensitivity of price to fluctuations in the interest rate. The longer the period for which your money is tied up, the greater the loss, and correspondingly the greater the drop in the bond price. 47 Bond Pricing between Coupon Dates In principle, the fact that the bond is between coupon dates does not affect the pricing problem. The procedure is always the same: Compute the present value of each remaining payment and sum up. But if you are between coupon dates, there will be fractional periods remaining until each payment, and this does complicate the arithmetic computations. Bond prices are typically quoted net of accrued interest. These prices, which appear in the financial press, are called flat prices. The actual invoice price that a buyer pays for the bond includes accrued interest. Thus, Invoice price = Flat price + Accrued interest Bond Yields Most bonds do not sell for par value. But ultimately, barring default, they will mature to par value. Therefore, we would like a measure of rate of return that accounts for both current income and the price increase or decrease over the bond’s life. The yield to maturity is the standard measure of the total rate of return. However, it is far from perfect, and we will explore several variations of this measure. Yield to Maturity The yield to maturity (YTM) is defined as the interest rate that makes the present value of a bond’s payments equal to its price. This interest rate is often interpreted as a measure of the average rate of return that will be earned on a bond if it is bought now and held until maturity. To calculate the yield to maturity, we solve the bond price equation for the interest rate given the bond’s price. The bond’s yield to maturity is the internal rate of return on an investment in the bond. The yield to maturity can be interpreted as the compound rate of return over the life of the bond under the assumption that all bond coupons can be reinvested at that yield. Yield to maturity is widely accepted as a proxy for average return. Yield to maturity differs from the current yield of a bond, which is the bond’s annual coupon payment divided by the bond price. For premium bonds (bonds selling above par value), coupon rate is greater than current yield, which in turn is greater than yield to maturity. For discount bonds (bonds selling below par value), these relationships are reversed. Realized Compound Return versus Yield to Maturity Yield to maturity will equal the rate of return realized over the life of the bond if all coupons are reinvested at an interest rate equal to the bond’s yield to maturity. The yield to maturity is 10%. If the $100 coupon payment is reinvested at an interest rate of 10%, the $1,000 investment in the bond will grow after two years to $1,210. To summarize, the initial value of the investment is V0 = $1,000. The final value in two years is V2 = $1,210. The compound rate of return, therefore, is calculated as follows: With a reinvestment rate equal to the 10% yield to maturity, the realized compound return equals yield to maturity. 50 3. Liquidity ratios: the two most common liquidity ratios are the current ratio (current assets/current liabilities) and the quick ratio (current assets excluding inventories/current liabilities). 4. Profitability ratios: measures of rates of return on assets or equity. Profitability ratios are indicators of a firm’s overall financial health. The return on assets (earnings before interest and taxes divided by total assets) or return on equity (net income/ equity) are the most popular of these measures. 5. Cash flow-to-debt ratio: this is the ratio of total cash flow to outstanding debt. Bond Indentures A bond is issued with an indenture, which is the contract between the issuer and the bondholder. Part of the indenture is a set of restrictions that protect the rights of the bondholders. Such restrictions include provisions relating to collateral, sinking funds, dividend policy, and further borrowing. The issuing firm agrees to these protective covenants in order to market its bonds to investors concerned about the safety of the bond issue. Sinking Funds Bonds call for the payment of par value at the end of the bond’s life. This payment constitutes a large cash commitment for the issuer. To help ensure the commitment does not create a cash flow crisis, the firm agrees to establish a sinking fund to spread the payment burden over several years. The fund may operate in one of two ways: 1. The firm may repurchase a fraction of the outstanding bonds in the open market each year. 2. The firm may purchase a fraction of the outstanding bonds at a special call price associated with the sinking fund provision. The sinking fund call differs from a conventional bond call in two important ways. First, the firm can repurchase only a limited fraction of the bond issue at the sinking fund call price. At most, some indentures allow firms to use a doubling option, which allows repurchase of double the required number of bonds at the sinking fund call price. Second, while callable bonds generally have call prices above par value, the sinking fund call price usually is set at the bond’s par value. If interest rates fall and bond prices rise, firms will benefit from the sinking fund provision that enables them to repurchase their bonds at below-market prices. In these circumstances, the firm’s gain is the bondholder’s loss. One bond issue that does not require a sinking fund is a serial bond issue, in which the firm sells bonds with staggered maturity dates. As bonds mature sequentially, the principal repayment burden for the firm is spread over time, just as it is with a sinking fund. Subordination of Further Debt One of the factors determining bond safety is total outstanding debt of the issuer. If you bought a bond today, you would be understandably distressed to see the firm tripling its outstanding debt tomorrow. Your bond would be riskier than it appeared when you bought it. To prevent firms from harming bondholders in this manner, subordination clauses restrict the amount of additional borrowing. Additional debt might be required to be subordinated in priority to existing debt; that is, in the event of bankruptcy, subordinated or junior debtholders will not be paid unless and until the prior senior debt is fully paid off. 51 Collateral Some bonds are issued with specific collateral behind them. Collateral is a particular asset that the bondholders receive if the firm defaults on the bond. If the collateral is property, the bond is called a mortgage bond. If the collateral takes the form of other securities held by the firm, the bond is a collateral trust bond. In the case of equipment, the bond is known as an equipment obligation bond. Collateralized bonds generally are considered safer than general debenture bonds, which are unsecured, meaning they do not provide for specific collateral. Yield to Maturity and Default Risk Because corporate bonds are subject to default risk, we must distinguish between the bond’s promised yield to maturity and its expected yield. The promised or stated yield will be realized only if the firm meets the obligations of the bond issue. Therefore, the stated yield is the maximum possible yield to maturity of the bond. The expected yield to maturity must take into account the possibility of a default. To compensate for the possibility of default, corporate bonds must offer a default premium. The default premium is the difference between the promised yield on a corporate bond and the yield of an otherwise- identical government bond that is riskless in terms of default. If the firm remains solvent and actually pays the investor all of the promised cash flows, the investor will realize a higher yield to maturity than would be realized from the government bond. The corporate bond has the potential for both better and worse performance than the default-free Treasury bond. In other words, it is riskier. The pattern of default premiums offered on risky bonds is sometimes called the risk structure of interest rates. The greater the default risk, the higher the default premium. 52 Chapter 15 The Term Structure of Interest Rates 15.1 The Yield Curve Practitioners commonly summarize the relationship between yield and maturity graphically in a yield curve, which is a plot of yield to maturity as a function of time to maturity. The yield curve is one of the key concerns of fixed-income investors. Bond Pricing If yields on different-maturity bonds are not all equal, how should we value coupon bonds that make payments at many different times? For example, suppose that yields on zero-coupon Treasury bonds of different maturities are as given in Table 15.1. The table tells us that zero-coupon bonds with 1-year maturity sell at a yield to maturity of y1 = 5%, 2-year zeros sell at yields of y2 = 6%, and 3-year zeros sell at yields of y3 = 7%. Which of these rates should we use to discount bond cash flows? The answer: all of them. The trick is to consider each bond cash flow—either coupon or principal payment—as at least potentially sold off separately as a stand-alone zero-coupon bond. What then do we mean by “the” yield curve? In fact, in practice, traders refer to several yield curves. The pure yield curve refers to the curve for stripped, or zero-coupon, Treasuries. The on-the-run yield curve refers to the plot of yield as a function of maturity for recently issued coupon bonds selling at or near par value. On-the-run Treasuries have the greatest liquidity, so traders have keen interest in their yield curve. 15.2 The Yield Curve and Future Interest Rates To start, consider a world with no uncertainty, specifically, one in which all investors already know the path of future interest rates. The Yield Curve under Certainty If interest rates are certain, what should we make of the fact that the yield on the 2-year zero coupon bond in Table 15.1 is greater than that on the 1-year zero? It can’t be that one bond is expected to provide a higher rate of return than the other. This would not be possible in a certain world—with no risk, all bonds (in fact, all securities!) must offer identical returns, or investors will bid up the price of the high-return bond until its rate of return is no longer superior to that of other bonds. Instead, the upward-sloping yield curve is evidence that short-term rates are going to be higher next year than they are now. To see why, consider two 2-year bond strategies. The first strategy entails buying the 2-year zero offering a 2-year yield to maturity of y2 = 6%, and holding it until maturity. The zero has face value $1,000, so it is purchased today for $1,000/1.062 = $890 and matures in two years to $1,000. The 2-year growth factor for the investment is therefore $1,000/$890 = 1.062 = 1.1236. 55 investment in a 2-year zero. Therefore, under certainty, The result in Example 15.5—that the forward rate exceeds the expected short rate— should not surprise us. We defined the forward rate as the interest rate that would need to prevail in the second year to make the long- and short-term investments equally attractive, ignoring risk. But when we account for risk, short-term investors will shy away from the long-term bond unless its expected return exceeds that of the 1-year bond. Therefore, the risk-averse investor would be willing to hold the long-term bond only if the expected value of the short rate is less than the break-even value, f2, because the lower the expectation of r2, the greater the anticipated return on the long-term bond. Therefore, if most individuals are short-term investors, bonds must have prices that make f2 greater than E(r2). The forward rate will embody a premium compared with the expected future short-interest rate. This liquidity premium compensates short-term investors for the uncertainty about the price at which they will be able to sell their long-term bonds at the end of the year. If all investors were long-term investors, no one would be willing to hold short-term bonds unless rolling over those bonds offered a reward for bearing interest rate risk. This would cause the forward rate to be less than the expected future spot rate. 15.4 Theories of the Term Structure The Expectations Hypothesis The simplest theory of the term structure is the expectations hypothesis. A common version states that the forward rate equals the market consensus expectation of the future short interest rate; that is, f2 = E(r2), and liquidity premiums are zero. If f2 = E(r2), yields on long-term bonds depend only on expectations of future short rates. Therefore, we can use the forward rates derived from the yield curve to infer market expectations of future short rates. For example, with (1 + y2)2 = (1 + r1) × (1 + f2) from Equation 15.5, according to the expectations hypothesis, we may also conclude that (1 + y2)2 = (1 + r1) × [1 + E(r2)]. An upward-sloping yield curve would be clear evidence that investors anticipate increases in interest rates. 56 Liquidity Preference Theory We’ve seen that short-term investors will be unwilling to hold long-term bonds unless the forward rate exceeds the expected short interest rate, f2 > E(r2), whereas long-term investors will be unwilling to hold short bonds unless E(r2) > f2. In other words, both groups of investors require a premium to hold bonds with maturities different from their investment horizons. Advocates of the liquidity preference theory of the term structure believe that short-term investors dominate the market so that the forward rate will generally exceed the expected short rate. The excess of f2 over E(r2), the liquidity premium, is predicted to be positive. 57 Interpreting the Term Structure We have seen that under certainty, 1 plus the yield to maturity on a zero-coupon bond is simply the geometric average of 1 plus the future short rates that will prevail over the life of the bond. This is the meaning of Equation 15.1, which we give in general form here: When future rates are uncertain, we modify Equation 15.1 by replacing future short rates with forward rates: Thus there is a direct relationship between yields on various maturity bonds and forward interest rates. The yield curve is upward-sloping at any maturity date, n, for which the forward rate for the coming period is greater than the yield at that maturity. This rule follows from the notion of the yield to maturity as an average (albeit a geometric average) of forward rates. If the yield curve rises as one moves to longer maturities, the extension to a longer maturity must result in the inclusion of a “new” forward rate higher than the average of the previously observed rates. Given that an upward-sloping yield curve implies a forward rate higher than the spot, or current, yield to maturity, we ask next what can account for that higher forward rate. The challenge is that there always are two possible answers to this question. Recall that the forward rate can be related to the expected future short rate according to: where the liquidity premium might be necessary to induce investors to hold bonds of maturities that do not correspond to their preferred investment horizons. Although it is tempting to infer from a rising yield curve that investors believe that interest rates will eventually increase, this does not necessarily follow. Indeed, Panel A in Figure 15.4 provides a simple counterexample. There, the short rate is expected to stay at 5% forever. Yet there is a constant 1% liquidity premium so that all forward rates are 6%. The result is that the yield curve continually rises, starting at a level of 5% for 1-year bonds, but eventually approaching 6% for long-term bonds as more and more forward rates at 6% are averaged into the yields to maturity. Therefore, while expectations of increases in future interest rates can result in a rising yield curve, the converse is not true: A rising yield curve does not in and of itself imply expectations of higher future interest rates. One very rough approach to deriving expected future spot rates is to assume that liquidity premiums are constant. An estimate of that premium can be subtracted from the forward rate to obtain the market’s expected interest rate. This approach has little to recommend it for two reasons. First, it is next to impossible to obtain precise estimates of a liquidity premium. The general approach to doing so would be to compare forward rates and eventually realized future short rates and to calculate the average difference between the two. However, the deviations between the two values can be quite large and unpredictable because of unanticipated economic events that affect the realized short rate. Second, there is no reason to believe that the liquidity premium should be constant. Still, very steep yield curves are interpreted by many market professionals as warning signs of impending rate increases. In fact, the yield curve is a good predictor of the business cycle as a whole, because long-term rates tend to rise in anticipation of an expansion in economic activity. 60 Duration Macaulay’s duration equals the weighted average of the times to each coupon or principal payment. The weight associated with each payment time clearly should be related to the “importance” of that payment to the value of the bond. In fact, the weight applied to each payment time is the proportion of the total value of the bond accounted for by that payment, that is, the present value of the payment divided by the bond price. We define the weight, wt, associated with the cash flow made at time t (denoted CFt) as: The numerator on the right-hand side of this equation is the present value of the cash flow occurring at time t while the denominator is the value of all the bond’s payments. These weights sum to 1.0 because the sum of the cash flows discounted at the yield to maturity equals the bond price. Using these values to calculate the weighted average of the times until the receipt of each of the bond’s payments, we obtain Macaulay’s duration formula: Duration is a key concept in fixed-income portfolio management for at least three reasons: 1. As we have noted, it is a simple summary statistic of the effective average maturity of the portfolio. 2. It turns out to be an essential tool in immunizing portfolios from interest rate risk. 3. Duration is a measure of the interest rate sensitivity of a portfolio, which we explore here. It can be shown that when interest rates change, the proportional change in a bond’s price can be related to the change in its yield to maturity, y, according to the rule: Practitioners commonly use Equation 16.2 in a slightly different form. They define modified duration as D* = D/(1 + y), note that Δ(1 + y) = Δy, and rewrite Equation 16.2 as: Because the percentage change in the bond price is proportional to modified duration, modified duration is a natural measure of the bond’s exposure to changes in interest rates. Actually, as we will see below, Equation 16.2, or equivalently, Equation 16.3, is only approximately valid for large changes in the bond’s yield. The approximation becomes exact as one considers smaller, or localized, changes in yields. 61 What Determines Duration? Malkiel’s bond price relations, which we laid out in the previous section, characterize the determinants of interest rate sensitivity. Duration allows us to quantify that sensitivity. Rules for Duration: 1. The duration of a zero-coupon bond equals its time to maturity. 2. Holding maturity constant, a bond’s duration is lower when the coupon rate is higher. This property corresponds to Malkiel’s fifth relationship and is due to the impact of early coupon payments on the weighted-average maturity of a bond’s payments. The higher these coupons, the higher the weights on the early payments and the lower the weighted average maturity of the payments. In other words, a higher fraction of the total value of the bond is tied up in the (earlier) coupon payments, whose values are relatively insensitive to yields, rather than the (later and more yield-sensitive) repayment of par value. 3. Holding the coupon rate constant, a bond’s duration generally increases with its time to maturity. Duration always increases with maturity for bonds selling at par or at a premium to par. This property of duration corresponds to Malkiel’s third relationship, and it is fairly intuitive. What is surprising is that duration need not always increase with time to maturity. It turns out that for some deep-discount bonds, duration may eventually fall with increases in maturity. However, for virtually all traded bonds, it is safe to assume that duration increases with maturity. In the weighted-average calculation of duration the distant payments receive greater weights, which results in a higher duration measure. 4. Holding other factors constant, the duration of a coupon bond is higher when the bond’s yield to maturity is lower. At higher yields a higher fraction of the total value of the bond lies in its earlier payments, thereby reducing effective maturity. Rule 4, which is the sixth bond-pricing relationship above, applies to coupon bonds. For zeros, of course, duration equals time to maturity, regardless of the yield to maturity. 5. The duration of a level perpetuity is: For example, at a 10% yield, the duration of a perpetuity that pays $100 once a year forever is 1.10/.10 = 11 years, but at an 8% yield it is 1.08/.08 = 13.5 years. 16.2 Convexity As a measure of interest rate sensitivity, duration clearly is a key tool in fixed-income portfolio management. Yet the duration rule for the impact of interest rates on bond prices is only an approximation. Equation 16.2, or its equivalent, Equation 16.3, which we repeat here, states that the percentage change in the value of a bond approximately equals the product of modified duration times the change in the bond’s yield: In other words, the percentage price change is directly proportional to the change in the bond’s yield. If this were exactly so, however, a graph of the percentage change in bond price as a function of the change in its yield would plot as a straight line, with slope equal to −D*. 62 The duration rule is a good approximation for small changes in bond yield, but it is less accurate for larger changes. Figure 16.3 illustrates this point. The curved line is the percentage price change for a 30-year maturity, 8% annual payment coupon bond, selling at an initial yield to maturity of 8%. The straight line is the percentage price change predicted by the duration rule. The slope of the straight line is the modified duration of the bond at its initial yield to maturity. The modified duration of the bond at this yield is 11.26 years, so the straight line is a plot of −D*Δy = −11.26 × Δy. Notice that the two plots are tangent at the initial yield. Thus for small changes in the bond’s yield to maturity, the duration rule is quite accurate. The duration approximation (the straight line) always understates the value of the bond; it underestimates the increase in bond price when the yield falls, and it overestimates the decline in price when the yield rises. This is due to the curvature of the true price-yield relationship. Curves with shapes like that of the price-yield relationship are said to be convex, and the curvature of the price-yield curve is called the convexity of the bond. The formula for the convexity of a bond with a maturity of T years making annual coupon payments is: As a practical rule, you can view bonds with higher convexity as exhibiting higher curvature in the price-yield relationship. 65 Immunization In contrast to indexing strategies, many institutions try to insulate their portfolios from interest rate risk altogether. Generally, there are two ways of viewing this risk. Some institutions, such as banks, are concerned with protecting current net worth or net market value against interest rate fluctuations. Other investors, such as pension funds, may face an obligation to make payments after a given number of years. What is common to all investors, however, is interest rate risk. The net worth of the firm or the ability to meet future obligations fluctuates with interest rates. Immunization techniques refer to strategies used by such investors to shield their overall financial status from interest rate risk. Bank liabilities are primarily the deposits owed to customers, most of which are short-term and, consequently, have low duration. Bank assets by contrast are composed largely of outstanding commercial and consumer loans or mortgages. These assets have longer duration, and their values are correspondingly more sensitive to interest rate fluctuations. When interest rates increase unexpectedly, banks can suffer serious decreases in net worth—their assets fall in value by more than their liabilities. For pension funds as interest rates fell, the value of their liabilities grew even faster than the value of their assets. The lesson is that funds should match the interest rate exposure of assets and liabilities so that the value of assets will track the value of liabilities whether rates rise or fall. In other words, the financial manager might want to immunize the fund against interest rate volatility. Consider, for example, an insurance company that issues a guaranteed investment contract, or GIC, for $10,000. (Essentially, GICs are zero-coupon bonds issued by the insurance company to its customers. They are popular products for individuals’ retirement savings accounts.) If the GIC has a 5-year maturity and a guaranteed interest rate of 8%, the insurance company promises to pay $10,000 × 1.085 = $14,693.28 in five years. Suppose that the insurance company chooses to fund its obligation with $10,000 of 8% annual coupon bonds, selling at par value, with six years to maturity. As long as the market interest rate stays at 8%, the company has fully funded the obligation, as the present value of the obligation exactly equals the value of the bonds. Table 16.4, Panel A, shows that if interest rates remain at 8%, the accumulated funds from the bond will grow to exactly the $14,693.28 obligation. Over the 5-year period, year-end coupon income of $800 is reinvested at the prevailing 8% market interest rate. At the end of the period, the bonds can be sold for $10,000; they still will sell at par value because the coupon rate still equals the market interest rate. Total income after five years from reinvested coupons and the sale of the bond is precisely $14,693.28. If interest rates rise, the fund will suffer a capital loss, impairing its ability to satisfy the obligation. The bonds will be worth less in five years than if interest rates had remained at 8%. However, at a higher interest rate, reinvested coupons will grow at a faster rate, offsetting the capital loss. In other words, fixed income investors face two offsetting types of interest rate risk: price risk and reinvestment rate risk. Increases in interest rates cause capital losses but at the same time increase the rate at which reinvested income will grow. If the portfolio duration is chosen appropriately, these two effects will cancel out exactly. When the portfolio duration is set equal to the investor’s horizon date, the accumulated value of the investment fund at the horizon date will be unaffected by interest rate fluctuations. For a horizon equal to the portfolio’s duration, price risk and reinvestment risk are precisely offsetting. 66 Panel B shows that if interest rates fall to 7%, the total funds will accumulate to $14,694.05, providing a small surplus of $.77. If rates increase to 9% as in Panel C, the fund accumulates to $14,696.02, providing a small surplus of $2.74. When interest rates fall, the coupons grow less than in the base case, but the higher value of the bond offsets this. When interest rates rise, the value of the bond falls, but the coupons more than make up for this loss because they are reinvested at the higher rate. As interest rates change, the change in value of both the asset and the obligation is equal, so the obligation remains fully funded. For greater changes in the interest rate, however, the present value curves diverge. This reflects the fact that the fund actually shows a small surplus in Table 16.4 at market interest rates other than 8%. This example highlights the importance of rebalancing immunized portfolios. As interest rates and asset durations change, a manager must rebalance the portfolio to realign its duration with the duration of the obligation. Moreover, even if interest rates do not change, asset durations will change solely because of the passage of time. Without rebalancing, durations will become unmatched. Obviously, immunization is a passive strategy only in the sense that it does not involve attempts to identify undervalued securities. Immunization managers still proactively update and monitor their positions. 67 Even if a position is immunized, however, the portfolio manager still cannot rest. This is because of the need for rebalancing in response to changes in interest rates. Moreover, even if rates do not change, the passage of time also will affect duration and require rebalancing. 70 the return that investors will require of any other investment with equivalent risk. We will denote this required rate of return as k. If a stock is priced “correctly,” it will offer investors a “fair” return, that is, its expected return will equal its required return. An underpriced stock will provide an expected return greater than the required return. Suppose that rf = 6%, E(rM) − rf = 5%, and the beta of ABC is 1.2. Then according to the CAPM, the value of k is: The expected holding-period return, 16.7%, therefore exceeds the required rate of return based on ABC’s risk by a margin of 4.7%. Naturally, the investor will want to include more of ABC stock in the portfolio than a passive strategy would indicate. Another way to see this is to compare the intrinsic value of a share of stock to its market price. The intrinsic value, denoted V0, is defined as the present value of all cash payments (on a per share basis) to the stockholder, including dividends as well as the proceeds from the ultimate sale of the stock, discounted at the appropriate risk-adjusted interest rate, k. If the intrinsic value, or the investor’s own estimate of what the stock is really worth, exceeds the market price, the stock is considered undervalued and a good investment. For ABC, using a 1-year investment horizon the intrinsic value is: At the current price of $48, the stock is underpriced compared to intrinsic value. At this price, it provides better than a fair rate of return relative to its risk. If the intrinsic value turns out to be lower than the current market price, investors should buy less of it than under the passive strategy. In market equilibrium, the current market price will reflect the intrinsic value estimates of all market participants. A common term for the market consensus value of the required rate of return, k, is the market capitalization rate, which we use often throughout this chapter. To sum up, there are two ways: 1. Compare expected HPR with k 2. Compare the current price with V0 18.3 Dividend Discount Models Consider an investor who buys a share of Steady State Electronics stock, planning to hold it for one year. The intrinsic value of the share is the present value of the dividend to be received at the end of the first year, D1, and the expected sales price, P1. We will henceforth use the simpler notation P1 instead of E(P1) to avoid clutter. Keep in mind, though, that future prices and dividends are unknown, and we are dealing with expected values, not certain values. We’ve already established: More generally, for a holding period of H years, we can write the stock value as the present value of dividends 71 over the H years, plus the ultimate sale price, PH: Note the similarity between this formula and the bond valuation formula. The key differences in the case of stocks are the uncertainty of dividends, the lack of a fixed maturity date, and the unknown sales price at the horizon date. Indeed, one can continue to substitute for price indefinitely, to conclude: Equation 18.3 states that the stock price should equal the present value of all expected future dividends into perpetuity. This formula is called the dividend discount model (DDM) of stock prices. Don’t forget that the price at which you can sell a stock in the future depends on dividend forecasts at that time. The DDM asserts that stock prices are determined ultimately by the cash flows accruing to stockholders, and those are dividends. The Constant-Growth DDM Equation 18.3 as it stands is still not very useful in valuing a stock because it requires dividend forecasts for every year into the indefinite future. To make the DDM practical, we need to introduce some simplifying assumptions. A useful and common first pass is to assume that dividends are trending upward at a stable growth rate that we will call g. For example, if g = .05, and the most recently paid dividend was D0 = 3.81, expected future dividends are and so on. Using these dividend forecasts in Equation 18.3, we solve for intrinsic value as: This equation can be simplified to: Equation 18.4 is called the constant-growth DDM, or the Gordon model. If dividends were expected not to grow, then the dividend stream would be a simple perpetuity, and the valuation formula would be V0 = D1/k. The constant-growth DDM is valid only when g is less than k. If dividends were expected to grow forever at a rate faster than k, the value of the stock would be infinite. If an analyst derives an estimate of g greater than k, that growth rate must be unsustainable in the long run. The appropriate valuation model to use in this case is a multistage DDM such as those discussed below. The constant-growth DDM is so widely used by stock market analysts that it is worth exploring some of its implications and limitations. 72 The constant-growth rate DDM implies that a stock’s value will be greater: 1. The larger its expected dividend per share. 2. The lower the market capitalization rate, k. 3. The higher the expected growth rate of dividends. Another implication of the constant-growth model is that the stock price is expected to grow at the same rate as dividends. The DDM implies that when dividends grow at a constant rate, the rate of price appreciation in any year will equal that growth rate, g. For a stock whose market price equals its intrinsic value (V0 = P0), the expected holding-period return will be: This formula offers a means to infer the market capitalization rate of a stock, for if the stock is selling at its intrinsic value, then E(r) = k, implying that k = D1/P0 + g. By observing the dividend yield, D1/P0, and estimating the growth rate of dividends, we can compute k. This equation is also known as the discounted cash flow (DCF) formula. Convergence of Price to Intrinsic Value Now suppose that the current market price of ABC stock is only $48 per share and, therefore, that the stock is undervalued by $2 per share. In this case the expected rate of price appreciation depends on an additional assumption about whether the discrepancy between the intrinsic value and the market price will disappear, and if so, when. One fairly common assumption is that the discrepancy will never disappear and that the market price will trend upward at rate g forever. 75 Life Cycles and Multistage Growth Models As useful as the constant-growth DDM formula is, you need to remember that it is based on a simplifying assumption, namely, that the dividend growth rate will be constant forever. In fact, firms typically pass through life cycles with very different dividend profiles in different phases. In early years, there are ample opportunities for profitable reinvestment in the company. Payout ratios are low, and growth is correspondingly rapid. In later years, the firm matures, production capacity is sufficient to meet market demand, competitors enter the market, and attractive opportunities for reinvestment may become harder to find. In this mature phase, the firm may choose to increase the dividend payout ratio, rather than retain earnings. Dividend growth slows because the company has fewer investment opportunities. To value companies with temporarily high growth, analysts use a multistage version of the dividend discount model. For example, a two-stage dividend discount model allows for an initial high-growth period before the firm settles down to a sustainable growth trajectory. The combined present value of dividends in the initial high-growth period is calculated first. Then, once the firm is projected to settle down to a steady growth phase, the constant-growth DDM is applied to value the remaining stream of dividends. Value Line projects fairly rapid growth in the near term, with dividends rising from $1.04 in 2017 to $1.60 in 2020. This growth rate cannot be sustained indefinitely. We can obtain dividend inputs for this initial period by using the explicit forecasts for 2017 and 2020 and linear interpolation for the years between: Now let us assume the dividend growth rate levels off in 2020. Value Line forecasts a dividend payout ratio of .53 and an ROE of 19.5%, implying long-term growth will be: g = ROE × b = 19.5% × (1 − .53) = 9.17% Value Line rounds this estimate off to 9%, so we too will set g = 9%. Our estimate of GE’s intrinsic value using an investment horizon of 2020 is therefore obtained from Equation 18.2, which we restate here: P2020 according to the constant-growth DDM, should be: The only variable remaining to be determined to calculate intrinsic value is the market capitalization rate, k. One way to obtain k is from the CAPM. Given beta = 1.1, risk-free rate = 2.5% and market risk premium = 8%. Therefore, we can solve for the market capitalization rate as: and today’s estimate of intrinsic value is: We know from the Value Line report that GE’s actual price was $30.98. Our intrinsic value analysis indicates that the stock was considerably underpriced. 76 Arbitrage Definitions In finance theory an “arbitrage” is defined as: – a trading strategy that generates a completely riskless profit, that is, – a trading strategy that generates a positive cash flow at some time and non-negative cash flows at all times. Arbitrage Pricing Important insight in finance: – there cannot be arbitrage opportunities – if there were, arbitrageurs would trade aggressively to exploit the arbitrage. – called the “No-Arbitrage Condition” Perhaps surprisingly, using this restriction alone we can – compute restrictions on security prices – compute explicitly prices of derivatives. Implications of No Arbitrage 1. If two securities have the same payoffs, they must have the same price 2. If a portfolio has the same payoff as a security, the price of the security must be equal to the price of the portfolio (replicating portfolio) 3. If a self-financing trading strategy has the same final payoff as a security, the price of the security must be equal to the cost of the strategy (dynamic hedging strategy) 77 Chapter 20 Options Markets: Introduction 20.1 The Option Contract A call option gives its holder the right to purchase an asset for a specified price, called the exercise or strike price, on or before some specified expiration date. On the expiration date, if the stock price is greater than the exercise price, the call value equals the difference between the stock price and the exercise price; but if the stock price is less than the exercise price, the call expires worthless. The net profit on the call is the value of the option minus the price originally paid to purchase it. The purchase price of the option is called the premium. It represents the compensation the call buyer pays for the right to exercise only when exercise is desirable. A put option gives its holder the right to sell an asset for a specified exercise or strike price on or before some expiration date. Whereas profits on call options increase when the asset price rises, profits on put options increase when the asset price falls. A put will be exercised only if the price of the underlying asset is less than the exercise price, that is, only if its holder can deliver for the exercise price an asset with a lesser market value. An option is described as in the money when its exercise would produce a positive cash flow. Therefore, a call option is in the money when the asset price is greater than the exercise price, and a put option is in the money when the asset price is less than the exercise price. A call is out of the money when the asset price is less than the exercise price; no one would exercise the right to purchase for the strike price an asset worth less than that amount. A put option is out of the money when the exercise price is less than the asset price. Options are at the money when the exercise price and asset price are equal. American options allow its holder to exercise the right to purchase (if a call) or sell (if a put) the underlying asset on or before the expiration date. European options allow for exercise of the option only on the expiration date. American options, because they allow more leeway than their European counterparts, generally will be more valuable. Adjustments in Option Contract Terms To account for a stock split, the exercise price is reduced by a factor of the split, and the number of options held is increased by that factor. For example, each original call option with exercise price of $150 would be altered after a 2-for-1 split to two new options, with each new option carrying an exercise price of $75. A similar adjustment is made for stock dividends of more than 10%; the number of shares covered by each option is increased in proportion to the stock dividend, and the exercise price is reduced by that proportion. Cash dividends do not affect the terms of an option contract. 80 Notice that potential losses are limited. Figure 20.7 compares the profits for the two strategies. The profit on the stock is zero if the stock price remains unchanged and ST = S0. It rises or falls by $1 for every dollar swing in the ultimate stock price. The profit on the protective put is negative and equal to the cost of the put if ST is below S0. The profit on the protective put increases one for one with increases in the stock price once ST exceeds S0. Protective put strategies provide a form of portfolio insurance. The cost of the protection is that if the stock price increases, your profit is reduced by the amount you expended on the put, which turned out to be unneeded. Covered Calls A covered call position is the purchase of a share of stock with a simultaneous sale of a call option on that stock. The call is “covered” because the potential obligation to deliver the stock can be satisfied using the stock held in the portfolio. Writing an option without an offsetting stock position is called by contrast naked option writing. The value of a covered call position at expiration, presented in Table 20.2, equals the stock value minus the value of the call. The call value is subtracted because the covered call position involves writing a call to another investor who may exercise it at your expense. The solid line in Figure 20.8, Panel C, is the payoff. You see that the total position is worth ST when the stock price at time T is below X and rises to a maximum of X when ST exceeds X. In essence, the sale of the call options means the call writer has sold the claim to any stock value above X in return for the initial premium (the call price). Therefore, at expiration, the position is worth at most X. 81 Straddle A long straddle is established by buying both a call and a put on a stock, each with the same exercise price, X, and the same expiration date, T. Straddles are useful strategies for investors who believe a stock will move a lot in price but are uncertain about the direction of the move. The worst-case scenario for a straddle is no movement in the stock price. The value of the portfolio at expiration, while never negative, still must exceed the initial cash outlay for a straddle investor to clear a profit. The profit line lies below the payoff line by the cost of purchasing the straddle, P + C. It is clear from the diagram. The profit line lies below the payoff line by the cost of purchasing the straddle, P + C. It is clear from the diagram that the straddle generates a loss unless the stock price deviates substantially from X. The stock price must depart from X by the total amount expended to purchase the call and the put for the straddle to clear a profit. Strips and straps are variations of straddles. A strip is two puts and one call on a security with the same exercise price and expiration date. A strap is two calls and one put. 82 Spreads A spread is a combination of two or more call options (or two or more puts) on the same stock with differing exercise prices or times to maturity. Some options are bought, whereas others are sold, or written. A money spread involves the purchase of one option and the simultaneous sale of another with a different exercise price. A time spread refers to the sale and purchase of options with differing expiration dates. Collars A collar is an options strategy that brackets the value of a portfolio between two bounds. Suppose that an investor currently is holding a large position in FinCorp stock, which is currently selling at $100 per share. A lower bound of $90 can be placed on the value of the portfolio by buying a protective put with exercise price $90. This protection, however, requires that the investor pay the put premium. To raise the money to pay for the put, the investor might write a call option, say, with exercise price $110. The call might sell for roughly the same price as the put, meaning that the net outlay for the two options positions is approximately zero. Writing