Fundamentals of Investments Lecture Notes PDF

FUNDAMENTALS OF INVESTMENTS Lecture notes Petri Jylhä Aalto University School of Business November 2, 2024 Forewords These lecture notes contain the condensed core readings for the FIN-A0104 – Fundamentals of Investments course at Aalto University School of Business. A typical introductory investments textbook is hundreds, if not over a thousand, pages long. These lecture notes are an attempt to condense the central topics relevant for the course in a very small amount of pages. The natural consequence is that this document contains less explanations and examples than a traditional textbook. A reader looking for a more thorough treatment of any of the topics covered here will be well served by any edition of Investments by Bodie, Kane, and Marcus. Contents I Fundamentals 1 1 Assets and markets 2 2 Prices and returns 10 II Portfolio theory 19 3 Portfolios 20 4 Portfolio optimization 30 III Asset pricing 40 5 Asset pricing theory 41 6 Asset pricing empirics 51 IV Applications 59 7 Fixed income basics 60 8 Fixed income portfolios 70 9 Security analysis 77 10 Derivative securities 87 11 Active portfolio management 97 12 Sustainable investing 105 V Appendix 109 Appendix A Time value of money 110 Appendix B Random variables 118 Part I Fundamentals 1 Chapter 1 Assets and markets This chapter introduces the basic concepts and terminology of investing and fi- nancial assets and markets. 1.1 Investing, assets and securities Investing is the process of committing resources today for the acquisition of assets that are expected to generate some benefits in the future. For example, this can be buying stocks today to enjoy dividends and price appreciation in the future, or sacrificing time and money today on education in order to enjoy higher lifetime earnings and greater job satisfaction in the future. Assets can be divided into two broad categories. Real assets are things like land, buildings, machinery, and knowledge that we use to generate the net income of our economy. Financial assets are things like stocks and bonds. These are financial contracts about who gets what share of the income generated by the real assets and who controls the real assets. As such, financial assets do not generate income. However, they facilitate the accumulation of the income-generating real assets. Securities are financial assets that are tradable and fungible. Securities can be 2 exchanged easily for cash or other financial securities. Historically, securities were in the form of a paper certificate, but nowadays they mainly take the form of a digital book entry. The two main categories of financial securities are debt and equity. Debt securities or fixed income securities are a form of securitized borrowing. The borrower (for example a government or a firm) borrows money from a lender by issuing a debt security. The lender gives money to the borrower and receives the debt security in return. The borrower promises to make fixed payments to the holder of the security: periodic coupon payments and par value at maturity. The lender can either hold the security until maturity or sell it to another investor who then inherits the rights to receive the payments. Debt securities typically do not come with any control rights over the assets of the issuer. The market for fixed income securities is divided in two segments. The market for debt securities with one year or less to maturity is known as money market. Money market securities issued by governments are called government bills, whereas those issued by banks are called certificates of deposit and those issued by firms are known as commercial paper. Fixed income securities with more than one year to maturity are known as bonds. Sometimes bonds with less than ten years to maturity are called notes. There are two main sources of risk associated with investing in fixed income secu- rities, interest rate risk and credit risk. Interest rate risk refers to the fact that when interest rates rise, the value of fixed income securities declines. Typically, the longer the maturity of the security, the greater the risk. Credit risk refers to the risk that the issuer will not have sufficient funds available to pay the coupons or the par value when due. Credit risk is typically relatively low in government bonds but varies widely across issuers in corporate bonds. Chapters 7 and 8 provide a more detailed analysis of fixed income investments. Equity securities are basically common stocks of corporations. They provide their holders two kinds of rights. Cash flow rights refer to the equityholders’ right to receive, as dividends, their share of the firm’s profits. Note that the 3 amount of dividend is not fixed, can vary substantially over time, and can also be zero for long periods of time. Equityholders also control the firm; this is known as control rights. Equityholders exercise this control by appointing the board of directors and by voting on various issues brought to the general meeting of the shareholders. Indexes make tracking the overall development of the market, or a segment of the market, easier. An index is basically a basket of securities chosen following a predetermined rule. There are bond indexes tracking the developments of bond markets and there are equity indexes tracking the development of equity markets. Under the umbrella of equity indexes, there are global indexes, regional indexes, sector indexes, style indexes, and so on. 1.2 Financial markets Financial markets are markets for financial securities. Buyers and sellers of fi- nancial securities meet in financial markets to exchange securities and cash under predetermined rules. Financial markets can be seen to serve a number of roles in a well-functioning economy: Facilitation of trade. Financial markets match the demand for securities by buyers with the supply of securities by sellers. This results in transactions (i.e. securities changing ownership) as well as prices for the securities. Production of information. Investors seeking to maximize the investment profits have strong incentives to buy securities when they are too cheap and sell when too expensive. This results in the prices reflecting the future prospects of the issuer. By facilitating efficient trade in financial securities, financial markets ensure that the price signals are as informative as possible. Allocation of resources. Financial markets, and market participants, use the information in prices to allocate resources to their highest valued uses. 4 Timing of consumption. Investors can use the financial markets to move consumption in time. An individual with surplus resources now, can invest in financial securities and use the proceeds from these investments to finance consumption in the future when she has less resources otherwise. Saving for retirement is an example of this. Sharing of risks. Through well-functioning financial markets, investors and entrepreneurs can diversify their risks so that they hold multiple securities rather than just a few. Separation of ownership and control. Some individuals may be very good at managing firms but do not possess the financial resources to start a firm. Others may have the financial resources to finance a firm but not the skills to manage one. Financial markets enable the first group to seek financing for their firm from the second group ensuring that every individual in the economy does what they are best at. Financial markets are typically thought to be very competitive. There are large amounts of investors seeking the best possible investments. Due to the potentially huge financial rewards, a lot of talent and resources are dedicated to identify- ing those best investments. This has the important implication that we are very unlikely to find free lunches on the financial markets. Due to the competitive mar- ket, prices of financial securities are likely to reflect all relevant information. Thus, finding too cheap or too expensive securities is impossible. As a result, expected profits only depend on the risks taken in investing, not on any informational ad- vantages. We discuss this concept of efficient markets hypothesis in more detail in Section 2.4. Financial markets have two sub-markets, primary and secondary. The term pri- mary market is used to denote when securities (like stocks and bonds) are initially created and sold. In primary markets, issuer (like firms and governments) issue, or sell, the securities to investor. Investors receive the securities and the issuer receives the money. The primary market is where issuers raise money from the investors. The secondary market is where investors trade already issued secu- 5 rities among themselves. In the secondary market, securities and cash move from investors to other investors. The issuer is not a counterparty to transactions in the secondary market.1 While the issuer does not directly benefit from the trans- actions in the secondary market, an active and liquid secondary market facilitates the functioning of the primary market. Thus, issuer benefit indirectly from the secondary market activity. The main participants in the financial markets are financial intermediaries, house- holds who are suppliers of capital (investors), and firms and governments who are net demanders of capital (issuers). Some firms and governments also invest through financial markets but on, aggregate, these sectors are net demanders. Financial intermediaries are institutions that help bring the suppliers and de- manders of capital together in the market, such as commercial banks, investment banks, investment companies, investment funds, pension funds, brokers, financial advisers, and securities exchanges. 1.3 Trading financial securities Broadly speaking, markets in general can be classified to be of one of four different types: In a direct search market, buyers and sellers find each other directly without a formalized market mechanism. In a brokered market, a broker is used to find and match buyers and sellers. Primary markets for financial securities are organized as brokered markets with, typically, an investment bank acting as the broker. In a dealer market, a dealer acts as the market-maker buying from the sell- ers and then selling to prospective buyers. For example, secondary markets for bonds are organized as dealer markets. 1 There are exceptions to this. For example, a firm may buy back its shares on the open market. A vast majority of secondary market transactions are between investors. 6 In an auction market, buyers and sellers gather in one venue (either phys- ical or virtual) to buy and sell through an auction mechanism. Secondary markets for equities are typically organized as continuous auction markets where buyers submit order to buy and sellers submit orders to sell. When there is a match in price of a buy and a sell order, a transaction is happens. In a continuous auction market, such as most secondary markets for equities, there are continuously many outstanding orders to buy and sell a security. Orders to sell are known as ask orders and orders to buy are known as bid orders. The collection of all currently outstanding asks and bids is known as an order book. Traders can submit two main types of orders. A market order is an order to buy at the lowest available ask price or to sell at the highest available bid price. A limit order is an order to buy or sell at a specific price. The limit orders go in the order book to wait for a market order to match. The market orders match to existing limit orders in the order book. A trader incurs transaction costs from trading. A bid-ask spread is the difference between the lowest ask price and the highest bid price, essentially the difference between the highest price that a buyer is willing to pay for an asset and the lowest price that a seller is willing to accept. The wider this spread, the higher the implicit costs of trading are. The trader also has to pay a fee to her broker for facilitating the trade. This fee is known as a commission. Having some money left over and buying some financial securities (such as stocks or bonds) seems quite natural. However, in modern financial markets, investors can also borrow to transact. Buying on the margin refers to an arrangement where an investor borrows part of the total purchase price of a security. The investor’s own money makes up only a part of the total purchase price, so she borrows the rest, typically from her broker. She, of course, has to pay back the borrowed amount, plus interest, back at a later date. Short selling is even more exotic and amounts to selling a financial security that the investor does not own. Say, the investor thinks that a stock will decline in value in the future. To benefit from this view, she could sell the stock. But she does not own any of the stock. Not a problem, she can borrow the stock through her broker, sell the stock on the 7 market, and buy back and return to the lender at a later date. If she turns out to be right, and the price declines, she profits from this transaction. 1.4 Portfolio management Much of what follows focuses on portfolios rather than individual assets. An investor’s portfolio is simply the collection of all her investments. A portfolio can contain just one asset or multiple assets. Portfolio allocation refers to the composition of the portfolio, what fraction of the portfolio is invested in each asset (or asset class). There are two big decisions every investor has to make when it comes to deciding their portfolio allocation. Asset allocation is the decision of how much to invest in each asset class, such as equities, bonds, and money market securities. Secu- rity selection is the decision about which particular assets hold and in which proportions. For example, an investor may decide on an asset allocation of 70% in equities and 30% in bonds, and within the equity part she may decide to invest 5% in the GM stock and not at all in the Tesla stock. Finally, there are two broad approaches to portfolio construction. The top-down approach starts from the asset allocation into broad asset classes and then selects securities in each class. Going the other way, the bottom-up approach focuses on selecting the best securities to hold, and the asset allocation is merely an outcome of the security selection. Most investing wisdom favors the top-down approach. Summary of Chapter 1 Investing is the process of committing resources today to receive some ben- efits in the future. In the context of this course, investing is the purchase of financial assets in order to enjoy cash flows and price appreciation in the future. 8 Real assets generate the net income of the economy, whereas financial asset determine the distribution of this income. Securities are financial assets that are tradable and fungible. The main classes of securities are debt (or fixed income) securities and equity securities. Financial markets are markets for financial securities. In addition to facili- tating trade, they serve many other useful purposes. Firms and governments issue securities in the primary market. Investors trade securities in the secondary market. Buying on margin and short selling allow investors to borrow either cash or securities for the purpose of trading. A portfolio is the collection of all an investors financial assets. 9 Chapter 2 Prices and returns This chapter establishes the general concepts of return, expected return, and risk of an investment, and discusses why prices of financial assets behave the way they do. 2.1 Price A key ingredient in any analysis of investments and investment opportunities is the price of an asset. Typically, the price is set on the market as a result of matching investors’ willingness to buy and sell the asset. Pt denotes the price at which an investor can buy or sell a particular asset at time t. Theoretically, the price of an asset should equal the present value of the expected cash flows generated by the asset: ∞ X E [CFt+τ ] Pt = , (2.1) τ =1 (1 + k)τ where E [CFt+τ ] are the expected future cash flows and k is the appropriate dis- count rate. In the chapters below, we will figure out which cash flows go in the numerator and which discount rate goes in the denominator. For now, we can 10 treat prices as given. The law of one price states that two assets that have exactly the same future cash flows should also have exactly the same price today. This makes perfect sense. For example, all shares of Apple have the same price as they are claims to exactly the same cash future flows. If the law of one price would for some reason not hold, this would create an opportunity to make instant riskless profits. If two assets have the same future cash flows but different prices, an investor could buy the cheaper asset and short the more expensive one. She would earn the price difference as profit today. In the future, her cash flows from the two positions would exactly offset and her net cash flow in all future periods would be zero. This is known as an arbitrage, and the investor engaged in this activity is known as an arbitrageur. A central assumption in financial economics is that the law of one price holds and there are no arbitrage opportunities. 2.2 Return The rate of return, or simply return, is the relative profit on an investment. It comprises any change in value of the investment and any cash flows (interest, dividend, or other) received by the investor. Assume an investor buys a stock at time t − 1 for price Pt−1. At time t, she receives a dividend Dt and sells the stock for price Pt. The investor’s profit from holding the stock is Pt + Dt − Pt−1. It is easier, and more intuitive, to work with return, which is simply the profit scaled by the initial investment: Pt + Dt − Pt−1 rt = , (2.2) Pt−1 where rt denotes return during the period ending at time t. This can be rewritten as Pt − Pt−1 Dt rt = + , (2.3) Pt−1 Pt−1 where the first term is known as the capital gain rate (often also written as Pt /Pt−1 − 1) and the second term is known as the dividend yield. Dividend yield is always non-negative, as stocks pay either positive dividends or no dividends at 11 all. Stock prices, however, fluctuate and the capital gain rate may be positive or negative. Equation (2.2) can of course also be used to calculate returns of assets other than stocks. Any cash flows the investor receives from the asset, such as coupons from a bond or rent from real estate, are captured by the Dt term. The equation can also be used in the case the investor decided not to sell the asset at the end of time t. In such a case, the capital gain rate is not realized but the return calculation appropriately accounts for the appreciation of the asset price. The return over multiple time periods can be calculated as the product (not the sum) of single-period returns. An asset’s return from period t to period t + τ is given by τ Y rt:t+τ = (1 + rt ) × (1 + rt+1 ) ×... × (1 + rt+τ ) − 1 = (1 + rt+i ) − 1. (2.4) i=0 It is important to draw a distinction between two related but very different con- cepts: realized and future returns. Realized return is a return that has already happened. It is a matter of historical record and there is no uncertainty associated with it. Future returns are returns that have not yet been realized. Hence, there is uncertainty, or risk, associated with them. While realized returns are important for the purposes of keeping records of investment performance, here we are more interested in future returns, especially risks and expected returns. Appendix B provides a primer on random variables which may be useful background for the analysis of future returns as random variables. 2.3 Expected return and risk As asset prices fluctuate in an unpredictable manner, the future return from hold- ing an asset is random. An investor will not know what her future return on an investment will be. The fact that future returns are random does not mean that we know nothing about them. We can have information about the most likely 12 future returns and the uncertainty related to the future returns. These are called expected return and risk, respectively, and are the two most important properties of an investment. Expected return is the expected value of the future return. It is the most likely return to be realized in the future. If the investor holds the investment for a very long (infinite) period of time, her average return will be equal the expected return. Expected return is commonly denoted by E[r] or µ (Greek letter mu). The uncertainty related to the future returns is known as risk. An investors may have an estimate of the expected return, but the actual return may end up being higher or lower than the expectation. Risk measures by how much the realized returns can deviate from the expected return. As is described in Appendix B, standard deviation is an intuitive measure of the degree of uncertainty associ- ated with a random variable. In the finance context, the standard deviation of returns is often referred to as volatility. A useful rule of thumb is that if returns are normally distributed, 68% of the realized returns will lie within one times the volatility from the average realized return and 95% of realized returns will lie within two times the volatility from the average. A high volatility means that the potential future returns of an investment are dispersed widely around the most likely return, i.e. expected return. Such a high volatility investment hence has a high risk. The potential future returns of a low volatility, i.e. low risk, investment are clustered relatively close to the expected return. An investment that has zero volatility has no risk, as the return will for sure be equal to the expected return. Volatility is commonly denoted by SD[r] or σ (Greek letter sigma). Variance is another measure of risk. Variance is simply the square of standard deviation. Because of this, there is a one-to-one mapping between standard de- viation and variance, and variance offers no additional information over standard deviation. Variance is also even less intuitive than standard deviation. However, in some applications below it is easier to work with variance than with standard deviation. Variance is commonly denoted by V ar[r] or σ 2. When the investment horizon grows the standard deviation of returns, or volatility, 13 grows as well. However, volatility only grows by the square root of the time √ horizon. Hence, if the one-period volatility is σ, the T -period volatility is σ × T. For example, if the standard deviation of monthly returns is 5% then the standard √ deviation of annual returns is 5% × 12 ≈ 17.3%, which is only about 3.5 (not 12, as one might guess) times the monthly figure. Figure 2.1: Risk-expected return graph. 20 Expected return, µ (%) 15 10 P 5 0 0 10 20 30 40 Standard deviation, σ (%) It is often useful to think about portfolios graphically plotting their expected returns against their risks. Figure 2.1 provides an example. In the figure, portfolio P has an expected return of 10% and a volatility of 20%. Dots higher up (lower down) in the graph represent portfolios with higher (lower) expected returns. Dots more to the right (left) represent portfolios with higher (lower) risk. 2.4 Efficient market hypothesis What moves asset prices? Why prices sometimes go up, sometimes down, and sometimes hardly move at all? As we note above in Section 2.1, the price of an 14 asset equals the present value of the expected cash flows generated by the asset: ∞ X E [CFt+τ ] Pt = , (2.5) τ =1 (1 + k)τ where E [CFt+τ ] are the expected future cash flows and k is the appropriate dis- count rate. Hence, prices change when either the expected cash flows change or the discount rate changes. The efficient market hypothesis states that asset prices reflect all relevant information instantaneously. This means that when any new information relevant to the expected cash flows or the discount rate arrives, the price of the asset moves immediately to reflect this new information. This immediate price reaction is the result of a very large number of investors constantly analyzing a very large number of information sources and trading in order to take advantage of any information they may have. At the arrival of positive (negative) news, these investors rush to buy (sell) the asset pushing its price up (down) until the price on the market equals the fair value under the new piece of information. With a large number of investors competing to be the fastest to gather and process new information and trade accordingly, the price reaction to new information is instantaneous. There are three versions of the efficient market hypothesis: weak, semi-strong, and strong. The weak version says that asset prices reflect all the information that can be derived from past trading data such as historical prices and trading volumes. The semi-strong version says that prices reflect all publicly available information. Finally, the strong version says that prices reflect all relevant information, even information that is not publicly available. Figure 2.2 illustrates the price reaction to new information under efficient markets and two alternatives. The figure plots the price of an asset over time. The dotted vertical line marks the arrival of new information. Before the new information arrives, the price does not change as all inputs to Equation 2.5 remain unchanged. When the new information arrives the price moves. The efficient market hypothesis states that the price should immediately jump to its new fair value and remain there until the next piece of new information arrives. The price reaction may 15 also be inefficient. The two most obvious cases of inefficient price reaction are underreaction and overreaction. In the case of an underreaction, the price moves immediately in the right direction, but not sufficiently much. In the case of an overreaction, the price reaction overshoots and the price moves in the right direction by too much. In both cases, the inefficient initial price reaction is followed by a drift towards the fair value over time. Figure 2.2: Price reaction to information. Overreaction Efficient Price Underreaction Time In the real world, asset price charts do not look like any of the three lines in Figure 2.2. Prices are in constant motion up and down in patterns that can best be described as random. This, however, is perfectly consistent with the efficient markets hypothesis. New information arrives almost continuously. And the content of this new information is random by definition. Hence, also the price changes are continuous and seem random. The efficient market hypothesis has some very strong implications for investors. If markets are efficient, there is no point in active portfolio management strategies such as trying to buy undervalued assets. As prices reflect all information, no asset is undervalued. An investor may think that she has found an undervalued asset, an asset whose market price is less than the investor’s own assessment of the value. This, however, is not a result of the market price being too low. On the contrary, the investor’s assessment of the value is is too high due to too optimistic cash flow expectations or a too low discount rate. If markets are efficient, the 16 expected return of any asset, investment, or investment strategy depends only on its riskiness, not on any advantages in information processing.2 A key building block underlying the efficient market hypothesis is that investors on aggregate are rational. This means that they process information correctly and make correct decisions given the information. In the world of the efficient market hypothesis, individual investors can make mistakes in processing informa- tion but these mistakes cancel out when all the investors’ actions are aggregated in the market to form asset prices. For example, some investors may interpret new information too positively whereas other interpret it too negatively. However, in the process of trading the assets, these mistakes cancel out and the resulting asset prices reflect the new information correctly. 2.5 Behavioral finance While the efficient market hypothesis relies on aggregate rationality of the in- vestors, this might not be an accurate description of how real humans process information and make decisions under uncertainty. Behavioral finance studies systematic deviations from the assumptions of unbiased information processing and rational decision making. When processing information, people, for example, put too much weight on the most recent evidence (memory bias), overestimate the precision of their own belief (overconfidence), are too slow to update their beliefs (conservatism), and treat small samples as too representative of the pop- ulation (sample size neglect). When making decisions, people react differently to the same information depending on how it is presented (framing), segregate fi- nancial decisions into various categories (mental accounting), and make decisions with a purpose of avoiding feeling regret in the future (regret avoidance). These, and many more biases documented in investor behavior, are strong enough to have significant negative impacts on investors’ own financial outcomes. If many investors behave in the same biased manner, and there are not enough rational 2 We will return to the relation between risk and expected return in Part III where we discuss asset pricing. 17 investors to correct their mistakes, the biases can also affect prices and returns on the aggregate level. Summary of Chapter 2 Return measures the relative (percentage) profit or loss on an investment. Expected return is the expected value of a future return. Risk is the degree of uncertainty related to a future return. Risk is often measured by the standard deviation of returns, also known as volatility. The price of an asset equals the present value of the future cash flows gener- ated by the asset. Prices change with the arrival of new information about the cash flows or the discount rate. As new information is random, changes in prices of financial assets are also random. Efficient markets hypothesis states that prices reflect relevant information instantaneously. If markets are efficient, investors cannot earn higher returns by gathering and processing information and trying to identify undervalued assets. Behavioral finance studies the real-world deviations from the assumption of rational information processing and decision making. 18 Part II Portfolio theory 19 Chapter 3 Portfolios Investors typically hold (or it would optimal for them to hold) diversified port- folios of risky assets. This means that they hold more than one risky asset. This is done to reduce the risk of the overall portfolio. This chapter introduces the basics of portfolios with focus on the expected returns and, especially, the risks of portfolios. 3.1 Portfolios of two assets Let us start with a two-asset portfolio.3 An investor holds portfolio P consisting of two risky assets: A and B. The investor invests the proportion wA in A and the remaining proportion, wB = 1 − wA , she invests in B. Expected return of this portfolio is given by µP = w A µA + w B µB , (3.1) where µA and µB are the expected returns of A and B, respectively. This equation 3 Investors really should not hold portfolios of just two assets. However, this simplified setting serves as a useful starting point to learn the basics of portfolio theory. We extend the analysis to more than two assets in Section 3.2. 20 makes intuitive sense: fraction wA of the portfolio is expected to earn return µA while fraction wB is expected to earn µB. The total portfolio expected return is the just a weighted average of the expected returns of the two assets. The risk of the portfolio is a bit more complicated, and interesting, than the expected return.4 The variance of the portfolio returns is given by σP2 = wA 2 2 2 2 σ A + wB σB + 2 wA wB σA σB ρA,B. (3.2) The standard deviation of the portfolio returns, or the portfolio volatility, is given simply by taking a square root of the variance, q 2 2 2 2 σP = wA σA + wB σB + 2 wA wB σA σB ρA,B. (3.3) In Equations 3.2 and 3.3, σA and σB are the volatilities of A and B, respectively. But the most interesting term in these equations is the correlation between the returns of A and B, denoted by ρA,B (Greek letter rho). The lower the return correlation between the two assets, the lower the portfolio volatility. Correlation is always between −1 and 1 and measures the degree to which the two assets move in relation to each other. A perfect positive correlation means that the correlation coefficient is exactly 1. This implies that as one asset moves, either up or down, the other stock moves in lockstep, in the same direction. A perfect negative correlation (-1) means that two assets always move in opposite directions, while a zero correlation implies no relation at all. Equation 3.3 implies that an investor can reduce the volatility, or riskiness, of her portfolio by investing in assets that have a low correlation with each other, for example stocks in unrelated industries. Reducing portfolio risk by investing in less-than-perfectly correlated asset is known as diversification. Figure 3.1 provides and illustration of diversification. There are two assets, A and B, with some expected returns and volatilities. The line connecting the assets plots all the possible portfolios made out of the two assets, assuming that the correlation 4 Appendix B discusses the calculation of variance and standard deviation of a sum of two random variables. 21 between the two assets is not perfect (positive or negative). One might intuitively expect that the line representing the possible portfolios is straight. That, however, is not the case. Rather, the line depicting the possible portfolios curves to the left, towards lower risk. This curving is due to the less-than-perfect correlation between the asset returns. Figure 3.1: Portfolios of two assets. B Expected return, µ A Standard deviation, σ Think about this situation from the perspective of an investor who currently has all her wealth invested in asset A. She could sell some of her holdings in A and buy some of asset B instead. This would move her portfolio along the line toward asset B. It is possible for her to construct a portfolio that has a higher expected return and a lower volatility than her current portfolio. Who would not want that? This is exactly the point of diversification; by holding multiple assets, an investor can construct a portfolio that is better in terms of reward-to-risk ratio than any individual asset on its own.5 Figure 3.2 extends Figure 3.1 by plotting the possible portfolios made out of assets A and B, with different levels of correlation between the asset returns. Let us inspect the different lines in the graph from right to left. First, the straight line running from A to B represents the possible portfolios when the returns of two assets are perfectly correlated (ρA,B = 1). In this case there is no curve to the line 5 We will return to the construction of optimal portfolios in Chapter 4. 22 and no benefits of diversification. This is because the perfect correlation means that returns of the two assets will always move in lockstep. Second from the right, the slightly curvy line represents the possible portfolios when the return correlation between the two assets is 0.6. The third line is identical to that in Figure 3.1 and represents the possible portfolios when the correlation is 0.2. On the fourth line, the correlation is −0.4. Finally, the fifth line, the combination of two straight lines, represents the situation where the asset returns are perfectly negatively correlated (ρA,B = −1). This figure shows quite simply that the smaller the correlation between the two assets the more effective diversification is. This is also easy to see from Equations 3.2 and 3.3 where plugging in a smaller value for ρA,B lowers the portfolio variance and volatility. Figure 3.2: Portfolios of two assets with different correlations. B Expected return, µ A Standard deviation, σ Besides correlation, another way of measuring the comovement between asset re- turns is covariance. Covariance, commonly denoted by Cov [rA , rB ] or σA,B , is simply the product of the volatilities of the two assets and the correlation between the assets, σA,B = σA σB ρA,B. (3.4) Covariance is not bounded like correlation and is thus less intuitive. However, it offers a useful, and more compact, way of expressing the product of volatilities and correlation that appears often in the context of portfolio theory. 23 3.2 Portfolios of many assets The results above results extend easily for portfolios of more than two assets. Let us assume a portfolio made out of N assets, numbered from 1 to N. The expected return of this N -asset portfolio is given by N X µP = w i µi , (3.5) i=1 where wi is the portfolio weight of asset i and µi is the expected return of asset i. The variance of the N -asset portfolio is given by N X N X −1 N X σP2 = wi2 σi2 + 2 wi wj σi σj ρi,j , (3.6) i=1 i=1 j=i+1 where σi2 is the variance of asset i and ρi,j is the correlation between the returns of asset i and asset j. The variance formula in Equation 3.6 may look complicated and messy but it is just a generalization of the 2-asset portfolio variance formula in Equation 3.2.6 There is a variance term for each asset i (wi2 σi2 ) and a covariance term for each pair of assets i and j (2 wi wj σi σj ρi,j ). The volatility of the N -asset portfolio is then simply the square root of the variance. One way to visualize the portfolio variance is to think of an N × N matrix like in Table 3.1. Here, each line represents an asset from 1 to N and each column represents an asset from 1 to N. In the elements of the matrix are the variance and covariance terms associated with the assets represented by the corresponding row and column. The total portfolio variance is just the sum of all the element of the matrix. In the diagonal elements of the matrix are all the variances and in the off-diagonal elements are all the covariances. Finally, it is worth noting that when the number of assets grows larger, the number 6 To be exact, Equation 3.2 is a special case of Equation 3.6 where N = 2. 24 Table 3.1: N -asset portfolio variance. Asset 1 Asset 2 Asset 3 ··· Asset N Asset 1 w1 w1 σ1,1 w1 w2 σ1,2 w1 w3 σ1,3 · · · w1 wN σ1,N Asset 2 w2 w1 σ2,1 w2 w2 σ2,2 w2 w3 σ2,3 · · · w2 wN σ2,N Asset 3 w3 w1 σ3,1 w3 w2 σ3,2 w3 w3 σ3,3 · · · w3 wN σ3,N.................. Asset N wN w1 σN,1 wN w2 σN,2 wN w3 σN,3 ··· wN wN σN,N of parameters that need to be estimated in order to calculate the portfolio variance grows even faster. To estimate the variance of a portfolio with N assets, one needs N variances (one for each asset) and N (N2−1) correlations. For example, with just two assets a total of three parameters (two variances and one correlation) are needed, whereas with 100 assets a one needs a total of 5,050 parameters (100 variances and 4,950 pairwise correlations). Figure 3.3: Portfolios of three assets. Expected return, µ Standard deviation, σ Plotting the possible portfolios of many assets is a bit more complicated than in the case of just two assets above. Rather than a line in the risk-expected return graph, the possible portfolios now form an area. Figure 3.3 illustrates the case with three assets. The assets are represented by the three dots. Using the formulas in Equations 3.5 and 3.6, one can calculate the expected returns and volatilities of all possible portfolios made out of these three assets by varying the weights of the assets. There are infinitely many such portfolios and together they form the 25 shaded area of the graph. Hence, an investor could pick any point in the shaded area and construct a portfolio whose expected return and volatility matches that particular point. Many of the portfolios are inefficient. For many portfolios, actually for all the portfolios in the inner region of the shaded area, there exists another possible portfolio that has the same risk but a higher expected return. No investor should ever invest in such inefficient portfolios. Hence, we are mainly interested in the upper part of the border of the shaded area, plotted by the solid line. This line represents all the efficient portfolios that have the highest expected return for a given level of risk. Hence, it is known as the efficient frontier. A rational investor will only invest in an efficient portfolio. It is important to note that none of the three original assets are efficient. They all lie far below the efficient frontier. This is, again, diversification at work. By diversifying across multiple assets, investors can construct portfolios that offer better return-to-risk ratios than any of the assets individually. The portfolio with lowest possible risk is known as the global minimum variance portfolio. This portfolio lies at the leftmost tip of the shaded area in the figure and is the starting point of the efficient frontier. Figure 3.4: Portfolios of five assets. Expected return, µ Standard deviation, σ Adding more assets to the mix naturally affects the set of possible portfolios. 26 Figure 3.4 presents the result of adding two more assets, marked by circles, to the three-asset example of Figure 3.3. The dashed line presents the frontier of the area of possible portfolios with just the three assets and the solid line plots the frontier with five assets. It is clear from the figure that adding the two additional assets moves the frontier to the left. This means that having access to more assets improves diversification as investors can construct even more efficient portfolios out of five assets than they could do out of three assets. It is important to note that the two additional assets are similar to the original three in terms of their expected returns and volatilities. Hence, the improvement in diversification here is not due to the new assets having particularly high expected return or low risk. The improvement is due to the less-than-perfect correlation between the returns of the assets. 3.3 Limits of diversification The result in Figure 3.4 raises the question that what happens if the number of assets is increased to very large, or infinite. Will all the risk be diversified away such that the global minimum variance portfolio has a zero variance? To see the answer, consider a simple case with N assets, all having the same expected return µ and the same volatility σ. Also all pairs of assets have the same correlation ρ. Let us construct a simple equal-weighted portfolio of the N assets. An equal-weighted portfolio invests the same weight in each asset. With N assets, the portfolio invests N1 in each asset. Applying the formula in Equation 3.6 and plugging in the volatilities and correlations, the variance of the equal-weighted portfolio is N N −1 N X 1 2 X X 1 1 σP2 = σ + 2 σ σ ρ. (3.7) i=1 N2 i=1 j=i+1 NN Noting that the summations are just repeating the same terms over and over again, the portfolio variance can be written as 1 2 N −1 2 σP2 = σ + σ ρ. (3.8) N N 27 Figure 3.5: Equal-weighted portfolios. Portfolio standard deviation, σP 0.5 0.4 0.3 0.2 0.1 0 1 10 100 1,000 Number of assets, N Figure 3.5 plots the standard deviation of the equal-weighted portfolio returns as a function of the number of assets N , calculated using Equation 3.8.7 The solid line depicts the case when the volatility of each individual asset is 0.5 and the return correlation between all asset pairs is 0.4. In this case, increasing the num- ber of assets initially lowers the portfolio volatility. However, the marginal effect of increasing N becomes smaller and smaller as N grows. Adding more assets beyond 100 has practically no impact on the portfolio volatility. The dashed line shows what happens if the pairwise correlation between the assets is zero. Now, increasing the number of assets lowers the portfolio volatility until, eventually, the portfolio becomes practically riskless when the number of assets becomes really large. Finally, the dotted line depicts the case when the assets are perfectly corre- lated (ρ = 1) and diversification does not lower the portfolio volatility at all. This is to be expected as diversification requires less-than-perfect correlation to work. Why does the solid line in Figure 3.5 not go to zero? Letting N in Equation 3.8 grow infinitely large yields the limit of the portfolio variance as lim σP2 = σ 2 ρ. (3.9) N →∞ 7 Note that the horizontal axis is on a logarithmic scale. 28 As long as the pairwise correlation between the assets is positive, the variance of an infinitely well-diversified portfolio will also be positive. Hence, while diversification can eliminate some risk, it cannot eliminate all risk. If the correlation is zero, then the limit of the portfolio variance is also zero, as seen in the dashed line in Figure 3.5. In the real world, the returns of different assets are, on average, positively corre- lated. This means that there is a limit on how much investors can reduce their portfolio risk by diversification. This distinction between diversifiable and non- diversifiable risk turns out to be very important for the determination of expected returns. This will be the topic of Chapter 5. Summary of Chapter 3 Correlation measures the degree of comovement between the returns of two assets. Numerically, correlation is always between −1 and 1. Low correlation between asset returns results in diversification benefits. An investor can improve the expected return-to-risk ratio of her portfolio by investing in assets that have a low correlation with the assets she already holds. Portfolios that have the highest possible expected return for a given level of volatility are called efficient portfolios. Efficient frontier is the collection of all the efficient portfolios. There are limits of diversification if assets, on average, are positively corre- lated. In such a case, not all risk can be eliminated by diversifying and some risk always remains even in infinitely diversified portfolios. 29 Chapter 4 Portfolio optimization This chapter works through the process of finding an optimal portfolio from the menu of infinitely many possible portfolios. The end result is surprisingly simple single portfolio of risky assets that is optimal to all investors. 4.1 Utility To find an optimal portfolio, we first have to know what aspects of the portfolio an investor cares about. For simplicity, let us assume that an investor only cares about the expected return and the riskiness of her portfolio.8 Given two equally risky mutually exclusive portfolios, the investor would prefer the one with higher expected return. Similarly, she would prefer the portfolio with lower risk when choosing between two portfolio with the same expected returns. But what if the investor can choose between two portfolios where one has a higher expected return and a higher risk than the other? We need a way to map the two dimensions of a portfolio, expected return and risk, to a single value in order to compare the two portfolios. Such single value is known as utility. We can think 8 This assumption ignores any non-monetary motives for investing, such as investing in a sports club to support the team rather than to earn financial returns. 30 of utility as the investor’s degree of happiness with the portfolio. The higher the utility the happier the investor is with the portfolio. An optimal portfolio is one that gives her the highest utility. A utility function provides the functional form of calculating the utility a particular investor experiences from holding a particular portfolio. Let us assume that an investor’s utility from holding a portfolio is given by 1 U = µ − A σ2, (4.1) 2 where µ is the expected return of the portfolio, σ 2 is the variance of the portfolio returns, and A is the investor’s coefficient of risk aversion. If an investor’s coefficient of risk aversion is positive (A > 0), she is said to be risk-averse. This means that she dislikes risk and would willing to trade off some of the expected return in order to reduce risk. Most investors are assumed to be risk-averse. The higher the coefficient of risk aversion, the more the investor dislikes risk, and the more expected return they are willing to give up to decrease portfolio risk by a small amount. If an investor’s coefficient of risk aversion is zero (A = 0), she is said to be risk- neutral. A risk-neutral investor does not care about the riskiness of her portfolio, only about its expected return. If an investor’s coefficient of risk aversion is nega- tive (A < 0), she is said to be risk-loving. Such an investor wants her portfolio to have more, rather than less risk. While risk-loving behavior among investors seems odd, it can explain why people play the lottery even though it has negative expected return and high risk. One way to represent the utility function graphically are indifference curves that plot all the portfolios that give the investor the same amount of utility. Figure 4.1 illustrates. It plots five portfolios (dots) and three indifference curves (lines). Assume that the investor is comparing portfolio P to the other portfolios. The solid line shows the investor’s indifference curve associated with portfolio P. All dots along this line, such as portfolios Q and R, represent portfolios that give the investor the same level of utility as portfolio P. If the investor would have to choose one of these portfolios, she would like them all equally much. Another way to put is that she is indifferent between portfolios P , Q, and R. 31 Figure 4.1: Portfolios and indifference curves. R Expected return, µ T P S Q Standard deviation, σ Any point below the solid line, such as portfolio S, is a portfolio that gives the investor less utility than portfolio P. Any point above the line, such as portfolio T is a portfolio that gives the investor more utility than portfolio P. The investor would prefer P over S, and she would prefer T over P. Portfolios S and T also have indifference curves associated with them, plotted in the figure by the dashed and the dotted line, respectively. The slope of the indifference curve is determined by the investor’s coefficient of risk aversion, A in Equation 4.1. As illustrated by Figure 4.2, a higher coefficient of risk aversion results in steeper indifference curves. The solid line presents the indifference curve of an investor with a relatively low coefficient of risk aversion whereas the dashed line is that of an investor with a higher risk aversion. 4.2 Portfolio optimization with one risky asset The goal of portfolio theory is to eventually construct an optimal portfolio for an investor out of multiple assets. For simplicity, let us start with a case where the investor is constructing a portfolio out of just two assets: a risky asset P and a risk-free asset. 32 Figure 4.2: Risk aversion and indifference curves. Expected return, µ (%) P Standard deviation, σ (%) A risk-free asset is a special asset that offers its holder a certain return of rf and has a volatility of zero. One can think of this as a bank account. Investing a positive amount in the risk-free asset is equivalent to depositing money in the bank. Investing a negative amount in the risk-free asset (i.e. shorting it) is equivalent to borrowing money from the bank. Now, the investor invests fraction w of her wealth in the risky asset P which has expected return µP and volatility σP. The remainder of her wealth, fraction 1 − w, she invests in the risk-free asset which, as mentioned above, has expected return rf and volatility 0. This results in a portfolio that can be denoted by C. Plugging this information into Equation 3.1 gives the expected return of portfolio C as µC = w µP + (1 − w) rf. (4.2) This is quite intuitive. A fraction w of the portfolio is expected to earn a return of µ and a fraction 1 − w earns rf. It turns out to be useful to rearrange the terms such that risky asset weight w appears only once and write the portfolio expected return as µC = rf + w(µP − rf ). (4.3) The difference between expected return of a risky asset and the risk-free rate of 33 return, µP − rf , is known as a risk premium. It measures how much more expected return a risky asset offers as compensation for risk relative to the risk- free alternative. It makes intuitive sense to focus on the risk premium rather than the expected return of a risky asset when thinking of compensation for risk. For example, a risky asset offering a 6% expected return looks very differently attractive if the risk-free rate of return is 1% than what it does if the risk-free rate is 5%. The expected return of portfolio C in Equation 4.3 is equal to the risk free rate of return plus the weight invested in the risky asset times the risk premium of the risky asset. The variance of portfolio C can be calculated by plugging in the necessary infor- mation into the formula in Equation 3.2: σC2 = w2 σP2. (4.4) Note that since the variance of the risk-free asset is zero, the last two terms of the variance formula in Equation 3.2 are zero and can be dropped. Taking a square root of the variance gives the volatility of portfolio C as σC = w σP. (4.5) This is also quite intuitive. Only a fraction w of the portfolio is invested in the risky asset whose volatility is σP. The investor’s possible portfolios form a straight line in the risk-expected return graph. But why? To see this, rearrange Equation 4.5 as σC w= (4.6) σP and plug this into Equation 4.3 to get µP − rf µC = rf + σ C. (4.7) σP This shows that the expected return of portfolio C (µC ) is linear in its volatility (σC ). It is important to note that this linear relation between a portfolio’s expected 34 return and volatility is not a general result that would apply to all portfolios of two assets. Section 3.1 shows that typically the relation is non-linear and depends on the correlation between the returns of the two assets. Here, the linearity arises because one of the assets is risk-free. The line in the risk-expected return graph showing the possible portfolios made out of a risky and a risk-free asset is known as the capital allocation line. Figure 4.3 provides an illustration. If the investor invests all her wealth in the risky asset P (w = 1), portfolio C will naturally have the same expected return and volatility as asset P. If the investor invests all her wealth in the risk-free (w = 0), portfolio C will naturally have expected return of rf and volatility zero. If the investor invests a part of her wealth in the risky asset and a part in the risk-free asset (0 < w < 1), portfolio C lies somewhere on the straight line between rf and P. Finally, if the investor borrows money in the risk-free asset and invests in risky asset P (w > 1), portfolio C lies somewhere on the line up and right of P. Figure 4.3: Capital allocation line. Expected return, µ (%) P rf Standard deviation, σ (%) The slope of the capital allocation line, µP − rf SP = , (4.8) σP turns out to be a very useful measure of the risk-adjusted expected return of the risky asset P. It measures the ratio of the risk premium of P to the volatility of 35 P. This measure is commonly referred to as the Sharpe ratio of asset P. How does the investor then choose the portfolio that is best for her among all the possibilities along the capital allocation line? Let us find the solution first graphically and then analytically. Figure 4.4 presents the process of finding the optimal portfolio, C ∗ , graphically. The investor must choose one portfolio from the solid line. She does this by finding the portfolio that gives her the highest utility. This portfolio is found at the point where the highest possible indifference curve just touches the capital allocation line. Any other point along the capital allocation line would be on a lower indifference curve and would give the investor a lower utility than C ∗ does. Hence, C ∗ must be the optimal portfolio that gives the investor the highest possible utility. Figure 4.4: Optimal portfolio. 20 Expected return, µ (%) 15 P 10 C∗ 5 rf 0 0 10 20 30 40 Standard deviation, σ (%) The optimal portfolio composition can also be solved for analytically. Remember that the investor invests w in the risky asset P , 1 − w in the risk-free asset giving her a portfolio with expected return µC = rf +w(µP −rf ) and volatility σC = w σP. She now has to decide which w gives her the highest utility. To do this, we can plug the expected return and volatility into the utility function in Equation 4.1: 1 UC = rf + w(µP − rf ) − A (w σP )2. (4.9) 2 36 Next, we calculate the derivative of the utility with respect to w, ∂UC = µP − rf − A w σP2 , (4.10) ∂w and find the optimal w∗ by setting the derivative equal to zero: 1 µP − rf µP − rf − A w∗ σP2 = 0 ⇒ w∗ =. (4.11) A σP2 This solution implies that the investor invests more of her wealth in the risky asset P when i) she is less risk-averse (her A is smaller), ii) the risky asset offers a higher risk premium (µP −rf is larger), and iii) the risky asset is less risky (σP is smaller). All these make intuitive sense. 4.3 Portfolio optimization with many risky asset Above we analyze the situation where the investor allocates between the risk-free asset and only one risky asset. But what about the more realistic case where she has access to multiple risky assets in addition to the risk-free asset? How should she allocate her portfolio? The easiest way to solve this problem is to break it in two separate steps. In the first step we find the optimal portfolio consisting of only the risky assets. In the second step we find the optimal allocation between this risky portfolio and the risk-free asset. Figure 4.5 illustrates the first step. Like in Figure 3.3, there are many risky as- sets (not displayed) from which the investor could form infinitely many different portfolios. The parabola plots the frontier of the set of possible portfolios (equiv- alent to the shaded area in Figure 3.3). We only need to concern ourselves with the frontier portfolios as all the portfolios within the set are dominated by some frontier portfolios. There is also a risk-free asset. The investor’s first problem is to pick the best possibly risky portfolio (a portfolio made out of risky assets only) to combine with the risk-free asset. Remember, the portfolios made out of the risk-free asset and any risky portfolio form a straight 37 Figure 4.5: Optimal risky portfolio. Expected return, µ Tangency portfolio rf Standard deviation, σ line in the volatility-expected return plot. Hence, whatever risky portfolio the investor picks to combine with the risk-free asset, will result in a set of possible portfolios represented by a straight line running from the risk-free asset through the investor’s chosen portfolio. As the investor wishes to have the highest total portfolio expected return for any level of portfolio volatility, she wants to pick a risky portfolio that results in a capital allocation line with the highest possible slope. This particular risky asset portfolio is known as the tangency portfolio because the straight line drawn from the risk-free asset through it just touches, or tangents, the efficient frontier. Another way of saying the tangency portfolio results in the highest slope of the capital allocation line is to say that the tangency portfolio has the highest Sharpe ratio among all the possible risky asset portfolios. Having now found the best possible risky portfolio, the investor decides the optimal allocation between the risk-free asset and the tangency portfolio. This however, is equivalent to allocating between the risk-free asset and one risky asset discussed above. The investor chooses the total portfolio that maximizes her utility, just like above in Section 4.2. It is noteworthy that the first step, finding the tangency portfolio, does not involve the investor’s preferences at all. Whatever her risk aversion happens to be, the tangency portfolio will always be the same. This is because the efficient frontier 38 and the location of the tangency portfolio only depends on the expected returns and risks of the assets available. All investors who agree on the expected returns, volatilities, and correlations will invest in the same tangency portfolio. In the second step the investor finds the optimal mix between the risk-free asset and the tangency portfolio. This decision depends on her risk aversion. More (less) risk-averse investor invests less (more) in the tangency portfolio. Summary of Chapter 4 Utility measures how satisfied an investor is with a portfolio that has a given expected return and a given level of risk. Higher utility is better. Risk aversion measures an investor’s dislike for risk. A coefficient of risk aversion is needed in a utility function to relate expected returns and risks. An indifference curve plots all the portfolios that give the investor the same level of utility. Portfolio optimization problem can be broken into two steps. In the first step the investor finds the risky asset portfolio with the highest Sharpe ratio, known as the tangency portfolio. In the second step, she allocates her wealth between the risk-free asset and the tangency portfolio to maximize her utility. 39 Part III Asset pricing 40 Chapter 5 Asset pricing theory This chapter discusses asset pricing, the process of determining expected returns of assets from the theoretical point-of-view. 5.1 Asset pricing So far, our perspective has been that of a very small investor who takes asset expected returns, volatilities, and correlations as given. However, from the per- spective of the market as a whole, expected returns are really not exogenously given. Rather, expected returns depend on the investors’ decisions to buy and sell assets. To illustrate the mechanism, remember that the price of an asset must equal the present value of the expected future cash flows generated by the asset, appropri- ately discounted: ∞ X E (CFt ) P = t. (5.1) t=1 (1 + µ) In this equation, the discount rate equals the expected return of the asset µ. Now assume that the asset becomes more popular among investors without any changes to the expected cash flows. As more investors flock to buy the asset, its 41 price increases. As the pricing identity must hold always, an increase in the price must be matched by a decrease in the expected return. Hence, investors’ decision to buy the asset result in it having a lower expected return. Asset pricing studies the determination of expected returns as a function of in- vestors’ demand for assets. The typical chain of logic goes as follows: i) investors dislike risk, ii) investors do not wish to hold a risky asset, iii) demand for the risky asset is low, iv) the price of the risky asset is low, and v) the expected return of the risky asset is high. Conversely, investors like safe assets, which results in safe assets having high prices and low expected returns. As an investor can earn the risk-free rate of return rf with no risk, it makes sense to focus on the risk premium, the expected return of a risky asset over the risk free rate: µ − rf. In standard asset pricing models, risk premium of an asset is linearly related to the risk of the asset. But what risk? 5.2 Systematic and idiosyncratic risk Figure 3.5 in Section 3.3 plots the development of portfolio volatility as more assets are added to and equal-weighted portfolio. If the correlation between the asset returns is greater than zero, some risk always remains no matter how many assets are added to the portfolio. This means that only a part of the risk of a portfolio can be diversified whereas some part of the risk is non-diversifiable. This gives us a decomposition the risk of any portfolio (composed of one or many assets) into two distinct components: diversifiable and non-diversifiable risk. Non-diversifiable risk, also known as systematic risk, is that part of the total risk that cannot be diversified away by adding more assets to the portfolio. This part of the risk is associated with the variation in returns that is common to all assets. The diversifiable risk, also known as idiosyncratic risk, is the part of the risk that can be diversified away. It is related to the asset-specific variation in returns. For example, returns of all stocks react to news about macroeconomic conditions, such as inflation. This is systematic risk that cannot be diversified. Conversely, the 42 rejection of a key patent application is very bad news for a small pharmaceutical company and its stock is likely to fall in value as a result. However, other companies are unaffected by this news, and the risk associated with the patent application is firm-specific and diversifiable. What part of the total risk affects risk premiums? It is easiest to intuit this by thinking of the risk premium as compensation for investors for bearing the risk related to holding an asset. The investors can always diversify away all idiosyn- cratic risk. Hence, they do not need to be compensated for bearing it. However, they cannot, by definition, diversify away the systematic risk. As investors always have to bear the systematic risk, they need to receive a compensation for it in the form of a risk premium. Hence, risk premium depends on the systematic risk, not on the idiosyncratic risk. Now that we have established that the risk premium should only depend on the systematic risk, two big questions remain. How do we measure the systematic risk of an asset? And, how are systematic risk and risk premium related? To answer these questions, we need an asset pricing model. An asset pricing model is a simplified portrayal of how investors perceive risk and how the financial markets price assets and determine expected returns. An asset pricing model provides a methodology for quantifying the systematic risk and translating it into a risk premium. The most commonly used asset pricing model is called (somewhat con- fusingly) the capital asset pricing model. 5.3 Capital asset pricing model The capital asset pricing model, or simply the CAPM, is a bedrock of asset pric- ing. Introduced in the early 1960’s, it is still widely used in various applications and it has spurred a large literature developing more complex models to price assets. A key advantage of the CAPM is its simplicity. It yields a simple and intuitive measure of systematic risk and a simple formula for connecting the sys- tematic risk to a risk premium. Basically, the CAPM provides the answer to the 43 following question: What what would the expected returns be if all investors were rational mean-variance optimizers and behaved as we prescribe above in Section 4.3? Let us start with the main assumptions underlying the CAPM: There are two kinds of assets: a risk-free asset and a number of risky assets. All investors can buy and sell all assets at market prices. All investors can lend and borrow at the risk-free rate. Investors incur no transaction costs or taxes. There are no other frictions. All investors hold efficient portfolios. They hold portfolios that yield the highest return for a given level of volatility. Some investors hold portfolios with higher volatility, some hold portfolios with lower volatility, depending on their individual preferences. All investors have identical expectations regarding the expected returns, volatilities, and correlations of all securities. The investors do not disagree. Solving for the market equilibrium (the investors’ portfolios and the prices of the securities) under these assumptions yields three main results. First, and least surprisingly given what we know from Section 4.3, all investors hold a combination of the risk-free asset and the tangency portfolio, or market portfolio, of the risky asset. While the investors’ total portfolios differ—as they hold different amounts of the risk-free asset depending on their preferences—they all hold the same portfolio of risky assets.9 In practical applications, the market portfolio is typically proxied with a value-weighted stock market index (such as the S&P 500 index in the United States). Second, and most importantly, the CAPM produces a very simple formula for the 9 This is not the case in the real world, as investors hold very different portfolios of risky assets. This is sometimes used as an argument against the CAPM and in favor of more complicated asset pricing models. 44 risk premium of an asset. In the CAPM, the risk premium of asset i is given by µi − rf = βi (µM − rf ). (5.2) The risk premium equals the amount of systematic risk ,βi , times the compensation for bearing one unit of systematic risk price of risk, µM − rf. µM stands for the expected return of the market portfolio of risky assets. µM − rf is known as the market risk premium. An asset’s beta, βi , is the measure of its systematic, non-diversifiable, risk. Beta is defined as σi,M σi βi = 2 = ρi,M , (5.3) σM σM where σi,M is the covariance between the returns of asset i and the market portfolio, ρi,M is the correlation between the returns of asset i and the market portfolio, σi is the volatility of asset i, and σM is the volatility of the market portfolio. Since the correlation can have either sign and the volatility can be very large, beta can (theoretically) take any real number value. Why is an asset’s risk premium given by the product of its beta and the market risk premium? The CAPM assumptions imply that all investors hold the market portfolio of risky assets. It is easy to see that the contribution of asset i to the total risk premium of the market portfolio is wi (µi − rf ), where wi is the weight of asset i in the market portfolio. It also turns out that the contribution of asset i to the total variance of the market portfolio is wi σi,M. Hence, the reward-to-risk ratio of asset i, from the perspective of an investor holding the market portfolio, is given by wi (µi − rf ) µi − rf =. (5.4) wi σi,M σi,M In an equilibrium, all assets must have the same reward-to-risk ratio. If one asset offers a higher ratio, all investors would buy that asset pushing up its price and pushing down its expected return. This would go on until the reward-to-risk ratio of the asset is in line with other assets. If one asset has a particularly low reward-to- risk ratio, investors would sell that asset causing its price to drop and its expected 45 return to increase, again, until the ratio is in line with other assets. Hence, all assets, including the market portfolio, must have the same reward-to-risk ratio. For the market portfolio, the relevant reward-to-risk measure is simply its risk premium relative to its variance. Since all assets have the same reward-to-risk ratio, it must hold for asset i and the market portfolio that µi − rf µM − rf = 2. (5.5) σi,M σM We can multiply both sides of the equation by σi,M to get σi,M µi − rf = 2 (µM − rf ). (5.6) σM σ So, there it is! Risk premium of asset i is given by the product of σi,M 2 , which is M exactly equal to beta in Equation 5.3, and the market risk premium. Next, let us try to understand the intuition behind Equations 5.2 and 5.3. First, think of an asset that has beta equal to zero. This asset has no systematic risk at all; all its risk is diversifiable. As such, it does not contribute at all to the total riskiness of the investors’ portfolios. Hence, investors are willing to pay a relatively high price for this asset. This is equivalent to saying that the investors require a relative low expected return for the asset. Indeed, the risk premium of a zero-beta asset is zero and its expected return equals the risk-free rate of return. This makes sense, because from a well-diversified investor’s perspective this asset does not carry any relevant, non-diversifiable, risk. Next, think of an asset that has beta equal to one. This asset is as risky as the market portfolio, as the market portfolio also has a beta of one.10 As this asset is as risky as the market portfolio, it will also have the same risk premium as the market portfolio. An asset with beta greater than one has more systematic risk than the market portfolio. The returns of this asset are positively correlated with the market port- 10 It is easy to see that the market portfolio beta equals one. Plug i = M in Equation 5.3 to get βM = ρM,M σσM M = 1. 46 folio and its volatility is higher than the market volatility. This is not a particularly nice asset from the investors’ perspective as it contributes a lot to the riskiness of their portfolios. Hence, the investors are only willing to pay a relatively low price for the high-beta asset, which is equivalent to saying that they demand a relatively high risk premium to hold the asset. This is clear from Equation 5.2 where the risk premium of a high-beta asset is higher than that of the market portfolio. An asset with beta between zero and one correlates positively with the market portfolio but has a lower systematic risk and, hence, a lower risk premium. Assets with negative betas are interesting. To have a negative beta, the asset’s returns need to be negatively correlated with the market portfolio returns. This means that when the investors’ portfolios have a negative returns, the negative- beta asset is likely to have a positive return. Hence, the asset actually decreases the overall riskiness of the investors’ portfolios. The investors like this insurance- like feature. Hence, they are willing to pay a very high price for the negative-beta asset. Equivalently, they are willing to accept a very low expected return, and a negative risk premium, to hold a negative-beta asset. Empirically, beta can be estimated as the covariance between the asset’s return and the market return divided by the market return variance. Identically, it can be estimated as the correlation of the asset’s return with the market return times the ratio of the return volatilities (asset divided by market). By a lucky coincidence, beta corresponds to the slope coefficient in a regression of the asset’s excess return on the market excess return. Excess return refers to the return of the asset minus the risk-free rate of return. We can denote excess return over time period t by Rt and define it as Rt = rt − rf. Thus, one additional way to estimate an asset’s beta is to collect realized excess returns of the asset, the market index, and the risk-free asset and estimate the regression Ri,t = αi + βi RM,t + ei,t. (5.7) This regression directly provides an estimate of the beta.11 11 Typically, either daily, weekly, or monthly returns are used to estimate the regression. As the risk-free rate of return varies over time, the regression needs to be estimated on excess returns, 47 Third, and finally, the CAPM provides a formula for the price of risk, i.e. the amount of risk premium investors earn for one unit of beta. As can be seen from Equation 5.2, increasing an asset’s beta by one increases its expected return by an amount equal to the market risk premium, µM − rf , the risk premium of the market portfolio. In an equilibrium, all investors’ borrowing in the risk-free asset must be matched by other investors investing in the risk free asset. This market clearing condition tells us that the market risk premium in the CAPM is equal to 2 the variance of the market portfolio (σM ) times the average degree of risk aversion of the investors (Ā): 2 µM − rf = Ā σM. (5.8) This, however, is not very operational as there is no way to measure the investors’ average degree of risk aversion. Hence, the typical approach to estimating the market risk premium is to rely on the average realized market return, in excess of the risk-free rate, over a very long period of time, or surveys of investors’ market return expectations. 5.4 Factor models It is useful to note the connection between the CAPM and something known as factor models. A factor model posits that an asset’s return (realized return, not the expected return) over the risk-free rate can be expressed as a sum of three components: a time-invariant constant return, a return due to exposure to a systematic factor, and an idiosyncratic (or asset-specific) return. Factor models differ in terms of which factors are used to model the systematic component of returns. The simplest example of a factor model is the single-index model. In a single- index model, the systematic factor is the return of a market index. Hence, in the i.e. the returns minus the risk-free rate of return each period. The maturity of the risk-free rate should match the frequency of the return observations. For example, if the observations are monthly returns, the risk-free rate should be the interest rate of a riskless one-month deposit or government bill. 48 single-index model, the excess return of asset i in time period t (Ri , t) is composed of the time-invariant component (αi ), an exposure to the market excess return (βi RM,t ), and an idiosyncratic component (ei,t ): Ri,t = αi + βi RM,t + ei,t. (5.9) A single-index model (or factor models more generally) is a convenient way to model asset returns in ways that capture many stylized facts of returns in the real world. First of all, since all assets have some beta exposure to the same market index return, the asset returns, on average, are positively correlated. Just like in the CAPM, assets with (low) high betas comove more (less) strongly with the market index. The idiosyncratic returns (ei,t ) are uncorrelated across assets. This results in correlations between assets to be less than perfect. The idiosyncratic return is also assumed to have an expected value of zero. Finally, the time-invariant alpha (αi ) represents the non-zero average excess return of asset i which is not due to the beta exposure to the market index return. Taking expectations of both sides of Equation 5.9, and noting that the expected excess return is the same thing as risk premium, yields µi − rf = αi + βi (µM − rf ). (5.10) This is almost the same as the CAPM in Equation 5.2. The only difference is that αi appears here but not in the CAPM. This is an important insight. The CAPM really just implies that all expected returns are due to the beta exposures to the market return and all alphas are zero. We will use this insight to test the empirical validity of the CAPM in Section 6.1. The variance of asset return in the single-index model is given by σi2 = βi2 σM 2 + σe2i , (5.11) where βi2 σM 2 is the systematic variance and σe2i is the idiosyncratic variance. Portfolios are also easy to analyze in the single-index model. A portfolio P consists 49 of N assets with wi denoting the weight of asset i. The excess return of the portfolio is modeled as RP,t = αP + βP RM,t + eP,t. (5.12) The portfolio’s alpha, beta, and idiosyncratic return are simply given by N X N X N X αP = wi αi , βP = w i βi , and eP,t = wi ei,t. (5.13) i=1 i=1 i=1 Since the idiosyncratic returns of different assets are by definition uncorrelated, the idiosyncratic variance of the portfolio is N X σe2P = wi2 σe2i. (5.14) i=1 Summary of Chapter 5 Asset pricing studies how asset prices and expected returns are set in a market equilibrium. Investors can diversify away all idiosyncratic risk and, hence, do not need to be compensated for bearing it. Systematic risk cannot be diversified away and investors need to receive a compensation, in terms of a risk premium, for bearing systematic risk. The capital asset pricing model (CAPM) provides a simple way to measure an asset’s systematic risk and to translate it into a risk premium or an expected return. In the CAPM, systematic risk is quantified by beta which measures the covariance of the asset’s returns with the market portfolio returns relative to the variance of the market returns. Factor models provide a simple way to model realistic asset returns. The CAPM is closely related to the single-index model, which a commonly used simple factor model. 50 Chapter 6 Asset pricing empirics Chapter 5 above focuses on the theory of asset pricing, especially the CAPM. In this chapter, we are going to study the empirical validity of the CAPM in describing expected returns and then look into some other empirical regularities related to asset prices. 6.1 Testing the CAPM The CAPM is a very popular model for calculating expected returns, mainly due to its simplicity and intuitive appeal. But how well does the model describe investors’ return expectations? The answer is: not that well. The CAPM, in Equation 5.2, states that the risk premium depends on beta, and only beta, and the price of risk equals the market risk premium. We can use data on asset returns to test whether these predictions are true in the real world. To test the CAPM, we take the following steps: 1. Collect historical returns of N test assets, the market portfolio, and the risk- free rate asset. The test assets can be individual assets (like stocks or bonds) or portfolios of assets. We are going to need long time series of returns. It 51 is common to use decades of monthly returns. Denote by T the number of return observations (typically months) we have for each test asset. Denote the excess return of a test asset i (i = 1,... , N ) in time period t (t = 1,... , T ) by Ri,t and the market portfolio excess return by RM,t. 2. Estimate beta for each test asset. The betas are easiest to calculate by estimating a regression where the test asset excess return is the dependent variable and the market excess return is the independent variable: Ri,t = αi + βi RM,t + ei,t. (6.1) This regression is estimated for each test asset separately. This step produces N beta estimates, one for each test asset. 3. Calculate the average average excess return for each test asset. Asset i’s average excess return is T 1X R̄i = Ri,t. (6.2) T t=1 This step produces N average returns, one for each test asset. 4. Relate the average excess returns to the betas. In the cross-section, estimate a regression where the average return is the dependent variable and the estimated beta is the independent variable: R̄i = γ0 + γ1 βi + ui. (6.3) There are N observations in this regressions, one for each test asset, and γ0 and γ1 are the parameters to be estimated. The regression in Equation 6.3 is an empirical version of the theoretical CAPM in Equation 5.2. On the left-hand side of the theoretical equation, we have the risk premium. In the empirical equation we substitute it with the long-run average excess return as there is no way to credibly measure investors’ expectation directly. Hence, the left-hand sides of the two equations are basically equivalent. On the right-hand side of the empirical equation we have the γ0 coefficient plus the γ1 52 coefficient times beta, whereas the right-hand side of the theoretical equation has beta times the market risk premium. From this we can see that the CAPM implies that γ0 should equal zero and γ1 should equal the average excess return of the market portfolio. Researchers have tested this prediction over and over again and the findings are very consistent: in the data γ0 is significantly greater than zero and γ1 is signif- icantly less than the average excess return of the market portfolio. This implies that investors do not seem to earn as much extra risk premium for holding higher- beta assets as the CAPM implies. A graph depicting risk premium (on the vertical axis) as a function of beta (on the horizontal axis) is commonly called the security market line. The empirical evidence shows that the true security market line is much flatter predicted by the CAPM. Since the difference between the theoretical prediction and the empirical reality is both statistically and economically very sig- nificant, this is evidence that the CAPM is not the true asset pricing model and does not appropriately describe how investors set return expectations and price assets. Another way to investigate the empirical validity of the CAPM is to test whether other factors besides beta affect risk premiums. The CAPM predicts that only beta affects the risk premium. If we find empirically that also other factors affect risk premiums, this is direct evidence against the CAPM being the true asset pricing model. We can easily augment the regression model in Equation 6.3 to include other factors. For example, we could use data on stock returns to test the CAPM and include the market capitalization of the stock as an independent variable in the cross-sectional regression.12 Fama and French (1992)13 do exactly this and report that the coefficient of the market capitalization is negative and statistically significant. This means that larger firms tend to have lower stock returns than smaller firms. This finding might be very intuitive and reasonable but it is in 12 Market capitalization measures the market value of a firm’s equity and is calculated simply as the share price multiplied by the numbers of shares outstanding. 13 Eugene F. Fama and Kenneth R. French, “The Cross-Section of Expected Stock Returns,” Journal of Finance, 1992. 53 direct contradiction of the CAPM predictions. The CAPM predicts that only beta matters, but the evidence suggests that firm size matters as well. This, again,

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