Foundations of Finance Notes on CAPM and Empirical Asset Pricing PDF

Foundations of Finance Notes on CAPM and Empirical Asset Pricing Prof. Simone Lenzu NYU Stern October 3, 2023 Introduction and instructions In class we discussed one reason why the...

Foundations of Finance Notes on CAPM and Empirical Asset Pricing Prof. Simone Lenzu NYU Stern October 3, 2023 Introduction and instructions In class we discussed one reason why the very elegant theory of portfolio selection is not very practical: The quadratic programming problem that needs to be solved to find optimal portfolio weights becomes very difficult to solve, even for the most powerful computers, when the number of assets we are considering is large. In fact, it is even hard to produce and frequently update all the the inputs needed to solve the quadratic programming problem (expected returns, standard deviations, correlations). The CAPM allows us to go around these very practical problems, by offering an empirically parsimonious and theoretically sound approach to asset pricing. This handout provides further learning material on the Capital Asset Pricing Model (CAPM) and on empirical asset pricing. Contents 1 Portfolio Variance with Many Risky Securities: Idiosyncratic and System- atic Risk 2 2 Risk-Reward relationship and the Security Market Line 3 2.1 A First Risk-Reward Relationship........................ 3 2.2 A Second Risk-Reward Relationship and the Security Market Line..... 5 1 3 Testing CAPM: The Fama-McBeth approach 6 3.1 Market price of a risk factor and exposure to a risk factor........... 7 3.2 Fama-MacBeth procedure............................ 7 4 Improving CAPM: Fama-French 3 factor model 9 1 Portfolio Variance with Many Risky Securities: Idiosyn- cratic and Systematic Risk 1. Case 1: Non-systematic risk only. Recall that when the correlation 𝜌 between two securities equals zero, the portfolio variance is given by: 𝜎𝑝2 = 𝑤 12𝜎12 + 𝑤 22𝜎22. A simple generalization of this formula holds for many securities provided that 𝜌 = 0 between all pairs of securities: 𝜎𝑝2 = 𝑤 12𝜎12 + 𝑤 22𝜎22 + · · · + 𝑤 𝑁2 𝜎𝑁2. (1) We will prove the following result. As 𝑁 → ∞, the portfolio standard deviation 𝜎𝑝 → 0. To make the notation simpler, assume that 𝜎1 = 𝜎2 = · · · = 𝜎𝑁 = 𝜎. This means that each asset is equally risky. Under those circumstances, we try the simple diversification strategy of dividing our wealth equally among each asset such that 𝑤𝑖 = 1/𝑁. These assumptions allow us to rewrite expression (1) as 2 2 2 1 1 1    𝜎𝑝2 = 2 𝜎 + 2 𝜎 +··· 𝜎 2. (2) 𝑁 𝑁 𝑁 There are 𝑁 identical terms in expression (2), which means: 1 2 2 1 2 𝜎𝑝2 = 𝑁 (3)
𝑁 𝜎 = 𝑁𝜎. The expression in (3) shows that as 𝑁 grows larger and larger, the variance of the portfolio declines. As 𝑁 → ∞, the variance goes to zero. 2. Case 2: Systematic and non-systematic risk. In fact, U.S. stocks do not have zero correlation with one another. We can capture the positive correlation of U.S. stocks with a factor model. Let 𝑅𝑖 denote the return on an individual stock, and 𝑅𝑀 the 2  return on a broad market index like the S&P 500. One way to capture the common source of variation is to run a regression of the values of 𝑅𝑖 on 𝑅𝑀 : 𝑅𝑖 = 𝛼𝑖 + 𝛽𝑖 𝑅𝑀 + 𝜀𝑖. (4) As we have seen in the notes for the Stats boot camp II, running a linear regression is equivalent to fitting a line through a scatter plot of pairs of returns (𝑅𝑀 , 𝑅𝑖 ). The slope of the line equals 𝛽𝑖. You may have heard in your statistics class that 𝐶𝑜𝑣 (𝑅𝑖 , 𝑅𝑀 ) 𝛽𝑖 = 2. (5) 𝜎𝑀 The coefficient 𝛽𝑖 is a measure of how much the stock moves together with the market index 𝑅𝑀. The error term, 𝜀𝑖 , measures the variability in 𝑅𝑖 that is independent of all other securities in 𝑅𝑀. Using (4), we can decompose the variance of a stock into its systematic and non-systematic components: 𝜎𝑖2 = 𝛽𝑖2𝜎𝑀 2 + 𝜎𝜀2𝑖 (6) Total risk = Systematic risk + Idiosyncratic risk. Here, 𝜎𝑀 2 is the variance of 𝑅. Equation (6) follows from the fact that 𝜀 and 𝑅 are 𝑀 𝑀 independent random variables (this is achieved by running the regression). Equation (6) shows that U.S. stocks have both systematic risk 𝛽𝑖2𝜎𝑀 2 and non-systematic risk 𝜎𝜀2𝑖. The argument from Case 1 demonstrates that the non-systematic component of stock risk goes away in a well-diversified portfolio (i.e. a portfolio with a large number of securities 𝑁 ). Only the systematic component remains. 2 Risk-Reward relationship and the Security Market Line 2.1 A First Risk-Reward Relationship. We showed that market forces combined with a search by investors for efficient portfo- lios would produce the following relationship for each security 𝑖, one of 𝑁 securities in a portfolio: 𝐸 [𝑅𝑖 ] − 𝑅 𝑓 𝐸 [𝑅𝑚 ] − 𝑅 𝑓 = , (7) 𝜕𝜎𝑚 /𝜕𝜔𝑖 𝜎𝑚 3 where 𝜔𝑖 is the portfolio weight of security 𝑖 and 𝜎𝑚 is the volatility of the market portfolio. This expression can be rewritten as: 𝜕𝜎𝑀 𝐸 [𝑅𝑀 ] − 𝑅 𝑓    𝐸 [𝑅𝑖 ] = 𝑅 𝑓 +. (8) 𝜕𝜔𝑖 𝜎𝑀 Expression (8) says that the equilibrium expected return on security 𝑖, 𝐸 [𝑅𝑖 ], should equal the risk-free rate 𝑅 𝑓 plus the market price of risk, 𝐸 [𝑅𝑀 ] − 𝑅 𝑓 /𝜎𝑚 , times 𝜕𝜎𝑀 /𝜕𝑤𝑖. What
is 𝜕𝜎𝑀 /𝜕𝑤𝑖 ? It measures by how much the volatility of the market changes if we change the portfolio weight of asset 𝑖 by an infinitesimal amount. Intuitively, it is the risk contri- bution of security 𝑖 to portfolio risk 𝜎𝑚. In other words, it is a measure of the quantity of risk. Mathematically, it is a partial derivative. The next paragraph shows how to obtain this derivative. You are not responsible for this derivation. The result of this calculation is: 1 ∑︁ 𝑁 𝜕𝜎𝑀 (9)
= 𝑤 𝑗 𝐶𝑜𝑣 𝑅𝑖 , 𝑅 𝑗. 𝜕𝑤𝑖 𝜎𝑀 𝑗=1 This expression states that the risk contribution of security 𝑖 to the portfolio depends on the covariance of security 𝑖 with each and every other asset. this makes considerable sense because we know that only systematic risk matters in a portfolio and systematic risk is measured by covariance. Optional notes: Calculating 𝜕𝜎𝑀 /𝜕𝑤𝑖. First, we recognize that the derivative of the portfolio variance with respect to 𝑤𝑖 equals the derivative of the standard deviation w.r.t. 𝑤𝑖 , multiplied by 2𝜎𝑀 : 2 𝜕𝜎𝑀 𝜕𝜎𝑀 = 2𝜎𝑀. (10) 𝜕𝑤𝑖 𝜕𝑤𝑖 Second, recall the definition of portfolio variance (see the notes for the Stats boot camp I): 2 = Í𝑁 Í𝑁 𝑤 𝑤 𝐶𝑜𝑣 𝑅 , 𝑅. Taking the derivative of this variance with respect to the
𝜎𝑀 𝑖=1 𝑗=1 𝑖 𝑗 𝑖 𝑗 portfolio weight 𝑤𝑖 gives: 2 𝜕𝜎𝑀 𝑁 ∑︁ = 2 (11)
𝑤 𝑗 𝐶𝑜𝑣 𝑅𝑖 , 𝑅 𝑗. 𝜕𝑤𝑖 𝑗=1 2 /𝜕𝑤. Setting them equal to each other, Equations (10) and (11) are two expressions for 𝜕𝜎𝑀 𝑖 4 and after simplifying we get: 1 ∑︁ 𝑁 𝜕𝜎𝑀 (12)
= 𝑤 𝑗 𝐶𝑜𝑣 𝑅𝑖 , 𝑅 𝑗 , 𝜕𝑤𝑖 𝜎𝑀 𝑗=1 which is what we needed to show. 2.2 A Second Risk-Reward Relationship and the Security Market Line The problem with equation (9) is that it is not operational; there are too many covariances needed to measure the quantity of risk of security 𝑖. We can simplify the measurement problem by recalling the regression equation (the Security Characteristic Line, SCL) re- lating the return on an individual security, 𝑅𝑖 , to the return on an index, 𝑅𝑚 , consisting of all other securities in the market. In particular, we have: 𝑅𝑖 = 𝛼𝑖 + 𝛽𝑖 𝑅𝑀 + 𝜀𝑖. (13) Note that 𝑅𝑚 is defined as the weighted average return on all securities in the market: 𝑁 ∑︁ 𝑅𝑀 = 𝑤𝑖 𝑅𝑖. (14) 𝑖=1 As we have seen in the notes for the Stats boot camp II, the definition of the regression coefficient in expression (13) is: 𝐶𝑜𝑣 (𝑅𝑖 , 𝑅𝑀 ) 𝛽𝑖 = 2. (15) 𝜎𝑀 5 This means that we can take the expression for 𝑅𝑀 in (14) and substitute it into expression (15) to get the following:  Í  𝐶𝑜𝑣 𝑅𝑖 , 𝑁𝑗=1 𝑤 𝑗 𝑅 𝑗 𝛽𝑖 = 2 (16) 𝜎𝑀 Í𝑁
𝑗=1 𝑤 𝑗 𝐶𝑜𝑣 𝑅𝑖 , 𝑅 𝑗 = 2 (17) 𝜎𝑀
1 𝑗=1 𝑤 𝑗 𝐶𝑜𝑣 𝑅𝑖 , 𝑅 𝑗 Í𝑁 = (18) 𝜎𝑀 𝜎𝑀 1 𝜕𝜎𝑀 =. (19) 𝜎𝑀 𝜕𝑤𝑖 The second equality uses a property of the covariance (see the notes for the Stats boot camp I), the third equality uses that the variance is the square of the standard deviation, and the fourth equality uses expression (9). We can now use this last expression for 𝛽𝑖 and substitute it into equation (8), to get a second risk-reward relationship: (20)
𝐸 [𝑅𝑖 ] = 𝑅 𝑓 + 𝛽𝑖 𝐸 [𝑅𝑀 ] − 𝑅 𝑓. This is the Security Market Line (SML). It says that, in equilibrium, the expected return on security 𝑖, 𝐸 [𝑅𝑖 ], is equal to the risk-free rate 𝑅 𝑓 plus the excess return on the market portfolio times the beta of security 𝑖. The risk-free rate is a compensation for the time value of money. The second piece, 𝛽𝑖 𝐸 [𝑅𝑀 ] − 𝑅 𝑓 , is a compensation of risk. Just as in
equation (8), the compensation for risk is the product of the price of risk and the quantity of risk. In equation (8), the quantity of (systematic) risk was 𝜕𝜎𝑀 /𝜕𝑤𝑖 and the price of risk was 𝐸 [𝑅𝑀 ] − 𝑅 𝑓 /𝜎𝑀. Now, in equation (18) 𝛽𝑖 measures the quantity of risk of security
𝑖. More precisely, it measures the quantity of systematic risk. This makes sense because systematic risk is measured by covariance, and we see from its definition in equation (15), that 𝛽𝑖 is a measure of how security 𝑖 covaries with all other securities. The price of risk, in units of 𝛽, is 𝐸 [𝑅𝑀 ] − 𝑅 𝑓. 3 Testing CAPM: The Fama-McBeth approach As we have seen in class, theories of asset pricing frequently use “risk factors” to explain asset returns. These factors can range from financial factors (from example, the market factor like CAPM, or firm size) to macroeconomic factors (for example, inflation rate or the 6 unemployment rate). As we discussed in class, Fama-MacBeth (henceforth FMB) in 1973 proposed a procedure that : 1. allows us to jointly estimate the the market price of risk factor 𝑗 (𝜆 𝑗 ) and exposure of a particular asset 𝑖 to that risk factor (𝛽𝑖 ). 2. offers a simple and practical way of testing if and how these factors describe portfolio or asset returns. This is what we used in class to test whether CAPM holds in the data and whether there are other risk factors (besides the market factor) that can explain stock returns (we found the Value-Growth factor and Small-Large factor). In what follows, I am going to briefly refresh what 𝜆 and 𝛽 are. Then I am going to offer a friendly summary of the Fama-MacBeth procedure to estimate them. 3.1 Market price of a risk factor and exposure to a risk factor. The market price of a given risk factor 𝑗, denoted by 𝜆 𝑗 , tells us how much excess return an investor should demand in order to purchase a stock that has a one-hundred percent exposure to a given risk factor. The exposure of a given stock 𝑖 to a particular risk factor 𝑗, denoted by 𝛽𝑖 , tells us how much the returns of stock 𝑖 can be explained by changes in the price of risk. The key equation is the following: 𝐸 [𝑅𝑖 ] − 𝑅 𝑓 = 𝜆 𝑗 𝛽𝑖 | {z } Expected Risk Premium for Stock 𝑖 The question is, how do I estimate 𝜆 and 𝛽𝑖 at the same time? This is where the Fama- MacBeth approach comes in. To illustrate the approach, we are going to apply it to the asset pricing model we are very familiar with: CAPM. 3.2 Fama-MacBeth procedure As you know very well, CAPM is a one-risk factor model, according to which stocks re- turns are explained purely by their exposure to the aggregate stock market (market risk). According to CAPM, the market excess return, 𝐸 [𝑅𝑀 ] − 𝑅 𝑓 , is the systematic risk factor for 7 which risk-averse investors demand a risk premium. So 𝐸 [𝑅𝑀 ] − 𝑅 𝑓 is the risk factor under CAPM. Data — We are going to collect a panel dataset with historical asset excess returns {𝑅𝑖𝑡 }𝑖=1,𝑡=1 𝑁 ,𝑇 historical realizations of our risk factors {𝑅𝑀𝑡 − 𝑅 𝑓 𝑡 }𝑇𝑡=1 and risk-free rate {𝑅 𝑓 𝑡 }𝑇𝑡=1 , where 𝑖 = 1,....𝑁 are the stocks we want to include in our analysis (e.g., all stocks listed in the S&P 500) and 𝑡 = 1,...𝑇 are the month-years in our sample. Step 1 — Run a time-series regression for each stock 𝑖 to recover 𝛽ˆ𝑖 For each stock 𝑖, run a separate time-series regression to recover an estimate of exposure of stock 𝑖 to the market risk-factor, 𝛽ˆ𝑖. Basically, for each stocks 𝑖, we take the Security Characteristic Line (SCL) to the data: 𝑅𝑖𝑡 − 𝑅 𝑓 𝑡 = 𝑎𝑖 + 𝛽𝑖 × (𝑅𝑀𝑡 − 𝑅 𝑓 𝑡 ) + 𝜖𝑖𝑡 |{z} | {z } ↑ ↑ Coefficient to be Regressor estimated We can estimate this via Ordinary Linear Regression (OLS), as we have seen in the notes for the Stats boot camp II. We will end up with a vector of estimated exposures, one for each stock 𝑖: { 𝛽ˆ𝑖 }𝑖=1 𝑁 Step 2 — Run a cross-sectional regression to recover 𝜆ˆ𝑀 Now we are going to treat the estimated exposures { 𝛽ˆ𝑖 }𝑖=1 𝑁 as data. Then we are going to estimate for each time period 𝑡 a cross-sectional regression. A cross-sectional regression asks whether firms with a higher 𝛽 give investors higher returns: This is the premise of CAPM. Practically, for each periods t, we take the Security Market Line (SML) to the data: 𝑅𝑖𝑡 − 𝑅 𝑓 𝑡 = 𝛼𝑡 + 𝜆𝑀𝑡 × 𝛽ˆ𝑖 + 𝑒𝑖𝑡 (21) |{z} |{z} ↑ ↑ Coefficient to be Regressor estimated We will end up with a vector of estimated factor exposure, one for each period 𝑡: ˆ {𝜆𝑀𝑡 }𝑇𝑡=1. Now define the risk premium for exposure to market risk as the sample aver- 8 age of all calculated 𝜆ˆ𝑀𝑡 : 1 ∑︁ ˆ 𝑇 𝜆ˆ𝑀 = 𝜆𝑀𝑡 (22) 𝑇 𝑡=1 Testing the empirical relevance of a risk factor — Now we can test whether investors receive a premium (higher expected returns) for being exposed to a given factor. In the case of CAPM, we want to test whether in fact stocks with higher 𝛽s have higher expected returns. To do so, we can do a t-test to see if 𝜆ˆ𝑀 in equation (22) is significantly different from zero. Formally: 𝜆ˆ𝑀 𝑡 𝜆𝑀 = √ 𝑠𝑡𝑑 (𝜆ˆ𝑀𝑡 )/ 𝑇 where 𝑠𝑡𝑑 (𝜆ˆ𝑀𝑡 ) is the sample standard deviation of the estimated 𝜆𝑀𝑡. This calcu- lated t-value is then compared against a value obtained from a critical value table (the T-Distribution Table). As a rule of thumb, a large t-score indicates that the null hypothesis (𝜆ˆ𝑀 = 0) is rejected and therefore the risk factor is indeed a risk factor that can explain the returns. If CAPM is a good asset pricing model, 𝜆ˆ𝑀 > 0. There is another test of CAPM that comes out of the FMB approach: the intercept. According to CAPM, what should the intercept of equation (21) be? If CAPM is correct, the answer is “zero”. We can design a test similar to the one above to test this second prediction of CAPM. Formally: 𝛼ˆ 𝑡𝛼 = √ 𝑠𝑡𝑑 (𝛼ˆ𝑡 )/ 𝑇 where 𝛼ˆ = 𝑇1 𝑇𝑡=1 𝛼ˆ𝑡. Here if CAPM is a a “good” asset pricing model, we should not Í reject the null hypothesis that 𝛼ˆ = 0. 4 Improving CAPM: Fama-French 3 factor model In the CAPM, the only systematic risk factor is the market portfolio and the only measure of systematic risk is beta. There is no additional role for other variables measuring sources of systematic risk, and therefore, no role for other variables as systematic explanatory factors of assets’ returns. As we have seen in class, turns out that this is not quite true: we have seen that there are “anomalies”. That is, some patterns in the data that suggest that other factors might actually systematically explain why some stocks have higher returns that 9 others. Two important factors are the size factor and the value factor. In an homework assignment you will be asked to show that indeed it is the case that the portfolios that have high loadings on small market cap firms and on firms with high book to market have returns that are persistently higher than the returns CAPM would predict. So the logical next step is to build a model that incorporates these two factors. This is what Fama and French did in their 1993 paper by introducing the SMB and HML factors. The Fama-French factors — In order to construct the factors, Fama and French use a double sort of firms into portfolios. Along the size dimension, stocks are sort into two groups: small or big. Along the value dimension, stocks are sort into three groups: value stocks and growth stocks. Using the classification above, FF construct two risk factors: SMB (Small Minus Big): 𝑅𝑆𝑀𝐵 = 𝑅𝑆𝑚𝑎𝑙𝑙 − 𝑅 𝐵𝑖𝑔 where 𝑅𝑆𝑚𝑎𝑙𝑙 is the return on a diversified portfolio that includes small cap stocks and 𝑅 𝐵𝑖𝑔 is the return on a diversified portfolio that includes large cap stocks. HML (High Minus Low): 𝑅 𝐻 𝑀𝐿 = 𝑅𝑉 𝑎𝑙𝑢𝑒 − 𝑅𝐺𝑟𝑜𝑤𝑡ℎ where 𝑅𝑉 𝑎𝑙𝑢𝑒 is the return on a diversified portfolio that includes high Book-to-Market stocks and 𝑅 𝐵𝑖𝑔 is the return on a diversified portfolio that includes low Book-to- Market stocks. As you can see, the factors are constructed by a long/short strategy on well diversified port- folios. For example, the HML factor involves buying value stocks and selling growth stocks. The motivation behind the portfolio approach is that any unwanted risks are taken out from the factor: by building a portfolio, we diversify away the idiosyncratic risk in individual stocks. The motivation for the long-short approach is that this approach hedges out the exposure to the market risk and therefore address the concern that a factor might be just picking up exposure to market risk (the CAPM factor). 10 The FF regression — Once we construct the SMB and HML factor for each month/year in our sample, we are can now test whether a CAPM-style regression, augmented with the SMB and HML factor actually does a better job explaining firms-returns. What does a better job mean? We should like to see if these factors are relevant we are able to explain a larger fraction of the variation in stocks’ returns as measured by the 𝑅 2 of the regression. In particular, the FF three-factor regression for any stock or portfolio 𝑖 is:
𝑅𝑖𝑡 − 𝑅 𝑓 𝑡 = 𝛼𝑖 + 𝛽𝑖𝐶𝐴𝑀𝑃 𝑅𝑀𝑡 − 𝑅 𝑓 𝑡 + 𝛽𝑖𝑆𝑀𝐵 𝑅𝑡𝑅𝑆𝑀 + 𝛽𝑖𝐻 𝑀𝐿 𝑅𝑡𝐻 𝑀𝐿 + 𝜖𝑖𝑡 Notice that we’ve put the two new factors, 𝑅𝑡𝑅𝑆𝑀 and 𝑅𝑡𝐻 𝑀𝐿 , in the regression and label the corresponding slope coefficients to be 𝛽𝑖𝑆𝑀𝐵 and 𝛽𝑖𝐻 𝑀𝐿. If you like, you can think of them as the “beta” on SMB and HML. In the homework assignment you will be asked to show that as we move along the size dimension from portfolio of small cap firms to portfolios of large cap firms, the estimated 𝛽ˆ𝑖𝑆𝑀𝐵 move from positive to negative. Likewise, as we move along the value dimension, low B/M to high B/M portfolios, the estimated 𝛽ˆ𝑖𝐻 𝑀𝐿 move from negative to positive. It tells us that indeed there is commonality in movement among small stocks that is different from large stocks. Regressing returns of small stocks on the SMB factor picks up this co-movement. Similarly, values stocks co-move together in ways that are different from growth stocks. Hence the HML factor. The FF three-factor model — Borrowing from the theory of CAPM, we can use the factors to build a new pricing model, the so called Fama-French Three-factor model. The pricing relation in the FF three-factor model is pretty straightforward:         𝐸 𝑅𝑖𝑡 − 𝑅 𝑓 𝑡 = 𝛼𝑖 + 𝛽𝑖𝐶𝐴𝑀𝑃 𝐸 𝑅𝑀𝑡 − 𝑅 𝑓 𝑡 + 𝛽𝑖𝑆𝑀𝐵 𝐸 𝑅𝑡𝑅𝑆𝑀 + 𝛽𝑖𝐻 𝑀𝐿 𝐸 𝑅𝑡𝐻 𝑀𝐿 + 𝜖𝑖𝑡 The risk-premium of individual stocks or portfolio 𝐸 𝑅𝑖𝑡 − 𝑅 𝑓 𝑡 should be proportional   to the exposure of that stock or portfolio to three systematic risk factors — the market risk factor 𝐸 [𝑅𝑀𝑡 − 𝑅 𝑓 𝑡 ] (as in CAPM) but also the size and value factors 𝐸 𝑅𝑡𝑅𝑆𝑀 and 𝐸 𝑅𝑡𝐻 𝑀𝐿     — and the exposures of the stock or portfolio to these factors 𝛽𝑖𝐶𝐴𝑀𝑃 𝛽𝑖𝑆𝑀𝐵 , 𝛽𝑖𝐻 𝑀𝐿. How does the three-factor model do? The Fama-French three-factor has been tested over time on a number of markets and time periods. Overall, it does a pretty good job in the sense that the returns predicted by the model line up fairly closely with the ones observed in the data for a large number of asset classes. Or, in other words, 𝛼s are much closer to zero than the 𝛼𝑠 estimated using the CAPM. In the homework assignment you will 11 be asked to use the FF three-factor model to evaluate the performance of a mutual fund. 12