Pricing Risk Chapter 10 Outline • Measuring risk and return – Expected return and return Variance – Realized versus expected return – Empirical distribution of returns • The risk return tradeoff – Computing average returns and volatility of returns from historical data • Systematic versus Idiosyncratic risk and Diversification • Pricing risk • The CAPM in practice Measuring Risk and Return A First Look at Risk and Return Consider the performance of the following portfolios of securities: Standard and Poor’s (S&P 500): A portfolio, constructed by Standard and Poor’s, of 500 U.S. leading stocks. Small Stocks: A portfolio of stocks of U.S. firms whose market values are in the bottom 10% of all stocks traded on the NYSE. World Portfolio: A portfolio of international stocks from all of the world’s major stock markets in North America, Europe, and Asia Corporate Bonds: A portfolio of long-term, AAA rated U.S. corporate bonds with maturity of approximately 20 years Treasury Bills: An investment in three-month U.S. Treasury Bills Comparing Portfolios of different Risk A First Look at Risk and Return An investment of $100 in small stocks in 1925 would be worth over $8 million in 2005! An average of 15.2% annual return An investment of $100 in Treasury bills 1925 would be worth about $2000. An average of 3.78% annual return. While small stocks realized the highest return, it was followed by S&P 500 (10.2%), international stocks (9%), corporate bonds (6.2%) and finally Treasury bills. Comparing to how prices changed over these years (CPI index) all of the investments grew faster than inflation (average 3%) How much do investors demand (in terms of higher expected return) to bear a given level of risk? Probability Distribution Measures of Risk and Return Expected (Mean) Return E[R] = å pR R R Variance and Standard Deviaiton V[R] = E(R - E(R))2 = å pR (R - E[R])2 R SD[R] = s R = Var(R) Expected Return and Variance for BFI E[R] = 0.25´ 40%+ 0.5´10%+ 0.25´ (-20%) =10% Var[R] = 0.25´(40%-10%)2 + 0.25´(-20%-10%)2 = 0.045 s R = 0.045 = 21.2% Realized versus Expected Performance Distinguish between realized returns and expected returns YTD Apple’s stock price went from $532 to $441 and paid two dividends of $2.65 and $3.05 Calculating Realized Returns Realized Return over year t, t+1 Divt+1 + Pt+1 Divt+1 Pt+1 - Pt Rt+1 = -1 = + Pt Pt Pt With quarterly dividends 1+ Rt+1 = (1+ RQ1 )(1+ RQ2 )(1+ RQ3 )(1+ RQ4 ) Quarterly returns calculation (example for second quarter): DivQ2 + PQ2 RQ2 = -1 PQ1 Calculating Realized Return for GM Realized Returns for S&P500, GM, and T bills The Empirical Distribution 1926-2004 Risk and Return Tradeoff The Empirical Distribution Average Annual Return last T years 1 R = ( R1 + R2 +... + RT ) T Variance Estimate 1 T 2 sR = (R R) å t T -1 t=1 Empirical Distributions of different Portfolios Excess Returns Excess Return: the difference between the return for the investment and the return fro Treasury bills (a risk free investment) Historical Tradeoff Between Risk and Return in Large Portfolios 1926-2004 Investments with higher volatility have rewarded investors with higher average returns. Historical Tradeoff Between Risk and Return 500 individual stocks 1926-2004 Is volatility a reasonable measure of risk for individual stocks? Systematic versus Idiosyncratic risk and Diversification Common Versus Independent Risk The risk of an individual security differs from the risk of a portfolio composed of similar assets. Insurance Example To illustrate this difference consider two types of insurance: theft insurance and earthquake insurance. Each year there is about a 1% chance that a given home in the San Francisco area will be robbed and a 1% chance it will be damaged by an earthquake Suppose that the insurance company writes 100,000 policies of each type for homeowners in San Francisco Common Versus Independent Risk The expected number of theft claims is 1000 (or 1% out of all policies issued). This is also the expected number of claims from an earthquake (1% chance that the earthquake hits San Francisco, in that case all 100,000 policy holders will file a claim). Earthquake and theft portfolios lead to very different risk characteristics! In the case of earthquake insurance, the insurance company needs to be ready to cover 100000 claims in a single year! With theft insurance, the insurance company can hold funds sufficient to cover around 1200 claims since the number of claims will almost always be between 875 and 1125 Common Versus Independent Risk The earthquake affects all houses simultaneously, so the risk is perfectly correlated across homes. We call risk that is perfectly correlated common risk Thefts in houses are more or less not related to each other, so the risk of theft is uncorrelated and independent across homes. We call this type of risk independent risk. When risks are independent, the overall number of claims is quite predictable. This averaging out of independent risks in a large portfolio is called diversification Diversification in Stock Portfolios Firm specific risk (or “idiosyncratic”, “unique”, “diversifiable”) • News about the individual company Market wide risk (or “systematic”, “undiversifiable”) • News that affects all stocks, such as the news about the economy Diversification in Stock Portfolios Diversification in Stock Portfolios Common Versus Independent Risk Example Consider three firm types Type S firms are affected only by the strength of the economy, a systematic risk which has 50% chance of being either strong or weak. If the economy is strong, type S stocks will earn a return of 40%; if weak, their return will be -20%. Type I firms are affected only by idiosyncratic risks. There returns are equally likely to be 35% or -25% based on factors specific to each firm’s local market. Common Versus Independent Risk What is the volatility of the average return of ten type S firms? Or type I firms? Diversification in Action Pricing Risk Pricing Risk The risk premium for diversifiable risk is zero, so investors are not compensated for holding firmspecific risk – Why can’t diversifiable risk carry a positive risk premium in efficient markets? The risk premium of a security is determined by its systematic risk and does not depend on its diversifiable risk – Stock’s volatility, which is a measure of total risk (that is, systematic plus diversifiable) is not especially useful in determining the risk premium that investors will earn. Measuring Systematic Risk A security’s systematic risk is measured by the extent to which its return is sensitive to economic conditions We assume that the changing state of the economy must be reflected in the return on the market portfolio - the market wide portfolio contains only systematic risk (all firm specific risk has been diversified) In practice, we do not know return data for many bonds and small stocks. It is common practice to use the S&P 500 index as the market portfolio We measure systematic risk of stock i by its beta Cov(ri , rmkt ) bi = Var(rmkt ) The return on the S&P 500 index is considered the return on the market “rmkt” Values of Beta in the data Advanced Micro Devices, Inc. Advanced Micro Devices, Inc. Cisco Systems, Inc. Estimating the Risk Premium Market Risk Premium: the excess return from holding the market portfolio Market Risk Premium = E(Rmkt ) - rf Estimating a Traded Security’s Expected Return (this is the “Capital Asset Pricing Model” or CAPM) E(Ri ) - rf = bi ´ éëE(Rmkt ) - rf ùû Expected excess return on stock “i” Systematic risk of stock “i” Market risk premium Special Cases A stock with beta of one: E(Ri ) = E(Rmkt ) A stock with beta of zero: E(Ri ) = rf A stock with negative beta: E(Ri ) < rf Is this a good investment? Project Cost of Capital A project’s cost of capital is given by the rate of return required by investors or their opportunity cost of capital. From the CAPM the cost of capital “r” is: r = rf + b ´ éëE(Rmkt ) - rf ùû Project Cost of Capital The CAPM in Practice Measuring Beta Beta is the expected percent change in the excess return of the security for a 1% change in the excess return of the market portfolio (S&P 500 Index) Ri - rf = bi ´ éëRmkt - rf ùû + ei Measuring Beta Forecasting Beta • Time Horizon: what data do we use? A short horizon leads to weak statistical power. A long horizon includes outdated data. • Market Proxy: The theoretical market portfolio includes all risky investments. Often the S&P 500 index is used, other proxies include the NYSE Composite Index (a value weighted index of all NYSE stocks), or when considering an international investment one can use a country or international market index. • Beta Extrapolation: Often adjustments are made to the beta estimate to reduce estimation error. Measuring Beta Forecasting Beta • The Risk-Free Interest Rate: used in the CAPM equation on the left-hand side (to calculate the excess return on the asset) is the current YTM on U.S. Treasury with maturity similar to our project’s horizon • The Market Risk Premium: is estimated from the historical excess return on the market. The difference between the average return on the market and the average return on the risk-free asset. Market Risk Premium is around 4%-5%.