The Capital Asset Pricing ModelSummaryRisk and ReturnSummary: Risk and Return for PortfoliosDiversification Example: Writing Maritime InsuranceSlide 6Slide 7Diversification and InsuranceCovariances and Correlations: The Keys to Understanding DiversificationCorrelation CoefficientsSlide 11What risk return combinations would be possible with different weights?What risk return combinations would be possible with a different correlation between A and B?Symbols: Variance of a Two-Asset PortfolioFor General PortfoliosRisk in N-asset PortfoliosWhat happens when the number of assets in the portfolio, N, becomes large?Slide 18Slide 19DiversificationNonsystematic/diversifiable risksSystematic/Nondiversifiable riskMeasuring Systematic RiskSome Additional Insight: Restate Beta in terms of correlations.Betas and PortfoliosRecap: What is Beta?How To Get Beta.Slide 28Slide 29Slide 30The Capital Asset Pricing Model (CAPM)CAPM IntuitionThe CAPM Intuition FormalizedApplying the CAPM to select a discount rateSlide 35Slide 36Slide 37Slide 39Slide 40Slide 41Slide 42The Capital Asset Pricing ModelThe Risk Return Relation FormalizedSummary•E(r) = rf + . = amount of risk  premium per unit risk.•Define the market portfolio as having one unit of risk, Var(rm) = 1 unit of risk.•Risk premium per unit of risk is then E(rm) – rf.•Any asset contributes to Var(rm) (in a w.d. port.) not by Var(ri) or STD (ri) but Cov(ri, rm).•Standardize Cov(ri, rm) by Var(rm) (to get it as the number of “units”) and get i as the measure of risk of any asset i.•Then: E(ri) = rf + i (E(rm) – rf) the famous SML.Risk and Return•When there is only one asset its risk and return can be measured as we discussed using expected return and variance or standard deviation.•If there is more that one asset (and so we can form portfolios) risk and return are more complex.•We will show there are two types of risk for individual assets:–Diversifiable/nonsystematic risk/idiosyncratic–Nondiversifiable/systematic/market risk•We can eliminate diversifiable risk without cost by combining assets into portfolios.–It is usually possible to improve the risk/return tradeoff offered by any single asset by diversifying.Summary: Risk and Return for Portfolios•(i) A portfolio’s expected return is always a weighted average of the expected returns of the portfolio's component assets.•(ii) The risk of a portfolio's return, as measured by the standard deviation of its returns, is almost always less than the weighted average of the total risks of its component assets. This is the powerful force of diversification.•(iii) Risk generally declines when new assets are added to a portfolio. Risk declines more the lower the correlations between the new asset's payoff (return) and the payoffs (returns) to the assets already in the portfolio.•(iv) For individual assets only the risk that contributes to the risk of a well diversified portfolio requires a premium.•Suppose we are in the business of insuring oil tankers. Each tanker is insured for $10.3 million, and the owner pays us a $300,000 insurance premium per boat. The probability of a tanker sinking is 1.1%.Event Probability Payoff ($mill)Sink0.011 -10.3+0.3=-10Don’t Sink0.989 0.3ExpectedValue0.1867StandardDeviation1.0743Diversification Example: Writing Maritime Insurance• Our expected gain is positive but it is a highly risky position!•Now suppose we insure 1/2 of two separate boats instead of all of one boat.Event Probability Payoff2 Boats Sink 0.0112-10.3+0.3= -101 Boat Sinks 2(0.989)(0.011)=0.021762(0.15)-5.15= - 4.85No Boats Sink 0.98920.3ExpectedValue0.1867StandardDeviation0.7596•What is happening here?If we insure 1/30 of 30 boats–Expected payoff = 0.1867–Standard Deviation = 0.196If we insure 1/100 of 100 boats–Expected payoff = 0.1867–Standard Deviation = 0.107If we insure a tiny part of each of an infinite number of boats? •An important feature of the example is that outcomes, whether boats sink, were assumed to be uncorrelated across boats. •More generally, the degree of diversification benefit depends on the correlation among the payoffs on the different assets. Diversification and InsuranceDiversification and InsuranceDiversification and Insurance•What would happen if we insure two boats but both boats sailed the same route, chained together?•In financial markets the returns on most assets are positively correlated and so we cannot get rid of all of the risk with diversification. However, we can get rid of quite a bit.Covariances and Correlations: The Keys to Understanding Diversification•When thinking in terms of probability distributions, the covariance between the returns of two assets’ (A and B) Covariance = Cov(A,B) = AB =•When estimating covariances from historical data, the estimate is given by:•Note: An asset’s variance is its covariance with itself.p])E[R-R])(E[R-R(iBBiAAin1=i)R-R)(R-R(1T1BBtAAtT1=tCorrelation Coefficients•Covariances are difficult to interpret. Only the sign is really informative. Is a covariance of 20 big or small? The correlation coefficient, , is a normalized version of the covariance given by:•Correlation = CORR(A,B) = •The correlation will always lie between 1 and -1.–A correlation of 1.0 implies ...–A correlation of -1.0 implies ...–A correlation of 0.0 implies ... BAABBA=Cov(AB)Return (%) Deviation from mean Squared Dev. fr. MeanProbability A B P A B P A*B A B P0.2 18 25 21.5 2 13 7.5 26 4 169 56.250.2 30 10 20 14 -2 6 -28 196 4 360.2 -10 10 0 -26 -2 -14 52 676 4 1960.2 25 20 22.5 9 8 8.5 72 81 64 72.250.2 17 -5 6 1 -17 -8 -17 1 289 64Mean 16 12 14 21 191.6 106 84.9Risk and Return in Portfolios: Example•Two Assets, A and B•A portfolio, P, comprised of 50% of your total investment •invested in asset A and 50% in B.•Only five equally probable future outcomes, summarized below.In this case:•VAR(A) = 191.6, STD(A) = 13.84, and E(rA) = 16%.•VAR(B) = 106.0, STD(B) = 10.29, and E(rB) = 12%.•COV(A,B) = 21•CORR(A,B) = 21/(13.84*10.29) = .1475.•VAR(P) = 84.9, STD(P) = 9.21, E(rp) = ½ E(rA) + ½ E(rB) = 14%•Var(P) or STD(P) is less than that of either component!What risk return combinations would be possible with different weights?CORR(AB)0.1475-0.5Risk and Return in