Slide 1Discounting Risky Cash FlowsSlide 3Returns for Different Types of SecuritiesRisk, More FormallySlide 6Slide 7Expected ReturnVariance and Standard DeviationAverage Return and VarianceHistory for US Portfolios (1926 – 2011)Slide 12Slide 13Slide 14The Returns of Individual StocksGoing ForwardGoing ForwardGoing ForwardGoing ForwardDiversificationDiversification ExampleExample…Example…Example…Slide 25Correlation CoefficientsSlide 27Risk/return pairs with different weightsRisk return pairs with different correlationsSlide 30For General PortfoliosFor General PortfoliosIn A Picture (N = 2)In A Picture (N = 3)In A Picture (N = 10)In A Picture (N = 20)Slide 37Implications of DiversificationSlide 39The Pricing Of RiskUnderstanding the Relation between Risk and ReturnDiscounting Risky Cash FlowsHow should the discount rate change in the NPV calculation if the cash flows are not riskless?The question is more easily answered from the “other side.” How must the expected return on an asset change so you will be happy to own it if it is a risky rather than a riskless asset?Risk averse investors say that to hold a risky asset they require a higher expected return than they require for holding a riskless asset. E(rrisky) = rf + .Note that we now have to start to talk about expected returns since risk has been explicitly introduced.Note also that this captures the two basic “services” investors perform for the economy.The Answer!!ExpectedReturnRiskBut, how should risk be measured? at what rate does the line slope up? is the relation linear?Lets look at some simple but important historical evidence.“E(r) = rf + θ”Returns for Different Types of SecuritiesRisk, More FormallyMany people think intuitively about risk as the possibility of an outcome that is worse than was expected. Must be incomplete.Also: for those who hold more than one asset, is it the risk of each asset they care about, or the risk of their whole portfolio? A useful construct for thinking rigorously about risk:The “probability distribution.”A list of all possible outcomes and their probabilities.Very importantly we think about the moments of the distribution.Example: Two Probability Distributions on Tomorrow's Share Price.The expected price is the same.Which implies more risk?The Empirical Distribution of Annual Returns for U.S. Large Stocks (S&P 500), Small Stocks, Corporate Bonds, and Treasury Bills, 1926–2011.Expected ReturnExpected (Mean) ReturnCalculated as a “weighted average of the possible returns,” (where the weights correspond to the associated probabilities).[ ]Expected Return = = ��RRE R P RVariance and Standard DeviationVarianceThe expected squared deviation from the meanStandard DeviationThe square root of the variance: commonly called volatility in finance jargonBoth are measures of the risk or volatility associated with a probability distribution( ) ( ) =SD R Var R[ ]( )[ ]( )2 2( ) � �= - = � -� ��RRVar R E R E R P R E RAverage Return and VarianceWhere Rt is the realized return of a security in year t, for the years 1 through TThe estimate of the volatility/standard deviation is the square root of the estimate of variance.( )1 2 11 1 == + + + =�LTT ttR R R R RT T( )2 11( ) 1TttVar R R RT== --�History for US Portfolios (1926 – 2011)PortfolioAverage AnnualReturnExcess Return:Average Return in Excess of T-BillsReturn Volatility(Standard Deviation)Small Stocks 18.7% 15.1% 39.2%S&P 500 11.7% 8.1% 20.3%Corporate Bonds6.6% 3.0% 7.0%Treasury Bonds3.6% 0.0% 3.1%Using Past Returns to Predict the Future: Let’s Remind Ourselves of Estimation ErrorStandard Error of the Estimate of Expected ReturnA statistical measure of the degree of estimation error 95% Confidence IntervalFor the S&P 500 (1926–2011)Or a range from 7.3% to 16.1%. So not a great deal of accuracy, and this is a large portfolio that has been around for a long time.Historical Average Return (2 Standard Error)� �20.3%11.7% 2 11.7% 4.4%86� �� = �� �� �SD(Individual Risk)Standard ErrorNumber of Observations=The Historical Tradeoff Between Risk and Return in Large Portfolios, 1926–2011Note: a positive linear relationship between volatility and average returns for large portfolios.Historical Volatility and Return for 500 Individual Stocks, by Size, Updated Quarterly, 1926–2011The Returns of Individual StocksIs there a positive linear relationship between volatility and average returns for individual stocks?As shown on the last slide, there is no precise relationship between volatility and average return for individual stocks. For a given level of average return, larger stocks tend to have lower volatility than smaller stocks.All stocks tend to have higher risk for a given average return relative to large portfolios. There must be something magical going on with portfolios.Volatility doesn’t seem to be an adequate measure of risk to explain the expected return of individual stocks.Can we deal with this and resurrect our simple idea?Going ForwardAs we discussed, the “market” pays investors for two services they provide: (1) surrendering their capital and forgoing current consumption and (2) sharing in the aggregate risk of the economy.The first gets you the time value of money.The second gets you a risk premium.From this we wrote E(r) = rf + θThe size of the risk premium should depend on the amount of aggregate risk you take on.So we refine this relation to E(r) = rf + Units × PriceIn other words the premium cannot be the same for all assets. If you take more risk (more units of risk) you require more of a premium.Going ForwardWe need a reference for measuring risk and choose the risk the market must distribute across investors, the aggregate total risk or the risk of the “market portfolio” as that reference.The market portfolio is defined to have one unit of risk (Var(rm) = 1 unit of risk). Other assets will be evaluated relative to this definition of one unit of risk.From E(r) = rf + Units × Price we can see that “Price” = {E(rm) – rf}. (Note: 1 unit of risk for the market.)Therefore, we also defined the price per unit risk (the market risk premium) once we select this reference for measuring risk.Going ForwardThe hard part is to show