Risk and Return

1. Risk and Return Portfolio Returns Investment Risk Historical Record Risk, Return and Dominance Diversification and Portfolio Risk Asset Allocation – A Risk Free and Risky Asset Asset Allocation – Two Risky Assets Efficient Frontier The Capital Asset Pricing Model Single Index Model

2. Quick Summary • Every financial instrument must be judged on: • expected return; and, • risk. But notindividually rather in the context of the entireportfolio. Why? • Diversification: Portfolio risk reduction is the primary motivation. • by adding assets that are not perfectly correlated reward to volatility will be enhanced.

3. Introduction • Risk/Return Tradeoff: higher expected returns are associated with greater risk. • Investors will differ in their choice of investments because of their willingness to trade off expected returns against risk. • risk averse: the willingness to take risk only when there is a reasonable expectation of being rewarded for it. • People have different degrees of risk aversion. • Every financial instrument must be judged on: • expected return; and, • risk.

4. Investment Risk • Risk is the chance that the actual return from an investment may differ from what is expected. • In general the broader the range of of possible of possible returns, the greater the investment risk and vice versa. • Sources of risk: • Business risk • Financial risk • Purchasing power risk • Interest rate risk • Default risk

5. Measuring Investment Risk • Standard deviation: the square root of the variance. • Historical returns. • Use average returns using sample data. • Variance of historical returns or variance of sample data becomes:

6. Measuring Expected Return & Risk • Uncertainty surrounding the portfolio investments • Suppose uncertainty about returns can be characterized as uncertainty about which of “S” States occur in the future. • First give probabilities, or weights, to each State, then estimate the returns for each State: let:

7. Measuring Expected Returns Expected return can be written as: The expected return for Asset “A” is the sum of the weighted-by-the-probabilities of the returns given each “S” State:

8. Measuring Expected Risk The Risk of the expected return can be written as: The risk of Asset “A” is the Root of the sum of the weighted-by-the-probabilities of the deviations squared| for each State:

9. Expected Return and Risk Calculations • Consider two assets each costing $100. • What are the expected returns? • Which is riskier?

10. The Standard Deviation of Returns Calculation for Asset 1

11. The Standard Deviation of Returns Calculation for Asset 2

12. Expected (Return) • Another Example • E(r) = 14 percent

13. Expected (Risk) • Risk of a single asset can bemeasured by dispersion of r across states, or the variance of the returns. • Variance: the expected value of the squared deviation from the mean. example:

14. Risk Premium: the difference between expected rate of return and risk free.

16. Diversification and Portfolio Risk • Consider a portfolio consisting of: • 1 stock full effect of that stocks risk component • 2 stocks if firm specific risks are not perfectly correlated there will be some dilution of risk . . . • N stocks: gradually declining risk.

17. total risk Nondiversifiable risk 20 40 number of securities Diversification and Portfolio Risk • As portfolio size increases, total portfolio risk, on average, declines. After a certain point, however, the marginal reduction in risk from the addition of another security is modest.

18. wherewi = the proportion invested in security i Calculating Portfolio Risk and Return • The expected return of a portfolio is a weighted average of the component expected returns.

19. Example E(Rp) = wx*0.10+wy*0.14+wz*0.16+wm*0.15= 14.6%

20. Asset Allocation and Two Risky Assets • Now consider a portfolio invested in two risky assets: stocks WS = Proportion of funds in stocks E(rs)= Expect return in stocks bonds WB = Proportion of funds in bonds E(rB )= Expected return in bonds

21. Asset Allocation and Two Risky Assets • portfolio return (rP) rP= WSrS+WBrB = WSrS+ (1-WS)rB • Portfolio variance (sP2) sP2= WS2sS2 + WB2sB2 + 2WSWB rSBsSsB • ss2 = variance of stock; sB2 = variance of bond • rSB= correlation between rSand rB

22. Asset Allocation and Two Risky Assets • Calculating the risk-return tradeoff. • Draw the set of feasible risk-return combinations (investment opportunity set). • get estimates of sssB and sSB from the data. • vary WS and calculate rP and sP.

23. Asset Allocation and Two Risky Assets • There are benefits to diversification whenever asset returns are less than perfectly correlated. • Historical data for stocks and bonds: • rS = 17 percent • rB = 10 percent • sS = 25 percent • sB = 12 percent • rSB = 20 percent • What is WS which minimizes the portfolio variance?

25. Asset Allocation and Two Risky Assets • Deriving minimum variance portfolio (MVP) First Order Conditions

26. Asset Allocation and Two Risky Assets • Deriving minimum variance portfolio. Solve the first order condition for WS. The choice of WS which minimizes the portfolio variance. • Notice that even though bonds are less risky than stocks the minimum variance portfolio will typically put some money into stocks.

27. Efficient frontier impossible portfolios expected return dominated portfolios risk Efficient Frontier

28. Efficient frontier: Rf to M to C impossible portfolios C expected return M dominated portfolios Rf Risk Capital Market Line • A risk free investment complements risky securities.

29. Diversification and Portfolio Risk • Two types of risk • Market-specific, systematic or nondiversifiable risk: risk factors common to the entire economy. For example, macroeconomic considerations. • Firm-specific, nonsystematic or diversifiable risk: risk factors that can be eliminated by diversification. For example, factors that affect one particular firm or perhaps industry. • Evidence indicates about three-fourths of total risk can be eliminated through proper diversification.

30. Diversification, Portfolio Risk and CAPM • The only risk that matters for an individual security is the risk it brings to the market portfolio. • Systematic risk is measured by CAPM’s beta. • A security with average market risk has a beta equal to one. • Riskier securities have a beta greater than one. • Less risky securities have a beta less than one. • Key to CAPM: Investors are only rewarded for bearing necessary risk.

31. Capital Asset Pricing Model • A pairwise comparison of the thousands of stocks in existence would be an unwieldy task. To get around this problem, the CAPM compares all securities to a benchmark measure. • The CAPM provides us with a tool to predict the correct price of a risky asset within the framework of the mean-variance setting. • Developed independently by Sharpe, Lintner and Mossin in the mid-1960s. • Although several extensions have been proposed the original CAPM remains a central tenet of modern financial economics.

32. Capital Asset Pricing Model • Model • The expected excess rate of return of an asset is proportional to the expected excess rate of return of the market portfolio and the proportionality factor is b. • An asset’s beta is all that we need to know about the asset’s risk characteristics to use the CAPM.

33. Capital Asset Pricing Model • Beta is the statistic relating an individual security’s returns to those of the market index.

34. Capital Asset Pricing Model • The CAPM can be expressed in graphical form by regarding the formula as a linear relationship. • Security Market Line (SML) specifies the equilibrium relationship between risk and return of assets according to the CAPM. • Fairly priced assets plot exactly on the SML.

36. Single Index Model • The single index model represents the operationalization of the CAPM. • The single index model relates security returns to their betas, thereby measuring how each security varies with the overall market.

37. Single Index Model • a model of stock return using a market index to represent common factors. • Market risk • Firm specific risk