CHAPTER 4 Risk and Rates of Return Stand-alone risk Portfolio risk Risk & return: CAPM / SML

Risk and Return Risk is the concept of fluctuations. This fluctuations can be (i) a deviation of the actual return from the expected return, or (ii) a deviation of average return from the year to year return. Higher the fluctuations, higher is the risk. Measures of risk are: i. Standard Deviation ii. Coefficient of Variance iii. Beta

Normal Distribution: A normal distribution looks like a bell-shaped curve. Probability Mean=12.2% Standard Deviation=20.5% – 3s – 49.3% – 2s – 28.8% – 1s – 8.3% 0 12.2% + 1s 32.7% + 2s 53.2% + 3s 73.7% 68.26% 95.44% 99.74%

Probability distributions With the same average return more standard deviation means more risk. Shown graphically. Note that as risk increases height goes down and width increases. Expected Rate of Return Rate of Return (%) 100 15 -70 Firm X Firm Y

Calculation of Risk-Return

Calculation of Risk-Return (Historical Data) Year Return (%) Dev. (Ri-E(R)) Dev. Square 2000 20 7 49 2001 5 -8 64 2002 -5 -18 324 2003 15 2 4 2004 30 17 289 Mean Return= 13% Sum of Dev sq= 730 Stand.Deviation2   730/(5-1)= 182.5 Stand.Deviation=  Square root (182.5)= 13.5%

Calculation of Risk-Return (Probability Distribution) Weather Probability (Pi) Return (Ri) (%) Exp. Value (Pi*Ri) Deviation (Ri-E(R)) Square Dev sq* Pi Rain 0.25 25 6.25 (25-13.25) =11.75 (11.25)2 =138.0625 (138*.25) =34.52 Moderate 0.5 14 7 (14-13.25) =0.75 (0.75)2 =0.5625 (.56*.5) =0.28 Dry (0-13.25) =-13.25 (-13.25)2 =175.5625 (175*.25) =43.89   E(R)= 13.25% Stand Dev2= 78.69 Risk= 8.9% Stand Dev 8.87

Comments on standard deviation as a measure of risk Standard deviation (σi) measures total, or stand-alone, risk. The larger σi is, the lower the probability that actual returns will be closer to expected returns. Larger σi is associated with a wider probability distribution of returns. It is difficult to compare standard deviations, because return has not been accounted for.

Coefficient of Variation (CV) A standardized measure of dispersion about the expected value, that shows the risk per unit of return. When, both return and risk increase then coefficient of variance (CV) should be used.

Use of coefficient of variance Example: We have 2 alternatives to invest. Security A has a mean return of 10% and a standard deviation of 6%, and security B has a mean return of 13% with a standard deviation of 8%. Which investment is better. So, security A is better as the Coefficient of variance of A is less than the that of B.

Investor attitude towards risk Risk aversion – assumes investors dislike risk and require higher rates of return to encourage them to hold riskier securities. Risk premium – the difference between the return on a risky asset and less risky asset, which serves as compensation for investors to hold riskier securities.

Effectiveness of Diversification of Portfolio Climate Probability Return of Umbrella Return Ice-cream Portfolio Rainy 0.25 25 -5 10 Moderate 0.5 14 12 Dry 15 7.5 E(R) =   13.3% 7.5% 10.4% Risk 8.9% 1.9%

Portfolio Effects on Risk and Return Feasible Set of 2 security case

Derivation of Optimum Portfolio and CML (2 security case): Given the feasible set highest possible utility function gives us O.P. and the tangency at that point is CML. Return IC1 CML O.P. Feasible set . Risk (σ)

Different points of CML: Optimum portfolio, Lending and Borrowing . Borrowing Return Optimum portfolio Lending Risk (σ)

Derivation of Optimum portfolio and CML (Case of many securities): . CML U1 O.P Borrowing Return Feasible set Lending Risk (σ) Given the feasible set highest possible utility function gives us Optimum Portfolio and the tangency at that point is CML

Role of correlation: Returns distribution for two perfectly negatively correlated stocks (ρ = -1.0) 25 15 -10 Stock W Stock M -10 Portfolio WM 25 25 15 15 -10

Role of correlation: Returns distribution for two perfectly positively correlated stocks (ρ = 1.0) Stock M 15 25 -10 Stock M’ 15 25 -10 Portfolio MM’ 15 25 -10

Creating a portfolio: Beginning with one stock and adding randomly selected stocks to portfolio σp decreases as stocks added, because they would not be perfectly correlated with the existing portfolio. The benefit of diversification gradually diminishes in such a manner that adding securities to a portfolio of around 20 does not reduce much risk.

Illustrating diversification effects of a stock portfolio # Stocks in Portfolio 10 20 30 40 2,000+ Company-Specific Risk Market Risk 20 Stand-Alone Risk, sp sp (%) 35

Breaking down sources of risk Stand-alone risk = Market risk + Firm-specific risk Firm-specific risk/Unique risk/Unsystematic/Diversifiable risk – portion of a security’s stand-alone risk that can be eliminated through proper diversification. Sources: factory being fired, stocks being stolen, technology obsoletes, market lost, CEO quits the firm. Market risk/Systematic risk/Non-diversifiable risk – portion of a security’s stand-alone risk that cannot be eliminated through diversification. Sources: Interest rate increases, inflation, changes in monetary, fiscal and commercial policy, changes in political government, natural disaster, etc. Measured by beta.

Capital Asset Pricing Model (CAPM) Since every body knows that a considerable extent of risk (i.e., unique risk) can be diversified by making a portfolio of 15-20, so all the investors assume a diversified portfolio. Risk premium is then due only for market risk, represented by beta (rather than total risk represented by sigma). Beta risk measures the volatility of security return in response to the volatility of the market return. βj= COV(Rj,Rm)/ σ2Rm In an equilibrium situation expected return of each security is composed of the risk free return and the risk premium. E(Rj)=Rf+(Rm-Rf) βj

Calculation of CAPM Return Economic state Probability Return j Market Return Depression 0.25 5 7 Normal 0.5 10 Inflation 20 15 COV(Rj,Rm)=∑(Rj-Rj)(Rm-Rm)(Pi)=15.6 σ2Rm=∑(Rm-Rm)2(Pi)=8.25 βj= COV(Rj,Rm)/ σ2Rm=15.6/8.25=1.9 E(Rj)=Rf+(Rm-Rf) βj=5%+(10.5%-5%)(1.9)=20.4%

Work Sheet of CAPM Economic State Probability Return J Market Return Ex.Value j Ex. Value M [Rj-E(Rj)] [Rm-E(Rm)] Dev (Rj)*Dev(Rm) Dev(Rj)*Dev(Rm)*Pi Dev(Rm)*Dev(rm) *Pi Depression 0.25 5 7 1.25 1.75 -6.25 -3.5 21.875 5.46875 3.0625 Normal 0.5 10 -1.25 -0.5 0.625 0.3125 0.125 Inflation 20 15 3.75 8.75 4.5 39.375 9.84375 5.0625   11.25 10.5 15.6 8.25

Security Market Line (SML) Return SML E(Rj) E(Rm) Rf=5% Systematic Risk /Beta Risk βm=1 βj=1.9

Comments on beta If beta = 1.0, the security is just as risky as the average stock. If beta > 1.0, the security is riskier than average. If beta < 1.0, the security is less risky than average. Most stocks have betas in the range of 0.5 to 1.5.