Managerial Finance:Chapter 13—Return, Risk & the Security Market Line OVU-ADVANCE Notes prepared by D. B. Hamm Updated January 2006
Expected Return (1) Most investments carry some degree of risk. Generally only U.S. securities (specifically T-bills) are considered risk free [Rf] because the Federal government can raise taxes or borrow as necessary to avoid default.
Expected Return (2): • Suppose Investment A has probable returns as follows: • In the previous "go-go" market, it had earned 12%. • In the recent market slump, it earned only 4%. • If we project a 60% probability of renewed boom and a 40% probability of bust, then the expected return of A [ E(RA) ] is as follows: • E(RA) = (.60 x .12) + (.40 x .04) • = .072 + .016 • = .088 or 8.8%
Risk Premium: Risk Premium is the difference between the expected return on the proposed investment and the risk free rate. If U.S. security G is earning 4% then the risk premium for investment A (from previous slide, E(R) = 8.8%) is: RiskA = E(RA) - Rf = .088 - .04 = .048 or 4.8%
Variance & Standard Deviation The Variance, or squared deviations from the expected return gives us a measurement of how much risk movement is in an investment. For Investment A: 2A = [prob1 x (return1 - E(RA)2] + [prob2 x (return2 - E(RA)2] 2A = [.60 x (.12 - .088)2] + [.40 x (.04 - .088)2] = [.60 x .001024 ] + [.40 x .002304 ] = [.00036864] + [.0009216] = .00129024 The Standard deviation is the square root of the variance. For A: A = SQRT of .00129024 =+-0.03592 = + or - 3.59% This gives some idea of the potential movement in Investment A
Investment Portfolios A portfolio of investments enables us to diversify and therefore minimize the portion of risk that relates to "surprises" or unexpected movement in individual securities. A portfolio won't remove risk related to the market as a whole ("market risk").
Portfolio Illustration Suppose we mix a portfolio of 40% in Investment A (previous) + 40% in Investment B, which may earn only 7% in a good market but booms to 14% in a recession, and we put the other 20% in government investment G earning 4%. Portfolio Expected Return for Portfolio "P" : E(RP) = [.40 x E(RA)] + [.40 x E(RB)] + [.20 x E(RG)] Where E(RA) =8.8% , E(RB) =9.8% , and E(RG) = 4% (the risk-free rate) E(RP) = ( .40 x .088) + (.40 x .098) + (.20 x .04) E(RP) = .0824 or 8.24%
Portfolio Illustration (continued): Note: The percentage weights are based on the total dollars invested in each security. If we invested $100,000 as follows: $40,000 in A, $40,000 in B, and $20,000 in G, then we would have the 40%-40%-20% mix above. The variance of this portfolio is 0.00000434062 and the standard deviation is .0020736 or about + or - 2/10 of 1%. In other words, diversifying eliminated almost all of the diversification risk or unexpected return.
Risk & Beta (1): • Total risk of any investment = both • the market risk (which can't be diversified) and • the diversifiable risk, which can be minimized or eliminated by diversification in a portfolio. • The market risk is called systematic and the diversifiable risk is called unsystematic. • Total risk = Systematic risk + Unsystematic risk • (market risk) (diversifiable risk)
Risk & Beta (2): Total risk = Systematic risk + Unsystematic risk (market) (diversifiable) The unsystematic risk is asset-specific and relates to individual investments which can be minimized through diversification. The systematic risk, or market risk, can affect all market investments. A recession or a war, for example, might impact all investments in a portfolio. Since we can usually eliminate the unsystematic risk, we focus primarily on the systematic risk. Expected return of any asset , or E(Rasset), depends only on the asset's systematic risk. We measure the systematic risk by the beta coefficient, or .
Risk & Beta (3): The Beta of an asset = Covariance of asset returns with The market index portfolio Variance with the market portfolio I don't want to figure that out--do you? There are people on this planet who live for this stuff and do that for most publicly traded assets. (Your facilitator is NOT one of them!) Therefore we will assume the Beta is given for any investment we work with. The general rule for is as follows: If = 1.0 then the investment has "normal" market risk If < 1.0 then the investment has below normal market risk (for example U.S. securities' = 0 or zero risk) If > 1.0 then the investment has a greater than normal market risk (higher risk)
Some Sample Betas (as of 1/31/07) • Ford Motor Co (recent financial concerns, stock has dipped from $13.17 to $8.08/share over 2 yrs) = 1.83 • Wal-Mart (solid, $47.19/sh)= 0.17 • GE (also solid, $36.11/sh) = 0.51 • CVS Corp. (near mkt average, $33.31/sh)= 0.94 • Microsoft (solid, but rolling out Windows Vista, $30.41/sh) = 0.71 • Trump Entertainment Resorts (considerable fluctuation, $17.57/sh) = 1.96 • NutriSystem, Inc. (also wildly fluctuates, $45.83/sh)= 2.06 (stock has recently endured a 12% drop)
Portfolio Beta: If we have the Beta coefficient for each of the individual investments in our portfolio, we can evaluate the overall risk in our entire portfolio. Using the earlier example, let's make the following assumptions: 40% + 40% + 20% = Portfolio P Investment A Investment B Investment G A = 1.40 B = .90 G = 0 (risk free) P = (.40 x 1.40) + (.40 x .90) + (.20 x 0) = .56 + .36 + 0 = .92 (slightly below normal systematic risk) (As we calculated earlier, the expected return E(R) on portfolio P: E(RP) = 8.24%. Since the portfolio Beta is slightly < 1, we assume its E(R) to be slightly < the market rate)
The Security Market Line (SML) When we mix a portfolio of assets, we find a linear ( positive correlation) relationship between the individual assets' expected returns and their Betas. Assets with a higher Beta generally have a higher expected return to compensate for the higher systematic (market) risk. (General concept of risk vs. return--the higher the potential return, the higher the potential risk.)
The Security Market Line (SML) (2) This linear relationship between expected return and Beta is called the Security Market Line (SML). The slope of the SML is as follows: E(RA) - Rf Slope of SML for Asset A = A Or the difference between expected return and risk free return divided by the beta coefficient.
Security Market Line (SML) (3) E(RA) - Rf Slope of SML for Asset A = A .088 - .04 For our Investment A = 1.40 = .0343 or 3.4% For our Investment B = .098 - .04 .90 = .0644 or 6.4% This is the reward-to-risk ratio. Here investment B is more attractive, although neither is particularly high in a “bull” market ( remember B was better in a “bear” market).
Security Market Line (SML) (4) In an organized market, this difference in reward-to-risk would not persist because buyers and sellers would bid up investment B over investment A which would lower B's return and increase A's return. We therefore assume the reward to risk ratio is the same for all assets in the market and can therefore be plotted on the SML.
Market Risk Premium If we create a theoretical portfolio of all securities in the market, which would therefore have a Beta of the market average M = 1.0 we can evaluate the entire market risk premium as Market Risk Premium = E(RM) - Rf Risk premium = Expected market return – risk free rate Example: If the “going” market rate were 11.5% and the T-bill (risk free) rate were 4%, then the market risk premium is the difference of 7.5%
Capital Asset Pricing Model (CAPM) If we select any asset "i" in this market and assume that trading in the market's assets has "normalized" the expected return so that it equals the same reward to risk, then the equation for the SML of any asset "i" in the market is Expected return = risk free rate + (risk premium x Beta) E(Ri) = Rf + [E(RM) - Rf] x i. This is called the Capital Asset Pricing Modelor CAPM.
CAPM Illustration (1): If the Rf = 4% and the E(RM)=11.5% Suppose we select an asset "i" with a i = .7 The expected return on this asset is therefore (using CAPM) E(Ri)= Rf + [E(RM) - Rf] x i = .04 + [.115 - .04] x .7 = .04 + (.075 x .7) = .04 + .0525 = .0925 or 9.25% Because the Beta is low risk (less than market), the expected return is less than the market rate.
CAPM Illustration (2): Expected Return = risk free rate + (risk premium) x Beta E(Ri)= Rf + [E(RM) - Rf] x I (Where Rf= 4%, E(RM)= 11.5%) If the = 1.0 then the expected return = 11.5% (the market rate) If the = 1.5 then the expected return = 15.25 % If the = 2.0 then the expected return = 19% (this is double the market risk!) If the = .5 then the expected return = 7.75% If the = 0 then the expected return = 4% (the risk-free rate)
CAPM (3): • As long as we have the following variables: • The risk free rate • The current market rate • The asset’s Beta • Then we can estimate the expected return for any asset (investment). • If we have the E(R) of an asset and any two of the above, we can work backward and find the missing variable. Example-if we knew the return on an asset over time, we could estimate what its Beta should be.
CAPM (conclusion): Assumptions of the Capital Asset Pricing Model (CAPM) • The pure time value of money This is the risk- free rate, or the rate you could expect to earn over time if you accepted no (zero) risk (govt. securities) • The reward for bearing systematic risk, or the risk premium (asset rate in excess of the risk free rate) • The amount of systematic risk in the market, or the Beta value