Calculating the Rate of Return on Assets Introduction o Risk & Return Econ 422: Investment, Capital & Finance University of Washington Summer 26 August 5, 26 Denote today as time the price of the asset by the price of the dividend or cash flow expected he rate of return of the asset and next period by time = the asset next period by P he rate of return r is calculated as P P + D P P D r = = + P P P = = P next period by D the capital gain (appreciation)/capital loss + the dividend or income yield Note : r = random variable as P and D are unknown Asset Returns Focusing now specifically on financial assets, first singly then in combination with other assets, we will address how investors evaluate and select assets (or determine their portfolio of assets). hink about your own experience via the investment project assignment. A return on an asset relies on the future cash flow and future price. In the absence of perfect foresight knowing the future cash flow and price with certainty asset returns are random variables. Populations versus Samples he sources of information for the Distribution of a random variable X can be from: A. Knowledge of the Population Knowledge of the Probability Distribution for the Random Variable X Recall: Population is the set of all conceivable or hypothetically possible observations of a particular phenomenon B. Sample Data for Random Variable X, i.e. a sample: x,x 2,,x n drawn from the X distribution. Recall the distribution of a random variable provides important information about the random variable, in particular possible outcomes and the probability of such outcomes. Recall: A set of data that consists only a part of the Population or only a part of all possible observations is a sample. he population may be unknown with a sample. What we will care most about are the first two moments of a random variable distribution.
Population Moments When a population/probability distribution is known we calculate the first two moments as follows: E(X) = Σ x i P(X = x i ) i for all possible values of x he expected value of X = the probability weighted sum of x values. V(X) = Σ( x i E(X)) 2 P(X = x i ) i for all possible values of x he variance of X = the probability weighted sum of squared dispersion terms. E(X) and V(X) are referred to as the parameters of the population distribution. Sample Statistics Observed data sample: x,, x When instead we have sample data available, we calculate the sample statistics : x = xi i= 2 2 s x = ( xi x) i= he sample mean and variance are estimates of the unknown population expected value and variance Advanced: You can show that the expected value of each sample statistic is the population parameter, i.e., in repeated sampling the statistics of the sample approach the population parameters. Sample Statistics Continued For a bivariate sample, (x,y ),, (x,y ), the sample covariance and correlation are defined as s = ( x x)( y y) = x y x y r xy i i i i i= i= xy sxy = s s x y Population Parameter: Expected Return Joint probability distribution for Asset A returns and Asset B returns as follows: State of the World/Outcomes Probability p s R A R B Bull Market.25.5.5 Normal.5.. Bear Market.25.5 Expected return on an asset: E( R) = Σ R s p s for all possible states of the world s For Assets A and B: E( R A ) = Σ R s p s =.5 *.25 +.*.5 + *.25 =.75 or 7.5% E( R B ) = Σ R s p s =.5 *.25 +.*.5 +.5*.25 =. or.% What properties of the asset returns do you notice? 2
Population Parameter: Variance and SD Joint probability distribution for Asset A returns and Asset B returns as follows: State of the World/Outcomes Probability p s R A R B Bull Market.25.5.5 Normal.5.. Bear Market.25.5 he variance of an asset return: V( R) = Σ (R s E(R)) 2 p s for all possible states of the world For assets A and B: V( R A ) = Σ (R As E(R A )) 2 p s = (.5-.75) 2 *.25 + (.-.75) 2 *.5 + (-.75) 2 *.25 =.36875 SD(R A ) = (.36875) /2 =.92 = 9.2% V( R B ) = Σ (R Bs E(R B )) 2 p s = (.5-.) 2 *.25 + (.-.) 2 *.5 + (.5-.) 2 *.25 =.25 SD(R B ) = (.24) /2 =.352 = 3.52% Population Parameter: Covariance Joint probability distribution for Asset A returns and Asset B returns as follows: State of the World/Outcomes Probability p s R A R B Bull Market.25.5.5 Normal.5.. Bear Market.25.5 he variance of an asset return = the covariance of an asset return with itself. he covariance of an asset return: Cov( R i, R j ) = Σ (R is E(R i ))(R js E(R j )) p s for all possible states of the world s, for assets i and j For assets A and B: Cov( R A, R B ) = Σ (R As E(R A ))(R Bs E(R B )) p s = (.5-.75)(.5-.)*.25 + (.-.75)(..)*.5 + (-.75) (.5.)*.25 = -.625 CORR(R A, R B ) = Cov(R A, R B )/(SD(R A )*SD(R B ) = -.626/(.92*.352) = -.93 Does this confirm your initial observation about assets A and B? Sample Statistics: Arithmetic Mean Returns Consider the following sample: Year Return to asset A Return to asset B 999 5% 3% 2 % 2% 2 5% 2% 22 2% 5% Sample average = arithmetic mean R is the arithmetic mean return calculated over periods and R t is the holding-period return over period t: R = (/) [R + R 2 + R 3 +... + R ] R A = (/4) [.5 + +.5 +.2] =. = % R B = (/4) [.3.2 +.2 +.5] =.2 = 2% How does this track with your initial thinking regarding the relative desirability of asset A versus asset B? Sample Statistics: Sample Variance Consider the following sample: Year Return to asset A Return to asset B 999 5% 3% 2 % 2% 2 5% 2% 22 2% 5% Sample variance: s 2 (R) = [/(-)]Σ( R i R) 2 t =,... s 2 (R A ) = [/3]{(.5.) 2 + (.) 2 +(.5.) 2 +(.2.) 2 } =.8333 s 2 (R B ) = [/3]{(.3.2) 2 + ( -.2.2) 2 +(.2.2) 2 +(.5.2) 2 } =.86667 Note: the sample standard deviation = (s 2 ( R)) /2 = s( R) = ( [/(-)]Σ( R i R) 2 ) /2 s(r A ) = (.8333) /2 =.93 = 9.3%; s(r B ) = (.86667) /2 =.2944 = 29.24% Does this change your initial thinking regarding the relative desirability of asset A versus asset B? 3
Sample Statistics: Sample Covariance Consider the following sample: Year Return to asset A Return to asset B 999 5% 3% 2 % 2% 2 5% 2% 22 2% 5% Sample covariance and correlation: s xy ( xi x )( yi y) xi yi x y i = i= = 4 (.5)(.3) + ()(.2) + (.5)(.2) + (.2)(.5) (.) (.2) =.875.875 rxy = =.93 (.93)(.2944) ( ) Excel Functions AVERAGE» Compute sample average VAR» Compute sample variance SDEV» Compute sample standard deviation COVAR, CORREL» Compute sample covariance and correlation ools/data Analysis/Covariance» Compute Covariance Matrix he Value of an Investment of $ in 926 he Value of an Investment of $ in 926 S&P Small Cap Corp Bonds Long Bond Bill 642 2587 64. S&P Small Cap Corp Bonds Long Bond Bill Real Value 66 267 Index 48.9 6.6 Index 6.6 5..7. 925 94 955 97 985 2 Source: Ibbotson Associates Year End. 925 94 955 97 985 2 Source: Ibbotson Associates Year End 4
Percentage Return Rates of Return 926-2 6 4 26 3 35 4 45 5 55 6 65 7 75 8 85 9 2-2 -4-6 Source: Ibbotson Associates Common Stocks Long -Bonds -Bills Year 95 2 otal Annual Returns, 926-2 Risk Premium Arithmetic (relative to U.S. Standard Series Mean reasury bills) Deviation Distribution Common Stocks 3.% 9.% 2.2% Small Company Stocks 7.3 3.4 33.4 Long-erm Corporate Bonds 6. 2. 8.7 Long -erm Government Bonds 5.7.8 9.4 Intermediate- erm Government Bonds 5.2.3 5.7 U.S. reasury Bills 3.9 -- 3.2 Inflation 3. -- 4.6 * he 993 small company stock total return was 42.9 percent -9% % +9% Slide 9-3 * Distribution for Large Company Stock Returns 926-23 Probability 68.26% 95.44% 99.74% 3σ 2σ σ µ +σ +2σ +3σ 49.3% 28.8% 8.3% 2.2% 32.7% 53.2% 73.7% Return on stocks In the case of a normal distribution, there is a 68.26 percent possibility that a return will be within one standard deviation of the mean. In this example, there is a 68.26 percent probability that a yearly return will be between 8.3 percent and 32.7 percent. here is a 95.44 percent probability that a return will be within two standard deviations of the mean. In this example, there is a 95.44 percent probability that a yearly return will be between 28.8 percent and 53.2 percent. Finally, there is a 99.74 percent probability that a return will be within three standard deviations of the mean. In this example, there is a 99.74 percent E. Zivot probability 25 that a yearly return will be between 49.3 percent and 73.7 percent. Risk Premium he average excess return over a safe investment (e.g. -Bills) is called the risk premium Risk Premium = Average Return on Risky Asset Return on Safe Asset Average Return on Risky Asset = Return on Safe Asset + Risk Premium 5
Risk premium, % 9 8 7 6 5 4 3 2 Average Market Risk Premia (999-2) 4.3 5. 6 6. 6. 6.5 6.7 7. 7.5 8 8.5 9.9 9.9 Den Bel Can Swi Spa UK Ire Neth USA Swe Aus Ger Fra Jap It Expected Return for the Market Portfolio From our definition of risk premium: Expected market return = risk free rate + expected market risk premium E(r mt ) = r ft + expected market risk premium If the expected annual market risk premium is estimated by its historical long run average of about.75 or 7.5% then E(r mt ) = r ft +.75 Country Expected Return for Capital Budgeting he risk of a project may be more of less than the average market risk. (We have not yet said how risk is measured.) For projects with the same degree of risk as the market, we can use the expected market return as the discount rate in our NPV calculations. Risk Measures for Individual Assets From our previous illustration, historically the annual return for a broadly diversified portfolio of large company stocks had a standard deviation about.23 Returns for individual stocks generally have standard deviations that are larger, for example, if we consider the standard deviation of monthly returns (rolling): Martha Stewart Living.9 Petco. Sears.4 IBM. Average, S&P5.5 How can groupings of several stocks have a lower standard deviation or asset return variability than a single asset? Diversification holding more than a single asset-- can reduce or eliminate much of the variability associated with individual assets because not every asset value fluctuates in the identical manner. 6
Risk Measures for Individual Assets Risk Measures for Individual Assets How can groupings of several stocks have a lower standard deviation or asset return variability than a single asset? Diversification holding more than a single asset-- can reduce or eliminate much of the variability associated with individual assets because not every asset value fluctuates in the identical manner. Diversification Eliminates Asset Specific Risk Portfolio Standard Deviation Risk Measures for Individual Assets, cont. Although standard deviation is a reasonable measure of the risk of a diversified portfolio, it is not a satisfactory measure of the risk for individual assets.» If individuals can hold assets in diversified portfolios, then much of an asset s total risk measured by standard deviation can be eliminated through diversification otal Risk Unique Risk Market Risk he standard deviation of a diversified portfolio approaches the standard deviation of the market portfolio. his is the remaining risk that cannot be diversified away. # of securities in portfolio 7
Risk Measures for Individual Assets, cont. In the lecture following next, we will see that the Capital Asset Pricing Model (CAPM) demonstrates that a better risk measure for an individual asset is provided by an asset s beta. A stock s beta measures the tendency of the stock s return to covary with that of the market. Beta is related to a measure of the contribution of the stock to the risk of a well diversified portfolio. Beta for the market is.. Beta for assets that have more than average sensitivity to market return movements will be greater than., those that are less sensitive will be less than.. 8