Holding Period Return. Return, Risk, and Risk Aversion. Percentage Return or Dollar Return? An Example. Percentage Return or Dollar Return? 10% or 10?

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1 Return, Risk, and Risk Aversion Holding Period Return Ending Price - Beginning Price + Intermediate Income Return = Beginning Price R P t+ t+ = Pt + Dt P t An Example You bought IBM stock at $40 last month. The price of IBM stock is $45 today. IBM paid $ dividend yesterday. What is your holding period return? Percentage Return or Dollar Return? It is more convenient to use the percentage return, or the rate of return. A $,000 return is A 0% return implies that Percentage Return or Dollar Return? So the rate of return presents the complete picture. In this course, return almost means the rate of return. 0.0% is called. Return is not unitless. We like to returns. Returns are bounded below at 0% or 0? We like to express return in percentage terms. 0% is the same as 0. but not the same as 0.

2 Averaging Returns Suppose you invested half of your money in X and half of your money in Y. The return of X is 00% and the return of Y is -50%. What is the average return? Averaging Returns Suppose you invested all of your money in X The return of X is 00% last year and -50% this year. What is your average return over the two years? Return is uncertain R P t+ t+ = Pt + Dt P Investments are risky. P t+ is not known at time t; hence R t+ is not known at time t. How do we capture the randomness of R t+? t Return and Risk Under uncertainty, we measure reward by using the expected (or average) return. We measure risk by using the variance of returns µ X Expected Return and Variance = E ( X ) = p X + p2 X 2 + L+ p X = Var 2 ( X ) = σ X = pi[ X i E( X )] i= 2 i= p X i i Sample Mean and Variance Var ˆ X = X i i= ( ) ( ) 2 X = X i X i= 2

3 Risk Aversion People like higher expected returns because they prefer more to less. People like lower risks because they are risk averse. How do we know that people are risk averse? Risk Aversion In the Who wants to be a millionaire show, suppose you have won $32,000. And you are 50% sure about the next question. Do you stop or continue? Risk Aversion In the Who wants to be a millionaire show, you have won $6,000. And you are 50% sure about the next question. Do you stop or continue? Risk Aversion Suppose you have won $6,000. If you are wrong on the next question, you go home with $0,000. If you are right on the next question, you get $ 23,000. Again, you are 50% sure. Do you stop or continue? Risk and Return Risk and return go hand in hand with each other. We can think of each investment having two dimensions: risk and return. The objective of this course is to pick the investments that have the best risk/return combination. Risk and Return There should be a relationship between risk and (expected) return. High Risk Current Price Future Return 3

4 Risk Premium Risk Premium or Excess Return = Expected Return Risk-free Rate To determine the appropriate risk premium, we need an asset pricing model such as CAPM. Portfolio People are often interested in the expected returns and variance (or standard deviation, volatility) of portfolios, as opposed to individual securities. After all, investors ultimately care about the return and risk of her entire portfolio, as opposed to any individual asset in the portfolio. Expected Return of A Portfolio So how do we go from individual asset to portfolios? The expected return of a portfolio is simply the weighted average expected returns of all the individual assets in the portfolio. Expected Return of a Portfolio w i = value of asset #i in the portfolio / total value of the portfolio E(R p ) = w E(R ) + w 2 E(R 2 ) + w n E(R n ) Variance of A Portfolio Variance is more complicated If there are two assets in the portfolio, Var(w X +w 2 Y) = w 2 Var(X) + w 22 Var(Y) + 2w w 2 Cov(X,Y) where Cov(X,Y)=Corr(X,Y)σ(X)σ(Y) Standard deviation is the square root of variance An Example Your portfolio consists of 40% X and 60% Y. The expected returns of X and Y are 0% and 2%, respectively. The standard deviations of X and Y are 20% and 30%. The correlation between X and Y is 0.5. What is the expected return of your portfolio? What is the standard deviation of your portfolio? 4

5 Formulas E(R p ) = w E(R ) + w 2 E(R 2 ) + w n E(R n ) Var(w X +w 2 Y) = w 2 Var(X) + w 22 Var(Y) + 2w w 2 Cov(X,Y) where Cov(X,Y)=Corr(X,Y)σ(X)σ(Y) An Example E(R)= Var(R)= Stdev(R)= Calculating the expected return and standard deviation of a portfolio with three assets are similar but more complicated More on Covariance ow consider the three asset case. Var(w X + w 2 Y + w 3 Z) = w 2 Var(X) + w 22 Var(Y) +w 32 Var(Z) + 2w w 2 Cov(X,Y)+ 2w w 3 Cov(X,Z)+ 2w 2 w 3 Cov(Y,Z) The formula gets very complicated when there are many assets in the portfolio. Annualize Return Sometimes you need to convert a monthly return to an annual return There are two ways you can do that Suppose R is the per period return and T is the number of periods per year. APR = R T EAR = ( + R ) T - Annualize Variance Variance is proportional to time Annualized Variance = σ 2 T Standard deviation is proportional to the square root of time Annualized Standard Deviation = σ T 0.5 Annualize the Variance Consider the return over two successive periods, e.g., January and February. Let R represent the January return and R2 represent the February return. Ignoring compounding, the two-month return over January and February is R+R2. 5

6 Annualize Variance The variance for each month is the same Var(R) = Var(R2) = Var (R). What is the variance for the two-month return, namely, R+R2? Successive stock returns are uncorrelated, i.e. Cov(R,R2)=0 Annualize the Variance Therefore, Var (R+R2) = Var (R) + Var(R2) + 2Cov(R,R2) = Var(R)+Var(R2) = 2 Var(R) Generalize the above result, Var(R+R2+ +R2)=2 Var (R) And, Stdev R + R + + R 2 Stdev R ( ) ( ) 2 2 = Historical Performance Historical Performance Average Standard Series Annual Return Deviation Distribution $0,000 Small Stocks Large Stocks Corp. Bonds $4,557 Large Company Stocks 2.7% 20.3% $,000 LT Gov't Bonds T-Bills Inflation $,37 Small Company Stocks Corporate Bonds $00 $48.3 $33.9 Government Bonds $0 $3.5 $8.8 U.S. Treasury Bills Inflation $ $0 90% 0% + 90% Objective Investors like higher expected returns and lower risks Having two objectives is not convenient to work with How do we put them into a single objective function? Risk Aversion & Utility Utility Function U = E ( r ) A σ 2 Or if you express the expected return and the standard deviation in percent. U = E ( r ) A σ 2 6

7 Risk Aversion U = E ( r ) A σ 2 A, the risk aversion parameter, is positive for risk averse investors. The higher the A is, the more risk averse an investor is. Experiments show that A is positive and generally under 0. Mean-Variance Utility Function Mean-Variance Utility Function U = E ( r ) A σ 2 You d like to maximize your utility, U. Everything else being equal, you like expected return, E (r). Everything else being equal, you like variance, σ 2 You can t compare the values of U s across individuals. Dominance Principle Expected Return Variance or Standard Deviation 2 dominates ; has a higher return 2 dominates 3; has a lower risk 4 dominates 3; has a higher return A choice problem Consider a risk-averse investor with coefficient of relative risk aversion A = 2. Suppose she was offered the opportunity to invest in one of two opportunities: Option A: CD with a 5% sure rate of return Option B: A stock with expected return = 20% and variance (σ 2 ) = 0.5 Which would she choose to invest in? Choice Problem Utility from investing in the stock: U stock Utility from investing in the CD: U CD = The investor is between the stock investment and a certain return of 5% Utility and Indifference Curves Represent an investor s willingness to tradeoff return and risk Example (A=4) Exp Ret St Deviation U=E ( r ) - 0.5Aσ 2 0% 20.0% % 25.5% % 30.0% % 33.9%

8 Indifference Curve Expected Return Risk aversion and the slope of the indifference curve Steeper indifference curve indicates that for an increase of one unit of risk (measured by standard deviation), the investors need to be compensated with higher additional expected returns to stay indifferent. Increasing Utility Standard Deviation Utility and Risk Aversion A B y y2 x 8