Computing Returns Return and Risk Return = An Example Percentage Return or Dollar Return? You bought IBM stock at $40 last month. The price of IBM stock is $45 today. IBM paid $1 dividend yesterday. What is your holding period return? Dollar return per share: $45-$40+$1=$6 Rate of return: $6 / $40 = 15% Percentage Return or Dollar Return? So the rate of return presents the complete picture. In this course, return means the rate of return. 0.01% is called a basis point. Return is not unitless. We tend to annualize returns. Returns are bounded below at -100%. Ending Price - Beginning Price + Dividends Beginning Price It is more convenient to use the percentage return, or the rate of return. A $1,000 return is quite good for an initial investment of $1,000, but not so impressive if the initial investment is $1 million. A 10% return implies that if you invest $1000 you would make $100 and if you invest $1 million you will make $100,000. 10% or 10? We like to express return in percentage terms. 10% is the same as 0.1 but not the same as 10. P = $1 / (10 - 5) = $0.20 P = $1 / (10% - 5%) = $20 1 Averaging Returns Averaging Returns Suppose you invested half of your money in X and half of your money in Y. The return of X is 100% and the return of Y is -50%. What is the average return? Future Return is unknown Suppose you invested all of your money in X The return of X is 100% last year and -50% this year. What is your average return over the two years? Return and Risk P − P + Dt Rt +1 = t +1 t Pt Under uncertainty, we measure “reward” by using the expected (or average) return. We measure “risk” by using the variance of returns Investments are risky. Pt+1 is not known at time t; hence Rt+1 is not known at time t. How do we capture the randomness of Rt+1? Expected Return and Variance Sample Mean and Variance N µ X = E ( X ) = p1 X 1 + p2 X 2 + L + p N X N = ∑ pi X i X= i =1 N Var ( X ) = σ = ∑ pi [ X i − E ( X )] 2 X i =1 2 Vaˆr ( X ) = 1 N N ∑X i =1 i 2 1 N ∑ (X i − X ) N − 1 i =1 2 Annualize Return There are two ways you can annualize returns. Suppose R is the per period return and T is the number of periods per year. APR = R × T EAR = ( 1 + R ) T - 1 Excess Returns The raw return includes compensation for both the time value of money and the risk of the security. An excess return or risk premium is the compensation for risk bearing alone. Excess Return or Risk Premium = Ri − R f = Rie Variance is proportional to time Annualized Variance = σ2 × T Standard deviation is proportional to the square root of time Annualized Standard Deviation = σ × T0.5 Expected Return of A Portfolio Adding more assets to a portfolio does not make the calculation of expected return more difficult. E(Rp) = w1E(R1) + w2E(R2) + … wnE(Rn) This is the raw return less the risk-free rate. Variance of A Portfolio Annualize Variance We are often interested in the variance of a portfolio. If there are two assets in the portfolio, Var(w1X +w2 Y) = w12Var(X) + w22Var(Y) + 2w1w2Cov(X,Y) More on Covariance Var(w1X + w2Y + w3Z) = w12Var(X) + w22Var(Y) +w32Var(Z) + 2w1w2Cov(X,Y)+ 2w1w3Cov(X,Z)+ 2w2w3Cov(Y,Z) where Cov(X,Y)=Corr(X,Y)σ(X)σ(Y) Now consider the three asset case. The formula gets very complicated when there are many assets in the portfolio. 3 Matrix Notation R is a column vector of expected returns W is a column vector of portfolio weights ∑ is the covariance matrix Example: 0.01 ⎤ ⎡ 0.1 ⎤ ⎡0.4⎤ ⎡0.04 0.01 R = ⎢⎢ 0.2 ⎥⎥ W = ⎢⎢0.3⎥⎥ Σ = ⎢⎢ 0.01 0.09 − 0.01⎥⎥ ⎢⎣0.15⎥⎦ ⎢⎣0.3⎥⎦ ⎢⎣ 0.01 − 0.01 0.04 ⎥⎦ Portfolio Return and Risk Using matrix notation for portfolio return and risk, E (R p ) = w1 R1 + w2 R2 + ... + wn Rn = w′R Var (R p ) = w12σ 12 + w22σ 22 + ... + wn2σ n2 + 2w1w2σ 1, 2 + 2w1w3σ 1,3 + ... = w' Σw Mean Standard Deviation Excel Functions Excel Functions 4 Excel Functions Covariance Correlation Data Analysis Covariance Covariance Matrix 5