Expected Return and Variance Risk and Return Expected Return and Variance

Probability of State of Economy Stock Returns if State Occurs Expected Return  State of Economy  Probability of State of Economy Stock Returns if State Occurs Stock A Stock B Boom 50% 0.70 0.10 Recession -0.20 0.30 Which one would you like?

Expected Return 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑅𝑒𝑡𝑢𝑟𝑛 = 𝑃 𝑏𝑜𝑜𝑚 × 𝑅 𝑏𝑜𝑜𝑚 + 𝑃 𝑅𝑒𝑐𝑒𝑠𝑠𝑖𝑜𝑛 × 𝑅 𝑟𝑒𝑐𝑒𝑠𝑠𝑖𝑜𝑛 𝑃 𝑏𝑜𝑜𝑚 = Probability of boom 𝑅 𝑏𝑜𝑜𝑚 = Return if boom occurs 𝑃 𝑟𝑒𝑐𝑒𝑠𝑠𝑖𝑜𝑛 = Probability of recession 𝑅 𝑟𝑒𝑐𝑒𝑠𝑠𝑖𝑜𝑛 = Return if recession occurs

Stock A and B Investors expect to earn from A 50%×0.70+50%×(−0.20)=0.25 How much do investors expect to earn from B? 0.20

Problem 11-6 Calculating Expected Return Consider the following information: Calculate the expected return.  State of Economy  Probability of State of Economy Stock Returns if State Occurs Recession 0.22 -0.12 Normal 0.48 0.14 Boom 0.30 0.33 E(R) = .22(–.12) + .48(.14) + .30(.33) = .1398, or 13.98%

How Volatile Stock Return Is? Variance measures the average squared difference between the actual returns and the expected return Standard deviation is the square root of variance The larger the variance or standard deviation is, the more spread out the returns will be

Variance and Standard Deviation 𝑉𝑎𝑟= 𝑝 𝑏𝑜𝑜𝑚 𝑅 𝑏𝑜𝑜𝑚 −𝑅 2 + 𝑝 𝑟𝑒𝑐𝑒𝑠𝑠𝑖𝑜𝑛 𝑅 𝑟𝑒𝑐𝑒𝑠𝑠𝑖𝑜𝑛 −𝑅 2 𝑃 𝑏𝑜𝑜𝑚 = Probability of boom 𝑅 𝑏𝑜𝑜𝑚 = Return if boom occurs 𝑃 𝑟𝑒𝑐𝑒𝑠𝑠𝑖𝑜𝑛 = Probability of recession 𝑅 𝑟𝑒𝑐𝑒𝑠𝑠𝑖𝑜𝑛 = Return if recession occurs 𝑅= Expected return Standard Deviation (𝑆𝐷) 𝑆𝐷= 𝑉𝑎𝑟

Variance Stock A 0.10 stock B

Problem 11-7 Calculating Standard Deviation Consider the following information: Calculate the expected return and standard deviation for two stocks  State of Economy  Probability of State of Economy Stock Returns if State Occurs Stock A Stock B Boom 0.24 0.055 -0.34 Normal 0.64 0.135 Recession 0.12 0.230 0.47 E(RA) = .24(.055) + .64(.135) + .12(.23) = .1272, or 12.72% E(RB) = .24(–.34) + .64(.24) + .12(.47) = .1284, or 12.84% σA^2 = .24(.055 – .1272)^2 + .64(.135 – .1272)^2 + .12(.23 – .1272)^2 = .00256 σA = (.00256)^1/2 = .0506, or 5.06% σB^2 = .24(–.34 – .1284)^2 + .64(.24 – .1284)^2 + .12(.47 – .1284)^2 = .07463 σB = (.07463)^1/2 = .2732, or 27.32%

𝑊 𝑖 = $ 𝑖𝑛𝑣𝑒𝑠𝑡𝑒𝑑 𝑜𝑛 𝑎𝑠𝑠𝑒𝑡 𝑖 𝑇𝑜𝑡𝑎𝑙 $ 𝑖𝑛𝑣𝑒𝑠𝑡𝑒𝑑 𝑜𝑛 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 Portfolio Portfolio = Collection of individual assets Weight of individual assets 𝑖 in the portfolio 𝑊 𝑖 = $ 𝑖𝑛𝑣𝑒𝑠𝑡𝑒𝑑 𝑜𝑛 𝑎𝑠𝑠𝑒𝑡 𝑖 𝑇𝑜𝑡𝑎𝑙 $ 𝑖𝑛𝑣𝑒𝑠𝑡𝑒𝑑 𝑜𝑛 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 Total $ invested on portfolio = Sum of $ invested on all individual assets in the portfolio 𝑊 𝑖 = The fraction of portfolio invested on each individual asset

If We Invest on A and B… If we invest $200 on stock A and invest $800 on B, what is the portfolio weight of on two stocks? 𝑊 𝐴 = $200 $200+$800 = 200 1000 =0.2 𝑊 𝐵 = $800 $200+$800 = 800 1000 =0.8 Expected return of stocks A is 0.25 and that of B is 0.20, what is the portfolio’s expected return? 0.2×0.25+0.8×0.20=0.21

Problem 11-1 Determining Portfolio Weights What are the portfolio weights for a portfolio that has 150 shares of Stock A that sell for $87 per share and 125 shares of Stock B that sell for $94 per share? Total value = 150($87) + 125($94) = $24,800 xA = 150($87) / $24,800 = .5262 xB = 125($94) / $24,800 = .4738

Portfolio Expected Return 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑅𝑒𝑡𝑢𝑟𝑛= 𝑊 1 𝑅 1 + 𝑊 2 𝑅 2 +…+ 𝑊 𝑛 𝑅 𝑛 𝑊 𝑛 = Weight of stock 𝑛 𝑅 𝑛 = Expected return of stock 𝑛

Portfolio Expected Return Suppose we had following investments: What is the expected return of this portfolio? Security Amount Invested Expected Return Stock A 1,000 8% Stock B 2,000 12% Stock C 3,000 15% Stock D 4,000 18%

Portfolio Expected Return 14.6 1.16

Problem 11-4 Portfolio Expected Return You have $13,000 to invest in a stock portfolio. Your choices are Stock X with an expected return of 15% and Stock Y with an expected return of 9%. Assume your goal is to create a portfolio with an expected return of 12.35%. How much money will you invest in Stock X and Stock Y? E(Rp) = .1235 = .15wX + .09(1 – wX) .1235 = .15wX + .09 – .09wX .0335 = .06wX wX = .5583 Investment in X = .5583($13,000) = $7,258.33 Investment in Y = (1 – .5583)($13,000) = $5,741.67

Portfolio Variance and Standard Deviation If we invest $200 on stock A and invest $800 on B, what is the portfolio variance?  State of Economy  Probability of State of Economy Stock Returns if State Occurs Stock A Stock B Boom 50% 0.70 0.10 Recession -0.20 0.30

Portfolio Variance and Standard Deviation

Problem 11-9 Portfolio Variance Consider the following information: What is the variance of a portfolio invested 22 percent each in A and B and 56 percent in C?  State of Economy  Probability of State of Economy Stock Returns if State Occurs Stock A Stock B Stock C Boom 0.67 0.10 0.04 0.35 Bust 0.33 0.24 0.30 -0.15 Boom: Rp = .22(.10) +.22(.04) + .56(.35) = .2268, or 22.68% Bust: Rp = .22(.24) +.22(.30) + .56(–.15) = .0348, or 3.48% E(Rp) = .67(.2268) + .33(.0348) = .1634, or 16.34% σp^2 = .67(.2268 – .1634)^2 + .33(.0348 – .1634)^2 = .00815