Question

Consider the following returns of two stocks in conjunction with the market M Std. dev. of stock1 Std. dev. of stock 2 Std. dev. of market Expected return on the market rM 1096 Corr. of stock 1 with the market piM Corr. of stock 2 with the market p2M 0.7 Risk free rate T1 2096 T2 3096 15% 0.4 . According to the CAPM, what should the expected return of stock 1 and stock 2 be? (Note: Your answer should be a number in percentage form. Do not enter "%) Expected return of stock 1 Expected return of stock 2 2. Suppose that the correlation between the returns of stock 1 and the returns of stock 2 is 0.5. What is the expected return and standard deviation of a portfolio that has 30% invested in stock 1 and 70% in stock 2" (Note: Your answer should be a number in percentage form. Do not enter %.) Expected return = Standard deviation = 3. Try to construct a new portfolio using a combination of the market portfolio and the risk free asset that has same returns as above (part 2) but with the lowest standard deviation. What is the standard deviation of this portfolio? (Note: Your answer should be a number in percentage form and negative numbers are allowed for the weights. Do not enter %.) Portfolio weight in the market Portfolio weight in the risk free asset- Standard deviation

Answer 1

1.

Beta_stock1 = Cov(stock1,M)/sigma_M² = correl(stock1,M) * sigma_stock1/sigma_M = 0.4*0.2/0.15 = 0.53333

using CAPM, E[R_stock1] = R_f + Beta_stock1 [E(R_m)-R_f] = 6% + 0.53333 * (10%-6%) = 8.13333% ...(1) {expected return of stock 1}

Beta_stock2 = Cov(stock2,M)/sigma_M² = correl(stock2,M) * sigma_stock2/sigma_M = 0.7*0.3/0.15 = 1.4

using CAPM, E[R_stock2] = R_f + Beta_stock2 [E(R_m)-R_f] = 6% + 1.4 * (10%-6%) = 11.6% ...(2) {expected return of stock 2}

2. given, w₁=0.3, w₂?=0.7, correl(stock1,stock2)=0.5

E(R_{portfolio_question2}) =  w₁*E(R_stock1) +w₂*E(R_stock2) = 0.3*8.13333% + 0.7*11.6% = 10.56%.

sigma_portfolio = [w_1² * sigma_{1² +} w_2² * sigma_2² + 2 * w₁ * w₂ * sigma₁ * sigma₂ * correlation(stock1,stock2)]^0.5

= [0.3²*0.2² + 0.7²*0.3² + 2*0.3*0.7*0.2*0.3*0.5]^0.5 =(0.0603)^0.5= 24.55606%

3. E(R_{portfolio_question3}) =  w₁*E(R_m) +(1-w₁)*E(R_f) =same as E(R_{portfolio_question2}) = 10.56%.

implies w₁*E(R_m) +(1-w₁)*E(R_f) = 0.1056

implies w₁*0.1 +(1-w₁)*0.06 =0.1056 ...(3)

implies w₁*0.1 +0.06 -w₁*0.06 =0.1056

implies 0.04 * w₁ = 0.0456

implies w₁ = 1.14 = 114% (market weight of portfolio)

weight of risk free asset = 1-1.14 = -0.14 = -14% (risk free asset weight of portfolio, negative means short position)

{Note, we have only one value of w₁ satisfying (3). this should be the minimum variance given constraint}

Alternatively, we can use solver in excel-

Minimize, variance = sqrt(w_1² sigma_m2) implies minimize vaiance = w₁ * 0.15

subject to constraints-

w₁ + w₂ = 100% ...(A)

w₁*0.1 +(1-w₁)*0.06 =0.1056 ...(B)

we get, w₁ = 114%.

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Variance = [w_1² * sigma_{market² +} w_2² * sigma_riskfree² + 2 * w₁ * w₂ * sigma_market * sigma_riskfree * correlation(market,riskfree)]^0.5 {note w₂= 1-w₁}

= [1.14² * 0.15²+ 0 + 0]^0.5 = 0.171 = 17.1% (standard dev. value)

(when we use solver, excel will also give above variance value)