Chapter 7:

14)

Investment in stock I: x1: 60% or 0.6

Investment in stock J: x2: 40% or 0.4

S.D of return on I: δ 1 = 10%

S.D of return on J: δ 2 = 20%

Let variance of portfolio be δ3

a) Correlation ρ = 1.0 δ3 = (0.6)^2(10)^2+(0.4)^2(20)^2+2(0.6*0.4*1*10*20) = 196

Variance of Portfolio = 196

b) Correlation ρ = 0.5 δ3 = (0.6)^2(10)^2+(0.4)^2(20)^2+2(0.6*0.4*0.5*10*20) = 148

Variance of Portfolio = 148

c) Correlation ρ = 0 δ3 = (0.6)^2(10)^2+(0.4)^2(20)^2+2(0.6*0.4*0*10*20) = 100

Variance of Portfolio = 100

19)

β=-0.25

a) As the beta is negative, the rate of return on the stock will go down when the market return goes up.

If market rises by an extra 5%, the stock goes down by -0.25*5= -1.25%. Stock return will be down by -1.25%.

If market falls by an extra 5%, the stock goes up by 0.25*5 = 1.25%. Stock return will be up by 1.25%.

b) Invested $1M in well-diversified portfolio.

We are looking to invest in the safest overall portfolio return i.e., the option that reduces our risk to the lowest value. Reducing risk is nothing but reducing beta. Investing in stocks with beta= 1 (option 2) is not a wise option because the overall portfolio return is equally impacted by the rate of market return. Investment in Treasury bills with beta =0 (option 1) is though not bad as option 2, it removes any sort of impact of the market return. However, there is neither gain nor loss due to market changes. Option 2 is only an option if someone is not willing to take any risk .i.e.., only looking for the risk free return. Option 3 with beta -02.5 is the best option as the stock return moves in opposite direction to the market and even a slightest drop of 0.25% market return can produce 0.25% increase of overall portfolio return. And with increase of 0.25% market return, the overall portfolio return will drop only by 0.25%. Therefore, the risk is lowered by option 3 and that is the best option of the three.

20)

Let, the expected return of the overall portfolio = rP = 12% or 0.12

Expected return on stock A = rA =10% or 0.1

Expected return on stock A = rB = 15% or 0.15

Weighted Average of asset A = xA

Weighted Average of asset B = xB

We know, rP = rA xA + rB xB (1) 0.12= 0.1xA + 0.15xB

The portfolio has only two assets A & B i.e., xB = 1-XA (2) Then we can reduce the formula to => 0.12 = 0.1xA +0.15(1-xA)

12 = 10xA+15-15xA

5xA = 3 => xA = 0.6 or 60% (3) xB = 0.4 or 40% (4)

Variance of the portfolio δp^2 = δA^2xA^2 + δB^2xB^2+2ρ δAxA δBxB (5) δp^2 = 20^2 * 0.6^2 + 40^2 * 0.4^2 + 2*0.5*20*0.6*40*0.4 = 592 δp = = 24.33%

S.D of portfolio = δp = (592)^1/2 = 24.33

21) Given, the standard deviation of the return on the market (assumption) is 15 % β for Dell: 1.41 and that for McDonald’s: 0.77

S.D for Dell: 30.9% and that for McDonald’s: 17.2%

a) Dell and McDonald's correlation coefficient (ρ) =0.31

In general: Variance of the Portfolio δp^2 = δA^2xA^2 + δB^2xB^2+2ρ δAxA δBxB

Thus: δp^2 = 0.5^2*30.9^2 + 0.5^2 *17.2^2 +2*0.31*0.5*0.5*30.9*17.2 δp^2 = 395.0419

Standard deviation of the portfolio δp= 19.875%

b) Investing in U.S Treasury bills is zero-risk and therefore, the standard deviation of that asset is zero.

Variance of the portfolio δp^2 = δA^2xA^2 + δB^2xB^2+2ρ δAxA δBxB δp^2 = (1/3)^2 (30.9)^2 + (1/3)^2 (17.2)^2 +2*(1/3)*(1/3)*(0.31)*(30.9)*(17.2) δp^2 = 175.574

Standard deviation of the portfolio δp = 13.25%

c) Margin of 50 %, twice the money of investor’s portfolio. Therefore, the risk in this case is twice that of the original risk. From question a, we derived S.D. of original portfolio as 19.875%

Standard deviation of this portfolio = 2 * 19.875 % = 39.75 %

Standard deviation of the portfolio δp = 39.75 %

c) The standard deviation of the portfolio is based on the average securities portfolio covariance and standard market investment.

Therefore, for 100 stocks like Dell with β = 1.41, the portfolio…