1. A share of stock sells for $38 today. The beta of the stock is 1, and the exp
ID: 2715828 • Letter: 1
Question
1. A share of stock sells for $38 today. The beta of the stock is 1, and the expected return on the market is 17 percent. The stock is expected to pay a dividend of $1.10 in one year. If the risk-free rate is 3.7 percent, what should the share price be in one year? (Round your answer to 2 decimal places. Omit the "$" sign in your response.)
Share price
$
2. A stock has a beta of 0.7 and an expected return of 9 percent. A risk-free asset currently earns 4 percent.
a.
What is the expected return on a portfolio that is equally invested in the two assets? (Round your answer to 2 decimal places. Omit the "%" sign in your response.)
Expected return
%
b.
If a portfolio of the two assets has a beta of 0.6, what are the portfolio weights?(Round your answers to 2 decimal places. Omit the "%" sign in your response.)
Weight
xS
%
xrf
%
c.
If a portfolio of the two assets has an expected return of 8 percent, what is its beta? (Round your answer to 2 decimal places.)
Beta
d.
If a portfolio of the two assets has a beta of 2.80, what are the portfolio weights? (Negative amounts should be indicated by a minus sign. Omit the "%" sign in your response.)
Weight
xS
%
xrf
%
3. Stock Y has a beta of 1.5 and an expected return of 12 percent. Stock Z has a beta of 0.8 and an expected return of 8 percent. What would the risk-free rate have to be for the two stocks to be correctly priced relative to each other? (Round your answer to 2 decimal places. Omit the "%" sign in your response.)
Risk-free rate
%
1. A share of stock sells for $38 today. The beta of the stock is 1, and the expected return on the market is 17 percent. The stock is expected to pay a dividend of $1.10 in one year. If the risk-free rate is 3.7 percent, what should the share price be in one year? (Round your answer to 2 decimal places. Omit the "$" sign in your response.)
Explanation / Answer
Answer (1)
Expected Price in one year = $ 43.43
Current Price = $ 38
Beta of the stock = 1
Expected return on the market rm = 17%
Risk-free rate rf = 3.7%
Expected Dividend in one year = $ 1.10
Expected return (Er) = rf + Beta * (rm – rf)
= 3.7% + 1 * (17% - 3.7%) = 3.7% + 1* 13.3% = 17%
Growth rate of dividends g = expected return – (Expected Dividend / Current Price)
Growth rate g = 0.17 – ($ 1.10/$38) = 0.17 – 0.028947 = 0.1410526 or 14.11% (rounded off)
Expected dividend in year 2 = $ 1.10 * (1+0.1411) = $ 1.10 * 1.1411 = $ 1.25521
Expected Price Next year = Dividend in year 2 / (Expected return – growth rate)
= $ 1.25521 / (0.17 – 0.1411) = $1.25521/0.0289 = $ 43.4328
Answer (2-a)
Expected return on a equally weighted portfolio = 6.50%
Answer (2-b)
Weight of Risk-free asset Xrf = 14.29%
Weight of stock XS = 85.71%
Answer (2-c)
Beta of stock = 1.12
Answer (2-d)
Xrf = -300%
XS = 400%
Answer (3)
Risk-free rate = 3.43%
working
Portfolio Beta = 2.80
Stock beta = 0.7
Let X be the weight of risk-free asset then (1-X) will be the weight of stock
Portfolio beta = weight * beta of risk-free asset + weight * beta of stock
2.80 = X * 0 + (1-X) * 0.7
2.80 = 0.7 – 0.7 *X
-0.7 * X = 2.80 – 0.7
-0.7 * X = 2.10
X = -2.10/0.7 = -3
Weight risk-free asset = -300%
Weight of stock = 100-(-300%) = 400%
Risk-free return = 4%
Stock Beta = 0.7
Stock expected return = 9%
Expected return on a equally weighted portfolio of stock and risk-free asset
Weight of stock= weight of risk-free asset = 0.50
Expected return on portfolio
E(P) = weight of stock * return on stock + weight of risk free asset * risk free rate
E(P) = 0.50 * 9% + 0.5 * 4%
E(P) = 4.5% + 2% = 6.5%
Portfolio Beta = 0.6
Weight of risk-free asset = x
Weight of stock = (1-x)
Portfolio Beta = weight of risk-free asset * beta of risk-free asset + weight of stock * beta of stock
0.60 = x * 0 + (1-x) *0.7
0.60 = 0.7 – 0.7 x
- 0.7*x = 0.60 – 0.70
- 0.7 * x = -0.10
X = 0.10 / 0.70 = 0.14285 or 14.29%
Weight of risk-free asset = 14.29%
Weight of stock = 100-14.29% = 85.71%
Expected return on market based on risk-free asset and stock
Expected return = rf + beta * (rm-rf)
9% = 4% + 0.7 * (rm – 4%)
9% = 4% + 0.7 * rm – 2.8%
9% = 1.2% +0.7*rm
0.7 *rm = 9% - 1.2% = 7.8%
rm = 7.8% / 0.7 = 11.14285% or 11.14% (rounded off)
Portfolio return = 8%
Let us assume that the portfolio is equally weighted between risk-free asset and the stock. Let r be the rate of return on stock
8% = 0.5 *4% + 0.5 * x%
8% - 2% = 0.5 * x%
X = 6% /0.5 = 12%
Expected return on stock = 12%
rm = 11.14% (taken from above)
rf = 4%
Expected return = rf + beta * (rm-rf)
12% = 4% + beta * (11.14% - 4%)
12% = 4% + 7.14% * Beta
7.14% * Beta = 12% - 4%
Beta = 8% /7.14% = 1.12044 or 1.12 (rounded off)
Stock Y
Beta = 1.5
Expected return = 12%
Stock Z
Beta = 0.8
Expected return = 8%
Let X be Risk-free rate to be for stocks to be priced correctly relative to each other
Expected return of Stock Y = rf + 1.5 * (rm-rf)
0.12 = rf +1.5 * (rm-rf)
0.12 = rf + 1.5*rm – 1.5*rf
0.12 = 1.5*rm -0.5*rf --- Equation (1)
Similarly Expected return of stock Z = rf + 0.8 * (rm – rf)
0.08 = rf +0.8 * rm – 0.8 * rf
0.08 = 0.8 *rm + 0.2 * rf
0.8*rm = 0.08 – 0.2*rf
rm = (0.08-0.2*rf)/0.8
Substituting value of rm in equation (1)
0.12 = 1.5 * (0.08-0.2*rf)/0.8 - 0.5 * rf
0.12 = 1.875 *(0.08 – 0.2*rf) – 0.5*rf
0.12 = 0.15 – 0.375 * rf – 0.5*rf
0.12 = 0.15 – 0.875*rf
0.12-0.15 = -0.875*rf
-0.03 = -0.875*rf
Rf = 0.03/0.875 = 0.034285 or 3.43%
rm = (0.08 – 0.2*0.0343)/0.8 = (0.08-0.00686)/0.8 = 0.091425 or 9.14% (rounded off)
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