MBA 702 MODULE 4, PRACTICE PROBLEM SOLUTIONS 2025 Louisiana State University, Shreveport
MODULE 4, PRACTICE PROBLEM SOLUTIONS
Expected Stock
1 Return Given the following returns under various states of the economy, what is the expected return on this stock?
Probability of State Rate of Return if
State of the Economy Probability * rate
of the Economy State Occurs
of return
Boom 10% 16% 1.60% =.10 * .16 = .016 or 1.6%
Normal 60% 9% 5.40% =.60 * .09 = .054 or 5.4%
Recession 30% -15% -4.50% =.30 * (-0.15 ) =-0.045 or -4.5%
2.50%
Hint on calculations: The probabilities must be entered as decimals but in this case, the stock returns can be entered as decimals
or whole numbers -- just be consistent!
E(r) = (.10 * 16%) + (.60 * 9%) + (.3 * -15%) = 2.5%
This is the same as: E(r) = (.10 * 0.16) + (.60 * 0.09) + (.3 * -0.15) = 0.025 which is 2.5%
Expected portfolio
You own a portfolio that has $52,000 invested in Stock A and $8,500 invested in Stock B. The expected returns on these stocks are
2 return
13 percent and 6.5 percent, respectively. What is the expected return on the portfolio?
Step 1, what I the total amount invested? 52000+8500 = 60500. Which means 52000/60500 in A and 8500/60500 in B
E(r) = (52000/60500 * 13%) + (8500/60500 * 6.5%) = 12.09%
Expected portfolio 52000/60500 = 85.95% in Stock A and 8500/60500 = 14.05% in Stock B
3 return What is the expected return on this portfolio?
Number of $ value
Expected return Stock price
Stock shares invested % value invested in each
A 15% 270 $17 $4,590 32.81%
B 7% 500 $6 $3,000 21.44%
C 9% 200 $32 $6,400 45.75%
$13,990 100.00%
Step 1, what is the amount invested in each stock and the total for the portfolio?
A = 270 shares * $17/share = $4,590
B = 500 shares * $6/share = $3000
C = 200 shares * $32/share = $6,400
Total invested = 4590 + 3000 + 6400 = 13990
E(r) = (4590/13990* 15%) + (3000/13990 * 7%) + (6400/13990 * 9%) = 10.54%
You are considering two different stocks for your portfolio and are concerned because their standard deviations and returns are
so different from each other. You are risk averse and want to compare the risk and return on a relative basis. Given the following,
4 calculate the coefficient of variation (CV) of the two stocks.
Standard Deviation Expected or
CV Stock (%) mean return
Alpha 11% 5.8%
Beta 28% 13.5%
CV for Alpha = 11%/5.8% = 1.90
CV for Beta = 28%/13.5% = 2.07
Given the relative coefficients of variation, personallyy, I would prefer Alpha, which has the lower risk to reward ratio.
return, The risk-free rate of return is 2.8 percent and the market risk premium is 7.1 percent. What is the required rate of return on a
5 required
CAPM stock with a beta of 0.98?
R = Rf + (beta * MRP) This is the same as R = Rf + (beta * (market return - Rf))
Rf = 2.80%
MPR = 7.10%
beta 0.98
, required return = 9.76% = 2.80% + (.98 * 7.10%)
return, The risk-free rate of return is 3.7 percent and the overall market return is 14.5%. What is the required rate of return on a stock
6 required
CAPM
with a beta of 1.3?
R = Rf + (beta * (market return - Rf)) This is the same as R = Rf + (beta * MRP)
Rf = 3.70%
Rm (market return) 14.50% Recall that the market risk premium (MRP) = Rm - Rf
beta 1.3
R= 17.74% = 3.7% + (1.3*(14.50% - 3.7%))
The beta here is 1.3, that means that this particular "risky" asset is 1.3 times as volatile or reactive to systematic (market-wide)
risk as the "average risky asset". For stocks, we use the overall market, often defined as the S&P 500 as the "average risk". By
definition, the beta of the overall market, or the "average risky asset" = 1.0
The risk-free rate is 3%. The market is expected to earn 11%. The firm’s stock has a beta of 1.4 and is
expected to earn 15%. S
6b
buy this stock?
We know from the lecture notes that we need to compare the "expected" return to the "required return".
Step 1 - required return: R = Rf + (beta * (market return - Rf)) Note that we were given the return on the market, not MRP he
R = 3 + (1.4 * (11-3)) = 3 + (1.4 * 8) = 14.20%
Step 2 - compare required and expected returns: required = 14.20% and expected is GREATER THAN THAT, at 15%
Buy the stock, because expected is >= required (an expected return of 14.20 would also mean we would buy the stock)
If the expected return had been less than 14.20%, we would NOT buy the stock
7 portfolio beta What is the beta of the following portfolio
Stock Amount invested Beta % of portfolio % of portfolio * beta
R $73,500 1.56 =73,500/176,500 0.42 = 0.42 * 1.56 = 0.65
S $46,000 1.15 =46,000/176,500 0.26 = 0.26 * 1.15 = 0.3
T $57,000 0.65 =57,000/176,500 0.32 =0.32 * 0.65 = 0.21
Total amount invested: $176,500 Add the weighted beta factors: 1.16
a) What is the total amount invested? 73500 + 46,000 + 57000 = $176,500
b) Portfolio beta = (73500/176500 * 1.56) + (46000/176500 * 1.15) + (57000/176500 * 0.65) = 1.16
8 Portfolio return What is the expected return on a portfolio if the weight in Stock A is 70% and Stock B is 30%?
Rate of Return if Rate of Return if
State Occurs State Occurs
Probability of
State of the Economy State of the Stock A Stock B
Economy
Boom 20% 13% 15%
Normal 80% 7% 9%
Stock A return: (.2 * 13%) + (.8 * 7%) = 8.2%
Stock B return: (.2 * 15%) + (.8 * 9%) = 10.2%
Now that we have the expected returns of each stock, we weight the returns by the stock holdings.
Portfolio return = (.7 * 8.2%) + (.3 * 10.2%) = 8.8%
Note that above, we have been looking at "expected" returns. That is based on no "unexpected" information, which would result
in "unexpected" returns. Over time, all firms have unexpected returns. Do we expect Elon Musk to go on Twitter and get in trouble
with the SEC? No, of course not. That unexpected news caused the stock price to drop. The point with unexpected (and
unpredictable) returns, is that over time, the unexpected increases and decreases in stock price over the short term due to
"unexpected news" cancel out in the long term.
MODULE 4, PRACTICE PROBLEM SOLUTIONS
Expected Stock
1 Return Given the following returns under various states of the economy, what is the expected return on this stock?
Probability of State Rate of Return if
State of the Economy Probability * rate
of the Economy State Occurs
of return
Boom 10% 16% 1.60% =.10 * .16 = .016 or 1.6%
Normal 60% 9% 5.40% =.60 * .09 = .054 or 5.4%
Recession 30% -15% -4.50% =.30 * (-0.15 ) =-0.045 or -4.5%
2.50%
Hint on calculations: The probabilities must be entered as decimals but in this case, the stock returns can be entered as decimals
or whole numbers -- just be consistent!
E(r) = (.10 * 16%) + (.60 * 9%) + (.3 * -15%) = 2.5%
This is the same as: E(r) = (.10 * 0.16) + (.60 * 0.09) + (.3 * -0.15) = 0.025 which is 2.5%
Expected portfolio
You own a portfolio that has $52,000 invested in Stock A and $8,500 invested in Stock B. The expected returns on these stocks are
2 return
13 percent and 6.5 percent, respectively. What is the expected return on the portfolio?
Step 1, what I the total amount invested? 52000+8500 = 60500. Which means 52000/60500 in A and 8500/60500 in B
E(r) = (52000/60500 * 13%) + (8500/60500 * 6.5%) = 12.09%
Expected portfolio 52000/60500 = 85.95% in Stock A and 8500/60500 = 14.05% in Stock B
3 return What is the expected return on this portfolio?
Number of $ value
Expected return Stock price
Stock shares invested % value invested in each
A 15% 270 $17 $4,590 32.81%
B 7% 500 $6 $3,000 21.44%
C 9% 200 $32 $6,400 45.75%
$13,990 100.00%
Step 1, what is the amount invested in each stock and the total for the portfolio?
A = 270 shares * $17/share = $4,590
B = 500 shares * $6/share = $3000
C = 200 shares * $32/share = $6,400
Total invested = 4590 + 3000 + 6400 = 13990
E(r) = (4590/13990* 15%) + (3000/13990 * 7%) + (6400/13990 * 9%) = 10.54%
You are considering two different stocks for your portfolio and are concerned because their standard deviations and returns are
so different from each other. You are risk averse and want to compare the risk and return on a relative basis. Given the following,
4 calculate the coefficient of variation (CV) of the two stocks.
Standard Deviation Expected or
CV Stock (%) mean return
Alpha 11% 5.8%
Beta 28% 13.5%
CV for Alpha = 11%/5.8% = 1.90
CV for Beta = 28%/13.5% = 2.07
Given the relative coefficients of variation, personallyy, I would prefer Alpha, which has the lower risk to reward ratio.
return, The risk-free rate of return is 2.8 percent and the market risk premium is 7.1 percent. What is the required rate of return on a
5 required
CAPM stock with a beta of 0.98?
R = Rf + (beta * MRP) This is the same as R = Rf + (beta * (market return - Rf))
Rf = 2.80%
MPR = 7.10%
beta 0.98
, required return = 9.76% = 2.80% + (.98 * 7.10%)
return, The risk-free rate of return is 3.7 percent and the overall market return is 14.5%. What is the required rate of return on a stock
6 required
CAPM
with a beta of 1.3?
R = Rf + (beta * (market return - Rf)) This is the same as R = Rf + (beta * MRP)
Rf = 3.70%
Rm (market return) 14.50% Recall that the market risk premium (MRP) = Rm - Rf
beta 1.3
R= 17.74% = 3.7% + (1.3*(14.50% - 3.7%))
The beta here is 1.3, that means that this particular "risky" asset is 1.3 times as volatile or reactive to systematic (market-wide)
risk as the "average risky asset". For stocks, we use the overall market, often defined as the S&P 500 as the "average risk". By
definition, the beta of the overall market, or the "average risky asset" = 1.0
The risk-free rate is 3%. The market is expected to earn 11%. The firm’s stock has a beta of 1.4 and is
expected to earn 15%. S
6b
buy this stock?
We know from the lecture notes that we need to compare the "expected" return to the "required return".
Step 1 - required return: R = Rf + (beta * (market return - Rf)) Note that we were given the return on the market, not MRP he
R = 3 + (1.4 * (11-3)) = 3 + (1.4 * 8) = 14.20%
Step 2 - compare required and expected returns: required = 14.20% and expected is GREATER THAN THAT, at 15%
Buy the stock, because expected is >= required (an expected return of 14.20 would also mean we would buy the stock)
If the expected return had been less than 14.20%, we would NOT buy the stock
7 portfolio beta What is the beta of the following portfolio
Stock Amount invested Beta % of portfolio % of portfolio * beta
R $73,500 1.56 =73,500/176,500 0.42 = 0.42 * 1.56 = 0.65
S $46,000 1.15 =46,000/176,500 0.26 = 0.26 * 1.15 = 0.3
T $57,000 0.65 =57,000/176,500 0.32 =0.32 * 0.65 = 0.21
Total amount invested: $176,500 Add the weighted beta factors: 1.16
a) What is the total amount invested? 73500 + 46,000 + 57000 = $176,500
b) Portfolio beta = (73500/176500 * 1.56) + (46000/176500 * 1.15) + (57000/176500 * 0.65) = 1.16
8 Portfolio return What is the expected return on a portfolio if the weight in Stock A is 70% and Stock B is 30%?
Rate of Return if Rate of Return if
State Occurs State Occurs
Probability of
State of the Economy State of the Stock A Stock B
Economy
Boom 20% 13% 15%
Normal 80% 7% 9%
Stock A return: (.2 * 13%) + (.8 * 7%) = 8.2%
Stock B return: (.2 * 15%) + (.8 * 9%) = 10.2%
Now that we have the expected returns of each stock, we weight the returns by the stock holdings.
Portfolio return = (.7 * 8.2%) + (.3 * 10.2%) = 8.8%
Note that above, we have been looking at "expected" returns. That is based on no "unexpected" information, which would result
in "unexpected" returns. Over time, all firms have unexpected returns. Do we expect Elon Musk to go on Twitter and get in trouble
with the SEC? No, of course not. That unexpected news caused the stock price to drop. The point with unexpected (and
unpredictable) returns, is that over time, the unexpected increases and decreases in stock price over the short term due to
"unexpected news" cancel out in the long term.
