20. For a portfolio that is equally invested in large-company stocks and long-term bonds:
21. We know that the reward-to-risk ratios for all assets must be equal (See Question 19). This can be
expressed as:
[E(RA) – Rf] / A = [E(RB) – Rf] / ßB
If the reward-to-risk ratios are the same, the ratio of the betas of the assets is equal to the ratio of the
risk premiums of the assets.
22. a. We need to find the return of the portfolio in each state of the economy. To do this, we will
multiply the return of each asset by its portfolio weight and then sum the products to get the
portfolio return in each state of the economy. Doing so, we get:
And the expected return of the portfolio is:
To calculate the standard deviation, we first need to calculate the variance. To find the variance,
we find the squared deviations from the expected return. We then multiply each possible
squared deviation by its probability, then add all of these up. The result is the variance. So, the
b. The risk premium is the return of a risky asset minus the risk-free rate. T–bills are often used as
the risk-free rate, so:
c. The approximate expected real return is the expected nominal return minus the inflation rate,
so:
To find the exact real return, we will use the Fisher equation. Doing so, we get:
1 + E(Ri) = (1 + h)[1 + e(ri)]
The approximate real risk-free rate is:
And using the Fisher effect for the exact real risk-free rate, we find:
1 + E(Ri) = (1 + h)[1 + e(ri)]
The approximate real risk premium is the approximate expected real return minus the risk-free
rate, so:
23. We know the total portfolio value and the investment of two stocks in the portfolio, so we can find
the weight of these two stocks. The weights of Stock A and Stock B are:
Since the portfolio is as risky as the market, the of the portfolio must be equal to one. We also
know the of the risk-free asset is zero. We can use the equation for the of a portfolio to find the
weight of the third stock. Doing so, we find:
So, the dollar investment in Stock C must be:
We also know the total portfolio weight must be one, so the weight of the risk-free asset must be one
minus the asset weight we know, or:
So, the dollar investment in the risk-free asset must be:
24. We are given the expected return of the assets in the portfolio. We also know the sum of the weights
of each asset must be equal to one. Using this relationship, we can express the expected return of the
portfolio as:
And the weight of Stock Y is:
The amount to invest in Stock Y is:
Investment in Stock Y = –.48571($100,000)
Investment in Stock Y = –$48,571.43
A negative portfolio weight means that you short sell the stock. If you are not familiar with short
25. The expected return of an asset is the sum of the probability of each state occurring times the rate of
return if that state occurs. So, the expected return of each stock is:
To calculate the standard deviation, we first need to calculate the variance. To find the variance, we
find the squared deviations from the expected return. We then multiply each possible squared
deviation by its probability, and then add all of these up. The result is the variance. So, the variance
and standard deviation of Stock A are:
2
And the standard deviation of Stock B is:
2
2
26. The expected return of an asset is the sum of the probability of each state occurring times the rate of
return if that state occurs. So, the expected return of each stock is:
To calculate the standard deviation, we first need to calculate the variance. To find the variance, we
find the squared deviations from the expected return. We then multiply each possible squared
deviation by its probability, and then add all of these up. The result is the variance. So, the variance
and standard deviation of Stock J are:
2
And the standard deviation of Stock K is:
2
To find the covariance, we multiply each possible state times the product of each asset’s deviation
from the mean in that state. The sum of these products is the covariance. So, the covariance is:
27. a. The expected return of the portfolio is the sum of the weight of each asset times the expected
return of each asset, so:
b. The variance of a portfolio of two assets can be expressed as:
2
2
2
2
2
P
2
28. a. The expected return of the portfolio is the sum of the weight of each asset times the expected
return of each asset, so:
The variance of a portfolio of two assets can be expressed as:
P
2
= X
A
2
A
2
+ X
B
2
B
2
+ 2XAXBABA,B
P
P
2
2
So, the standard deviation is:
b.
P
2
= X
A
2
A
2
+ X
B
2
B
2
+ 2XAXBABA,B
P
P
2
2
c. As Stock A and Stock B become less correlated, or more negatively correlated, the standard
29. a. (i) Using the equation to calculate beta, we find:
P
P
2
(ii) Using the equation to calculate beta, we find:
(iii) Using the equation to calculate beta, we find:
(iv) The market has a correlation of 1 with itself.
(v) The beta of the market is 1.
b. Using the CAPM to find the expected return of the stock, we find:
Firm A:
According to the CAPM, the expected return on Firm A’s stock should be 10.95 percent.
However, the expected return on Firm A’s stock given in the table is only 10 percent. Therefore,
Firm A’s stock is overpriced, and you should sell it.
Firm B:
According to the CAPM, the expected return on Firm B’s stock should be 14.80 percent.
However, the expected return on Firm B’s stock given in the table is 14 percent. Therefore,
Firm B’s stock is overpriced, and you should sell it.
Firm C:
According to the CAPM, the expected return on Firm C’s stock should be 12.96 percent.
30. Because a well-diversified portfolio has no unsystematic risk, this portfolio should lie on the Capital
Market Line (CML). The slope of the CML equals:
a. The expected return on the portfolio equals:
b. The expected return on the portfolio equals: