First set your calculator into 5 decimal places
2nd….format
Question 1
State of Probability of Stock A Stock B Stock C
Economy State of
Economy
Boom 0.25 0.25 0.25 0.45
Good 0.25 0.13 0.09 0.23
Poor 0.05 0.08 0.04 0.18
Bust 0.45 -0.02 -0.09 0.03
Invest : 14% in Stock A and C, 72% in Stock B
Feedback:
This portfolio does not have an equal weight in each asset. We first 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:
Boom: E(Rp) = 0.14(0.25) + 0.72(0.25) + 0.14(0.45)
E(Rp) = 0.278 or 27.8%
Good: E(Rp) = 0.14(0.13) + 0.72(0.09) + 0.14(0.23)
E(Rp) = 0.1152 or 11.52%
Poor: E(Rp) = 0.14(0.08) + 0.72(0.04) + 0.14(0.18)
E(Rp) = 0.0652 or 6.52%
Bust: E(Rp) = 0.14(-0.02) + 0.72(-0.09) + 0.14(0.03)
E(Rp) = -0.0634 or -6.34%
And the expected return of the portfolio is:
E(Rp) = 0.25(0.278) + 0.25(0.1152) + 0.05(0.0652) + 0.45(-0.0634)
E(Rp) = 0.07303 or 7.3%
Feedback:
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 sum. The result is the variance. So, the variance and standard
deviation of the portfolio is:
σp2 = 0.25(0.278 – 0.07303)2 + 0.25(0.1152 – 0.07303)2 + 0.05(0.0652 – 0.07303)2 +
0.45(-0.0634 – 0.07303)2
σp2 = 0.01933
σp = (0.01933)1/2
σP = 0.139 or 13.9%
Question 2:
Reward-to-Risk Ratios
Stock Y has a beta of 1.5 and an expected return of 14 percent. Stock Z has a beta of
0.9 and an expected return of 11 percent. If the risk-free rate is 5.5 percent and the
market risk premium is 6.3 percent, are these stocks correctly priced? (Input answers
as a percent rounded to 2 decimal places, without the percent sign.)
The reward-to-risk ratios for stocks Y and Z are 5.67 percent and 6.11 percent,
respectively.
Since the SML reward-to-risk is 6.30 percent, Stock Y is overvalued and Stock Z is
overvalued.
Feedback:
There are two ways to correctly answer this question. We will work through both. First, we
can use the CAPM. Substituting in the value we are given for each stock, we find:
E(RY)= risk-free rate + SML reward-to-risk * beta of Stock Y
E(RY) = 0.055 + 0.063(1.5)
E(RY) = 0.1495 or 14.95%
It is given in the problem that the expected return of Stock Y is 14 percent, but according to
the CAPM, the return of the stock based on its level of risk, the expected return should be
14.95 percent. This means the stock return is too low, given its level of risk. Stock Y plots
below the SML and is overvalued. In other words, its price must decrease to increase the
expected return to 14.95 percent. For Stock Z, we find:
E(RZ)= risk-free rate + SML reward-to-risk * beta of Stock Z
E(RZ) = 0.055 + 0.063(0.9)
E(RZ) = 0.1117 or 11.17%
The return given for Stock Z is 11 percent, but according to the CAPM the expected return
of the stock should be 11.17 percent based on its level of risk. Stock Z plots above the
SML and is overvalued. In other words, its price must decrease to increase the expected
return to 11.17 percent.
We can also answer this question using the reward-to-risk ratio. All assets must have the
same reward-to-risk ratio. The reward-to-risk ratio is the risk premium of the asset divided
by its β. We are given the market risk premium, and we know the β of the market is one, so
the reward-to-risk ratio for the market is 0.063, or 6.3 percent. Calculating the
reward-to-risk ratio for Stock Y, we find:
Reward-to-risk ratio Y = (expected return - risk-free rate) / beta of Stock Y
Reward-to-risk ratio Y = (0.14 – 0.055) / 1.5
Reward-to-risk ratio Y = 5.67
The reward-to-risk ratio for Stock Y is too low, which means the stock plots below the SML,
and the stock is overvalued. Its price must decrease until its reward-to-risk ratio is equal to
the market reward-to-risk ratio. For Stock Z, we find:
Reward-to-risk ratio Z = (expected return - risk-free rate) / beta of Stock Z
Reward-to-risk ratio Z = (0.11 – 0.055) / 0.9
Reward-to-risk ratio Z = 6.11
The reward-to-risk ratio for Stock Z is too high, which means the stock plots above the
SML, and the stock is overvalued. Its price must decrease until its reward-to-risk ratio is
equal to the market reward-to-risk ratio.
the SML reward-to-risk = the market risk premium
Question 3:
Portfolio Expected Return
You have $200,000 to invest in a stock portfolio. Your choices are Stock H with an expected return
of 13 percent and Stock L with an expected return of 8 percent. If your goal is to create a portfolio
with an expected return of 10.6 percent, you will invest $104000 (50%) in Stock H and $96000 (50%)
in Stock L. (Round your answers to 2 decimal places. Omit the "$" sign in your
Feedback:
Here, we are given the expected return of the portfolio and the expected return of the
assets in the portfolio and are asked to calculate the dollar amount of each asset in the
portfolio. So, we need to find the weight of each asset in the portfolio. Since we know
the total weight of the assets in the portfolio must equal 1 (or 100%), we can find the
weight of each asset as:
E[Rp] = 0.106 = 0.13wH + 0.08(1 - wH)
wH = 0.52
wL = 1 - wH
wL = 1 - 0.52
wL = 0.48
So, the dollar investment in each asset is the weight of the asset times the value of the
portfolio, so the dollar investment in each asset must be:
Investment in H = 0.52($200,000)
Investment in H = $104,000
Investment in L = 0.48($200,000)
Investment in L = $96,000
Question 4:
Analyzing a Portfolio
You have $100,000 to invest in either Stock D, Stock F, or a risk-free asset. You must invest all of
your money. Your goal is to create a portfolio that has an expected return of 12.1 percent. If D has
an expected return of 15.9 percent, F has an expected return of 10.7 percent, and the risk-free rate
is 5.8 percent, and if you invest $47,900 in Stock D, you will invest $29838.78 (100%) in Stock F.
(Round your answer to 2 decimal places. Omit the "$" sign in your respon
Feedback:
We know the expected return of the portfolio and of each asset, but only one portfolio
weight. We need to recognize that the weight of the risk-free asset is one minus the
weight of the other two assets. Mathematically, the expected return of the portfolio is:
E[Rp] = 0.121 = 0.479(0.159) + wF(0.107) + (1 - 0.479 - wF)(0.058)
0.121 = 0.479(0.159) + wF(0.107) + 0.058 -0.027782 - 0.058 wF
wF = 0.2984
So, the weight of the risk-free asset is:
wRf = 1 - 0.479 - 0.2984
wRf = 0.2226
And the amount of Stock F to buy is:
Amount of stock F to buy = 0.2984(100,000)
Amount of stock F to buy = $29,838.78
Question 5:
Portfolio Returns and Deviations
Consider the following information on a portfolio of three stocks:
Probability Stock A Stock B Stock C
State of of State of Rate of Rate of Rate of
Economy Economy Return Return Return
Boom 0.35 0.45 0.15 0.35
Normal 0.60 0.29 0.07 0.03
Bust 0.05 0.21 -0.22 -0.48
Required:
If your portfolio is invested 20 percent each in A and B and 60 percent in C, the
portfolio expected return is 15.50 (33%) percent. The variance is
0.02316 (33%) and standard deviation is 15.22 (33%) percent. (Round the
variance to 5 decimal places and input the other answers as a percent
rounded to 2 decimal places, without the percent sign. Negative amount
should be indicated with a minus sign.)
Similar to Question 1
Feedback:
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:
Boom: E(Rp) = 0.2(0.45) + 0.2(0.15) + 0.6(0.35)
E(Rp) = 0.33 or 33%
Normal: E(Rp) = 0.2(0.29) + 0.2(0.07) + 0.6(0.03)
E(Rp) = 0.09 or 9%
Bust: E(Rp) = 0.2(0.21) + 0.2(-0.22) + 0.2(-0.48)
E(Rp) = -0.29 or -29%
And the expected return of the portfolio is:
E(Rp) = 0.35(0.33) + 0.60(0.09) + 0.05(-0.29)
E(Rp) = 0.155 or 15.5%
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 sum. The
result is the variance. So, the variance and standard deviation of the portfolio are:
2p = 0.35(0.33 – 0.155)2 + 0.60(0.09 – 0.155)2 + 0.05(-0.29 – 0.155)2
2p = 0.02316
p = (0.02316)1/2
p = 0.1522 or 15.22%
(b) The expected T-bill rate is 3.7 percent; the expected risk premium on the
portfolio is 11.80 (100%) percent. (Input your answer as a percent
rounded to 2 decimal places, without the percent sign.)
Feedback:
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:
RPi = E(Rp) – Rf
RPi = 0.155 (the expected return of the portfolio ) – 0.037(expected T-bill rate)
RPi = 0.118 or 11.8%