FEDERICO VILLARREAL NATIONAL UNIVERSITY

FACULTY OF ECONOMIC SCIENCES

“YEAR OF THE FIGHT AGAINST CORRUPTION AND IMPUNITY”

COURSE: BUSINESS FINANCE II

PROFESSOR: ECONOMICS, JAIME BELLEZA

TOPIC: RESOLUTION OF CHAPTER 11 OF ROSS BUSINESS FINANCE

MEMBERS:

GARCIA VILLEGAS MARYCIELO

PINTO PRADO PIERRE

RAMOS ORTIZ SEBASTIAN

YEAR: 4TH

CLASSROOM: 47-C

SHIFT: NIGHT

1. Determining Portfolio Weights What are the weights of a portfolio that holds 95
shares of A selling for $53 each and 120 shares of B selling for $29 each?
SOLUTION:
Calculating the total value:
Total Value = 95(53) + 120(29) = 8515
Calculating portfolio asset weights:
WA =
95(53
) = 0.5913 = 59.13%,The weighting of Asset A in the portfolio is 59.13% 8515
120(29
WB = ) = 0.4087 = 40.87%,The weighting of Asset B in the portfolio is 40.87%

2. Expected Return of a Portfolio Suppose you own a portfolio that has $1,900
invested in stock A and $2,300 in stock B. If the expected returns of these stocks are
10% and 15%, respectively, what is the expected return of the portfolio?
SOLUTION:
The expected return of a portfolio is the sum of the weight of each asset by the expected
return of each asset. The total value of the portfolio is:
Total Value = 1900 + 2300 = 4200
So, the expected return of this portfolio is:
E(Rp) = (1900) 0.10 + (2300) 0.15 = 0.1274 = 12.74% 4200 4200
The expected return of the portfolio is 12.74%
3. Expected return of a portfolio You own a portfolio that has 40% invested in stock
X, 35% in stock Y, and 25% in stock Z. The expected returns for these three stocks
are 11%, 17% and 14%, respectively. What is the expected return of the portfolio?
SOLUTION:
The expected return of a portfolio is the sum of the weight of each asset by the expected
return of each asset. So, the expected return of the portfolio is:
E(Rp) = (0.40) (0.11) + (0.35) (0.17) + (0.25) (0.14) = 0.1385 = 13.85%
The expected return of the portfolio is 13.85%
4. Expected return of a portfolio You have $10,000 to invest in a stock portfolio.
Your options are stock X with an expected return of 16% and stock Y with an
expected return of 10%. If your goal is to create a portfolio with an expected return
of 12.9%, how much money will you invest in stock X? And what about Y shares?
SOLUTION:
Here we are given the expected return of the portfolio and the expected return of each asset
in the portfolio and we are asked to calculate the weight of each asset. We can use the
equation for the expected return of a portfolio to solve this problem. Since the total weight
of a portfolio must equal 1 (100%), the weight of stock Y must be one minus the weight of
stock X. Mathematically speaking, this means:
E(Rp) = 0.129 = 0.16 Wx + 0.10 (1 – Wx)
We can now solve this equation for the weight of X as:
0.129 = 0.16 Wx + 0.10 – 0.10 Wx
0.029 = 0.06 Wx
Wx = 0.4833

So the dollar amount invested in Stock X is the weight of Stock X multiplied by the total
value of the portfolio:
Investment in X = 0.4833 (10000) = 4833.33
And the dollar amount invested in Stock Y is:
Investment in Y = (1 - 0.4833) (10000) = 5166.67
5. Calculation of expected return Based on the following information, calculate the
expected return:
Probability of the State of the Rate of return if such a state
State of the Economy Economy occurs
Recession 0.20 -0.09
Normal 0.50 0.11
Boom 0.30 0.23
SOLUTION:
The expected return of a portfolio is the sum of the probability of each return occurring
multiplied by the probability of that return occurring. So, the expected return of the portfolio
is:
E(Rp) = (0.20) (-0.09) + (0.50) (0.11) + (0.30) (0.23) = 0.1060 = 10.60%
The expected return of the portfolio is 10.60%
6. Calculating returns and standard deviations Based on the following information,
calculate the expected return and standard deviation of two stocks:
Rate of return if such a state occurs
Probability of the state of
State of the economy the economy Action A Action B
Recession 0.15 0.06 -0.20
Normal 0.65 0.07 0.13
Boom 0.20 0.11 0.33
SOLUTION:

The expected return of an asset is the sum of the probability of each return occurring
multiplied by the probability of that return occurring. So the expected return on each asset
is:
E(RA) = 0.15(0.06) + 0.65(0.07) + 0.20(0.11) = 0.0765 = 7.65%
E(RB) = 0.15(–0.20) + 0.65(0.13) + 0.20(0.33) = 0.1205 = 12.05%
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 this all up. The result is the
variance. So the variance and standard deviation of each stock are:
σ2A
= 0.15(0.06 – 0.0765)2 + 0.65(0.07 – 0.0765)2 + 0.20(0.11 – 0.0765)2 = 0.00029

σA = √0.00029 = 0.0171 = 1.71%

σ2B 2 2 2
= 0.15(–0.20 – 0.1205) + 0.65(0.13 – 0.1205) + 0.20(0.33 – 0.1205) = 0.02424

σB = √0.02424 = 0.1557 = 15.57%

7. Calculating returns and standard deviations From the following information,
calculate the expected return and standard deviation:
Probability of the state of the Rate of return if such a state
State of the economy
economy occurs
Depression 0.10 -0.045
Recession 0.25 0.044
Normal 0.45 0.120
Boom 0.20 0.207
SOLUTION:

The expected return of an asset is the sum of the probability of each return occurring
multiplied by the probability of that return occurring. So, the expected return on the stock is:
E(RA) = 0.10(–0.045) + 0.25 (0.044) + 0.45(0.12) + 0.20(0.207) = .1019 = 10.19%
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 this all up. The result is the
variance. So, the variance and standard deviation are:

σ2= 0.10(–0.045 – 0.1019)2 + 0.25(0.044 – 0.1019)2 + 0.45(0.12 – 0.1019)2 + 0.20(0.207 –

0.1019)2 = 0.00535

σ = √0.00535 = 0.0732 = 7.32%

8. Calculation of expected returns A portfolio consists of 15% G shares, 65% J
shares, and 20% K shares. The expected returns on these stocks are 8%, 15% and
24%, respectively. What is the expected return of the portfolio? How do you
interpret his answer?
SOLUTION:
The expected return of a portfolio is the sum of the weight of each asset by the expected
return of each asset. So, the expected return of the portfolio is:
E(Rp) = 0.15(0.08) + 0.65(0.15) + 0.20(0.24) = 0.1575 = 15.75%
If we own this portfolio, we would expect to earn a return of 15.75 percent.
 9. Yields and standard deviations Consider the following information:
State of the Probability of the Rate of return if such a state occurs
economy state of the
economy Action A Action B Action C
Boom 0.80 0.07 0.15 0.33
Crisis 0.20 0.13 0.03 -0,06
a) What is the expected return of a portfolio consisting of these three stocks with equal
weights?
b) What is the variance of a portfolio distributed as follows: 20% in A, 20% in B and
60% in C?
SOLUTION:
a) To find the expected return of the portfolio, we need to find the return of the
portfolio in each state of the economy. This portfolio is a special case since all three
assets have the same weight. To find the expected return on an equally weighted
portfolio, we can add up the returns on each asset and divide by the number of

(0.07 + 0.15 + 0.33)

assets, so the expected return on the portfolio in each state of the economy is:

Boom: E(Rp) =
(0.13 + 0.03 0.06) = 0.1833 = 18.33%
Crisis: E(Rp) = = 0.0333 = 3.33%

To find the expected return of the portfolio, we multiply the return in each state of the
economy by the probability of that state occurring, and then add. By doing this, we find:
E(Rp) = 0.80(0.1833) + 0.20(0.0333) = 0.1533 = 15.33%
b) This portfolio does not have the same weight in each asset. We still need to find the
portfolio return in each state of the economy. To do this, we will multiply the return
of each asset by the weight of its portfolio and then add the products to obtain the
return of the portfolio in each state of the economy. By doing so, we get
Boom: E(Rp)= 0.20(0.07) +0.20(0.15) + 0.60(0.33) =0.2420 = 24.20%
Crisis: E(Rp) = 0.20(0.13) +0.20(0.03) + 0.60(0.06) = –0.0040 = –0.40%
And the expected return of the portfolio is:
E(Rp) = 0.80(0.2420) + 0.20(0 .004) = 0.1928 = 19.28%
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 this all up. The
result is the variance. So, the portfolio variance is:

σ2 = 0.80(0.2420 – 0.1928)2 + 0.20(0.0040 – 0.1928)2 = 0.00968

10. Yields and standard deviations Consider the following information:
State of the Probability of the Rate of return if such a state occurs
 state of the
economy economy Action A Action B Action C
Boom 0.20 0.30 0.45 0.33
Well 0.35 0.12 0.10 0.15
Deficient 0.30 0.01 -0.15 -0.05
Crisis 0.15 -0.06 -0.30 -0.09
a) Your portfolio is invested as follows: 30% in each of stocks A and C and
40% in B shares. What is the expected return of the portfolio?
c) What is the variance of this portfolio? What about the standard deviation?
SOLUTION:
This portfolio does not have the same weight in each asset. First we need to find the return
on the portfolio in each state of the economy. To do this, we will multiply the return of each
asset by the weight of its portfolio and then add the products to obtain the return of the
portfolio in each state of the economy. By doing so, we get:
Boom: E(Rp) = 0.30(0.30) + 0.40(0.45) + 0.30(0.33) = 0.3690 = 36.90%
Good: E(Rp) = 0.30(0.12) + 0.40(0.10) + 0.30(0.15) = 0.1210 = 12.10%
Poor: E(Rp) = 0.30(0.01) + 0.40(–0.15) + 0.30(–0.05) = –0.0720 = –7.20%
Crisis: E(Rp) = 0.30(–0.06) + 0.40(–0.30) + 0.30(–0.09) = –0.1650 = –16.50%
And the expected return of the portfolio is:
E(Rp) = 0.20(0.3690) + 0.35(0.1210) + 0.30(–0.0720) + 0.15(–0.1650) = 0.0698 = 6.98%
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 this all up. The result is the
variance. So, the variance and standard deviation of the portfolio is:

σ2= 0.20(0.3690 – 0.0698)2 + 0.35(0.1210 – 0.0698)2 + 0.30(–0.0720 – 0.0698)2 + 0.15(–

0.1650 – 0.0698)2= 0.03312

σ = √0.03312 = 0.1820 = 18.20%

11. Calculating portfolio betas You have a stock portfolio distributed as follows: 25%
in Q shares, 20% in R shares, 15% in S shares, and 40% in T shares. The betas of
these four stocks are .75, 1.90, 1.38 and 1.16, in each case. What is the beta of the
portfolio?
SOLUTION:
The beta of a portfolio is the sum of the weight of each asset multiplied by the beta of each
asset. So, the beta of the portfolio is:
βP= 0.25(0.75) + 0.20(1.90) + 0.15(1.38) + 0.40(1.16) = 1.24
12. Calculating Portfolio Betas You have a portfolio equally distributed between one
 risk-free asset and two stocks. If one stock has a beta of 1.85 and the entire portfolio
has the same risk as the market, what should be the beta of the other stocks in your
portfolio?
SOLUTION:
The beta of a portfolio is the sum of the weight of each asset multiplied by the beta of each
asset. If the portfolio is as risky as the market, it should have the same beta as the market.
Since the market beta is one, we know that the beta of our portfolio is one. We must also
remember that the beta of the risk-free asset is zero. It has to be zero since the asset has no
risk. Setting up the equation for our portfolio's beta, we get:
βP=1 = 1(0) + 1(1.85) + 1(βx)
Solving the beta of Asset X, we get: βx = 1.15
13. Using the CAPM A stock has a beta of 1.25, the expected market return is 12%, and
the risk-free rate is 5%. What should be the expected return on this stock?
SOLUTION:
CAPM establishes the relationship between the risk of an asset and its expected return:
E(Ri) = Rf + [E(RM) – Rf] × βi
Substituting the values given to us, we find:
E(Ri) = 0.05 + (0.12 – 0.05) (1.25) = 0.1375 = 13.75%
14. Using the CAPM A stock has an expected return of 14.2%, the risk-free rate is 4%,
and the market risk premium is 7%. What should be the beta of this stock?
SOLUTION:
We are given the values for the CAPM, except for stock. We need to substitute these values
into the CAPM, and solve for the stock. One important thing to keep in mind is that we are
given the market risk premium. The market risk premium is the expected market return
minus the risk-free rate. We must be careful not to use this value as the expected market
return. Using the CAPM, we find:
E(Ri) = 0.142 = 0.04 + 0.07βi
βi = 1.46
15. Using the CAPM A stock has an expected return of 10.5%, its beta is .73, and the
risk-free rate is 5.5%. What should be the expected market return?
SOLUTION:
Here we need to find the expected return of the market using CAPM. Substituting the given
values and solving for the expected market return, we find:
E(Ri) = 0.105 = 0.055 + [E(RM) – 0.055] (0.73)
E(RM) = 0.1235 = 12.35%
 16. Using the CAPM A stock has an expected return of 16.2%, its beta is 1.75, and the
expected market return is 11%. What should the risk-free rate be?
SOLUTION:
Here we need to find the risk-free rate using CAPM. Substituting the given values and
solving for the risk-free rate, we find:
E(Ri) = 0.162 = Rf + (0.11 – Rf) (1.75)

0.162 = Rf +0 .1925 – 1.75Rf

Rf = 0.0407 = 4.07%
17. Using the CAPM A stock has a beta of .92 and an expected return of 10.3%. A risk-
free asset currently earns 5%.
a) What is the expected return of a portfolio that is equally distributed between the two
assets?
b) If a portfolio of the two assets has a beta of .50, what are the portfolio weights?
c) If a portfolio of the two assets has an expected return of 9%, what is its beta?
d) If a portfolio of the two assets has a beta of 1.84, what are the portfolio weights?
How do you interpret the weightings of the two assets in this case? Explain your
answer.
SOLUTION:
a) Again, we have a special case where the portfolio is equally weighted, so we can add
up the returns of each asset and divide by the number of assets. The expected return
of the portfolio is:
(0.103 +
0.05)
E(Rp) = = 0.0765 = 7.65%

2
b) We need to find the portfolio weights that result in a portfolio with a value of 0.50.
We know that the risk-free asset is zero. We also know that the weight of the risk-
free asset is one minus the weight of the stock, since the portfolio weights must add
up to one, or 100 percent. So:
βi = 0.50 = wS(0.92) + (1 – wS) (0)
0.50 = 0.92wSwS
= 0.5435

And, the weight of the risk-free asset is:

WRf = 1 – 0.5435 = 0.4565
c) We need to find the portfolio weights that result in a portfolio with an expected
return of 9 percent. We also know that the weight of the risk-free asset
 d) is one minus the weight of the stock, since the portfolio weights must add up to one,
or 100 percent. So:
E(Rp) = 0.09 = 0.103wS + .05(1 – wS)
0.09 = 0.103wS + 0.05 – 0.05wSwS
= 0.7547
Then calculating the β:
β = 0.7547(0.92) + (1 – 0.7547) (0) = 0.694
e) Solving the portfolio as we did in part a, we find:

1.8
βp = 1.84 = wS (0.92) + (1 – wS) (0)

4
0.9
WS =

2
WRf = 1 – 2 = –1
The portfolio is invested 200% in stocks and -100% in risk-free assets. This represents
borrowing at a risk-free rate to buy more shares.
18. Using the LMV Asset W has an expected return of 13.8% and a beta of 1.3. If the
risk-free rate is 5%, complete the following table for the portfolios of asset W and a
risk-free asset. Illustrate the relationship between the portfolio's expected return and
its beta by graphing the expected returns against the betas. What is the slope of the
resulting line?
Percentage of portfolio Expected portfolio
invested in asset W performance Portfolio Beta
0%
25%
50%
75%
100%
125%
150%
SOLUTION:
First, we need to find the wallet. The value of the risk-free asset is zero, and the weight of
the risk-free asset is one minus the weight of the stock, that of the portfolio is:
ßp = wW (1.3) + (1 – wW) (0) = 1.3wW
So, to find the portfolio for any stock weight, we simply multiply the stock weight by its β
Although we are solving for the expected return of a portfolio of a stock and the risk-free
asset for different portfolio weights, we are really solving for the SML. Any combination of
this action and the risk-free asset will fall into the SML. For that matter, a portfolio of any
stock and risk-free asset, or any stock portfolio, will fall into the SML. We know that the
slope of the SML line is the market risk premium, so using the CAPM and the information
about this stock, the market risk premium is:
E(RW) = 0.138 = 0.05 + MRP (1.30)
MRP = 0.088 = 0.0677 = 6.77%
1.3
So now we know that the CAPM equation for any stock is
E(Rp) = 0.05 + 0.0677βp
The slope of the SML is equal to the market risk premium, which is 0.0677. Using these
equations to complete the table, we obtain the following results:
Percentage of portfolio Expected portfolio
invested in asset W performance Portfolio Beta
0% 0.05 0
25% 0.072 0.325
50% 0.094 0.65
75% 0.116 0.975
100% 0.138 1.3
125% 0.16 1.625
150% 0.182 1.95

19. Risk premium ratios Action Y has a beta of 1.35 and an expected return of 14%.
The Z has a beta of .85 and an expected return of 11.5%. If the risk-free rate is 5.5%
and the market risk premium is 6.8%, are these stocks correctly valued?
SOLUTION:
There are two ways to correctly answer this question. We will work through both. First, we
can use the CAPM. By replacing the value given to us for each action, we find:
E(RY) = 0.055 + 0.068(1.35) = 0.1468 = 14.68%
The problem states that the expected return on stock Y is 14 percent, but according to the
CAPM, the stock's return based on its risk level, the expected return should be 14.68
percent. This means that the return on the stock is too low, given its level of risk. Stock Y is
below the SML
 and is overvalued. In other words, its price must decrease to increase the expected return to
14.68 percent.
For Stock Z, we find: E(RZ) = 0.055 + 0.068(0.85) = 0.1128 = 11.28%
The given return for Stock Z is 11.5 percent, but according to the CAPM, the expected
return of the stock should be 11.28 percent based on its risk level. Stock Z is plotting above
the SML and is undervalued. In other words, its price must increase to decrease the expected
return to 11.28 percent.
We can also answer this question using the reward/risk ratio. All assets should have the
same reward/risk ratio, that is, all assets should have the same ratio of the asset's risk
premium to its beta. The reward/risk ratio is the asset's risk premium divided by its. This is
also known as the Treynor ratio or Treynor index. We are given the market risk premium,
and we know that the market's risk premium is one, so the reward/risk ratio for the market is
0.068, or 6.8 percent. Calculating the reward/risk ratio for stock Y, we find:
(0.14 –
0.055)
Reward/risk ratio Y = 0.0630

1.35
The reward/risk ratio for stock Y is too low, meaning the stock is trading below the LME
and the stock is overvalued. Its price must decrease until its reward/risk ratio equals the
market reward/risk ratio. For Stock Z, we find:
(0.115 – 0.055)
0.85
Reward/risk ratio Z = =
The reward to risk ratio for stock Z is too high, 0.706
meaning the stock is trading above the LME and the stock is undervalued. Its price must
increase until its reward/risk ratio equals the market reward/risk ratio.
20. Risk Premium Ratios In the previous problem, what would the risk-free rate have to
be for both stocks to be correctly valued?
SOLUTION:
We need to set the reward/risk ratios of the two assets equal to each other
(0.14 – ^f) (0.115 – ^f)
1.35 0.85
We can cross multiply to get:
0.85(0.14 – Rf) = 1.35(0.115 – Rf)
Solving for the risk-free rate, we find:
0.119 – 0.85Rf = 0.15525 – 1.35Rf
Rf = .0725 = 7.25%
21. Portfolio Returns Based on the information in the previous chapter about the history
of capital markets, determine the return on a portfolio that is equally distributed
between large company stocks and long-term government bonds. What is the return
on a portfolio that is equally distributed between small company stocks and Treasury
bills?
SOLUTION:

(11.7% + 6.1%)
For a portfolio that is invested equally in large company stocks and long-term bonds:

2
Return = = 8.95%
For a portfolio invested equally in small stocks and Treasury bills:
Return = (16.4% + 3.8%)/2 = 10.10%
22. CAPM Apply the CAPM to show that the ratio of the risk premiums of two assets is
equal to the ratio of their betas.
SOLUTION:

[E(RA)-Rf ]_[E(RB) – ^f]

We know that the reward to risk ratios for all assets must be equal. This can be expressed as
PA BB
The numerator of each equation is the risk premium of the asset, so
 RPA _ RPBBA
=7 B

^B
We can rearrange this equation to obtain:
- Rp
BPA RPA
If the reward-to-risk ratios are the same, the ratio of asset betas is equal to the ratio of asset
risk premiums.
23. Portfolio Returns and Deviations Consider the following information about three
stocks:
Probability of the Rate of return if such a state occurs
State of the state of the
economy economy Action A Action B Action C
Boom 0.35 0.20 0.35 0.60
Normal 0.40 0.15 0.12 0.05
Crisis 0.25 0.10 -0.25 -0.50
a) If your portfolio invests 40% in A, 40% in B and 20% in C, what is the expected return of
the portfolio? What is the variance? What about the standard deviation?
b) If the expected rate on Treasury bonds is 3.80%, what is the expected risk premium
of the portfolio?
c) If the expected inflation rate is 3.50%, what is the approximate and exact real return
expected from the portfolio? What is the approximate and exact real risk premium
expected from the portfolio?
SOLUTION:
We need to find the return on the portfolio in each state of the economy. To do this, we will
multiply the return of each asset by the weight of its portfolio and then add the products to
obtain the return of the portfolio in each state of the economy. By doing so, we get:
Boom: E(Rp) = 0.4(0.20) + 0.4(0.35) + 0.2(0.60) = 0.3400 = 34.00%
Normal: E(Rp) = 0.4(0.15) + 0.4(0.12) + 0.2(0.05) = 0.1180 = 11.80%
Crisis: E(Rp) = 0.4(0.01) + 0.4(–0.25) + 0.2(–0.50) = –.01960 = –19.60%
And the expected return of the portfolio is:
E(Rp) = 0.35(0.34) + 0.40(0.118) + 0.25(–0.196) = 0.1172 =11.72%
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 this all up. The result is the
variance. So, the variance and standard deviation of the portfolio is:
σ2= 0.35(0.34 – 0.1172)2 + 0.40(0.118 – 0.1172)2 + 0.25(–0.196 – 0.1172)2
σ2= 0.04190
Calculating the Standard Deviation:
σ= √0.04190 = 0.2047 = 20.47%
The risk premium is the return on a risky asset, minus the risk-free rate. T-bills are often
used as the risk-free rate, so
RPi = E(Rp) – Rf = 0.1172 – 0.038 = 0.0792 = 7.92%
The approximate expected real return is the expected nominal return minus the inflation
rate, so:
Approximate expected actual return = 0.1172 – 0.035 = 0.0822 = 8.22%
To find the exact actual yield, we will use the Fisher equation. By doing so, we get:
1 + E(Ri) = (1 + h)[1 + e(ri)]
1.1172 = (1.0350) [1 + e(ri)]
e(ri) = (1.1172/1.035) – 1 = .0794 = 7.94%
The approximate real risk premium is the expected return minus the inflation rate, therefore:
Approximate expected actual risk premium = 0.0792 - 0.035 = 0.0442 = 4.42%
To find the exact expected real risk premium, we use the Fisher effect. In doing so, we find:
Exact expected real risk premium = (1.0792 / 1.035) - 1 = 0.0427 = 4.27%
24.Portfolio Analysis You want to create a portfolio that has the same risk as the market
and you have $1 million to invest. Given this information, fill in the remaining part
of the following table:
Asset Investment Beta
Active A 180 000 0.75
Active B 290 000 1.30
Active C 1.45
Risk Free Asset

SOLUTION:
We know the total value of the portfolio and the investment of two stocks in the portfolio, so
we can find the weight of these two stocks. The weights of stocks A and B are:
180,000
WA = 1,000,000 .

290,000
1,000,00
0
WB = 0.29

Since the portfolio is as risky as the market, the portfolio must equal one. We also know that
the risk-free asset is zero. We can use the portfolio equation to find the weight of the third
stock. In doing so, we find:
βp = 1.0 = wA(.75) + wB(1.30) + wC(1.45) + wRf(0)
wC = 0.33655172
Investment in Asset C = 0.33655172($1,000,000) = 336 551.72
We also know that the total weight of the portfolio must be one, so the weight of the risk-
free asset must be one minus the weight of the asset we know, or:
1 = wA + wB + wC + wRf
1 = 0.18 + 0.29 + 0.33655172 + wRf
wRf= 0.19344828
Investment in Risk Free Assets = 0.19344828 (1,000,000) = 193448.28
Completing the picture:
Asset Investment Beta
Active A 180 000 0.75
Active B 290 000 1.30
Active C 336551.72 1.45
Risk Free Asset 193448.28

25. Portfolio Analysis You have $100,000 to invest in a portfolio containing stock X,
stock Y, and a risk-free asset. You must invest all of your money. Their goal is to
create a portfolio that has an expected return of 10.7% and only 80% of the risk of
the broader market. If X has an expected return of 17.2% and a beta of 1.8, Y has an
expected return of 8.75% and a beta of .5, and the risk-free rate is 7%, how much
money should you invest in stock X? How do you interpret his response?
SOLUTION:
We are given the expected return y of a portfolio and the expected return y of the assets in
the portfolio. We know that the risk-free asset is zero. We also know that the sum of the
weights of each asset must equal one. So the weight of the risk-free asset is one minus the
weight of Stock X and the weight of Stock Y. Using this relationship, we can express the
expected return of the portfolio as:
E(Rp) = 0.1070 = wX (0.172) + wY (0.0875) + (1 – wX – wY) (0.055)
And the β of the portfolio is: βp = 0.8 = wX (1.8) + wY (0.50) + (1 – wX – wY) (0)
We have two equations and two unknowns. Solving these equations, we find that:
wX = –0.11111 wY= 2.00000 WRf = –0.88889

The amount to invest in Asset X is:

Investment in shares X = –0.11111 ($100,000) = - $11,111.11
A negative portfolio weight means you are shorting the stock. If you are not familiar with
short selling, it means that you borrow a stock today and sell it. You must then purchase the
shares at a later date to pay for the borrowed shares. If you short sell a stock, you make a
profit if the stock decreases in value. The negative weight on the risk-free asset means that
we borrow money to invest.
 26. Covariance and correlation. Based on the following information, calculate the
expected return and standard deviation of each of the following actions. Suppose
that each state of the economy has the same probability of occurring. What is the
covariance and correlation between the returns of the two stocks?

States of the economy

Performance of share A Performance of share B
Downward .082 -.065
Normal .095 .124
On the rise .063 185
SOLUTION:

E(Ra) = .33(.082) + .33(.095) + 0.33(.063) = 0.0800 or 8.00%

E(Rb) = .33(–.065) + .33(.124) + 0.33(.185) = 0.0813 or 8.13%
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 square deviation by its probability, and then add it all up. The result is the variance.
So, the variance and standard deviation of A are:

σ2=.33(.082 – .0800)2 + .33(.095 – .0800)2 + .33(.063 – .0800)2 = .00017

σ= (.00017)1/2= .0131 or 1.31%
And the standard deviation of B is:

2 = 33(-.065 - .0813)22+ .33(.124 - .0813)2+ .33(.185 - .0813)2 = .01133

σ=(.01133)1/2 = .1064 or 1064%
So the covariance is:
Cov(A, B) = .33(.092 – .0800)(–.065 – .0813) + .33(.095 – .0800)(.124 – .0813) + .33(.063
– .0800 )(.185 – .0813)
Cov(A, B) = –.000472
And the correlation is:
–.000472 / (.0131) (.1064) =–.3373

27. Covariance and correlation. From the following information, calculate the
expected return and standard deviation of each of the following actions.
What is the covariance and correlation between the returns of the two stocks?
State of the Probability of the J Share K Share
economy state of the Performance Performance
economy
Downward .30 -.020 .034

Normal .50 .138 .062

On the rise .20 .218 .092
SOLUTION:
E(Ra) = .30(–.020) + .50(.138) + .20(.218) = .1066 or 10.66%

σ^2= =.30(–.020 – .1066)2 + .50(.138 – .1066)2 + .20(.218 – .1066)2 = .00778

E(Rb) = .30(.034) + .50(.062) + .20(.092) = .0596 or 5.96%

σ^= = (.00778)1/2 = .0882 or 8.82%

o2 = .30(.034- .0596)2 + .50(.062 - .0596)2 + .20(.092 - .0596)2 = .00041

Standard deviation of B is:

σB= (.00041)
1/2
= 0202 or 2.02%
Cov(A, B) = .30(–.020 – .1066)(.034 – .0596) + .50(.138 – .1066)(.062 – .0596) + .20(.218
– .1066 )(.092 – .0596)
Cov(A, B) = .001732
The correlation is:
.001732 / (.0882) (.0202) = .9701
28. Portfolio standard deviation. The value F has an expected return of 10% and a
standard deviation of 26% per year. Value G has an expected return of 17% and a
standard deviation of 58% per year.
SOLUTION:
a) What is the expected return of a portfolio composed of 30% of value F and 70% of
value G?
E(Rp) = Wp E(Rf) + Wa E(Rg)
E(Rp) = .30(.10) + .70(.17)
E(Rp) = .1490 or 14.90%
b) If the correlation between the returns of security F and security G is .25, what is the
standard deviation of the portfolio described in part a)?

op= w
of + w G o2 + 2wFwG σFσgPf,G o2= . 302 (. 262) + . 702 (. 582 ) + 2(.30) (.70)
(.26) (.58) (.25) a2 .18675
Standard deviation: σ
p= (.18675) 1/2= .4322 or 43.22%
29. Standard deviation of a portfolio. Suppose the expected returns and standard
deviations of stocks A and B are E(Ra)= .13, E(Rb) = .19, σ^=.38, and σB=.62,
respectively.
SOLUTION
a) Calculate the expected return and standard deviation of a portfolio consisting of 45% A
and 55% B when the correlation between the returns of A and B is .5.
E(Rp) = .45(.13) + .55(.19)
E(Rp) = .1630 or 16.30%
The variance of a two-asset portfolio can be expressed as: o2=. 452 (. 382 ) + . 552 (. 622)

+ 2(.45) (.55) (.38) (.62) (.50) o2 = .20383 σ

p=(.20383 )1/2= .4515 or 45.15%

b) Calculate the standard deviation of a portfolio consisting of 40% A and 60% B when the
correlation coefficient between the returns of A and B is -5.
Solution:

o2 = . 402 (. 382 ) + . 602 (. 622) + 2(.40) (.60) (.38) (.62) (-.50) o2 = .08721

σ=(.08721 )1/2= .2953 or 29.53%

c) How does the correlation between the returns of A and B affect the standard deviation
of the portfolio?
As stock A and stock B become less correlated or more negatively correlated, the standard
deviation of the portfolio decreases.
30. Correlation and beta. The following data have been provided about the securities
of three companies, the market portfolio and the risk-free asset:
Worth Expected Standard Correlation Beta
performance deviation
Company A .10 .27 (Yo) .85
B Company .14 (ii) .50 1.50
Company C .17 .70 .35 (iii)
Market Portfolio .12 .20 (iv) (v)

.05 (vi) (vii) (viii)

Risk-free asset
SOLUTION:
(I) PA = (PA, M ) (σ^)/ σM
0.85=(PA, M )(0.27) /0.20
(PA, M) =0.63
(II) PB=( pB ,M ) (σ B )/ σ M
1.50= (.50)( σB)/ 0.20 σB=0.60
(III) P. C= ( pC,M ) (σ C )/ σ M
= (.35) (.70) / (0.20)
P. C = 1.23
(IV) The market has a correlation of 1 with itself.
(V) The market beta is 1.
(VI) The risk-free asset has a zero standard deviation
(VII) The risk-free asset has zero correlation with the market portfolio
(VIII) The beta of the risk-free asset is 0.

b) Are Company A's shares correctly valued according to the capital asset pricing model
(CAPM)? What could be said about the shares of company B? What about company C? If
these stocks are not valued correctly, what is your investment recommendation for someone
with a well-diversified portfolio?
•
E(Ra) = Rf + PA [E(Rm) — R f]
For A:

E(Ra) = 0.05 + 0.85(0.12 – 0.05)

E(Ra) = .1095 or 10.95%
According to the CAPM, the expected return on Company A's stock should be 10.95
percent.
However, the expected return on Company A's stock listed in the table is only 10 percent.
Therefore, Company A's shares are too expensive and must be sold.
• For B:
E(Rb) = R f+ BB [E(Rm) - R f]
E(Rb) = 0.05 + 1.5(0.12 – 0.05)
E(Rb) = .1550 or 15.50%
According to the CAPM, the expected return on Company B's stock should be 15.50
percent.
However, the expected return on Company B's stock in the table is 14 percent. Therefore,
Company B's stock is too expensive, and it must be sold.

• For C
E(Rc) = Rf+ Bc [E(Rm) - R f]
E(Rc) = 0.05 + 1.23(0.12 – 0.05)
E(Rc) = .1358 or 13.58%
According to the CAPM, the expected return on Company C's stock should be 13.58
percent.
However, the expected return on Company C's stock in the table is 17 percent. Therefore,
Firm C shares are not expensive, and you should buy them.
31. LMC. The market portfolio has an expected return of 12% and a standard deviation
of 19%. The risk-free rate is 5%
SOLUTION
Because a well-diversified portfolio has no unsystematic risk, this portfolio should fall
within the Capital Market Line (CML). The slope of the CML is equal to:
Slope in^eCM L = [E(Rm) - RF] / σM
Pending in^eCM^= (0.12 – 0.05) / 0.19
I have left in^eCM^=0.36842
a) What is the expected return of a well-diversified portfolio with a standard deviation of
7%?
Solution:
E(Rp) = RF +SlopeCML(ap)
E(Rp) = .05 + .36842(.07)
E(Rp) = .0758 or 7.58%
b) What is the standard deviation of a well-diversified portfolio with an expected return of
20%?
Solution:

E(Rp) = RF +SlopeCML(ap)
.20 = .05 + 36842(0p)
(op) = .4071 or 40.71%
32. Beta and CAPM. A portfolio that combines a risk-free asset with the market
portfolio has an expected return of 9% and a standard deviation of 13%. The risk-
free rate is 5% and the expected return of the market portfolio is 12%. Suppose the
capital asset pricing model remains the same. What expected rate of return would a
security earn if it had a correlation of ?45 with the market portfolio and a standard
deviation of 40%?
SOLUTION:
First, we can calculate the standard deviation of the market portfolio using the Capital
Market Line. (CML). We know that the risk-free rate asset has a return of 5 percent and a
standard deviation of zero and the portfolio has an expected return of 9 percent and a
standard deviation of 13 percent.
These two points must be on the Capital Market Line. The slope of the capital market line is
equal to:
SlopeCML = Increase in expected performance/
Increase in standard deviation
CML Slope = (.09 - .05) / (.13 - 0)
CML Slope = .31
According to the capital market line:

E(Ri) = RF + SlopeCML ((^j)

Since we know the expected return of the market portfolio, the risk-free rate, and the slope
of the Capital Market Line, we can solve for the standard deviation of the market portfolio
which is:
E(Rm) = RF + SlopeCML(am)
.12 = .05 + (.31) (σ
m)
(σ
m)=
(σ m)= .2275 or 22.75%
(.12 – .05) / .31

Pi=(pi, m) (^i)/ (0m)

B = (.45) (.40) / .2275

Pi= 0.79
E(Ri) = RF + B, [E(Rm) -RF]
E(Ri) = 0.05 + 0.79(.12 – 0.05)
E(Ri) = .1054 or 10.54%
33. Beta and CAPM. Suppose the risk-free rate is 4.8% and the market portfolio has an
expected return of 11.4% and a variance of .0429. Portfolio Z has a correlation
coefficient with the market of .39 and a variance of .1783. According to the capital
asset pricing model, what is the expected return on portfolio Z?
SOLUTION:
First, we need to find the standard deviation of the market and the portfolio, which are:
(σ M) = (.0429)1/2
(σ
M) = .2071 or 20.71%
(σ^)= (.1783)1/2
(σ^)= .4223 or 42.23%
We can now use the equation for beta to find the beta of the portfolio, which is:

(β^)=(p^,M)(σ^)/ (σM)

(β^) = (.39)(.4223) / .2071

(β^)= .80
Now, we can use the CAPM to find the expected return of the portfolio, which is:

E(Rz) = RF +(B2)[E(Rm) - RF]

E(Rz) = .048 + .80(.114 – .048)
E(Rz) = .1005 or 10.05%
34. Systematic risk and non-systematic risk. Consider the following information
about actions I and II:
 State of th Probability of Rate of return if such a state occurs
economy e state of the
Action 1 Action 2
economy
Recession .15 .09 -.30
Normal .55 .42 .12
Superabundance .30 .26 .44
irrational
The market risk premium is 7.5%, and the risk-free rate is 4%. Which stock has the highest
systematic risk? Which has the highest unsystematic risk? Which stock is more “risky”?
Explain your answer
SOLUTION:
The amount of systematic risk is measured by that of an asset. How do we know the market
risk?
premium and the risk-free rate, if we know the expected return of the asset, we can use the
CAPM to
solve for the asset. The expected return on action 1 is:
E(Ri) = .15(.09) + .55(.42) + .30(.26) = .3225 or 32.25%
.3225 = .04 + .075 (β ) 1

(β^) = 3.77
The total risk of the asset is measured by its standard deviation, so we must calculate the
Standard deviation of action 1. Starting with the calculation of the variance of the stock, we
find:
o2= 15(.09 - .3225)2 + .55(.42 - .3225)2 + .30(.26 - .3225)2
q2= .01451

or,= 014511/2= .1205 or 12.05%

Using the same procedure for action 2, we find that the expected return is:
E(Rii) = .15(–.30) + .55(.12) + .30(.44) = .1530
.1530 = .04 + .075(β2)
(β2) =1.51
And the standard deviation of stock 2 is:
σ22= 15(–.30 – .1530)2 + .55(.12 – .1530)2 + .30(.44 – .1530)2
σ 2= .05609
2

σ2 = (05609)1/2= .2368 or 23.68%

Although stock 2 has more total risk than I, it has much less systematic risk, since its beta is
much smaller than 1. Therefore, 1 has a more systematic risk and 2 has a more non-
systematic and more total risk. Since unsystematic risk can be diversified away, 1 is actually
the "most dangerous" stock despite the lack of volatility in its returns. Action 1 has a higher
risk premium and a higher expected return.
35. LMV. Suppose you observe the following situation:
Worth Beta Expected performance
Pete Corp. 1.4 .150
Repeat Co. .9 .115
Assume that these securities are correctly valued. Based on the CAPM, what is the expected
return of the market? What is the risk-free rate?
SOLUTION
Here we have the expected return and beta for two assets. We can express the returns of the
two assets using CAPM. If the CAPM is true, then the security market line also holds,
meaning all assets have the same risk premium. Setting the reward/risk ratio of the assets
equal to each other and solving for the risk-free rate, we find:
(.15 – Rf)/1.4 = (.115 – Rf)/.90
90(.15 – Rf) = 1.4(.115 – Rf)
.135 – .9Rf = .161 – 1.4Rf
.5Rf = .026
Rf= .052 or 5.20%
Now, using CAPM to find the expected return on the market with both stocks, we find:
.15 = .0520 + 1.4(Rm – .0520)
Rm = .1220 or 12.20%
.115 = .0520 + .9(Rm – .0520)
Rm = .1220 or 12.20%
36. Portfolio covariance and standard deviation. There are three values in the market.
The following table shows their possible returns:
State Probability of Value Value Value
occurrence performance 1 Performance Performance 3
1 .15 .25 2.25 .10
2 .35 .20 .15 .15
3 .35 .20 .15 .15
4 15 .10 .10 .25
SOLUTION
a) What is the expected return and standard deviation of each security?
The expected return of an asset is the sum of the probability of each return occurring
multiplied by the probability of that return occurring. To calculate the standard deviation,
we first need to calculate the variance. To find the variance, we find the squared deviations
from the expected. We then multiply each possible squared deviation by its probability, and
then add all of these up. The result is the variance. So the expected return and standard
deviation of each stock are:
•
Value 1
E(R1) = .15(.25) + .35(.20) + .35(.15) + .15(.10) = .1750 or 17.50%
σ12=15(.25 – .1750)2 + .35(.20 – .1750)2 + .35(.15 – .1750)2 + .15(.10
– .1750)2 = .00213
1/2
σ1=(.002139) = .0461 or 4.61%
•
Value 2
E(R2) = .15(.25) + .35(.15) + .35(.20) + .15(.10) = .1750 or 17.50%
σ22=15(.25 – .1750)2 + .35(.15 – .1750)2 + .35(.20 – .1750)2 + .15(.10
– .1750)2 = .00213
1/2
σ2=(.00213) = .0461 or 4.61%
•
Value 3
E(R3) = .15(.10) + .35(.15) + .35(.20) + .15(.25) = .1750 or 17.50%
σ32=15(.10 – .1750)2 + .35(.15 – .1750)2 + .35(.20 – .1750)2 + .15(.25
– .1750)2 = .00213
1/2
σ3=(.00213) = .0461 or 4.61%
b) What are the covariances and correlations between pairs of values?
• Active 1 and Active 2:
Cov(1,2) = .15(.25 – .1750)(.25 – .1750) + .35(.20 – .1750)(.15 – .1750) + .35(.15 – .1750)
(.20 – .1750) + .15(.10 – .1750)(.10 – .1750)
Cov(1,2) = .000125

p1,2= Cov(1,2)/ σ1σ2

p1,2= .000125 / (.0461) (.0461)
p1,2= 5882
• Active 1 and Active 3:
Cov(1,3) = .15(.25 – .1750)(.10 – .1750) + .35(.20 – .1750)(.15 – .1750) + .35(.15 – .1750)
(.20 – .1750) + .15(.10 – .1750)(.25 – .1750)
Cov(1,3) = –.002125

p1,3= Cov(1,3)/ σ1σ3

p1,3= –.002125 / (.0461) (.0461)
= –1
p1,3

• Active 2 and Active 3:

Cov(2,3) = .15(.25 – .1750)(.10 – .1750) + .35(.15 – .1750)(.15 – .1750) + .35(.20 – .1750)
(.20 – .1750) + .15(.10 – .1750)(.25 – .1750)
Cov(2,3) = –.000125

p2,3= Cov(2,3)/ σ2σ3

p2,3=–.000125 / (.0461)(.0461)
p2,3=–.5882
c) What is the expected return and standard deviation of a portfolio with half of the funds
invested in security 1 and half in security 2?
E(Rp) = w1E(R1) + w2E(R2)
E(Rp) = .50(.1750) + .50(.1750)
E(Rp) = .1750 or 17.50%
The variance of a two-asset portfolio can be expressed as:
o2= w 12σ12 +
w22 σ22 + 2w1w2 σ1σ2p1,2
o2= . 502 (. 04612) + . 502 (. 04612 ) + 2(.50) (.50) (.0461) (.0461) (.5882)
a2= .001688
And the standard deviation of the portfolio is: σ
p = (.001688)
1/2

σ
p= .0411 or 4.11%
d) What is the expected return and standard deviation of a portfolio with half of the funds
invested in security 1 and half in security 3?
E(Rp) = w1E(R1) + w3E(R3)
E(Rp) = .50(.1750) + .50(.1750)
E(Rp) = .1750 or 17.50%
The variance of a two-asset portfolio can be expressed as:
o2= . 502 (. 04612) + . 502 (. 04612 ) + 2(.50) (.50) (.0461) (.0461) (-1)
o2= .000000
Since the variance is zero, the standard deviation is also zero.
e) What is the expected return and standard deviation of a portfolio with half of the funds
invested in security 2 and half in security 3?
E(Rp) = .50(.1750) + .50(.1750)
E(Rp) = .1750 or 17.50%
The variance of a two-asset portfolio can be expressed as:
o2= . 502 (. 04612) + . 502 (. 04612 ) + 2(.50) (.50) (.0461) (.0461) (-.5882)
o2= 000438
And the standard deviation of the portfolio is:
σ
p = ( 000438)1/2
σ
p = .0209 or 2.09%
f) What do your answers to parts a), c), d), and e) imply about diversification?
As long as the correlation between the returns of two securities is less than 1, there is a
benefit to diversification. A portfolio of negatively correlated stocks can achieve greater risk
reduction than a portfolio of positively correlated stocks, while holding the expected return
of each stock constant. Applying appropriate weights on perfectly negatively correlated
stocks can reduce portfolio variance to 0.
37. LMV. Suppose you observe the following situation:

State of the Probability of the Action A Action B

economy state
Crisis .15 -.08 -.05
Normal 70 .13 .14
Boom .29 .48 .29
SOLUTION:
a) Calculate the expected return of each stock
E(Ra) = .15(–.08) + .70(.13) + .15(.48) = .1510 or 15.10%
E(Rb) = .15(–.05) + .70(.14) + .15(.29) = .1340 or 13.40%
b) Assuming the capital asset pricing model holds and the beta of Stock A is .25 greater
than the beta of Stock B, what is the expected market risk premium?
We can use the expected returns we calculated to find the slope of the Security Market Line.
We know that the beta of Stock A is .25 greater than the beta of Stock B. Therefore, as a
beta
increases by .25, the expected return of a security increases by .017 (= .1510 - .1340). the
slope of the security market line (SML) is equal to:

pend i en^eSML = Increase in expected return / Increase in BKPA

Slope i entSM^ = (.1510 – .1340) / .25

Pending SM^= 0680 or 6.80%

Since the market beta is 1 and the risk-free rate has a beta of zero, the slope of the Safety
Line is equal to the expected market risk premium. So the expected market risk premium
should be 6.8 percent.
38. Standard deviation and beta. There are two stocks on the market: stock A and stock
B. Today, the price of the first one is $75. Next year, this price of share A will be
$63 if the economy is in recession, $83 if the economy is normal, and $96 if the
economy is expanding. The probabilities of recession, normal times, and expansion
are .2, .6, and .2, respectively. Additionally, this stock does not pay dividends and
has a correlation of .8 with the market portfolio. Stock B has an expected return of
13%, a standard deviation of 34%, a correlation with the market portfolio of .25, and
a correlation with Stock A of .48. The market portfolio has a standard deviation of
18%. Suppose the CAPM holds.
SOLUTION
a) If you are a typical risk-averse investor with a well-diversified portfolio, which stock
would you prefer? Because?
A typical risk-averse investor seeks high returns and low risks. For a risk-averse investor
with a well-diversified portfolio, beta is the appropriate measure of an individual security's
risk. To evaluate the two stocks, we need to find the expected return and beta of each of the
two stocks.
Since Stock A does not pay dividends, the yield on Stock A is simply: (P1 - P0) / P0. So the
return for each state of the economy is:
C = ($63 – 75) / $75 = –.160 or –16.0%
RNorm^= ($83 – 75) / $75 = .107 or 10.7%

RAUG^
= ($96 – 75) / $75 = .280 or 28.0%
E(Ra) = .20(–.160) + .60(.107) + .20(.280) = .0880 or 8.80%
And the variance:
σ^2= .20 (. –0.160 – 0.0882) + .60 (.107 – .088)2 ) + .20 (.280 – .088)2
σ^2 = 0.0199
σ^= (0.0199)1/2
σ^ = .1410 or 14.10%
Now we can calculate the beta of the stock, which is:
(β^)=(p^,M)(σ^)/ (σM)
(β^)= (.80) (.1410) / .18
(β^)= .627
For stock B, we can directly calculate beta from the information provided. So, the beta for
stock B is:
(β b)=(pB,M)(σB)/ (σM)
(β b)= (.25) (.34) / .18
(β b)= .472
The expected return on stock B is greater than the expected return on stock A. The risk of
Stock B, measured by its beta, is lower than the risk of Stock A. Therefore, a typical risk-
averse investor with a well-diversified portfolio will prefer Stock B. Note that this situation
implies that at least one of the stocks is mispriced since the higher risk (beta) stock has a
lower return than the lower risk (beta) stock.
b) What is the expected return and standard deviation of a portfolio consisting of 70% A
shares and 30% B shares?
E(Rp) = .70(.088) + .30(.13)
E(Rp) = .1006 or 10.06%
To find the standard deviation of the portfolio, we first need to calculate the variance.
The variance of the portfolio is:
o2= 702 (. 1412) + (30)2 (. 34)2 + .2(70) (.30) (.141) (.34) (.48)
a2= .02981

σ
p = (. 02981)
1/2

σ
p = .1727 or 17.27%

β
c) What is the beta of the portfolio in part b)?
p = 70(.627) + .30(0.472)
β
p = .580
39. Minimum variance portfolio. Suppose stocks A and B have the following
characteristics:
Action Expected performance Standard deviation
TO 9 22
B 15 45
The covariance between the returns of the two stocks is .001.
SOLUTION
a) Suppose an investor has a portfolio consisting only of stock A and stock B. Find the
portfolio weights, XA and XB, such that the variance of your portfolio is minimized. (Hint:
Remember that the sum of the two weights must equal 1.)
o= w2 o2+ w2 o2 + 2 w
^ σB σ^ Cov(A, B)
op= w2 o + (1 - WA) o2 + 2w 1 - w^) σB σ^ Cov(A, B)
^(

To find the minimum for any function, we find the derivative and set the derivative equal to
zero. By finding the derivative of the variance function with respect to the weight of asset A,
setting the derivative equal to zero and solving for the weight of asset A, we find:

w^= (. 45 – .001) / [.22 + .45 – 2(.001)]

wA = [ q2 - Cov(A, B)] / [ o2+ q2 - 2 Cov(A, B)]
2 2 2

W^ =.8096
wB = 1 – Wa wB = 1 – .8096
wB = .1904
b) What is the expected return of the minimum variance portfolio?
E(Rp) = .8096(.09) + .1904(0.15)
E(RP) = 0.1014 or 10.14%E(RP) = 0.1014 or 10.14%
c) If the covariance between the returns of the two stocks is 2.05, what are the minimum
variance weights?
wA = [ q2 - Cov(A, B)] / [ o2+ q2 - 2 Cov(A, B)]
w^= (. 452 +– .05) / [.222 + .452 – 2(-.05)]
w^= .7196
wB = 1 – wA
wB = 1 – .7196
wB = .2804
d) What is the variance of the portfolio in part c)?
o2= . 71962 (. 222) + (.2804)2 (. 45)2 + 2(.7196) (.2804) (.22) (.45) (-.05)
o2= .0208
And the standard deviation of the portfolio is:
σ
p = (. 0208)1/2

ΣP = .1442 or 14.42%