, Solution manual for Finance for Executives
Managing for Value Creation, 7th Edition Gabriel
Hawawini
Notes
1- The file is chapter after chapter.
2- We have shown you 10 pages.
3- The file contains all Appendix and Excel
sheet if it exists.
4- We have all what you need, we make
update at every time. There are many
new editions waiting you.
5- If you think you purchased the wrong file
You can contact us at every time, we can
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, Chapter 2
Answers to Review Problems
Finance for Executives – 7th Edition
1. Finding the implicit interest rate
If indifferent then the present values of the alternatives should be the same, that is,
, and thus (1 + k) 2 = $1,000 = 1.180 from which we get k = 8.63%.
$1,000 $1,180 $1,180
=
1+k (1+k) 3
2. APR versus effective interest rate
Using equation 2.4 we can write: 1 + k eff = 1.0617 = 1 + , thus:
APR 12
12
1
from which we get APR = 6%.
APR
(1.0617) 12 = 1.0050 = 1 + 12
With a financial calculator, enter N=12, PV=1, PMT=0, FV= −1.0617 and press I/YR. you
will find a monthly APR of 0.5% which multiplied by 12 gives you 6%.
3. Compounded value and compounded rate
a.
1 + 3% × 1 + 5% × 1 + 6% = $1.1464.
b.
(1 + k) 3 = 1.1464 from which we get k = 4.66%.
4. Alternative financing plans
PV(Plan 1) = $12,400 + $400 × ADF(T=35; k=6%/12) = $12,400 + $400 × 32.0354 = $25,214.
PV(Plan 2) = $492 × ADF(T=60; k=6%/12) = $492 × 51.7256 = $25,449.
The first plan is preferable because it is less expensive because it has a lower present value.
For use with Finance for Executives: Managing for Value Creation 7th edition
by Gabriel Hawawini and Claude Viallet, ISBN 9781473778917
© 2022 Cengage Learning EMEA
,5. Annuity versus perpetuity
The future value of the $100 a year for the next 10 years (see formula 2.13 for the future
value of an annuity) at the rate ‘k’ must be equal to the present value, at the end of 10, of a
$100 perpetuity at the same rate ‘k’, hence we have:
$100
(1 + k) 10 − 1 = $100 , and thus (1 + k) 10 − 1 = 1, from which we get (1 + k) 10 = 2.
k k
Using a financial calculator we find k = 7.18%. (Enter N=10, PV=1, PMT=0, FV=−2 and
press I/YR. you will find 7.18%.)
6. Valuing a loan
a.
The loan will generate fixed interest income of $800,000 (8% of $10 million) every year over
the next 4 years plus $10 million at the end of the fourth year. Its value is thus the sum of the
present value of 4-year, $800,000 annuity at 7 percent (the prevailing market rate) and the
present value of $10 million to be received in 4 years at 7 percent:
Value of loan = [$800,000 × ADF(T=4; k=7%)] + [$10,000,000 × DF(T=4; k=7%)]
Value of loan = [$800,000 × 3.3872] + [$10,000,000 × 0.7629] = $10,338,760.
b.
Value of loan = [$400,000 × ADF(T=8; k=3.5%)] + [$10,000,000 × DF(T=8; k=3.5%)]
Value of loan = [$400,000 × 6.8740] + [$10,000,000 × 0.7594] = $10,343,600.
7. Perpetual cash flows
a.
If the current membership is renewed every year in perpetuity with fees growing at 3 percent
annually, its the present value at 6 percent is PV = = $68,667. This is a
$2,000(1+3%) $2,060
6%−3%
= 0.03
higher amount than the proposed price of $65,000 for life-long family membership. The life-
long family membership is thus a better deal.
For use with Finance for Executives: Managing for Value Creation 7th edition
by Gabriel Hawawini and Claude Viallet, ISBN 9781473778917
© 2022 Cengage Learning EMEA
, b.
The interest rate that makes you indifferent is the one that equates the present value of the
two choices, that is, $65,000 = k−3% , from which we get k = 6.17%.
$2,060
c.
The annual fee, call it X, that makes you indifferent is giving by the equation: $65,000 =
, from which we get X = $1,893.20
X×1.03
6%−3%
d.
$68,667, which is the present value of the current membership if it were an annuity growing
at 3 percent.
8. American Airlines AAirpass
a.
𝑪𝑭 𝟏 𝟏
𝑷𝑽 = 𝟏−
𝒌 (𝟏 + 𝒌) 𝑻
𝑻
𝑪𝑭 𝟏 𝟏+𝒈
𝑷𝑽 = 𝟏−
𝒌−𝒈 𝟏+𝒌
We have the following information:
Payment amount (𝐶𝐹 1 ) = $1,000,000
Discount rate (k) = 10% or 0.10
Periods (T) = 2008 – 1987 = 21
The present value (PV) of these payments can be now calculated as:
$𝟏,𝟎𝟎𝟎,𝟎𝟎𝟎 𝟏
𝑷𝑽 = 𝟏− = $𝟖,𝟔𝟒𝟖,𝟔𝟗𝟒.𝟐𝟗
𝟎.𝟏𝟎 (𝟏 + 𝟎.𝟏𝟎) 𝟐𝟏
Hence, it costs the company $8,648,694.29 for the AAirpass and this should have been the
lowest value at which it should have been sold.
For use with Finance for Executives: Managing for Value Creation 7th edition
by Gabriel Hawawini and Claude Viallet, ISBN 9781473778917
© 2022 Cengage Learning EMEA
, b. The information required is as follows:
Payment amount (𝐶𝐹 1 ) = $1,100,000
Discount rate (k) = 10% or 0.10
Growth rate (g) = 5% or 0.05
Periods (T) = 20
𝟐𝟎
$𝟏,𝟏𝟎𝟎,𝟎𝟎𝟎 𝟏 + 𝟎.𝟎𝟓
𝑷𝑽 = 𝟏− = $𝟏𝟑,𝟑𝟐𝟑,𝟐𝟗𝟐.𝟒𝟔
𝟎.𝟏𝟎 − 𝟎.𝟎𝟓 𝟏 + 𝟎.𝟏𝟎
The price of the pass should therefore be ideally higher than $13,323,292.46.
9. Growing annuities versus growing perpetuities
a.
It is the present value, at 8 percent, of an annuity of $80 million growing at 3 percent for 5
years. Using formula 2.12 you get:
5
$80m 1.03
PV = 1− = $1,600m × 0.2110 = $𝟑𝟑𝟕.𝟔𝟎 𝐦𝐢𝐥𝐥𝐢𝐨𝐧
8% − 3% 1.08
b.
It is the present value, at 8 percent, of a perpetuity growing at 3 percent, that is, PV =
$80m
8%−3%
,
which is $1,600 million.
10. Mortgage loan
a.
The monthly mortgage payment, call it X, is the solution to the equation:
$80,000 = X × ADF[T=360; 8%/12] = X × 136.2783
from which we get X = $587.03.
b.
Interest payment in first installment = $80,000 × 12 = $533.33.
8%
For use with Finance for Executives: Managing for Value Creation 7th edition
by Gabriel Hawawini and Claude Viallet, ISBN 9781473778917
© 2022 Cengage Learning EMEA
, Principal repayment = $587.03 – $533.33 = $53.70.
c.
Total interest payments = Total payments – Principal repayment = ($587.03 × 360) –
$80,000
Total interest payments = = $211,330.80 − $80,000 = $131,330.80.
11. Retirement planning
a.
The capital needed at 65, call it X, is an immediate annuity such as:
X = $50,000 + $50,000 × ADF[T=19; 6%] = $50,000 + $50,000 × 11.1581 = $607,905.82
The lump sum needed today is the present value at DF (T=40; k=6%):
Lump sum = $609,905.82 × DF[T=40; k=6%] = $609,905.82 × 0.0972 = $59,088.45.
b.
Amount to invest every month is an annuity, call it X, whose present value (including the
immediate payment) must be equal to the lump sum $59,088.45, that is:
$59,088.45 = X + X × ADF[T=40×12; k=6%/12] = X + X × 181.7476,
from which we get X = $323.33.
12. Annuity versus annuity due
The relation for the future value of the net income will be calculated as per the formula.
However, since the rent is paid at the start of the month and each of the cash flows will be
compounded for one more period, we need to adjust the first cash flow by compounding it for
one period.
For Rent:
Discount rate (k) = 6% or 0.50% monthly = 0.005
CF1 = $2,000 × (1 + 0.005) = $2,010.00
T = 12
For use with Finance for Executives: Managing for Value Creation 7th edition
by Gabriel Hawawini and Claude Viallet, ISBN 9781473778917
© 2022 Cengage Learning EMEA
, $𝟐,𝟎𝟏𝟎.𝟎𝟎
𝑭𝑽 = (𝟏 + 𝟎.𝟎𝟎𝟓) 𝟏𝟐 − 𝟏 = $𝟐𝟒,𝟕𝟗𝟒.𝟒𝟖
𝟎.𝟎𝟎𝟓
The formula for the rent which is a type of annuity due can therefore be adjusted to:
𝑪𝑭 𝟏 (𝟏 + 𝒌)
𝑭𝑽 = (𝟏 + 𝒌) 𝑻 − 𝟏
𝒌
For net income, the future value can be calculated as:
Discount rate (k) = 0.005
CF1 = $10,000
T = 12
$𝟏𝟎,𝟎𝟎𝟎.𝟎𝟎
𝑭𝑽 = (𝟏 + 𝟎.𝟎𝟎𝟓) 𝟏𝟐 − 𝟏 = $𝟏𝟐𝟑,𝟑𝟓𝟓.𝟔𝟐
𝟎.𝟎𝟎𝟓
For use with Finance for Executives: Managing for Value Creation 7th edition
by Gabriel Hawawini and Claude Viallet, ISBN 9781473778917
© 2022 Cengage Learning EMEA
, Chapter 3
Answers to Review Problems
Finance for Executives – 7th Edition
1. Attitudes toward risk
a. She is either risk averse or risk neutral because stock A has a higher expected return with
less risk.
b. He is a risk seeker because stock B has a lower expected return with more risk.
c. A risk neutral investor will buy stock B if it offers more than 13 percent.
d. A risk-averse investor will buy stock B if its volatility is 25 percent.
2. Characteristics of a two-stock portfolio
a. E R P1 = 25% × 8% + 75% × 16% = 𝟏𝟒%
b. σ AB = ρ AB × σ A × σ B = 0.30 × 0.18 × 0.30 = 𝟎.𝟎𝟏𝟔𝟐
c. σP1 = (0.25 × 0.18)2 + (0.75 × 0.30)2 + 2 × 0.25 × 0.75 × 0.0162 = 0.0587 = 𝟐𝟒.𝟐𝟑%
σ 2B − σAB
d. w∗A =
2
(0.30) −0.0162
= = 82%
σ 2A + σ 2B − 2σAB 2 2
(0.18) +(0.30) − (2×0.0162)
E R MRP = 82% × 8% + 18% × 16% = 9.44%
σ MRP = (0.82 × 0.18)2 + (0.18 × 0.30)2 + 2 × 0.82 × 0.18 × 0.0162 = 0.02948 = 𝟏𝟕.𝟏𝟕%
e. No, portfolio P2 is not efficient because it has the same expected return as portfolio P1 with
more risk. To be efficient, portfolio P2 must have a volatility of 24.23 percent, the volatility
of portfolio P1.
For use with Finance for Executives: Managing for Value Creation 7th edition
by Gabriel Hawawini and Claude Viallet, ISBN 9781473778917
© 2022 Cengage Learning EMEA
, 3. Risk reduction through diversification
a. The expected return of the equally-weighted portfolio is the same as the expected return
of the three individual stocks, that is, E(RP) = (1/3)10% + (1/3)10% + (1/3)10% = 10%.
b. The variance of the 3-stock portfolio has the following structure: three variances each
equal to 20 percent and three covariances each equal to 0.0200 (0.50×0.20×0.20). Given
one-third investment in each stock, the variance of the portfolio is thus:
Var (RP) = [(1/3)2 × (0.20)2 + (1/3)2 × (0.20)2 + (1/3)2 × (0.20)2]
+ [2 × (1/3)2 × (0.02) + 2 × (1/3)2 × (0.02) + 2 × (1/3)2 × (0.02)] = 0.026667
Volatility of the portfolio is thus 0.026667 = 𝟏𝟔.𝟑𝟑%. The risk of the portfolio is thus
significantly lower than the risk of the individual stocks in the portfolio.
4. Correlations, covariances and betas
a. Re-arrange the SML equation, E(R) = 4% + 6%β, as β = Using the expected
E R −4%
6%
.
returns in Exhibit 3.12, row 5, you can infer the betas of the 5 stocks. We have β A =
= 𝟎.𝟑𝟎. Use the same procedure you get βB = 0.50, βC = 0.90, βD = 1.30 and βE =
5.8%−4%
6%
1.50.
b. We have β A = 0.30 = from which you get σ AM = β A × 0.04 = 0.0120. Using
σ AM σ AM
=
σ 2M (0.20) 2
the same procedure you get σ BM = 0.0200, σ CM = 0.0360, σ DM = 0.0520 and σ EM =
0.0600. We have ρ AM = σ = 0.15×0.20 = 0.40. Using the same procedure you get
σ AM 0.0120
A×σ M
ρ BM = 0.40, ρ CM = 0.60,ρ DM = 0.65 and ρ EM = 0.60.
β Aβ Bσ 2M
c.
Cov(R A ; R B) Cov(a A+β AR M+U A ; a B+β BR M+U B ) Cov(β AR M ; β BR M)
ρ AB = σ A×σ B
= σ A×σ B
= σ A×σ B
= σ A×σ B
σ σ
β Aβ Bσ 2Mρ AM σ A ×ρ BM σ B ×σ 2M
M M
ρ AB = σ A×σ B
= σ A×σ B
= ρ AM × ρ BM = 0.40 × 0.40 = 𝟎.𝟏𝟔.
d. β P = 0.30 + 0.50 + 0.90 + 1.30 + 1.50 = 0.90. The portfolio beta is lower than the
1
5
market beta because there are more stocks with betas lower than one than higher. Recall
that the market beta is a value-weighted portfolio, not an equally-weighted one.
For use with Finance for Executives: Managing for Value Creation 7th edition
by Gabriel Hawawini and Claude Viallet, ISBN 9781473778917
© 2022 Cengage Learning EMEA
Managing for Value Creation, 7th Edition Gabriel
Hawawini
Notes
1- The file is chapter after chapter.
2- We have shown you 10 pages.
3- The file contains all Appendix and Excel
sheet if it exists.
4- We have all what you need, we make
update at every time. There are many
new editions waiting you.
5- If you think you purchased the wrong file
You can contact us at every time, we can
replace it with true one.
Our email:
, Chapter 2
Answers to Review Problems
Finance for Executives – 7th Edition
1. Finding the implicit interest rate
If indifferent then the present values of the alternatives should be the same, that is,
, and thus (1 + k) 2 = $1,000 = 1.180 from which we get k = 8.63%.
$1,000 $1,180 $1,180
=
1+k (1+k) 3
2. APR versus effective interest rate
Using equation 2.4 we can write: 1 + k eff = 1.0617 = 1 + , thus:
APR 12
12
1
from which we get APR = 6%.
APR
(1.0617) 12 = 1.0050 = 1 + 12
With a financial calculator, enter N=12, PV=1, PMT=0, FV= −1.0617 and press I/YR. you
will find a monthly APR of 0.5% which multiplied by 12 gives you 6%.
3. Compounded value and compounded rate
a.
1 + 3% × 1 + 5% × 1 + 6% = $1.1464.
b.
(1 + k) 3 = 1.1464 from which we get k = 4.66%.
4. Alternative financing plans
PV(Plan 1) = $12,400 + $400 × ADF(T=35; k=6%/12) = $12,400 + $400 × 32.0354 = $25,214.
PV(Plan 2) = $492 × ADF(T=60; k=6%/12) = $492 × 51.7256 = $25,449.
The first plan is preferable because it is less expensive because it has a lower present value.
For use with Finance for Executives: Managing for Value Creation 7th edition
by Gabriel Hawawini and Claude Viallet, ISBN 9781473778917
© 2022 Cengage Learning EMEA
,5. Annuity versus perpetuity
The future value of the $100 a year for the next 10 years (see formula 2.13 for the future
value of an annuity) at the rate ‘k’ must be equal to the present value, at the end of 10, of a
$100 perpetuity at the same rate ‘k’, hence we have:
$100
(1 + k) 10 − 1 = $100 , and thus (1 + k) 10 − 1 = 1, from which we get (1 + k) 10 = 2.
k k
Using a financial calculator we find k = 7.18%. (Enter N=10, PV=1, PMT=0, FV=−2 and
press I/YR. you will find 7.18%.)
6. Valuing a loan
a.
The loan will generate fixed interest income of $800,000 (8% of $10 million) every year over
the next 4 years plus $10 million at the end of the fourth year. Its value is thus the sum of the
present value of 4-year, $800,000 annuity at 7 percent (the prevailing market rate) and the
present value of $10 million to be received in 4 years at 7 percent:
Value of loan = [$800,000 × ADF(T=4; k=7%)] + [$10,000,000 × DF(T=4; k=7%)]
Value of loan = [$800,000 × 3.3872] + [$10,000,000 × 0.7629] = $10,338,760.
b.
Value of loan = [$400,000 × ADF(T=8; k=3.5%)] + [$10,000,000 × DF(T=8; k=3.5%)]
Value of loan = [$400,000 × 6.8740] + [$10,000,000 × 0.7594] = $10,343,600.
7. Perpetual cash flows
a.
If the current membership is renewed every year in perpetuity with fees growing at 3 percent
annually, its the present value at 6 percent is PV = = $68,667. This is a
$2,000(1+3%) $2,060
6%−3%
= 0.03
higher amount than the proposed price of $65,000 for life-long family membership. The life-
long family membership is thus a better deal.
For use with Finance for Executives: Managing for Value Creation 7th edition
by Gabriel Hawawini and Claude Viallet, ISBN 9781473778917
© 2022 Cengage Learning EMEA
, b.
The interest rate that makes you indifferent is the one that equates the present value of the
two choices, that is, $65,000 = k−3% , from which we get k = 6.17%.
$2,060
c.
The annual fee, call it X, that makes you indifferent is giving by the equation: $65,000 =
, from which we get X = $1,893.20
X×1.03
6%−3%
d.
$68,667, which is the present value of the current membership if it were an annuity growing
at 3 percent.
8. American Airlines AAirpass
a.
𝑪𝑭 𝟏 𝟏
𝑷𝑽 = 𝟏−
𝒌 (𝟏 + 𝒌) 𝑻
𝑻
𝑪𝑭 𝟏 𝟏+𝒈
𝑷𝑽 = 𝟏−
𝒌−𝒈 𝟏+𝒌
We have the following information:
Payment amount (𝐶𝐹 1 ) = $1,000,000
Discount rate (k) = 10% or 0.10
Periods (T) = 2008 – 1987 = 21
The present value (PV) of these payments can be now calculated as:
$𝟏,𝟎𝟎𝟎,𝟎𝟎𝟎 𝟏
𝑷𝑽 = 𝟏− = $𝟖,𝟔𝟒𝟖,𝟔𝟗𝟒.𝟐𝟗
𝟎.𝟏𝟎 (𝟏 + 𝟎.𝟏𝟎) 𝟐𝟏
Hence, it costs the company $8,648,694.29 for the AAirpass and this should have been the
lowest value at which it should have been sold.
For use with Finance for Executives: Managing for Value Creation 7th edition
by Gabriel Hawawini and Claude Viallet, ISBN 9781473778917
© 2022 Cengage Learning EMEA
, b. The information required is as follows:
Payment amount (𝐶𝐹 1 ) = $1,100,000
Discount rate (k) = 10% or 0.10
Growth rate (g) = 5% or 0.05
Periods (T) = 20
𝟐𝟎
$𝟏,𝟏𝟎𝟎,𝟎𝟎𝟎 𝟏 + 𝟎.𝟎𝟓
𝑷𝑽 = 𝟏− = $𝟏𝟑,𝟑𝟐𝟑,𝟐𝟗𝟐.𝟒𝟔
𝟎.𝟏𝟎 − 𝟎.𝟎𝟓 𝟏 + 𝟎.𝟏𝟎
The price of the pass should therefore be ideally higher than $13,323,292.46.
9. Growing annuities versus growing perpetuities
a.
It is the present value, at 8 percent, of an annuity of $80 million growing at 3 percent for 5
years. Using formula 2.12 you get:
5
$80m 1.03
PV = 1− = $1,600m × 0.2110 = $𝟑𝟑𝟕.𝟔𝟎 𝐦𝐢𝐥𝐥𝐢𝐨𝐧
8% − 3% 1.08
b.
It is the present value, at 8 percent, of a perpetuity growing at 3 percent, that is, PV =
$80m
8%−3%
,
which is $1,600 million.
10. Mortgage loan
a.
The monthly mortgage payment, call it X, is the solution to the equation:
$80,000 = X × ADF[T=360; 8%/12] = X × 136.2783
from which we get X = $587.03.
b.
Interest payment in first installment = $80,000 × 12 = $533.33.
8%
For use with Finance for Executives: Managing for Value Creation 7th edition
by Gabriel Hawawini and Claude Viallet, ISBN 9781473778917
© 2022 Cengage Learning EMEA
, Principal repayment = $587.03 – $533.33 = $53.70.
c.
Total interest payments = Total payments – Principal repayment = ($587.03 × 360) –
$80,000
Total interest payments = = $211,330.80 − $80,000 = $131,330.80.
11. Retirement planning
a.
The capital needed at 65, call it X, is an immediate annuity such as:
X = $50,000 + $50,000 × ADF[T=19; 6%] = $50,000 + $50,000 × 11.1581 = $607,905.82
The lump sum needed today is the present value at DF (T=40; k=6%):
Lump sum = $609,905.82 × DF[T=40; k=6%] = $609,905.82 × 0.0972 = $59,088.45.
b.
Amount to invest every month is an annuity, call it X, whose present value (including the
immediate payment) must be equal to the lump sum $59,088.45, that is:
$59,088.45 = X + X × ADF[T=40×12; k=6%/12] = X + X × 181.7476,
from which we get X = $323.33.
12. Annuity versus annuity due
The relation for the future value of the net income will be calculated as per the formula.
However, since the rent is paid at the start of the month and each of the cash flows will be
compounded for one more period, we need to adjust the first cash flow by compounding it for
one period.
For Rent:
Discount rate (k) = 6% or 0.50% monthly = 0.005
CF1 = $2,000 × (1 + 0.005) = $2,010.00
T = 12
For use with Finance for Executives: Managing for Value Creation 7th edition
by Gabriel Hawawini and Claude Viallet, ISBN 9781473778917
© 2022 Cengage Learning EMEA
, $𝟐,𝟎𝟏𝟎.𝟎𝟎
𝑭𝑽 = (𝟏 + 𝟎.𝟎𝟎𝟓) 𝟏𝟐 − 𝟏 = $𝟐𝟒,𝟕𝟗𝟒.𝟒𝟖
𝟎.𝟎𝟎𝟓
The formula for the rent which is a type of annuity due can therefore be adjusted to:
𝑪𝑭 𝟏 (𝟏 + 𝒌)
𝑭𝑽 = (𝟏 + 𝒌) 𝑻 − 𝟏
𝒌
For net income, the future value can be calculated as:
Discount rate (k) = 0.005
CF1 = $10,000
T = 12
$𝟏𝟎,𝟎𝟎𝟎.𝟎𝟎
𝑭𝑽 = (𝟏 + 𝟎.𝟎𝟎𝟓) 𝟏𝟐 − 𝟏 = $𝟏𝟐𝟑,𝟑𝟓𝟓.𝟔𝟐
𝟎.𝟎𝟎𝟓
For use with Finance for Executives: Managing for Value Creation 7th edition
by Gabriel Hawawini and Claude Viallet, ISBN 9781473778917
© 2022 Cengage Learning EMEA
, Chapter 3
Answers to Review Problems
Finance for Executives – 7th Edition
1. Attitudes toward risk
a. She is either risk averse or risk neutral because stock A has a higher expected return with
less risk.
b. He is a risk seeker because stock B has a lower expected return with more risk.
c. A risk neutral investor will buy stock B if it offers more than 13 percent.
d. A risk-averse investor will buy stock B if its volatility is 25 percent.
2. Characteristics of a two-stock portfolio
a. E R P1 = 25% × 8% + 75% × 16% = 𝟏𝟒%
b. σ AB = ρ AB × σ A × σ B = 0.30 × 0.18 × 0.30 = 𝟎.𝟎𝟏𝟔𝟐
c. σP1 = (0.25 × 0.18)2 + (0.75 × 0.30)2 + 2 × 0.25 × 0.75 × 0.0162 = 0.0587 = 𝟐𝟒.𝟐𝟑%
σ 2B − σAB
d. w∗A =
2
(0.30) −0.0162
= = 82%
σ 2A + σ 2B − 2σAB 2 2
(0.18) +(0.30) − (2×0.0162)
E R MRP = 82% × 8% + 18% × 16% = 9.44%
σ MRP = (0.82 × 0.18)2 + (0.18 × 0.30)2 + 2 × 0.82 × 0.18 × 0.0162 = 0.02948 = 𝟏𝟕.𝟏𝟕%
e. No, portfolio P2 is not efficient because it has the same expected return as portfolio P1 with
more risk. To be efficient, portfolio P2 must have a volatility of 24.23 percent, the volatility
of portfolio P1.
For use with Finance for Executives: Managing for Value Creation 7th edition
by Gabriel Hawawini and Claude Viallet, ISBN 9781473778917
© 2022 Cengage Learning EMEA
, 3. Risk reduction through diversification
a. The expected return of the equally-weighted portfolio is the same as the expected return
of the three individual stocks, that is, E(RP) = (1/3)10% + (1/3)10% + (1/3)10% = 10%.
b. The variance of the 3-stock portfolio has the following structure: three variances each
equal to 20 percent and three covariances each equal to 0.0200 (0.50×0.20×0.20). Given
one-third investment in each stock, the variance of the portfolio is thus:
Var (RP) = [(1/3)2 × (0.20)2 + (1/3)2 × (0.20)2 + (1/3)2 × (0.20)2]
+ [2 × (1/3)2 × (0.02) + 2 × (1/3)2 × (0.02) + 2 × (1/3)2 × (0.02)] = 0.026667
Volatility of the portfolio is thus 0.026667 = 𝟏𝟔.𝟑𝟑%. The risk of the portfolio is thus
significantly lower than the risk of the individual stocks in the portfolio.
4. Correlations, covariances and betas
a. Re-arrange the SML equation, E(R) = 4% + 6%β, as β = Using the expected
E R −4%
6%
.
returns in Exhibit 3.12, row 5, you can infer the betas of the 5 stocks. We have β A =
= 𝟎.𝟑𝟎. Use the same procedure you get βB = 0.50, βC = 0.90, βD = 1.30 and βE =
5.8%−4%
6%
1.50.
b. We have β A = 0.30 = from which you get σ AM = β A × 0.04 = 0.0120. Using
σ AM σ AM
=
σ 2M (0.20) 2
the same procedure you get σ BM = 0.0200, σ CM = 0.0360, σ DM = 0.0520 and σ EM =
0.0600. We have ρ AM = σ = 0.15×0.20 = 0.40. Using the same procedure you get
σ AM 0.0120
A×σ M
ρ BM = 0.40, ρ CM = 0.60,ρ DM = 0.65 and ρ EM = 0.60.
β Aβ Bσ 2M
c.
Cov(R A ; R B) Cov(a A+β AR M+U A ; a B+β BR M+U B ) Cov(β AR M ; β BR M)
ρ AB = σ A×σ B
= σ A×σ B
= σ A×σ B
= σ A×σ B
σ σ
β Aβ Bσ 2Mρ AM σ A ×ρ BM σ B ×σ 2M
M M
ρ AB = σ A×σ B
= σ A×σ B
= ρ AM × ρ BM = 0.40 × 0.40 = 𝟎.𝟏𝟔.
d. β P = 0.30 + 0.50 + 0.90 + 1.30 + 1.50 = 0.90. The portfolio beta is lower than the
1
5
market beta because there are more stocks with betas lower than one than higher. Recall
that the market beta is a value-weighted portfolio, not an equally-weighted one.
For use with Finance for Executives: Managing for Value Creation 7th edition
by Gabriel Hawawini and Claude Viallet, ISBN 9781473778917
© 2022 Cengage Learning EMEA
