SEEM3590 Investment Science Tutorial 8
Solution to Homework 4
ZHU XIANHAO
November 20, 2023
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Outline
1 Capital Asset Pricing Model
2 Solution to Homework 4
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Capital Asset Pricing Model Capital Market Line
Capital Market Line
The Capital Market Line (CML) describes the trade-off between risk and
return for efficient portfolios.
It’s mathematically represented as:
E(Rp) = Rf +
E(Rm)−Rf
σm
σp
where:
• E(Rp) is the expected return on the portfolio
• Rf is the risk-free rate
• σp is the standard deviation of the portfolio’s returns
• E(Rm) is the expected return on the market
• σm is the standard deviation of the market’s returns
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Capital Asset Pricing Model Security Market Line
Security Market Line
The Security Market Line (SML) is a graphical representation of the CAPM
which plots the expected return of a security as a function of its systematic,
non-diversifiable risk (β).
This security may or may not be efficient.
The SML is represented mathematically as:
E(Ri) = Rf + βi(E(Rm)−Rf )
where:
• E(Ri) is the expected return on investment i
• Rf is the risk-free rate
• βi is the sensitivity of investment i’s returns to the market returns, reflecting
the systematic risk
• E(Rm) is the expected return on the market portfolio
The expected return in SML accounts only for the systematic risk, regardless
of the idiosyncratic risk.
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Capital Asset Pricing Model Frequently Used Formulae
Frequently Used Formulae
CML: E(Rp) = Rf +
E(Rm)−Rf
σm
σp
SML: E(Ri) = Rf + βi(E(Rm)−Rf )
βi =
σi,M
σ2M
= ρi,M
σi
σM
βp =
∑
wiβi
Systematic risk of portfolio i measured in variance: β2i σ
2
M ; the ratio of
systematic risk in portfolio i: ρ2i,M (derive it in your assignment).
Note that the systematic risk of portfolio p: β2pσ
2
M ̸=
∑
w2i β
2
i σ
2
M
Idiosyncratic risk of portfolio measured in variance: var(ϵi) = σ
2
i − β2i σ2M .
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Solution to Homework 4 Problem 1
Problem 1
(Capital market line) Assume that the expected rate of return on the market
portfolio is 3% and the rate of return on T-bills (the risk-free rate) is 1%. The
standard deviation of the market portfolio is 10%.
1 What is the equation of the capital market line?
2 (i) If an expected return of 6% is desired, what is the standard deviation of
this position? (ii) If you have $1,000 to invest, how should you allocate it to
achieve the above position?
3 If you invest $300 in the risk-free asset and $700 in the market portfolio, how
much money should you expect to have at the end of the year?
The portfolios are efficient because they consist solely of the market portfolio and
the risk-free asset.
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Solution to Homework 4 Problem 1
Solution 1
1 The equation of the capital market line is
r¯ = 1%+
3%− 1%
10%
σ = 1%+ 0.2σ.
2 Setting r¯ = 6%, we find
σ =
6%− 1%
0.2
= 25%.
To achieve this position, we need to allocate a portion w0 of the initial
wealth into the risk-free asset and the rest into the market portfolio, then
w0rf + (1− w0)rM = r¯,
where
w0 =
r¯ − rM
rf − rM =
6%− 3%
1%− 3% = −
3
2
.
Thus, we need to borrow $1,500 and invest the $2,500 into the market
portfolio.
3 The money we expect to have at the end of the year is
300× (1 + 1%) + 700× (1 + 3%) = $1, 024. 7 / 17
Solution to Homework 4 Problem 1
Graph of the CML in Problem 1
If E(Rm)−Rf > 0, then
0% 5% 10% 15% 20% 25% 30%
0%
2%
4%
6%
8%
σp
E
(R
p
)
Capital Market Line
Market Portfolio
Portfolio in 1.2
β < 1
β > 1
β = 1
β = 0
β < 0
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Solution to Homework 4 Problem 2
Problem 2
(Risk analysis) Assume that the market portfolio has expected rate of return
r¯M = 0.05 and standard deviation σM = 0.25. The risk-free rate is rf = 0.02.
Answer the following questions under the CAPM framework for three assets
A,B,C in this market.
1 Asset A has σA = 0.4 and its return correlation with the market portfolio
ρAM = 0.1. Find r¯A and βA.
2 Asset B has the same expected return as A but with σB = 0.6. What is the
idiosyncratic risk of B?
3 Asset C has ρCM = 0.3. What percentage of its return variance is
idiosyncratic risk?
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Solution to Homework 4 Problem 2
Solution 2
1 According to the formulas in page 10 of lecture notes 5,
βA =ρAM
σA
σM
= 0.1× 0.4
0.25
= 0.16,
r¯A =rf + βA(r¯M − rf ) = 0.02 + 0.16× (0.05− 0.02) = 2.48%.
2 From r¯B = r¯A = 0.0248 and
r¯B = rf + βB(r¯M − rf ),
we find βB = βA = 0.16. According to the formula on page 15 of lecture
notes 5,
σ2B = β
2
Bσ
2
M + var(ϵB).
Thus, the idiosyncratic risk of B is
var(ϵB) = σ
2
B − β2Bσ2M = 0.62 − 0.162 × 0.252 = 0.3584.
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Solution to Homework 4 Problem 2
Solution 2
3 Recall the formula βi =
σi,M
σ2M
= ρi,M
σi
σM
. The required percentage is
var(ϵC)
σ2C
=
σ2C − β2Cσ2M
σ2C
=1− (ρCM
σC
σM
)2σ2M
σ2C
=1− ρ2CM
=1− 0.32
=0.91.
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Solution to Homework 4 Problem 3
Problem 3
(A simple portfolio) Consider an individual who believes in the single-factor model.
In this model, the excess return of the market portfolio serves as the factor for the
entire collection of stocks.
The individual plans to invest in a portfolio that consists of three stocks, with
details as exhibited in the following table. In addition, the market portfolio has an
expected rate of return of 5% and a variance of (20%)2. The risk-free rate is 1%.
Simple Portfolio
Stock i βi Var(ϵi) wi
A 1.20 (8.0%)2 20%
B 0.80 (2.0%)2 50%
C 1.00 (1.0%)2 30%
1 What is the portfolio’s expected rate of return?
2 Assuming the factor model is accurate, what is the variance of the portfolio’s
rate of return? Within this variance, what are the systematic risk and
idiosyncratic risk of this portfolio, respectively?
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Solution to Homework 4 Problem 3
Solution 3
1 The beta of the portfolio is
β =wAβA + wBβB + wCβC
=0.2%× 1.2 + 50%× 0.8 + 30%× 1.0
=0.94.
Hence, the single factor model r = rf + β(rM − rf ) + ϵ together with the
fact that E[ϵ] = 0 gives rise to
r¯ = rf + β(r¯M − rf ) = 1% + 0.94× (5%− 1%) = 4.76%.
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Solution to Homework 4 Problem 3
2 Note that var(r) = β2var(rM ) + var(ϵ) + 2βcov(rM , ϵ) and the fact that
cov(rM , ϵ) = 0, then
var(r) = β2var(rM ) + var(ϵ).
Since ϵ = wAϵA + wBϵB + wCϵC , and
cov(ϵA, ϵB) = cov(ϵA, ϵC) = cov(ϵB , ϵC) = 0, we have
var(ϵ) =w2Avar(ϵA) + w
2
Bvar(ϵB) + w
2
Cvar(ϵC)
=(20%)2 × (8%)2 + (50%)2 × (2%)2 + (30%)2 × (1.0%)2
=0.000365.
Thus, the variance of the portfolio’s systematic risk is
β2σ2M = 0.94
2 ∗ 0.22 = 0.035344, and the variance of the portfolio return is
var(ϵ) + β2σ2M = 0.035709.
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Solution to Homework 4 Problem 4
Problem 4
While studying the performance of a particular fund, an investor found that the
historical estimates of the mean and variance of the fund’s monthly return are
3.0% and (5%)2, respectively. She also estimated the mean and variance of the
market portfolio’s monthly return to be 4.0% and (10%)2, respectively, and the
correlation between the return rates of the fund and the market is 0.60. The
monthly risk-free return rate is constant at 1% in the past. The loadings of the
portfolio for SMB and HML are estimated to be 0.30 and 0.50, respectively. The
expected payoffs for the zero-cost strategies associated with SMB and HML are
estimated to be 0.40% and 0.6%, respectively.
1 What is the beta of this fund?
2 According to the Fama-French three-factor model, is the strategy of this fund
a good one? Why?
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Solution to Homework 4 Problem 4
Solution 4
1 We compute β first:
β =
cov(r, rM )
var(rM )
=
ρσσM
σ2M
=
0.60× 5%× 10%
(10%)2
= 0.3.
2 The loadings of the strategy on the market risk factor, SMB, and HML, are
0.3, 0.30, and 0.50, respectively. The risk premiums of these three factors are
4.0%− 1.0%, 0.40%, and 0.60%, respectively. Thus, the model risk premium
of the strategy under the Fama-French three-factor model is
0.3× (4.0%− 1.0%) + 0.30× 0.40% + 0.50× 0.60% = 1.32%,
which is lower than the estimated risk premium 2%. Thus, this strategy is a
good one when benchmarked to the Fama-French three-factor model.
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Solution to Homework 4 Problem 4
Thank You!