FINM2003 Investments
Tutorial 5 Solution
Question 1
The following are estimates for the two stocks.
Stock
Expected return E(r)
Beta β
Firm-specific standard deviation σe
A
13%
0.8
30%
B
18%
1.2
40%
The market index has a standard deviation σM = 22% and the risk free rate rf = 8%
(a) What are the standard deviations of stocks A and B?
(b) Suppose that we were to construct a portfolio with proportions:
Asset
wi
Stock A
30%
Stock B
45%
Risk free asset
25%
Compute the expected return, standard deviation, beta, and firm-specific standard deviation
of the portfolio.
Answer
(a) The variance of a stock
2
σi2 = βi2 σM
+ σe2i
Therefore, the standard deviation
q
2
βA2 σM
+ σe2A
√
= 0.82 × 0.222 + 0.32
σA =
= 34.78%
1
q
2
σB = βB2 σM
+ σe2B
√
= 1.22 × 0.222 + 0.42
= 47.93%
(b) For the portfolio
The expected return is simply the weighted average expected returns
E(rP ) = wA E(rA ) + wB E(rB ) + wf rf
= 0.3 × 13% + 0.45 × 18% + 0.25 × 8%
= 14%
We solve the portfolio’s standard deviation in several steps
βP = wA βA + wB βB + wf βf
= 0.3 × 0.8 + 0.45 × 1.2 + 0.25 × 0
= 0.78
σe2P
=
N
X
2 2
2 2
wi2 σe2i = wA
σeA + wB
σeB + wf2 σe2f
i=1
= 0.32 × 30%2 + 0.452 × 40%2 + 0.252 × 02
= 0.0405
Then the portfolio’s standard deviation
q
2
βP2 σM
+ σe2P
√
= 0.782 × 0.222 + 0.0405
σP =
= 26.45%
2
Question 2
Consider the following two regression lines for stocks A and B in the following figure.
Assume two figures use the same scale.
B
A
(a) Which stock has higher alpha?
(b) Which stock has higher systematic (market) risk?
(c) Which stock has higher firm-specific risk?
(d) Which stock has higher R2 ?
(e) Which stock has higher correlation with the market?
Answer
(a) Alpha is the intercept of the SCL. Stock A has a higher alpha than stock B.
2
(b) Systematic risk (βi2 σM
) is quantified by β, which is the slope of the SCL. Stock B has a
higher slope than stock A, and therefore stock B has a higher β and higher systematic risk.
(c) Firm-specific risk is measured by the deviations of observations (sum of squares of residuals
σe2 ) from the fitted SCL. The observations of stock A are much worse fitted than stock B, and
therefore stock A has higher firm-specific risk than stock B.
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(d) Recall the definition of R2
R2 =
Systematic risk
β 2σ2
= 2 2i M 2
Total risk
βi σM + σei
Stock B has higher systematic risk but lower firm-specific risk, so stock B has higher R2 than
stock A.
(e) The correlation coefficient is defined as ρi,M =
√
R2 . As stock B has a higher R2 , it also
has a higher correlation with the market.
Question 3
Consider the two (excess return) index model regression results for A and B:
Stock
Estimated index model
R2
Residual standard deviation
A
1% + 1.2[E(rM ) − rf ]
0.576
10.3%
B
−2% + 0.8[E(rM ) − rf ]
0.436
9.1%
(a) Which stock has higher firm-specific risk?
(b) Which stock has higher market risk?
(c) For which stock does market movement explain a greater fraction of return variations?
Answer
(a) Residual standard deviation from regression of index model is the same as the firm-specific
standard deviation σe . As σeA = 10.3% > σeB = 9.1%, stock A has higher firm-specific risk.
(b) As βeA = 1.2 > βeB = 0.8, stock A has higher market risk.
2
2
(c) As RA
= 0.576 > RB
= 0.436, market movement explains a greater fraction of return
variations for stock A than stock B.
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Question 4
The estimated index models for stock A and B are as follows
rA − rf = 3% + 0.7(rM − rf ) + eA
rB − rf = −2% + 1.2(rM − rf ) + eB
2
2
With R-squared RA
= 0.2, RB
= 0.12
Assume the standard deviation of market index σM = 20%
(a) What is the standard deviation of each stock?
(b) Break down the variance of each stock into its systematic and firm-specific component
(please keep the variance form and no need to take square root.)
Answer
(a) The variance and R2 of a stock are defined as
2
σi2 = βi2 σM
+ σe2i
R2 =
2
2
βi2 σM
βi2 σM
=
2
βi2 σM
+ σe2i
σi2
Therefore
σi =
q
r
σi2 =
2
βi2 σM
βi σM
= √
2
R
R2
The standard deviation of each stock is then
βA σM
0.7 × 0.2
= 31.31%
σA = p 2 = √
0.2
RA
βB σM
1.2 × 0.2
σB = p 2 = √
= 69.28%
0.12
RB
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(b) Break down the variance of each stock
2
Systematic variance of stock A = βA2 σM
= 0.72 × 0.22
= 1.96%
2
Systematic variance of stock B = βB2 σM
= 1.22 × 0.22
= 5.76%
2
Firm-specific variance of stock A = σe2A = σA2 − βA2 σM
= 31.31%2 − 0.72 × 0.22
= 7.84%
2
Firm-specific variance of stock B = σe2B = σB2 − βB2 σM
= 69.28%2 − 1.22 × 0.22
= 42.24%
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