FM212 MT2014 Problem Set Solutions
Class 5
13.
a. Variance measures the total risk of a security and is a measure of stand-alone risk. Total
risk has both unique risk and market risk. In a well-diversified portfolio, unique risks tend
to cancel each other out and only the market risk is remaining. Beta is a measure of
market risk and is useful in the context of a well-diversified portfolio. Beta measures the
sensitivity of the security returns to changes in market returns. Market portfolio has a
beta of one and is considered the average risk.
b. If we hold long positions in both stocks: the correlation coefficient that gives the
maximum reduction in risk for a two-stock portfolio is -1. If one stock is sold short and
another stock is a long position in the portfolio then a correlation of +1 is actually best to
minimize portfolio risk.
c. Mean A = 8%, Mean M=16%, Cov(Ra, Rm)=0.0138, Var(Rm)=0.0084,
Beta=0.0138/0.0084=1.643.
d. Cov(Rb,Rm)= (0.8)(0.20)(0.35) = 0.056, Beta = 0.056/0.04 = 1.4.
14. Expected portfolio return = x
A
E[R
A
] + x
B
E[R
B
] = 12% = 0.12
Let x
B
= (1 – x
A
)
x
A
(0.10) + (1 – x
A
) (0.15) = 0.12 ⇒ x
A
= 0.60 and x
B
= 1 – x
A
= 0.40
Portfolio variance = x
A2
σ
A2
+ x
B2
σ
B2
+2 (x
A
x
B
ρ
AB
σ
A
σ
B
)
= (0.60
2
) (0.20
2
) + (0.40
2
) (0.40
2
) + 2(0.60)(0.40)(0.50)(0.20)(0.40) = 0.0592
Standard deviation =
24.33%0.0592σ==
15.
a. In general:
Portfolio variance = σ
P2
= x
12
σ
12
+ x
22
σ
22
+ 2x
1
x
2
ρ
12
σ
1
σ
2
Thus:
σ
P2
= (0.5
2
)(0.2932
2
)+(0.5
2
)(0.2927
2
)+2(0.5)(0.5)(0.59)(0.2932)(0.2927)
σ
P2
= 0.0682
Standard deviation = σ
P
= 26.12%
b. One of these securities, T-bills, has zero risk and, hence, zero standard deviation. Thus:
σ
P2
= (1/3)
2
(0.2932
2
) +(1/3)
2
(0.2927
2
)+2(1/3)(1/3)(0.59)(0.2932)(0.2927)
σ
P2
= 0.0303
Standard deviation = σ
P
= 17.41%
Another way to think of this portfolio is that it is comprised of one-third T-Bills and two-
thirds a portfolio which is half Dell and half Home Depot. Because the risk of T-bills is
zero, the portfolio standard deviation is two-thirds of the standard deviation computed in
Part (a) above:
Standard deviation = (2/3)(26.12%) = 17.41%
