Question

Consider the following securities: Risky security: E(R) = 10% and standard deviation= 20. Risk-free
security: Rf = 5%. You want to construct a portfolio combining the risky security
and the risk-free security such that you get an expected return of 15%.
(a) What weights would you need to put in the risky and the risk-free securities to
earn a 15%?
(b) What is the standard deviation of this portfolio? What is the reward-to-
variability ratio?
(c) Draw the capital allocation line (CAL). Label the points and the axes clearly.
(d) Now, suppose that instead of one risky security and one risk-free security, you
can invest in two risky securities. Security 1: E(R1) = 10% and standard deviation₁ = 20%.
Security 2: E(R2) = 6% and standard deviation₂ = 10% with p= 0:3. What weights would you
need to place in the two risky securities to earn a 15% expected return? What
is the standard deviation of this portfolio?
(e) Find the expected return and the standard deviation of the minimum-variance
portfolio (MVP) on the investment opportunity set.

Answer 1

Answer (a)
Let the weight of risky security = wa
weight of risk free security = wb =1-wa
Given E(R)=10%
portfolio return = wa*E(R) + wb*Rf
0.15 = wa*0.10 + (1-wa)*0.05
0.15 = 0.10*wa + 0.05 - 0.05wa
wa = 0.10 / 0.05 = 2
wb = 1-2 = -1

(b)
Sd of Risky security = 20%
Sd of Risk-free security = 0
Portfolio sd = wa*20% (short cut formula, when including risky free asset)
= 2*20%
=40%

Reward to Variabilty Ratio = {Rp - Rf}/Sd of Portfolio

= (0.15 - 0.05)/ 0.632 =0.158

(d)
E(R1) = 10%,    sd1 = 20%
E(R2) = 6%,    sd2 = 10%

Rp = wa*10% + wb*6%

Rp = wa*10% + (1 - wa)*6%

15% = 10%wa +6% - 6%wa

wa = 11% / 4% = 2.25
wb=1-2.25 = -1.25

wa= Investment in first Risky asset.
wb= Investment in second Risky asset.

Sd of portfolio = wa*sd1 + wb*sd2 +2*wa*wb*sd1*sd2*r

=√[(2.25*0.20)^2 +(0.10*-1.25)^2 + 2*2.75*-1.25*0.20*0.10*0.30]

=39.13%