## Problem Set II: Suggested Answers

### 11.2

a. Systematic Risk = 0.042 (4,480 - 4,416) - 1.4 (4.3% - 3.1%) - 0.67 (11.8% - 9.5%)
= -0.53%
b. Unsystematic Risk = -2.6%
c. Total Return = 9.5% - 0.53% - 2.6% = 6.37%

### 11.4

a. Stock A:
Ra = E[Ra] + ßa (Rm - E[Rm]) + ea
= 10.5% + 1.2 (Rm - 14.2%) + ea

Stock B:
Rb = E[Rb] + ß(Rm - E[Rm]) + eb
= 13.0% = 0.98 (Rm - 14.2%) + eb

Stock C:
Rc = E[Rc] +Bc (Rm - E[Rm]) + ec
= 15.7% + 1.37 (Rm - 14.2%) + ec

b.

RP = 0.30 Ra + 0.45Rb + 0.25RC
= 0.30 [10.5% + 1.2 (Rm - 14.2%) + ea]
+0.45[13.0% + 0.98(Rm - 14.2%) + eb]
+0.25[15.7% + 1.37 (Rm - 14.2%) + ec]
= 0.30 (1.05%) + 0.45(13%) + 0.25 (15.7%)
+[0.03 (1.2) + 0.45(0.98) + 0.25 (1.37)] (Rm - 14.2%)
+[0.03 (ea) = 0.45eb + 0.25ec
= 12.925% + 11.435(Rm - 14.2%)
+0.30ea + 0.45eb + 0.25Ec

c.
Ra = 10.5% + 1.2 (15% - 14.2%)
= 11.46%
Rb = 13% + 0.98 (15% - 14.2%)
= 13.7%
Rc= 15.7% + 1.37 (15% - 14.2%)
= 16.8%

ii.
Rp = 12.925% + 1.1435 (15% - 14.2%)
= 13.8398%

### 11.8

a.
Let X = the proportion of security of one in the portfolio and (1-X) = the proportion of security two in the portfolio.

Rpt = XR1t + (1-X) R2t
= x[e(R1t) + b11F1t +b12F2t] + (1-X) [E (R2t) + b21F1T + b22F2t]

The condition that the return of the portfolio does not depend on F1 implies:

Xb11 + (1-X) b21 = 0
X + (1-X) 0.5 = 0
Thus, P = (-1,2); i.e. sell short security one and buy security two.

E(Rp) = (-1) 20% +2 (20%) = 20%
b p2 = (-1) (1.5) + 2(2) = 2.5

b.
Following the same logic as in part a.

Xb31 + (1-X)b41= 0
X + (1-X)1.5 = 0
X = 3
Where X is the proportion of security three in the portfolio. Thus, sell short security four and buy security three.

E (Rp) = 3(10%) + (-2) (10%) = 10%
bp2 = 3(0.5) - 2(0.75) = 0
this is a risk free portfolio!

c.
The portfolio in part b provides a risk free return of 10% which is higher than the
5% return provided by the risk free security. To take advantage of this
opportunity, borrow at the risk free rate of 5% and invest the funds in a portfolio
built by selling short security four and buying security three with weights (3, -2).

d.
Assuming that the risk free security will not change the price of security four
(that everyone is trying to sell short) will decrease and the price of security three
(that everyone is trying to buy) will increase, hence the return of security four will
increase and the return of security three will decrease.

The alternative is that the prices of seurities three and four will remain the same,
and the price of the risk-free security drops until its return is 10%.

Finally, a combined movement of all security prices is also possible. The prices of security four and the risk-free security will decrease and the price of security four will increase until the opportunity disappears.