108156.pdf

Solutions 283

5.2First we putK 2 (ω 2 )=xand compute

Var(K 1 )=0. 001875 ,

Var(K 2 )=0.187 5x^2 +0. 015 x+0. 0003.

The two securities will have the same risk if Var(K 1 )=Var(K 2 ). This gives

the following equation

0 .0003 + 0.187 5x^2 +0. 015 x=0. 001875

with two solutionsx=− 0 .14 or 0.06. This means thatK 2 (ω 2 )=−14% or

6%.

5.3First we use the formula eki=1+Kifori=1,2 to compute the logarithmic

returns and then work out the variance of each return:

Scenario Probability K 1 K 2 k 1 k 2

ω 1 0. 510 .53% 7 .23% 10 .01% 6 .98%

ω 2 0. 513 .89% 10 .55% 13 .01% 10 .03%

Variance 0 .000282 0.000276 0.000224 0. 000232

We find that Var(K 1 )>Var(K 2 ), whereas Var(k 1 )<Var(k 2 ).

This is an interesting observation because it shows that greater risk as

measured by Var(K) does not necessarily mean greater risk in the sense of

Var(k). Nevertheless, when the rates of return are of the order of 10% or lower,

the differences between these two measures of risk are tiny and can simply

be ignored in financial practice. This is because the errors due to inaccurate

estimation of the parameters (the probabilities and values of return rates in

different scenarios) are typically greater than these differences.

5.4Letx 1 andx 2 be the number of shares of type 1 and 2 in the portfolio. Then

V(1) =x 1 S 1 (1) +x 2 S 2 (1) =V(0)

(

w 1 SS^1 (1)

1 (0)

+w 2 SS^2 (1)

2 (0)

)

= 100

(

0. 25 ×^4845 +0. 75 ×^3233

)

=99. 394.

5.5The return on the portfolio isKV=w 1 K 1 +w 2 K 2 .Thisgives

KV=0. 30 ×12%− 0. 7 ×4% = 0.8% in scenarioω 1 ,

KV=0. 30 ×10% + 0. 7 ×7% = 7.9% in scenarioω 2.

5.6The initial and final values of the portfolio are

V(0) =x 1 S 1 (0) +x 2 S 2 (0),

V(1) =x 1 S 1 (0)ek^1 +x 2 S 2 (0)ek^2

=V(0)

(

w 1 ek^1 +w 2 ek^2

)

.

As a result, the return on the portfolio is

ekV=VV(1)(0)=w 1 ek^1 +w 2 ek^2.