Question

1.

A pension fund manager is considering three assets. The first is a stock fund, the second is a long-term government and corporate bond fund, and the third is a T-bill yielding 0.05. The probability distribution of the risky funds is as follows:

	Expected ret.	std. dev.
Stock fund	0.19	0.25
Bond fund	0.09	0.13

The correlation between the fund returns is 0.17. An investor has a risk-aversion of 8. In her optimal complete portfolio (including stocks, bonds, and risk-free assets), what is the proportion of the risk-free asset?

2.

A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term government and corporate bond fund, and the third is a T-bill money market fund that yields a rate of 0.05. The probability distribution of the risky funds is as follows:

	Expected ret.	std. dev.
Stock fund	0.17	0.28
Bond fund	0.08	0.12

The correlation between the fund returns is 0.13.

Jessica has a risk aversion level of 6. In her optimal complete portfolio (including stocks, bonds, and risk-free assets), what is the proportion of the stock fund?

Answer 1

1) For finding the proportion of the risk-free asset first we need to calculate the Expected return of portfolio and risk of portfolio and weights of the portfolio;

Given:

Standard deviation of Stock fund:0.25 or 25%
Standard deviation of Bond fund:0.13 or 13%
correlation between the fund returns is 0.17
Expected Return of Stock Fund= 0.19 or 19%
Expected Return of Bond Fund= 0.09 or 9%

Weight of Portfolio:

Optimal Risky Portfolio formula=

Here = Standard deviation of Stock fund
   = Standard deviation of Bond fund
= Correlation of both funds

Now,

Weight of Stock Fund ( W_s) =

Weight of Stock Fund ( W_s) = 113.75 / 683.5

Weight of Stock Fund ( W_s) = 0.1664 or 16.64%

Weight of Bond Fund ( W_b) = 100 - 16.64
= 83.36% or 0.8336

Now,

Expected Return of Portfolio= R_s * W_s + R_b * W_b

₌ 0.19 * 0.1664 + 0.09 * 0.8336
= 0.0316 + 0.0750
= 0.1066 or 10.66%

Portfolio Risk=  
=
= 11.58% or 0.1158

Now,

optimal proportion of the complete portfolio to invest in the risky component=

Formula=

where,

= Expected Return on portfolio
= risk free return
A=  risk-aversion
= portfolio risk

now,

Proportion of complete portfolio to invest in risky component=

= 0.5279 or 52.79%

Proportion of Risk free aseet in the portfolio= 100 - 52.79
= 47.21%

the investor would place 52.79% of his wealth in Portfolio P and 47.21% in T-Bills.

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2) Given:

Standard deviation of Stock fund:0.28 or 28%
Standard deviation of Bond fund:0.12 or 12%
correlation between the fund returns is 0.13

The proportion of the stock fund:

Optimal Risky Portfolio formula=

Here = Standard deviation of Stock fund
   = Standard deviation of Bond fund
= Correlation of both funds

Now,

putting all of the given information in the question we, get

Proportion ( weight ) of stock fund =  

Proportion ( weight ) of stock fund = 100.33 / 840.64

Proportion ( weight ) of stock fund = .1193 or 11.93%

And proportion of Bond fund = 100 - 11.93 = 88.07%