1. Here are some characteristics of two securities: Security 1 E(R1) = .10 2 1 =

Question

1. Here are some characteristics of two securities:

Security 1 E(R1) = .10 2 1 = .0025

Security 2 E(R2) = .16 2 2 = .0064

Answer the following questions:

(a) Suppose the investor can only hold a single stock.

(i) Which security should she choose if she wants to maximize expected returns only?

(ii) Which security should she choose if she wants to minimize risk only?

(b) Suppose the correlation of returns is -1.0, what fraction of the investor’s net worth should be held in security 1 and in security 2 in order to produce a zero risk portfolio?

(c) What is the expected return on the portfolio in (c)? How does this compare with the riskless return on Treasury Bills of 10%? Would an investor who is risk averse and likes high expected return want to invest in Treasury Bills?

Explanation / Answer

Given Information:

12 = 0.0025

(a) Suppose the investor can only hold a single stock

1. In order to maximise his expected returns only, he must choose Security 2 as expected returns on Security 2 is higher than the expected returns on Security 1.

2. In oder to minimize risks, the investor should choose secutity 1 as the Variance of Security 1 is lesser than the variance of Security 2 meaning security 1 is less volatile than the Security 2.

(b)  Suppose the correlation of returns is -1.0, what fraction of the investor’s net worth should be held in security 1 and in security 2 in order to produce a zero risk portfolio?

Correlation of returns i.e. on Security1 and Security 2 is given as -1.

We can use the following formula in order to find the Covariance between the Returns of 2 securities.

COV(1,2) = Corr(1,2) * S1*S2 ; Where Cor(1,2) is Correlation b/w the Returns of 2 securities, S1*S ar-1e the Standard Deviation of Security 1 and 2 respectively.

Thus Cov(1,2) = (-1) * (0.05) * (0.08) = -0.004

Now Lets us assume W1 and W2 be the weights of securities 1 and 2 respectively in the portfolio.

Thus the Variance of Portfolio shall be given by :

p2 = (W1)2 * 12 + (W2)2 * 22 + 2 W1 *W2 * COV(1,2) = a2 (0.0025) + b2 (0.0064) + 2 (a) (b) (-0.004)

We know that W1+W2 = a+b = 1 or b = (1-a)

We need a portfolio with zero risk so variance of portfolio = 0

Thus solving equation written above putting p2 = 0 and Solve for a and b should give us a = ?and b = ?

You will get a quadtratic equation in a like : 169a2 - 208a +64 = 0

Thus solving this will give us a = 0.615 and b = 0.385

(c) Expected Return on Portfolio in Part (b)

E(P) = 0.615 * E(R1 ) + 0.385 * E(R2 )

= 0.615 *0.10 + 0.385 * 0.0064 = 0.064

Here we have Risk of Portfolio = 0 and Expected Return from Portfolio = 6.4%

Comparing this to a 10% Treasery bill, Treasury bill also have 0 risk but higher Rate of Return. So an Investor is who is risk averse, would want to invest in the Treasurey Bills.

12 = 0.0025

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