Question

Suppose Merck stock offers an expected rate of return of 12% and a standard deviation of 25%. The risk-free rate is 5%.

(a) Graph the Capital Allocation Lines (CAL) for Merck stock.

(b) What is the Sharpe (reward-to-volatility) ratio of Merck stock?

(c) If you have a portfolio of $15,000 with $5,000 invested in a risk free security and $10,000 in Merck stock, what is the expected rate of return of your portfolio? What is your portfolio's standard deviation? What is the reward-to-volatility ratio (Sharpe ratio) of your portfolio?

(d) If your friend is more risk averse than you are and that she also has a portfolio of $15,000 invested in a risk free asset and Merck stock. Do you think she has more or less than $10,000 in Merck stock? Why?

Answer 1

a.) Capital Allocation Line for Merck Stock

The capital allocation line show various rate of return at diffrent level of risk (i.e Standard deviation)

At X-axis STD. Dev(SD) is given in % and on Y-axis Expected return is given (E(r)).

Now here Risk free return is 5% and return on Merck stock is 12% with SD of 25%. So if you invest (capital allocted) whole money into risk free return then you have Zero risk and 5% return. That's why Capital allocation line started at 5% on Y-axis. And if you invest (capital allocated) whole money into Merck stock then you will have Excepted return of 12% with 25% of SD or risk. Now what happen if you invest part of money in risk free instrument and remaining in Merck stock?

The slope of the line (Capital Allocation line shows this) i.e Your Expected return with the level of risk.

for e.g you invested 50% in risk free instrument and 50% Merck stock. So how will calculated E(r) and risk?

E(r) of risk free instrument = R1 (5%)

E(r) of Merck Stock = R2 (12%)

Risk (SD) of risk free instrument = Sd1 (0%)

Risk (SD) of Merck Stock = Sd2 (25%)

Capital allocation in risk free instrument = W1 (50%)

Capital allocation in Merck Stock = W2 (50%)

Er(P) = Return on Portfolio

SD(P) = Risk of the portfolio

Er(P) = R1*W1 + R2*W1

= 5*50% + 12*50%

= 8.5%

SD(P) = W1*Sd1 + W2*Sd2

= 50%*0+ 50%*25

= 12.5%

For 50% - 50% allocation Er(P) excepted return on portfolio is 8.5% with risk (SD) is 12.5%. So the capital allocation line show this what will return on portfolio and risk on portfolio in different capital allocation.

b.) Sharpe ratio for merck stock:

Formual for calculating Sharpe ratio is return from the instrument minus risk free return divided by SD.

So Sharpe ratio for merck stock is

[12% (return of merck stock) - 5% (risk free return)] / 25% (SD merck stock)

= 0.28 is Sharpe ratio

c.) Total investment = $ 15,000, and in risk free = $ 5,000, in merck stock = $ 10,000

So 33.33% invested in risk free and 66.67 is invested in Merck stock

Er(P) Expected return on portfolio = W1*R1 + W2*R2

= 33.33%*5 + 66.67%*12

=9.67% is Er(P)

SD(P) risk of portfolio = W1*Sd1 + W2*Sd2

= 33.33%*0 + 66.67%*25

=16.67% is SD(P)

d.) If your friend is more risk averse (i.e not willing to take more risk) then he should invest more money in risk free instrument and secured instrument. In this case if he is ready to accept risk more 16.67% for his portfolio then he can invest more than $ 10,000 otherwise not.

But he is risk averse person then he should invest less than $ 10,000 in Merck stock.