1. A student’s NASDAQ portfolio from a previous class contained the following in
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Question
1. A student’s NASDAQ portfolio from a previous class contained the following investment characteristics. What is the expected return on this portfolio?
2. A pension fund manager is considering three funds, a stock fund, a T-bill money market fund that yields a risk-free rate of 5.5%, and a long-term government and corporate bond fund. The probability distributions of the risky funds are:
a.) What is the Sharpe ratio of the best feasible CAL?
b.) Suppose that your portfolio must yield an expected return of 12% and be efficient- that is, on the best feasible CAL. What is the standard deviation of your portfolio? What is the proportion invested in the T-bill fund and each of the two risky funds?
c.) If you were to use only the two risky funds and still require an expected return of 12%, what would be the investment proportions of your portfolio?
Return Standard
Deviation Portfolio
Weight Eli Lilly 15% 22% .5 Bond Pharmaceuticals 10% 8% .4 Richardson Research 6% 3% .1
Explanation / Answer
Answer 1)
Expected return on portfolio = Weight of Investment 1 * return on investment 2 + Weight of Investment 2 * return on investment 1 +Weight of Investment 3* return on investment 3
Expected return on portfolio = putting the values in formulae above = 12.1%
Answer 2)
Answer a) To calculate the sharpe ratio we first have to Calculate the mean and standard deviation of optimal risky portfolio.
First we need to find the proportion od stocks in optimal risky portfolio:
The formulae for which is give as :-
[(Ers – rf)* (St. dev of Bonds)^2 ] – [(Erb-rf)*covariance(Bond,stock)]
Divide by
(Ers – rf)* (St. dev of Bonds)^2 + (Erb-rf) )* (St. dev of Stock)^2 – [Ers-rf+Erb-rf]* covariance(Bond,stock)
Where,
Ers = Return of stock
Erb = Return of bond
Rf = risk free rate
Covariance (Bond,stock) = (St. dev of Bonds)* (St. dev of Stock)*correlation coefficient
When we put the values in the above formulae we get ,
Proportion of Stock in optimal risky portfolio as:- 0.6466
Proportion of Bonds in optimal risky portfolio as:- 0.3534
Now, we calculate the Return on the optimal risky portfolio
Erp =
(Weight of Stock * return on stock + weight of bond* return on bond )
= 0.096994132+0.031803521 = 0.128797653
St. dev of portfolio =[ (Weight of stock*St. dev of stock)^2 + (Weight of bond*St. dev of bond)^2 +( 2* Weight of bond*St. dev of bond* Weight of stock*St. dev of stock*correlation coefficient)]^1/2
= [0.042816224+ 0.006605734+ 0.005045288]^1/2
= [0.054467]^0.5
= 0.233382
Sharpe Ratio = Porfolio return - Risk free rate / Porfolio st. deviation
=(0.128797653 - 0.055) / 0.233382 = 0.316209
Answer b)
Now we require a Cal portfolio of mean return 12%, the corresponding st. deviation is given as
Erc = rf + [(Erp-rf)/ St.dev of portfolio ]* St.dev of of Cal portfolio
Where Erc = 12% 0r 0.12
Erp = 0.128797653
Rf = 0.055
St. dev of optimal risky portfolio = 0.233382
Solving the above formulae for St. Deviation of the Cal portfolio
0.12 = 0.055 + [(0.128797653- 0.055)/ 0.23332]* St. Deviation of the Cal portfolio
St. Deviation of the Cal portfolio = (0.09-0.03) / (0.316209449) = 0.205559955
Now to find the amount invested in T-bill we use the below formulae
Erc=Rf + y*(Erp – Rf) Where y is the amount invested in stocks and bond for a cal porfolio and 1-y is the amount invested in T-bills
0.12=0.055+y*(0.128797653-0.055)
Y = 0.8807868
Amount invested in stocks = 0.8807868*0.6466 = 0.569541021
Amount Invested in Bonds = 0.8807868*0.3534 = 0.311245801
Amount in T-bills = 1-y = 1-0.8807868 = 0.1192132
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