Question

Now consider the set of portfolios that can be obtained by combining the stock and the bond. Please show your detailed calculations or provide arguments to support your answers

Allocation to Stock	Allocation to Bond	Portfolio Mean	Portfolio Std Dev
0%	100%	10.0%	10.00%
25%	75%	11.25%	10.55%
50%	50%	12.50%	12.85%
75%	25%	13.75%	16.17%
100%	0%	15.0%	20.00%

a. What percentage of your wealth is allocated in the stock and bond at the minimum variance portfolio?

b. What percentage of your wealth is allocated in the stock and bond at the Sharpe -optimal portfolio?

c. Calculate the expected return and standard deviation for the Sharpe-optimal portfolio

d. What is the smallest expected loss for this portfolio over the coming year with a probability of 2.5% using the VaR?

Answer 1

Question a) - First of all find out the portfolio which is having minimum variance. Since, variance is square of standard deviation therefore, the portfolio with minimum variance will also have minimum standard deviation which is the first portfolio in the table above having portfolio standard deviation of 10%. Percentage of wealth allocated to stock is 0% and wealth allocated to bond is 100% in this portfolio. This is very much on the expected lines since bond gives fixed income stream in the form of coupon payments while stock is a high risk asset class with no certainty regarding the future income stream

Question b) talks about sharpe optimal portfolio. To find out this, we first need to understand what is sharpe ratio . Sharpe ratio is excess return over return of risk free assets divided by standard deviation of the portfolio.

To find out sharpe optimal portfolio, let us assume that the return of risk free asset is 5%

Now, let's find out sharpe ratio for reach of the portfolios given in the question

For first portfolio, we have sharpe ratio = (portfolio mean- return of risk free asset) / portfolio std dev

Therefore, we get, sharpe ratio= (10%-5%)/ 10%= 5%/10%= 0.5

second portfolio sharpe ratio =(11.25%-5%)/10.55%= 6.25%/10.55%=0.59

third portfolio sharpe ratio= (12.50%-5%)/12.85%= 0.58

fourth portfolio sharpe ratio= (13.75%-5%)/16.17%=0.54

fifth portfolio sharpe ratio= (15%-5%)/20%=0.50

As we can see from above, second portfolio has the best possible sharpe ratio of 0.59 and can thus be called as sharpe-optimal portfolio. In this portfolio, 25% allocation is given to stock while 75% is given to bonds

Question (c)- Since second portfolio has been identified as sharpe optimal portfolio therefore expected return of sharpe optimal portfolio is 11.25% (portfolio mean) and standard deviation is 10.55%

Question (d)- Smallest expected loss for the sharpe optimal portfolio over the coming year with the probability of 2.5% using VaR which can be said as 97.5% confidence interval

Now, z value for 97.5% confidence interval is -1.96

therefore the max loss that we can incur is portfolio standard deviation * z value for 97.5% confidence interval

max loss= 10.55%*1.96= 20.678%

Therefore the portfolio can go down by 20.678% (max expected loss) or up by 20.678% (max expected gain )

There is nothing called as minimum expected loss