Question

Show that E(RX) = 10.5%, σX = 3.12%; E(RZ) = 9.4%, σZ = 0.49%; & E(Rp) = 10.06%,
σp = 1.92%, for assets X & Z, predicted to return (15 & 10%) in booming, (10 & 9%)
in normal & (5 & 10%) in busting economy, if chances of boom, normal & bust
economy are 0.25, 0.6 & 0.15, & you invest $6,000 in asset X & $4,000 in asset Z.

I get E(Rp)= 10.06%, but when I try to get standard deviation I can't get 1.92%.

Answer 1

The information given in this case is various probabilistic scenarios and their corresponding returns .

In such a case , we apply the following formulas for Expected return and Expected Standard Deviation

  

where pi represents the individual probabilities in different scenarios

Ri represents the corresponding returns in different scenarios and

represents the expected return calculated as above.

Therefore , Expected Return of Stock X = 0.25*15%+ 0.6*10%+ 0.15*5%  = 10.5%

Similarly, Expected Return of Stock Z =0.25*10%+ 0.6*9%+ 0.15*10%  = 9.4%

Expected Standard Deviation of stock X

= sqrt[0.25*(0.15-0.105)2+0.6*(0.1-0.105)2+0.15*(0.05-0.105)2]

= sqrt (0.000975) = 0.031225 or 3.12%

Expected Standard Deviation of stock Z

= sqrt [0.25* (0.1-0.094)2 +0.6* (0.09-0.094)2 + 0.15* (0.1-0.094)2]

=sqrt (0.000024) = 0.004899 or 0.49%

· Covariance between returns of two stocks A and M is given by

·

· Where RAi are the individual return of the stock A for probability pi and

· RMi are the individual return of the Stock M for probability pi and and are the expected returns of Stock A and Stock M as calculated above

So, covariance between returns of X and Z

= 0.25*(0.15-0.105)*(0.1-0.094)+0.60*(0.10-0.105)*(0.09-0.094)+0.15*(0.05-0.105)*(0.1-0.094)

Now , The return of a portfolio is the weighted return of the component stocks. The portfolio is 60% invested in X and 40% in Z

So Return of this portfolio = 0.60 * 10.5% +0.40 *9.4% = 10.06%

The standard deviation of a portfolio is given by

Where Wi is the weight of the security i,

is the standard deviation of returns of security i.

and is the correlation coefficient between returns of security i and security j

So, standard deviation of portfolio

=sqrt (0.602*0.0312252+0.42*0.0048992+2*0.6*0.4*0.00003)

=sqrt(0.000369)

=0.019216 =1.92%