# Show that E(RX) = 10.5%, σX =

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 ofboom, normal & busteconomy are 0.25, 0.6 & 0.15, & you invest $6,000 in assetX & $4,000 in asset Z.

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

Answer:

Theinformation given in this case is various probabilistic scenariosand their corresponding returns .

In sucha case , we apply the following formulas for Expected return andExpected Standard Deviation

where pi representsthe individual probabilities in differentscenarios

Rirepresents the corresponding returnsin different scenarios and

represents the expected return calculated asabove.

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

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

ExpectedStandard 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%**

ExpectedStandard 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 betweenreturns of two stocks A and M is given by

·

· Where RAi are theindividual return of the stock A for probability piand

· RMi are the individualreturn of the Stock M for probability pi andandare theexpected returns of Stock A and Stock M as calculatedabove

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 thecomponent stocks. The portfolio is 60% invested in X and 40% inZ

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,

isthe standard deviation of returns of security i.

and isthe correlation coefficient between returns of security i andsecurity 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%**