8. Thr ee securities have the following expected returns and standard deviations

Question

8. Thr ee securities have the following expected returns and standard deviations: Security Parameter Expected return 115% 20% 35% Standard deviation 20% 40% 70% Also, their correlation coefficients are given by: P12 0.5, P13 0.7, and p23 0.9. (a) If you invest 30% of your wealth in security 1, 40% in security 2, and 30% in secu- (b) Suppose that you wanted to form a portfolio that has an expected return of 30% rity 3, what will be the expected return and the standard deviation of this portfolio? and a variance of 0.35, how would you proceed? NOTE: You do not have to solve for the portfolio weights; just explain what system of equations you would have to solve, and how you would use your solution to this system.

Explanation / Answer

Return of Security 1( R1) = 15% , Return of Security 2 (R2) = 20%,  Return of Security 3 (R3) = 35%
Standard Deviation of Security 1( 1 ) = 15% , Standard Deviation of Security 2( 2 )= 20%,  Standard Deviation of Security 3( 3 )= 35%
a) Weight w1 = 0.3 , w2 = 0.4 , w3 = 0.3
Expected return = w1* R1 + w2* R2 + w3* R3 = 0.3 * 15% + 0.4 * 20% + 0.3 * 35% = 0.23 or 23%
Standard deviation of portfolio
= [(w1* 1)2 + (w2* 2)2+ (w3* 3)2 + 2 w1* 1 *w2* 2 * 12 + 2 w3* 3 *w2* 2 * 23 + 2 w1* 1 *w3* 3 * 13]0.5 =  [(0.3* 20%)2 + (0.4* 40%)2+ (0.3* 70%)2 + 2 *0.3* 20% *0.4* 40% * 0.5 + 2 * 0.3* 70%*0.4* 40% * 0.9  + 2 * 0.3* 20% *0.3* 70% * 0.7]0.5 =.4013or 40.13 %

b)  
w1* R1 + w2* R2 + w3* R3 =w1 * 15% + w2* 20% + w3 * 35% = 30% (Equation 1)
Expected Standard Deviation =
[(w1* 1)2 + (w2* 2)2+ (w3* 3)2 + 2 w1* 1 *w2* 2 * 12 + 2 w3* 3 *w2* 2 * 23 + 2 w1* 1 *w3* 3 * 13]0.5 = 35%
[(w1* 0.2)2 + (w2* 0.4)2+ (w3* 0.7)2 + 2 w1* 0.2 *w2* 0.4 * 0.5 + 2 w3* 0.7  *w2* 0.4 * 0.9  + 2 w1* 0.2 *w3* 0.7 * 0.7]0.5 = 35%
0.04 w1  + 0.16 w2 + 0.49 w3 + 0.08 w1 w2 + 0.504 w2w3 + 0.196 w1w3 = 35% (equation 2)
w1 + w2 + w3   = 1 ( equation 3)

By finding 3 weights they can be placed in expected return and standard deviation formula to get expected return and standard deviation of portfolio.

Thank You. God Bless

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