Question

1. Risky Asset A and Risky Asset B are combined so that the new portfolio consists of 70% Risky Asset A and 30% Risky Asset B.  If the expected return and standard deviation of Asset A are 0.08 and 0.16, respectively, and the expected return and standard deviation of Asset B are 0.10 and 0.20, respectively, and the correlation coefficient between the two is 0.25:
2. 1. What is the expected return of the new portfolio consisting of Assets A & B in these proportions?

2. What is the standard deviation of this portfolio?

3. Assuming a riskless rate of 0.06, what are the proportions of these two securities in their optimal combination of risky assets? What is the expected return of this portfolio combination?

4. Assuming this optimal combination of risky assets is then combined with the riskless asset which has a return of 0.06, what standard deviation would you have to tolerate if you wanted to earn a rate of return of 0.09 from this new portfolio?

5. Again assuming this optimal combination of risky assets is combined with the riskless asset, suppose you have $100,000 to invest and you choose a preferred portfolio consisting of 60% risky assets and 40% riskless assets.  Under these parameters, how much of your $100,000 would you need to invest each in Asset A, Asset B, and the riskless asset?

please show work

Answer 1

a) Return A = 0.08; Return B = 0.1; S.D A = 0.16; S.D B = 0.2; Correlation coefficient = 0.25; Weight A = 0.7; Weight B = 0.3.

Expected return of the portfolio = (Weight A * Return A) + (Weight B * Return B)

= (0.7*0.08) + (0.3*0.1) = 0.056+0.03 = 0.086 or 8.6%

b) Standard deviation (SD) of the portfolio = Square root of {[(weight A)2*(SD A)2] + [(weight B)2*(SD B)2] + (2*SD A*SD B*Weight A*Weight B*Correlation coefficient)}

= Square root of {[(0.7)2 *(0.16)2] + [(0.3)2 *(0.2)2] + (2*0.16*0.2*0.7*0.3*0.25)}

= Square root of {(0.49*0.0256) + (0.09*0.04) + 0.00336}

= Square root of {0.012544 + 0.0036 + 0.00336}

= Square root of {0.019504}

= 0.1397 or 13.97%

c)

Weight A = [(0.2)2 - (0.25*0.16*0.2)]/[(0.16)2+(0.2)2-(2*0.25*0.16*0.2)]

= [0.04 - 0.008]/[0.0256+0.04-0.016]

= 0.032/0.0496

= 0.65 or 65%

Weight B = 1-weight A

=1-0.65 = 0.35 or 35%

Expected return of the optimal combination portfolio = (Weight A * Return A) + (Weight B * Return B)

= (0.65*0.08) + (0.35*0.1) = 0.052+0.035 = 0.087 or 8.7%

d) Required rate of return = 0.09; return on optimal combination of risky assets = 0.087; return on riskless asset = 0.06; Assumed weight of the optimal combination of risky assets = "x" weight of riskless asset = "1-x"

Required rate of return = (weight of optimal combination of risky assets * return on optimal combination of risky assets) + (weight of riskless asset * return on riskless asset)

0.09 = (x*0.087)+([1-x]*0.06)

0.09 = 0.087x + 0.06 - 0.06x

0.087x-0.06x = 0.09-0.06

0.027x = 0.03

x = 0.03/.027

x = 1.11

1-x = 1-1.11 = -0.11

Proportion of optimal combination of risky assets is 1.11 & proportion of riskless asset is -0.11 to earn 9% return

SD of optimal combination portfolio = Square root of {[(weight A)2*(SD A)2] + [(weight B)2*(SD B)2] + (2*SD A*SD B*Weight A*Weight B*Correlation coefficient)}

= Square root of {[(0.65)2 *(0.16)2] + [(0.35)2 *(0.2)2] + (2*0.16*0.2*0.65*0.35*0.25)}

= Square root of {(0.4225*0.0256) + (0.1225*0.04) + 0.00364}

= Square root of {0.010816 + 0.0049 + 0.00364}

= Square root of {0.019356}

= 0.1391 or 13.91%

SD to earn 9% return = Proportion of optimal combination of risky assets * SD of optimal combination of risky assets

= 1.11 * 0.1391

= 0.1544 or 15.44% would have to tolerate to earn 9% return on combination of riskfree assets.

e)

Total investment = $100,000

First to invest in riskfree asset = $100,000 * 40% (as given) =$40,000

Balance to invest in optimal combination of risky assets = $100,000 * 60% = $60,000

Investment in Asset A = $60,000 * 65% (refer part c) = $39,000

Investment in Asset B = $60,000 * 35% (refer part c) = $21,000