In: Finance
Your portfolio is invested 20 percent each in A and C, and 60 percent in B. What is the standard deviation of the portfolio?
Rate of Return if State Occurs | ||||
State of Economy | Probability of State of Economy | Stock A | Stock B | Stock C |
Boom | .15 | .37 | .47 | .27 |
Good | .45 | .22 | .18 | .11 |
Poor | .35 | -.18 | -.22 | -.08 |
Bust | .05 | -.04 | -.07 | -.05 |
29.78%
21.36%
25.63%
10.96%
17.26%
Step 1: Calculation of Expected Return & Variance of all the three stocks
1. Calculation of Expected Return & Variance of Stock A
State | Probability | Return | Return * prob | dA | dA^2 | dA^2 * prob |
Boom | 0.15 | 0.37 | 0.0555 | 0.2805 | 0.07868025 | 0.011802038 |
Good | 0.45 | 0.22 | 0.0990 | 0.1305 | 0.01703025 | 0.007663613 |
Poor | 0.35 | (0.18) | -0.0630 | -0.2695 | 0.07263025 | 0.025420588 |
Bust | 0.05 | (0.04) | -0.0020 | -0.1295 | 0.01677025 | 0.000838513 |
Total (ER) | 0.0895 | Total () | 0.04572475 |
(I) Expected Return (ER(A)) = 0.0895 or 8.95%
where dA = Return - ER
(II) 0.04572475
2. Calculation of Expected Return & Variance of Stock B
State | Probability | Return | Return * prob | dB | dB^2 | dB^2 * prob |
Boom | 0.15 | 0.47 | 0.0705 | 0.3990 | 0.15920100 | 0.02388015 |
Good | 0.45 | 0.18 | 0.0810 | 0.1090 | 0.01188100 | 0.00534645 |
Poor | 0.35 | (0.22) | -0.0770 | -0.2910 | 0.08468100 | 0.02963835 |
Bust | 0.05 | (0.07) | -0.0035 | -0.1410 | 0.01988100 | 0.00099405 |
Total (ER) | 0.0710 | Total () | 0.059859 |
(I) Expected Return (ER(B)) = 0.0710 or 7.10%
where dB = Return - ER
(II) 0.059859
3. Calculation of Expected Return & Variance of Stock C
State | Probability | Return | Return * prob | dC | dC^2 | dC^2 * prob |
Boom | 0.15 | 0.27 | 0.0405 | 0.2105 | 0.04431025 | 0.006646538 |
Good | 0.45 | 0.11 | 0.0495 | 0.0505 | 0.00255025 | 0.001147613 |
Poor | 0.35 | (0.08) | -0.0280 | -0.1395 | 0.01946025 | 0.006811088 |
Bust | 0.05 | (0.05) | -0.0025 | -0.1095 | 0.01199025 | 0.000599513 |
Total (ER) | 0.0595 | Total () | 0.01520475 |
(I) Expected Return (ER(C)) = 0.0595 or 5.95%
where dC = Return - ER
(II) 0.01520475
Step 2 : Calculation of Covariance between all the stocks
1. Calculation of Covariance between Stock A & Stock B
State | Probability | dA | dB | dA*dB | dA*dB* Prob |
Boom | 0.15 | 0.2805 | 0.399 | 0.1119195 | 0.016787925 |
Good | 0.45 | 0.1305 | 0.109 | 0.0142245 | 0.006401025 |
Poor | 0.35 | -0.2695 | -0.291 | 0.0784245 | 0.027448575 |
Bust | 0.05 | -0.1295 | -0.141 | 0.0182595 | 0.000912975 |
Total | 0.0515505 |
Cov (A,B) = 0.0515505
2. Calculation of Covariance between Stock B & Stock C
State | Probability | dB | dC | dB*dC | dB*dC* Prob |
Boom | 0.15 | 0.399 | 0.2105 | 0.0839895 | 0.012598425 |
Good | 0.45 | 0.109 | 0.0505 | 0.0055045 | 0.002477025 |
Poor | 0.35 | -0.291 | -0.1395 | 0.0405945 | 0.014208075 |
Bust | 0.05 | -0.141 | -0.1095 | 0.0154395 | 0.000771975 |
Total | 0.0300555 |
Cov (B,C) = 0.0300555
3. Calculation of Covariance between Stock A & Stock C
State | Probability | dA | dC | dA*dC | dA*dC* Prob |
Boom | 0.15 | 0.2805 | 0.2105 | 0.05904525 | 0.008856788 |
Good | 0.45 | 0.1305 | 0.0505 | 0.00659025 | 0.002965613 |
Poor | 0.35 | -0.2695 | -0.1395 | 0.03759525 | 0.013158338 |
Bust | 0.05 | -0.1295 | -0.1095 | 0.01418025 | 0.000709013 |
Total | 0.02568975 |
Cov (A,C) = 0.02568975
Step 3 : Calculation of Standard Deviation of Portfolio
Weight of A (wA) = 0.2
Weight of B (wB)= 0.6
Weight of C (wC)= 0.2
0.04572475
0.059859
0.01520475
Cov (A,B) = 0.0515505
Cov (B,C) = 0.0300555
Cov (A,C) = 0.02568975
0.21360 approx or 21.36%
21.36%