Question

In: Finance

Your portfolio is invested 20 percent each in A and C, and 60 percent in B....

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%

Solutions

Expert Solution

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%


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