Question

Suppose we have the following projects on two stocks. Assume the correlation among the two assets returns is 0.6

State of Economy	Probability	Stock A	Stock B
Recession	0.2	-2%	6%
Slow	0.4	4	8
Average	0.4	12	19

a. Find the expected return and standard deviation of shares A and B.

b. Find the investment percentage (weights) needed in A and B shares to create the minimum variance portfolio.

Answer 1

Part A:

Expected Ret:

Expected Ret = Sum [ Prob * ret ]

Stock A:

Scenario	Prob	Ret	Prob * Ret
Recesion	0.2000	(0.0200)	(0.0040)
Slow	0.4000	0.0400	0.0160
Average	0.4000	0.1200	0.0480
Expected Ret			0.0600

Stock B:

Scenario	Prob	Ret	Prob * Ret
Recesion	0.2000	0.0600	0.0120
Slow	0.4000	0.0800	0.0320
Average	0.4000	0.1900	0.0760
Expected Ret			0.1200

Strandard Deviation:

Standard deviation is a measure of amount of variation or dispersion of set of values. It spcifies the risk of set of values.

SD = SQRT [ SUm [ Prob * (X-AVgX)^2 ] ]

Stock A:

State	Prob	Ret (X)	(X-AvgX)	(X-AvgX)^2	Prob * (X-Avg X)^2
Recesion	0.2000	(0.0200)	(0.0800)	0.006400	0.00128
Slow	0.4000	0.0400	(0.0200)	0.000400	0.00016
Average	0.4000	0.1200	0.0600	0.003600	0.00144
Sum[ Prob * ( X-AvgX)^2 ) ]					0.00288
SD = SQRT [ [ Sum[ Prob * ( X-AvgX)^2 ) ] ] ]					0.05367

SD is 5.37%

Stock B:

State	Prob	Ret (X)	(X-AvgX)	(X-AvgX)^2	Prob * (X-Avg X)^2
Recesion	0.2000	0.0600	(0.0600)	0.003600	0.00072
Slow	0.4000	0.0800	(0.0400)	0.001600	0.00064
Average	0.4000	0.1900	0.0700	0.004900	0.00196
Sum[ Prob * ( X-AvgX)^2 ) ]					0.00332
SD = SQRT [ [ Sum[ Prob * ( X-AvgX)^2 ) ] ] ]					0.05762

SD is 5.76%

Minimum Variance Portfolio or Optimal Risky Portfolio:

A minimum variance portfolio is a collection of securities that combine to minimize the price volatility of the overall portfolio. with the given weights to securities/ Assets in portfolio, portfolio risk will be minimal.

Weight in A = [ [ (SD of B)^2] - [ SD of A * SD of B * r(A,B) ] ] / [ [ (SD of A)^2 ]+ [ (SD of B)^2 ] - [ 2* SD of A * SD of B * r (A, B) ] ]
Weight in B = [ [ (SD of A)^2] - [ SD of A * SD of B * r(A,B) ] ] / [ [ (SD of A)^2 ]+ [ (SD of B)^2 ] - [ 2* SD of A * SD of B * r (A, B) ] ]

Particulars	Amount
SD of A	5.37%
SD of B	5.76%
r(A,B)	0.6000

Weight in A = [ [ (SD of B)^2] - [ SD of A * SD of B * r(A,B) ] ] / [ [ (SD of A)^2 ]+ [ (SD of B)^2 ] - [ 2* SD of A * SD of B * r (A, B) ] ]
= [ [ (0.0576)^2 ] - [ 0.0537 * 0.0576 * 0.6 ] ] / [ [ (0.0537)^2 ] + [ ( 0.0576 )^2 ] - [ 2 * 0.0537 * 0.0576 * 0.6 ] ]
= [ [ 0.00331776 ] - [ 0.001855872 ] ] / [ [ 0.00288369 ] + [ 0.00331776 ] - [ 2 * 0.001855872 ] ]
= [ 0.001461888 ] / [ 0.002489706 ]
= 0.587173

Weight in B = [ [ (SD of A)^2] - [ SD of A * SD of B * r(A,B) ] ] / [ [ (SD of A)^2 ]+ [ (SD of B)^2 ] - [ 2* SD of A * SD of B * r (A, B) ] ]
= [ [ (0.0537)^2 ] - [ 0.0537 * 0.0576 * 0.6 ] ] / [ [ (0.0537)^2 ] + [ ( 0.0576 )^2 ] - [ 2 * 0.0537 * 0.0576 * 0.6 ] ]
= [ [ 0.00288369 ] - [ 0.001855872 ] ] / [ [ 0.00288369 ] + [ 0.00331776 ] - [ 2 * 0.001855872 ] ]
= [ 0.001027818 ] / [ 0.002489706 ]
= 0.412827

Weight of Investment in Stock A = 58.72%

Weight of Investment in Stock A = 41.28%