Question

State of Economy Boom Growth Stagnant Recession Probability of State 0.27 0.36 0.21 0.16 Return on Asset J in State 0.065 0.065 0.065 0.065 Return on Asset K in State 0.190 0.100 0.025 -0.150 Return on Asset L in State 0.280 0.200 0.050 -0.180

Answer 1

Parts a & b)

States	Probability	J	Probability Weighted Return	P(X - Expected return of J)^2
boom	0.27	6.50%	0.27x6.5%=1.755%	0.27(0.065-0.065)^2=0%
Growth	0.36	6.50%	0.36x6.5%=2.34%	0.36(0.065-0.065)^2=0%
Stagnant	0.21	6.50%	0.21x6.5%=1.365%	0.21(0.065-0.065)^2=0%
Recession	0.16	6.50%	0.16x6.5%=1.04%	0.16(0.065-0.065)^2=0%
Expected Return = sum of Probability Weighted Return			6.500%
Variance= sum of P(X - Expected return of J)^2				0.000%
Standard deviation =square root of variance				0.000%
States	Probability	K	Probability Weighted Return	P(X - Expected return of K)^2
boom	0.27	19.00%	0.27x19%=5.13%	0.27(0.19-0.09015)^2=0.2691906075%
Growth	0.36	10.00%	0.36x10%=3.6%	0.36(0.1-0.09015)^2=0.00349281%
Stagnant	0.21	2.50%	0.21x2.5%=0.525%	0.21(0.025-0.09015)^2=0.0891349725%
Recession	0.16	-1.50%	0.16x-1.5%=-0.24%	0.16(-0.015-0.09015)^2=0.17690436%
Expected Return = sum of Probability Weighted Return			9.015%
Variance =P(X - Expected return of K)^2				0.539%
Standard deviation = square root of variance				7.340%
States	Probability	L	Probability Weighted Return	P(X - Expected return of L)^2
boom	0.27	28.00%	0.27x28%=7.56%	0.27(0.28-0.1869)^2=0.23402547%
Growth	0.36	20.00%	0.36x20%=7.2%	0.36(0.2-0.1869)^2=0.00617796%
Stagnant	0.21	5.00%	0.21x5%=1.05%	0.21(0.05-0.1869)^2=0.39357381%
Recession	0.16	18.00%	0.16x18%=2.88%	0.16(0.18-0.1869)^2=0.000761760000000004%
Expected Return= sum of Probability Weighted Return			18.690%
Variance = sum of P(X - Expected return of L)^2				0.635%
Standard deviation = square root of variance				7.966%

Part c & d)

We need to find the covariance and the correlation, before we can calculate the portfolio expected return and standard deviation:

State	Probability	J	K	P(X - Expected return of J) x (X - Expected return of K)
boom	0.27	6.50%	19.00%	0.27(0.065-0.065)x(0.19-0.09015)=0%
Growth	0.36	6.50%	10.00%	0.36(0.065-0.065)x(0.1-0.09015)=0%
Stagnant	0.21	6.50%	2.50%	0.21(0.065-0.065)x(0.025-0.09015)=0%
Recession	0.16	6.50%	-1.50%	0.16(0.065-0.065)x(-0.015-0.09015)=0%
Expected return		6.50%	9.02%
Standard Deviation		0.00%	7.34%
Covariance = sum of (X - Expected return of J) x (X - Expected return of K)				0.000%
Correlation = Covariance/product of standard deviation)				0.00

State	Probability	J	L	P(X - Expected return of J) x (X - Expected return of L)
boom	0.27	6.50%	28.00%	0.27(0.065-0.065)x(0.28-0.1869)=0%
Growth	0.36	6.50%	20.00%	0.36(0.065-0.065)x(0.2-0.1869)=0%
Stagnant	0.21	6.50%	5.00%	0.21(0.065-0.065)x(0.05-0.1869)=0%
Recession	0.16	6.50%	18.00%	0.16(0.065-0.065)x(0.18-0.1869)=0%
Expected return		6.50%	18.69%
Standard Deviation		0.00%	7.97%
Covariance = sum of (X - Expected return of J) x (X - Expected return of L)				0.000%
Correlation = Covariance/product of standard deviation)				0.00

State	Probability	K	L	P(X - Expected return of K) x (X - Expected return of L)
boom	0.27	0.19	0.28	0.27(0.19-0.09015)x(0.28-0.1869)=0.250992945%
Growth	0.36	0.1	0.2	0.36(0.1-0.09015)x(0.2-0.1869)=0.00464526%
Stagnant	0.21	0.025	0.05	0.21(0.025-0.09015)x(0.05-0.1869)=0.187299735%
Recession	0.16	-0.015	0.18	0.16(-0.015-0.09015)x(0.18-0.1869)=0.01160856%
Expected return		0.09015	0.1869
Standard Deviation		0.073397735	0.079657956
Covariance = sum of (X - Expected return of L) x (X - Expected return of K)				0.45%
Correlation = Covariance/product of standard deviation)				0.78

Portfolio Expected return is calculated by solving the following equation:

Expected retur n portfolio = Weight jri+ Weightkrk + Weightır. Expected return portfolio = 0.2857 X 6.5 +0.7143 x 9.015 + 0.42 x 18.69 Expected return portfolio = 16.15%

Portfolio standard deviation is calculated by solving the following equation:

portfolio = V(Weight ,01)2+(Weight Ox)2+(Weight ++2Weight: 0; Weight KOK Xcorrelartion . +2Weight 0,Weight correlartion+2W eight KOK*Weight LOL Xcorrelartion KL

portfolio = V(0.2857 x0)2 +0.7143 x 7.34)2+(0.42x7.97)2 ++2(0.2857x0x0.7143 x 7.34 x 0)+2(0.2857 x0x0.42x7.97x 0)+2(0.42 x 7.97 x0.7143 x 7.34 x0.78)

Oportfolio = 8.13%