Question

Consider three scenarios with the probabilities given below. Let the returns on two different stocks in these scenarios be as follows: Scenario Probability return K1 return K2 ω1 0.2 −10% −30% ω2 0.5 0% 20% ω3 0.3 10% 50% If a portfolio has 60% of funds invested in stock 1 and 40% of funds invested in stock 2, find the risk σV for this portfolio. (Need explanation not just spreadsheet)

Answer 1

	Weights	Scenario 1	Scenario 2	Scenario 3
Stock 1	0.6	-10%	0%	10%
Stock 2	0.4	-30%	20%	50%
Probability		0.2	0.5	0.3

Before finding the Risk of portfolio we need to find the correlation between these 2 stocks , their own volatility & expected return

Expected return for (Stock1)=E(S1)=0.2(-0.1)+0.5(0)+0.3(0.1)=0.01
Expected return for (Stock2)=E(S2)=0.2(-0.3)+0.5(0.2)+0.3(0.5)=0.19

Variance of Stock 1=E(S1^2)-E(S1)^2=0.2(-0.1^2)+0.5(0)+0.3(0.1^2)-(0.01)^2=0.0049

Variance of Stock 2=E(S2^2)-E(S2)^2=0.2(-0.3^2)+0.5(0.2^2)+0.3(0.5^2)-(0.19)^2=0.0769

Covariance between S1 and S2=E[(S1-E(S1))*(S2-E(S2))]

S1-E(S1) S2-E(S2) (S1-E(S1))*(S2-E(S2))
-0.11 -0.49 0.0539
-0.01 0.01 -0.0001
0.09 0.31 0.0279

Covarience of S1 and S2=E(S1-E(S1)*S2-E(S2))=0.0191

Correlation between S1 and S2=Covarience/SD(S1)*SD(S2)=0.0191/(0.07*0.277)=0.985

Now we are ready to calculate standard deviation of portfolio

w(S1)=0.6 ; w(S2)=0.4

SD of Portfolio=(w(S1)^2*(0.0049)+w(S2)^2*(0.0769)+2*correlation coefficient*w(S1)*w(S2)*SD(S1)*SD(S2))^0.5

=(0.6^2*0.0049+0.4^2*0.0769+2*0.985*0.6*0.4*0.07*0.277)^0.5=0.1524

Therefore SD of Portfolio is 15.24%