Scenario Analysis Example
______State______ ___Probability___ Return Stock A Return Stock B
Boom 20% 15% 10%
Average 60% 5% -5%
Recession 20% -10% 20%
Step 1: Calculate the expected returns. What is the expected return formula we use?
Therefore,
E(rA) = (20%) * (15%) + (60%) * (5%) + (20%) * (-10%)
= 0.0300 + 0.0300 – 0.0200 = 0.0400
= 4.00 %
Similarly,
E(rB) = (20%) * (10%) + (60%) * (-5%) + (20%) * (20%)
= 0.0200 - 0.0300 + 0.0400 = 0.0300
= 3.00 %
Step 2: Calculate the variance and the standard deviation of returns.
I prefer the following formula:
We need to calculate E(rA2) and E(rB2) first.
E(rA2) = (20%) * (15%)2 + (60%) * (5%)2 + (20%) * (-10%)2
= 0.0045 + 0.0015 + 0.0020
= 0.0080
We know that Var (rA) = E(rA2) – (E(rA))2
Var (rA) = 0.0080 – (0.0400)2 = 0.0080 – 0.0016 = 0.0064
Stdev (rA) = [Var (rA)]1/2 = (0.0064)1/2 = 0.08 = 8%
Now let’s look at Stock B…
E(rB2) = (20%) * (10%)2 + (60%) * (-5%)2 + (20%) * (20%)2
= 0.0020 + 0.0015 + 0.0080
= 0.0115
We know that Var (rB) = E(rB2) – (E(rB))2
Var (rB) = 0.0115 – (0.0300)2 = 0.0115 – 0.0009 = 0.0106
Stdev (rB) = [Var (rB)]1/2 = (0.0106)1/2 = 0.1030 = 10.3%
Step 3: Calculate the covariance and the correlation of the two returns.
I prefer the following formula:
To calculate the covariance we need to first find E[(rA)*(rB)].
E[(rA)*(rB)]= (20%) * (15%) * (10%) + (60%) * (5%) * (-5%) + (20%) * (-10%) * (20%)
= 0.0030 - 0.0015 – 0.0040
= - 0.0025
Cov [ rA , rB ] = E[(rA)*(rB)] - E(rA) * E(rB) = -0.0025 – (0.0400) * (0.0300) =
= -0.0025 – 0.0012 = - 0.0037
Cov [ rA , rB ] = - 0.0037
What about the correlation?
ρ [ rA , rB ] = Cov [ rA , rB ] / [Stdev (rA) * Stdev (rB) ]
= 0.0038 / [ (0.08) * (0.103)] = -0.45