Question

Probability                              

State of           of each State                           Rate of Return if State Occurs

The Economy  Occurring                                Walmart          Apple   

Boom              0.10                                         20%                 35%                 

Normal            0.55                                         1%                   7%       

Bust                 0.35                                         -10%                -25%    

Beta                                                                 0.32                 1.17

• What is the probability of achieving greater than a 6 percent return on each investment alternative?

Answer 1

Let X be a random variable which denotes return on investment, then X is assumed to follow normal distribution with mean M and standard deviation S

The standard normal variable is Z=(X-M)/S

For Investment on Walmart

E( return) =Mean=M=SUM[ return*probability]

=20%*0.1+1%*0.55-10%*0.35 = -0.95%

Volatility of return = Standard deviation of return =

S=SQRT{ SUM[( return-M)^2*probability]}

=SQRT{(20%-(-0.95%))^2*0.1+(1%-(-0.95%))^2*0.55+(-10%-(-0.95%))^2*0.35}

=SQRT{(20%+0.95%)^2*0.1+(1%+0.95%)^2*0.55+(-10%+0.95%)^2*0.35}

=8.64%

Now, probability of achieving more than 6% return on Walmart investment is

P(X>6%) = P(Z>(X-M)/S) = P(Z>(6%-(-0.95%))/8.64%) = P(Z>(6%+0.95%)/8.64%)

=P(Z>0.80)=1-P(Z<=0.80) = 1-0.7881=0.2119

For Investment on Apple

E( return) =Mean=M=SUM[ return*probability]

=35%*0.1+7%*0.55-25%*0.35 = -1.4%

Volatility of return = Standard deviation of return =

S=SQRT{ SUM[( return-M)^2*probability]}

=SQRT{(35%-(-1.4%))^2*0.1+(7%-(-1.4%))^2*0.55+(-25%-(-1.4%))^2*0.35}

=SQRT{(35%+1.4%)^2*0.1+(7%+1.4%)^2*0.55+(-25%+1.4%)^2*0.35}

=19.14%

Now, probability of achieving more than 6% return on Apple investment is

P(X>6%) = P(Z>(X-M)/S) = P(Z>(6%-(-1.4%))/19.14%) = P(Z>(6%+1.4%)/19.14%)

=P(Z>0.39)=1-P(Z<=0.39) = 1-0.6517=0.3413