It is equally probable that stock A will have a +10% or -10% rate of return The only other possibility is that it will return 0%.The probability of 0% is twice that for a return of +10%. What is the expected return and Standard deviation of the return for stock A?
To calculate the expected return and standard deviation of the return for stock A, we need to use the probabilities associated with each possible return.
Let's denote the probability of a +10% return as P(+10%), the probability of a -10% return as P(-10%), and the probability of a 0% return as P(0%). We are given that the probability of a 0% return is twice that of a +10% return.
We can set up the following equations based on this information:
P(0%) = 2 * P(+10%) (equation 1)
P(0%) + P(+10%) + P(-10%) = 1 (equation 2)
Since there are only three possible returns, the sum of their probabilities must equal 1.
Now, let's solve the system of equations:
Substituting equation 1 into equation 2, we get:
2 * P(+10%) + P(+10%) + P(-10%) = 1
Combining like terms, we have:
3 * P(+10%) + P(-10%) = 1
Since the probability of a 0% return is twice that of a +10% return, we can rewrite equation 1 as:
P(+10%) = 0.5 * P(0%) (equation 3)
Substituting equation 3 into the previous equation, we get:
3 * (0.5 * P(0%)) + P(-10%) = 1
Simplifying, we have:
1.5 * P(0%) + P(-10%) = 1
Now, we can solve for P(0%) in terms of P(-10%):
P(0%) = 1 - P(-10%) (equation 4)
Substituting equation 4 into the previous equation, we get:
1.5 * (1 - P(-10%)) + P(-10%) = 1
Simplifying, we have:
1.5 - 1.5 * P(-10%) + P(-10%) = 1
Combining like terms, we get:
0.5 * P(-10%) = 0.5
Dividing by 0.5, we find:
P(-10%) = 1
Now that we have the probabilities for each return, we can calculate the expected return and standard deviation.
Expected return:
The expected return is calculated by multiplying each return by its respective probability and summing them up.
Expected return = (+10% * P(+10%)) + (0% * P(0%)) + (-10% * P(-10%))
= (+10% * 0.5) + (0% * 0.25) + (-10% * 0.25)
= +5% + 0% - 2.5%
= +2.5%
Therefore, the expected return for stock A is +2.5%.
Standard deviation:
The standard deviation of the return measures the volatility or dispersion of the returns around the expected return.
Standard deviation = sqrt( [ ( +10% - Expected return )^2 * P(+10%) ] + [ ( 0% - Expected return )^2 * P(0%) ] + [ ( -10% - Expected return )^2 * P(-10%) ] )
Plug in the values:
Standard deviation = sqrt( [ ( +10% - +2.5% )^2 * 0.5 ] + [ ( 0% - +2.5% )^2 * 0.25 ] + [ ( -10% - +2.5% )^2 * 0.25 ] )
Simplify and calculate:
Standard deviation = sqrt( [ ( 10% - 2.5% )^2 * 0.5 ] + [ ( -2.5% )^2 * 0.25 ] + [ ( -12.5% )^2 * 0.25 ] )
= sqrt( [ ( 7.5% )^2 * 0.5 ] + ( 2.5%^2 * 0.25 ] + [ ( 12.5% )^2 * 0.25 ] )
= sqrt( [ 0.5625 * 0.5 ] + [ 0.0625 * 0.25 ] + [ 1.5625 * 0.25 ] )
= sqrt( 0.28125 + 0.015625 + 0.390625 )
= sqrt( 0.6875 )
≈ 0.83
Therefore, the standard deviation of the return for stock A is approximately 0.83.