The average return for large-cap domestic stock funds over the three years 2009–2011 was 14.5%. Assume the three-year returns were normally distributed across funds with a standard deviation of 4.7%.
a. What is the probability an individual large-cap domestic stock fund had a three-year return of at least 20% (to 4 decimals)?
b. What is the probability an individual large-cap domestic stock fund had a three-year return of 10% or less (to 4 decimals)?
c. How big does the return have to be to put a domestic stock fund in the top 10% for the three-year period (to 2 decimals)?
Solution:
Given that,
= 0.145
= 0.047
a ) ( x 0.20 )
= 1 - p ( x 0.20)
= 1 - p ( x - /) (0.20 - 0.145 /0.047)
= 1 - p( z 0.055 / 0.047 )
= 1 - p ( z 1.17)
Using z table
= 1 - 0.8790
= 0.1210
Probability = 0.1210
b ) p ( x < 0.10)
= p ( x - /) < (0.10 - 0.145 /0.047)
= p( z < -0.045 / 0.047 )
= p ( z < - 0.96 )
Using z table
= 0.1685
Probability = 0.1685
c ) P(Z > z) = 10%
1 - P(Z < z) = 0.10
P(Z < z) = 1 - 0.10 = 0.90
P(Z < 1.282 ) = 0.99
z = 1.282
Using z-score formula,
x = z * +
x = 1.282 * 0.047 + 0.145
x = 0.205254
x = 0.21
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