Question

The average return for large-cap domestic stock funds over the three years 2009–2011 was 14.5%. Assume...

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)?

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Answer #1

Solution:

Given that,

  \mu = 0.145

\sigma = 0.047

a ) ( x  \geq 0.20 )

= 1 - p ( x  \leq 0.20)

= 1 - p ( x - \mu/\sigma)  \leq (0.20 - 0.145 /0.047)

= 1 - p( z  \leq 0.055 / 0.047 )

= 1 - p ( z \leq 1.17)   

Using z table

= 1 - 0.8790

= 0.1210

Probability = 0.1210

b ) p ( x <  0.10)

= p ( x - \mu/\sigma) <  (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 * \sigma + \mu

x = 1.282 * 0.047 + 0.145

x = 0.205254

x = 0.21

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