# Frequency curves and Histogram

For normal distribution with mean = 60 and standard deviation = 6, determine the probability content of the interval [62 to 67]

Question 2

The table below gives the deviation of a hypothetical portfolio's annual total return (gross of fees) from its benchmark's annual returns, for a 12-year period ending in 2003.

Portfolio's Deviation from benchmark return, 1992-2003

1992 -7.14%

1993 1.62%

1994 2.48%

1995 -2.59%

1996 9.37%

1997 -0.55%

1998 -0.89%

1999 -9.19%

2000 -5.11%

2001 -0.49%

2002 6.84%

2003 3.04%

(1) Calculate the frequency, cumulative frequency, relative frequency, and cumulative relative frequency for the portfolio's deviation from benchmark return, given the set of intervals in the table below

return interval frequency cumulative frequency relative frequency cumulative relative frequency

-9.19<A<-4.55

-4.55<B<0.09

0.09<C<4.73

4.73<D<9.37

(2) Construct a histogram using the data. Identify the modal interval of the grouped data. Tracking risk (also called tracking error) is the standard deviation of the deviation of a portfolio's gross-of-fees total returns from benchmark return. Calculate the tracking risk of the portfolio, stated in percent (give the answer to two decimal places).

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#### Solution Preview

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Question 1

For normal distribution with mean = 60 and standard deviation = 6, determine the probability content of the interval [62 to 67]

Answer

Let X be the random variable with mean 60 and standard deviation 6. That is, X ~ N (60, 6)

P (62 < X < 67) = 0.2478

Normal Probabilities

Common Data

Mean 60

Standard Deviation 6

Probability for a Range

From X Value 62

To X Value 67

Z Value for 62 0.333333333

Z Value for 67 1.166666667

P(X<=62) 0.6306

P(X<=67) 0.8783

P(62<=X<=67) 0.2478

Question 2

The table below gives the deviation of a hypothetical portfolio's annual total return (gross of fees) from its benchmark's annual returns, for a 12-year period ending in 2003.

Portfolio's Deviation from ...

#### Solution Summary

The solution provides step by step method for the calculation of frequency, cumulative frequency, relative frequency, cumulative relative frequency distributions and normal probabilities. The solution also provides step by step method for constructing histogram. Formula for the calculation and Interpretations of the results are also included.