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FAQs about StyleAdvisor Math

- Thomas Becker

Zephyr Associates, Inc. 2000

Purpose

The purpose of computing is insight, not

numbers. Richard Wesley Hamming

- Frequently asked questions often hint at deeper,

conceptual issues that cannot be explained in a

tech support conversation. - The purpose of this talk is to explore at least

one of your frequently asked questions to its

true depth.

Overview

- FAQs that I will try to address in this talk
- I have encountered two different definitions of

standard deviation. Which is the correct one, and

why? - I have encountered two different definitions of

annualized excess return. Which is the correct

one, and why?

Two Standard Deviations?

Which of the following two frequently encountered

definitions of standard deviation is the correct

one, or which is the better one under what

circumstances? stddev(x1, ?, xn) stddev(x1,

?, xn)

An Easy Example

Suppose we wish to gather statistical data on the

height of the people in this room. We would ask

everybody for their height to obtain a set of

numbers x1, ?, xn. The first and most obvious

statistic to compute is the mean, or average of

the data

Variability (Dispersion)

var(x1, ?, xn)

Variance is the mean of the squares of these

distances

?

?

?

?

mean

?

?

?

Sampling

Now suppose we wish to obtain statistical data

about the heights of the entire US population. It

is not realistic to obtain the height of every

person in the US. Instead, we must take one or

more random samples, calculate our statistics on

the sample data, and then draw conclusions on the

statistics for the entire population. Big

question What are the conclusions that can be

drawn from sample data? That is a very difficult

question, and it leads to a lot of highly

non-trivial mathematical theory.

Unbiased Estimators

Suppose we were to take every possible sample of

size 50 out of the entire US population. Suppose

further that we were to calculate the average

height for each of these samples, say m1, m2, ? .

Then the mean of all those sample means is the

true average height of the entire population. In

other words, if we measure peoples average

height on a larger and larger number of samples

of size 50, then in the long run, we will

approach the true average height of the

population. Mathematically The expected value of

the sample mean is the population mean. The

sample mean is an unbiased estimator of the

population mean.

Back To Variance

Suppose we were to take every possible sample of

size 50 out of the entire US population. Suppose

further that we were to calculate the variance of

peoples heights for each of these samples, say

m1, m2, ? . Then the mean of all those sample

variances is not the true variance of the entire

population. In other words, if we take a larger

and larger number of samples of size 50 and

calculate the variances, then on average, we will

not approach the true variance of the

population. The expected value of the sample

variance is not the population variance. The

sample variance is not an unbiased estimator of

the population variance.

Sample Vs. Population Variance

E(vars) varP wheren sample sizevars

variance of samplesvarP population variance In

other words, if we keep taking samples of size n

and calculating the variance on these samples,

then we will, on average, approach a number that

is a little smaller than the actual variance of

the entire population.

Unbiased Estimator For Variance

E(vars) varP E(vars) varP E(

vars) varP In other words, if we keep

taking samples of size n and calculating n/(n?1)

times the variance on these samples, then we

will, on average, approach the actual variance of

the entire population. The statistic n/(n?1)

times the variance is an unbiased estimator for

the population variance.

Putting It All Together

The one and only true definition of variance

isvar(x1, ?, xn) When the variance is

estimated via sampling, then the statistic that

must be calculated on the sample is var(x1, ?,

xn) ?

Variance In Portfolio Theory

When analyzing historical data of a portfolio

(money manager, mutual fund), which of the two

formulas should we use? The most widely held

point of view is that the sample variance should

be used (denominator n ? 1). However, there isnt

really any right or wrong here. The decision

which formula to use is a philosophical one.

Returns Sample or Population

The crucial question is When we look at

historical return data of a manager or mutual

fund, is this data

- the entire statistical population, or
- a sample of a larger population of data that

extends indefinitely into the future?

If your answer is a), then you must use the

formula for the actual variance (denominator n).

Else, you must use the formula for the sample

variance (denominator n ? 1).

Annualized Excess Return

The FAQ was I have encountered two different

definitions of annualized excess return. Which is

the correct one, and why?

Annualization

Let r1, ?, rn be a return series of any

periodicity. Then the total return (cumulative

return, compound return) of the series is r ?

The annualized return of the series is defined

as the constant annual return that would result

in the same compound return over the time period

covered by the series.

Annualized Excess Return

Let r1, ?, rn and s1, ?, sn be return series.

Then the excess return series is simply the

pointwise difference of the two (e1, ?, en) ?

(r1 ? s1, ?, rn ? sn) The following two

definitions of annualized excess return are

encountered AnnExRtn ? AnnExRtn ?

X

!

The Simplest Example Ever

Let us consider the monthly returns of a manager

and a benchmark for two months Obviously,

the annualized excess return of the manager over

the benchmark must be positive the manger made

money, the benchmark lost money!

Example (Cont.)

X

AnnExRtn ? ? (.9 ? 1.11)6 ? 1 ? ?

0.059850 AnnExRtn ? ((.91 ? 1.1)6 ? 1) ? ((1.01

? .99)6 ? 1) ? 0.0066

negative!

!