FAQs about StyleAdvisor Math

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FAQs about StyleAdvisor Math

• Thomas Becker

Zephyr Associates, Inc. 2002
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Overview

• FAQ that I will address in this talk
• How do we calculate alpha?

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How Do We Calculate Alpha?
When you try to replicate StyleAdvisors alpha
(e.g., in Excel), you may get a different value.
There are two possible reasons for that.

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Annualization
We annualize alpha. Alpha is by nature a return,
and therefore one can make it comparable across
analyses with different periodicities by
annualizing it

alphaann (1 alpha)NumPeriodsPerYear ? 1
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Subtracting Cash
In the following graphs and tables, we subtract
cash returns from the manager returns when
calculating alpha (vs. market benchmark and vs.
style benchmark)

• Risk/Return Table
• Manager Vs. Benchmark Graph and Table
• Manager Vs. Universe Graph and Table
• Custom Axis Graph

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More about Alpha
A managers alpha vs. his or her benchmark is
often intuitively viewed as a measure of the
excess return of the manager over the benchmark.
(Positive alpha is the essence of life)

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Positive Alpha Is The Essence
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Or Is It?
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Performance Graph
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Alpha Vs. Excess Return
In this example, the benchmark made about 2.5,
the manager lost almost 17, and yet the
managers alpha vs. the benchmark is a whopping
13.79. Does this mean that alpha is evil? No,
it does not. But there are two major caveats
about alpha.
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Linear Regression Intercept
Manager Returns
alpha
Benchmark Returns
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Alpha Requires a Good R2
If the points in the scatterplot do not roughly
lie on a straight line, then linear regression
line is basically a random line, and accordingly,
alpha and beta are random values. It can be shown
that the points in the scatterplot lie close to a
straight line if and only if the correlation
between the manager and the benchmark is high.
Caveat 1 Do not even look at alpha and beta
unless you have a good R2.
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Back to the Example
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Scatterplot of the Example
Total Stock Market Returns
T-Bill Returns
In this case, the regression line could lean to
the left (large positive alpha), or it could lean
to the right (large negative alpha). Alpha is
basically a random number.
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Calculation of Alpha and Beta 1
beta cov(manager, benchmark) / var(benchmark)
If we define the residual series as residual
manager ? beta benchmark then manager beta
benchmark residual Moreover, it can be shown
that corr(manager, residual) 0 and for all
constants c ? beta, var(residual) lt var(manager
? c benchmark)
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Calculation of Alpha and Beta 2
Alpha is defined as the arithmetic mean of the
residual series. Therefore
manager alpha beta benchmark
rest (1) where rest is a return series
with corr(manager, rest) 0 and mean(rest)
0 Moreover, alpha and beta are unique with these
properties.
Calculating alpha and beta is tantamount to a
decomposition of the manager series into three
summands a constant, a scalar multiple of the
benchmark, and an uncorrelated rest with zero
mean.
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Back To Linear Regression
Under the assumption that the rest term in
equation (1) above is small, we get manager ?
alpha beta benchmark (2) Moreover, as
mentioned above, beta is the number that
minimizes the variance of the rest term in
equation (1). Since the rest term has zero mean,
this is equivalent to minimizing the sum of the
squares of the rest term. Therefore, equation (2)
is the solution to the least squares linear
regression problem.
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Linear Regression And R2
manager alpha beta benchmark
rest (1) manager ? alpha beta
benchmark (2) It can be shown that (
corr(manager, benchmark) )2 1 ? ( stddev(rest)
/ stddev(manager) )2 Therefore, a low
correlation between manager and benchmark
corresponds to a high standard deviation of the
rest term. That is why in the presence of low
correlation between manager and benchmark, the
approximation (2) above is poor, and therefore,
alpha and beta are largely meaningless.
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Alpha Is an Arithmetic Mean
Equation (2) above states that manager ? beta
benchmark ? alpha Therefore, it is true that
alpha represents an excess return, namely, the
excess return of the manager over beta times the
benchmark. But
Caveat 2 Alpha is the arithmetic mean of the
excess return of manager over beta times the
benchmark. An arithmetic mean does not reflect
the actual compounding of money over time.
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The Meaning Of Alpha
The following statement describes the meaning of
alpha
If it is known that manager and benchmark returns
are well correlated (i.e., the historical returns
hug a straight line in the scatterplot), then it
is possible to extrapolate via linear regression
whenever the benchmark return is x in a
particular period, the manager return y is
approximately y ? alpha beta x Notice that
this extrapolation is not necessarily a
prediction of the future. Time is surpressed in
the linear regression view.
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Are Arithmetic Means Evil?
No. Arithmetic means of returns and excess
returns have a legitimate place in Modern
Portfolio Theory. However, it is important to
bear in mind that the arithmetic mean of a return
series or excess return series is never an
adaquate description of the actual, compounded
return or excess return that was achieved by a
manager. Arithmetic means often occur in
extrapolation methods. These methods can be
applied to historical returns to explore what
will happen in the future under certain
assumptions. Linear regression is an example of
such an extrapolation method.