Modern Portfolio Theory (MPT) Statistics

Beta (continued)
Beta is calculated as:
Cov
β
r
=
σ
rb
2
b
where:
β
r
= Beta of portfolio r
Cov
rb
= Covariance between the excess returns of the portfolio r and the benchmark b
σ = Variance of the excess returns of the benchmark
2
b
and:
Cov
=
1 n
e
e
e
rb ∑[(
R − )( −
e
i
R Bi
B )]
n -1 i=
1
where:
e
R
i
= Excess return of the portfolio for month i = R i - RF i , where R i is the portfolio return for
month i and RF i is the risk-free return for month i
e
R = Average monthly excess return of the portfolio over n periods (simple mean)
e
B
i
= Excess return of the benchmark for month i = B i - RF i , where B i is the benchmark
return for month i and RF i is the risk-free return for month i
e
B = Average monthly excess return of the benchmark index over n periods (simple mean)
n = number of periods (Morningstar typically uses 36 months)
e
R is the simple arithmetic average excess return for the portfolio:
R
e
=
1
n
n
∑
i=
1
R
e
i
The denominator for beta is the variance of the excess returns of the benchmark:
σ
2
b
=
1 n
e
∑(
B −
e
i
B
i=
1
n -1
)
2
2
A similar calculation can also be used for the variance of the portfolio, σ r
.
Standard deviation is the square root of variance.
Morningstar MPT Statistics| July 31, 2008
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