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After-Tax Risk as measured by standard deviation is
σAT2 = (1 – tI ) 2 σI2
+ (1 – TtG ) 2 σG2 + (1 – tI
) (1 – TtG ) σI σG ρIG
σAT=
Total after-tax
standard deviation
σI =
Income before-tax standard
deviation
σG =
Capital Gains before-tax standard deviation
ρIG =
Income and Capital
Gains before-tax correlation
ti =
Income tax rate
tG =
Capital Gains tax rate
T =
Turnover
From a practical standpoint, estimating the Income and
Capital Gains correlation,
ρIG is difficult and
often inaccurate. In most cases,
simplifying to a Price Risk model or Reinvestment Risk model yields more
accurate results.
The Price Risk model assumes only the capital gains
return is random. This assumption is
useful for assets with long investment horizons, variable market prices and
relatively fixed income streams such as stocks.
After-tax Risk is
σAT = (1 – TtG ) σT
σT =
Total after-tax standard deviation
σT =
Total before-tax standard
deviation
tG =
Capital Gains tax rate
T =
Turnover
The Reinvestment Risk model assumes only the income
return is random. This assumption is
useful for assets with short investment horizons, variable income reinvestment
rates and relatively fixed prices such as T-bills.
After-tax Risk is
σAT = (1 – tI ) σT
σAT =
Total after-tax standard deviation
σT =
Total before-tax standard
deviation
ti =
Income tax rate
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