The Windham Portfolio Advisor 99
Within-horizon exposure to loss
Both of these calculations pertain only to the distribution of values at the end
of the horizon and therefore ignore variability in value that occurs throughout
the horizon. To capture this variability, we use a statistic called first passage
time probability. 2 This statistic measures the probability ( P W ) of a first occur-
rence of an event within a finite horizon. It is equal to:
PN LT T
NLTTLW
[( ( ) ) /( )]
[( ( ) ) /( ( ) /ln
ln )]1
112μσ
μσ μσ√
√ ^^2 (4.3)^It gives the probability that an investment will depreciate to a particular value
over some horizon if it is monitored continuously.^3
Note that the first part of this equation is identical to the equation for the end
of period probability of loss. It is augmented by another probability multiplied by
a constant, and there are no circumstances in which this constant equals zero or
is negative. Therefore, the probability of loss throughout an investment horizon
must always exceed the probability of loss at the end of the horizon. Moreover,
within-horizon probability of loss rises as the investment horizon expands in
contrast to end-of-horizon probability of loss, which diminishes with time. This
effect supports the notion that time does not diversify all measures of risk and
that the appropriate equity allocation is not necessarily horizon dependent.
We can use the same equation to estimate continuous value at risk. Whereas
value at risk measured conventionally gives the worst outcome at a chosen
probability at the end of an investment horizon , continuous value at risk gives
the worst outcome at a chosen probability from inception to any time during
an investment horizon. It is not possible to solve for continuous value at risk
analytically. We must resort to numerical methods. We set Equation (4.3) equal
to the chosen confidence level and solve iteratively for L. Continuous value at
risk equals L times initial wealth.
Consider the implications of these risk measures on a hypothetical hedge
fund’s exposure to loss. Suppose this hedge fund employs an overlay strategy,
which has an expected incremental return of 4.00% and an incremental stand-
ard deviation of 5.00%. Further, suppose this hedge fund leverages the overlay
strategy. Table 4.1 shows the expected return and risk of the hedge fund and its
components for varying degrees of leverage.
These figures assume that the underlying asset is a government note with a
maturity equal to the specified 3-year investment horizon and that its returns are
uncorrelated with the overlay returns. Managers sometimes have a false sense of
security, because they view risk as annualized volatility, which diminishes with the
duration of the investment horizon. This view, however, fails to consider the like-
lihood that the fund’s assets may depreciate significantly during the investment
2 The first passage probability is described in Karlin and Taylor (1975).
3 See Kritzman and Rich (2001) for a discussion of its application to risk measurement.