Optimizing Optimization: The Next Generation of Optimization Applications and Theory (Quantitative Finance)

100 Optimizing Optimization

horizon and thereby trigger withdrawals from the fund. Exhibit 6 compares the
likelihood of a 10% loss at the end of the 3-year horizon with its likelihood at
some point within the 3-year horizon. It reveals that although the chance of a
10% loss at the end of the horizon is low, there is a much higher probability that
the fund will experience such a loss at some point along the way, which may trig-
ger withdrawals and threaten the fund’s solvency ( Figure 4.4 ).
The same issue applies if exposure to loss is perceived as value at risk. Figure 4.5
shows the hedge fund’s value at risk measured conventionally and continuously.
Whereas conventional value at risk for leverage ratios less than 10 to 1 is negative
(a gain) and still very low for leverage ratios up to 15 to 1, continuous value at
risk ranges from more than one-fourth the portfolio’s value to nearly half its value.
These examples illustrate rather vividly that investors are exposed to far
greater risk throughout their investment horizon than end-of-horizon risk
measures indicate. How should investors respond to this reassessment of risk?
Investors should not necessarily reduce risk, although such a course of action
may sometimes be warranted. Rather, they should at least be aware of the like-
lihood that they will be unable to sustain their investment strategies. Moreover,
investors should be informed of their within-horizon exposure to loss so that

0%

10%

20%

30%

40%

50%

60%

70%

80%

5:1 7.5:1 10:1 15:1

Leverage

End of period During period

Figure 4.4 Probability of insolvency, 3-year lockup period.

Table 4.1 Leveraged hedge fund expected return and risk

Underlying
asset

Overlay

strategy

Leverage

2

Leverage

4

Leverage

6

Leverage

8

Leverage

10

Expected
return

3.50% 4.00% 11.50% 19.50% 27.50% 35.50% 43.50%

Standard
deviation

3.00% 5.00% 10.44% 20.22% 30.15% 40.11% 50.09%