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Within-Horizon Probability of Loss

Asset returns vary throughout an investment time-horizon. Conventional probability of loss only estimates total loss at the end of an investment horizon without accounting for an asset’s losses from the investment inception to end.  The conventional approach to risk measurement ignores intolerable losses that might occur throughout an investment period. An investor therefore would be interested in knowing the probability of a certain level of loss at any given moment during the horizon.

Example (Kritzman and Rich 2002)
Each line below represents a possible path of an investment of $100 through four periods. The horizontal line at 90% represents the loss threshold of 10%. Only one of the five paths breaches this threshold at the end of the horizon. Thus, the likelihood of a 10% loss,  , is 20%. If instead we consider any point within the time-horizon, four of the five paths breach the investment time-horizon. The likelihood of a 10% loss,  , is 80%.

withinHorizonTime.png

To estimate within-horizon variability, we use a statistic called “first-passage time probability, “which estimates the probability that the asset will breach the value at risk threshold, L, within a finite time-horizon.

The likelihood of an end-of-horizon loss diminishes with time; the likelihood of a within-horizon loss never diminishes as a function of the length of the horizon (It increases at a decreasing rate but never decreases). Only the first breach in the threshold is counted; once a path crosses the threshold line it counts toward the probability of the investment breaching the threshold within the time-horizon.

Related Articles

  1. Kritzman, M. and Risk, D., The Mismeasurement of Risk, Financial Analysts Journal, May/June 2002


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