# Probability of Loss

Probability of loss is used to determine the likelihood of a specified loss or gain over an investment horizon. Instead of evaluating the monetary loss or gain at a given confidence, an investor determines the probability that a specified monetary loss or gain will occur.

Probability of loss uses the expected distribution of returns in order to estimate potential loss. We estimate probability of loss from a portfolio’s expected return and standard deviation under the assumption that the portfolio’s returns are log-normally distributed.

Absolute probability of loss estimates the likelihood of an investor’s portfolio incurring a specified absolute loss. Relative probability of loss estimates the likelihood of under performing the benchmark by a given amount.

### 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.

Each line in the illustration above 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%.

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 (Kritzman and Rich 2002).

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.

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