 ### Absolute Vs. Relative Loss

What is the difference between absolute loss and relative loss? What do they mean?

Absolute Loss refers to the probability of a given loss from the selected portfolio

Relative Loss is the probability of the selected portfolio underperforming the benchmark by a specified relative loss percentage. The benchmark can be defined in the setup section of the software. This measure may not be relevant to the individual client but it is common applications with money managers and institutional clients.

### Tracking Error

Tracking error measures the relative volatility of a portfolio to a specified benchmark. Similar to calculating absolute volatility, we use the portfolio’s active weight (excess weight to a benchmark) in the formula for portfolio variance.

### Probability of Loss (POL)

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 underperforming 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.

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%. 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.

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

### Value at Risk (VaR)

Value at risk (VaR) is a method of assessing risk that estimates the worst expected loss over an investment horizon at a given confidence level.

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

Absolute value at risk estimates the dollar value of under-performance of an investor’s portfolio and is a function of the portfolio’s return and risk. Relative value at risk estimates the dollar value of under-performance to a benchmark and is a function of excess return and risk (tracking error).

### Within – Horizon Value at Risk (VaR)

Asset returns vary throughout an investment time-horizon. Conventional value at risk only estimates total loss only at the end of an investment horizon without accounting for losses throughout the investment horizon. An investor may be very adverse to losses breaching a particular threshold and would therefore be interested in knowing the probability of breaching a certain level of loss at any moment during the horizon.

To estimate within-horizon variability, we use a statistic called “first-passage time probability”, which estimates the probability that an investment will breach a loss within a finite time-horizon. This method estimates the first time a particular path will breach the threshold. Multiple breaches of the threshold are considered only as a single breach.

Within-Horizon Value at Risk gives the worst outcome at a chosen probability from the beginning to any point in time during an investment horizon. There is no closed form (analytic) solution, the Windham Software uses a numerical approach to solve for this statistic.