# Value at Risk

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 underperformance 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 underperformance to a benchmark and is a function of excess return and risk (tracking error).

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Since Value at Risk is a measure of capital loss, a negative value describes a gain.
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![Value at Risk for a Model Moderate Portfolio](https://2662147689-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-MEd8w0azNaE8v7NnoDC%2F-MF1CL31IJg6oaIUGfWJ%2F-MF1CVXO_Vf4K21MuhET%2Fimage.png?alt=media\&token=ff4f7c80-7971-42d2-8a2b-2a14be0ad436)

### Within-Horizon Value at Risk

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 Portfolio Advisor uses a numerical approach to solve for this statistic.

## Tail Risk (Conditional Value at Risk)

Value at Risk (VaR) is a forward looking estimate of the amount of money a portfolio could lose over a specified time horizon at a given probability.  Where VaR falls short is its inability to provide insight about how much the portfolio could lose beyond that threshold. The Windham Portfolio Advisor includes a tail risk measure, Conditional Value at Risk, to provide insight into this shortfall.

Conditional Value at Risk (CVaR) is the average loss beyond VaR in the tail.  It is the weighted average of the VaR estimate and the expected losses beyond VaR, it is not a measure of the most extreme loss.  A CVaR estimate will always be more extreme than a VaR estimate.
