# 1.4 Value-at-Risk

Suppose an investment fund indicates that, based on the composition of its portfolio and on current market conditions, there is a 90% probability it will either make a profit or otherwise not lose more than USD 2.3MM over the next trading day. This is an example of a value-at-risk (VaR) measurement. For a given time period and probability, a value-at-risk measure purports to indicate an amount of money such that there is *that* probability of the portfolio not losing more than *that* amount of money over *that* time period. Stated another way, value-at-risk purports to indicate a quantile of the probability distribution for a portfolio’s loss over a specified time period.

To specify a **value-at-risk metric**, we must identify three things:

- The period of time over which a possible loss will be calculated—1 day, 2 weeks, 1 month, etc. This is called the
**value-at-risk horizon**. In our example, the value-at-risk horizon is one trading day. - A quantile of that possible loss. In the example, the portfolio’s value-at-risk is expressed as a .90 quantile of loss.
- The currency in which the possible loss is denominated. This is called the
**base currency**. In our example, the base currency is USD.

In this book, we measure time in units equal to the length of the value-at-risk horizon, which always starts at time 0 and ends at time 1. We adopt the following convention for naming value-at-risk metrics: the metric’s name is given as the horizon, quantile, and currency, in that order, followed by “VaR” or “value-at-risk”. If the horizon is expressed in days without qualification, these are understood to be trading days. The quantile *q* is generally indicated as a percentage. Based on this convention, the value-at-risk metric of the investment fund in our example above is one-day 90% USDvalue-at-risk. If a British bank calculates value-at-risk as the 0.99 quantile of loss over ten trading days, as required under the Basel Accords, this would be called 10-day 99% GBPvalue-at-risk.

###### 1.4.1 Probabilistic Metrics of Market Risk (PMMRs)

Value-at-risk is one example of a category of risk metrics that we might call **probabilistic metrics of market risk** (PMMRs). While this book focuses on value-at-risk, we shall see that the computations one performs to calculate value-at-risk are mostly identical to those you would perform to calculate any PMMR. In this section, we formalize the notion of value-at-risk metrics by first formalizing PMMRs. This will provide a general perspective for understanding value-at-risk in a context of other familiar market risk metrics.

Suppose a portfolio were to remain untraded for a certain period—say from the current time 0 to some future time 1. The portfolio’s market value at the start of the period is known. Its market value at the end of the period is unknown. It is a random variable. As a random variable, we may assign it a probability distribution conditional upon information available at time 0. We might quantify the portfolio’s market risk with some real-valued parameter of that conditional distribution.

Formally, we define a PMMR as a real-valued function of:

- the distribution of conditional on information available at time 0; and
- the portfolio’s current value .

Standard deviation of , conditional on information available at time 0, is an example:

[1.1]

Volatility, defined as the standard deviation of a portfolio’s simple return ^{1}*Z*, is a PMMR:

[1.2]

If we define portfolio loss as

[1.3]

then the conditional standard deviation of ^{1}*L* is a PMMR:

[1.4]

###### 1.4.2 Value-at-Risk as a PMMR

Let and denote cumulative distribution functions (CDFs) of ^{1}*P* and ^{1}*L*, conditional on information available at time 0. The preceding superscripts 1|0 are a convention to alert you that the distributions are “for random variables at time 1 but conditional on information available at time 0.”

If these conditional CDFs are strictly increasing, their inverses and exist and provide quantiles of ^{1}*P* and ^{1}*L*. As we have already indicated, value-at-risk metrics represent a *q*-quantile of loss ^{1}*L*, and this satisfies the definition of PMMR:

[1.5]

Recall that risk measures are categorized according to the metrics they support. Having defined value-at-risk metrics, we define value-at-risk as the category of risk measures that are intended to support value-at-risk metrics. If a risk measure is intended to support a metric that is a value-at-risk metric, then the measure is a **value-at-risk measure**. If we apply a value-at-risk measure to a portfolio, the value obtained is called a **value-at-risk measurement** or, less precisely, the portfolio’s value-at-risk.

To use a value-at-risk measure, we must implement it. We must secure necessary inputs, code software, and install the software on computers and related hardware. The result is a **value-at-risk implementation**.

###### Exercises

Which of the following represent PMMRs?

- conditional variance of a portfolio’s USD market value 1 week from today;
- conditional standard deviation of a portfolio’s JPY net cash flow over the next month.
- beta, as defined by Sharpe’s (1964) Capital Asset Pricing Model, conditional on information available at time 0.
- expected tail loss (ETL), which is defined as the expected value of a portfolio’s loss over a specified horizon, assuming the loss exceeds the portfolio’s value-at-risk for that same horizon.1

Using the naming convention indicated in the text, name the following value-at-risk metric: conditional 0.99 quantile of a portfolio’s loss, measured in GBP, over the next trading day.

As part of specifying a value-at-risk metric, we must indicate a base currency. This makes sense because value-at-risk indicates an amount of money that might be lost. It is measured in units of currency. But what about other PMMRs? Consider, for example, a 1-day standard deviation of simple return. A portfolio’s return is a unitless quantity; so is its conditional standard deviation of return. Must we specify a base currency for this PMMR?