# 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”. 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% USD VaR. 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% GBP VaR.

###### 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?