# 1.8 Value-at-Risk Measures

In the previous section, we described several market risk measures, most of them value-at-risk measures. Despite a disparity in modeling techniques, our treatment was standardized. Certain concepts recurred. You are now familiar with notation such as **ω**, , θ, , , and ^{1|0}**Σ**.

We have many value-at-risk measures to consider. Before long, we will stop describing entire value-at-risk measures and start describing stand-alone components of value-at-risk measures—much as auto enthusiasts might discuss types of brakes or fuel injectors without having a particular automobile in mind. In this sense, our discussions will have a “building block” quality. We don’t want every value-at-risk measure to be a unique monolith standing on its own. Instead, we will treat them as modular. Avoiding the top-down approach of discussing Toyotas, Fords, and Mercedes, we will take a bottom-up approach, discussing fuel injectors, suspension systems, and brakes. To this end, we must identify the essential components that make up any value-at-risk measure—indeed, any PMMR measure. In doing so, we will lay out a framework for much of this book.

###### 1.8.1 Risk Factors

A risk factor is any random variable whose value will be realized during the interval (0,1] and will affect the market value of a portfolio at time 1. A risk vector is a random vector of risk factors. If a risk vector reflects a future value of some time series, we may speak of its current value ^{0}* q* or historical values

^{0}

*,*

**q**^{–1}

*,*

**q**^{–2}

*,*

**q**^{–3}

*…*

**q**One particular risk factor and two risk vectors play important roles in value-at-risk measures. We give them special names and notation. These are:

- the portfolio’s future value ,
- the asset vector , and
- the key vector .

The **portfolio’s future value** represents the market value at time 1 of the portfolio for which value-at-risk is to be measured. The portfolio is assumed fixed in the sense that it will not be traded during the period [0,1] and no assets will be added or withdrawn. This does not preclude traders or portfolio managers from trading! It simply means that a value-at-risk measure quantifies the market risk of a portfolio based upon its composition at time 0. The value-at-risk measure can recognize changes in the portfolio’s composition during the period [0,1] due to planned events such as options expiring, dividends being paid, or scheduled payments being made on a swap. We are interested in the portfolio’s current value because value-at-risk metrics depends upon it. We generally do not consider or attempt to define prior historical portfolio values.

**Asset vector** has **asset values** as components. These represent accumulated values of specific assets that may make up a portfolio. Realizations may be negative, so our definition recognizes no accounting distinction between assets and liabilities. Accumulated value is denominated in the base currency employed by the value-at-risk metric. It may reflect such variables as capital gains, dividends, coupons, margin payments, reinvestment income, storage costs, insurance, financing, changes in exchange rates, leasing income, etc.

Mathematically, we define a **portfolio** as a pair (, ) where the constant is the portfolio’s current value, and the random variable is the portfolio’s future value. Similarly, we mathematically define an **asset** as a pair (, ), where is the asset’s current value, and is the asset’s future value.

We have considerable leeway in how we select what financial instruments to represent with assets. This may affect value-at-risk results. Consider an investor who borrows EUR 100,000 and invests it in Hoechst stock. We might model the portfolio three different ways:

- as comprising holdings in two assets whose values
^{1}*S*_{1}and^{1}*S*_{2}represent the accumulated values of the stock and the financing, respectively; - as comprising a single asset whose value
^{1}*S*_{1}represents the accumulated value of the stock less the accumulated value of its financing; - as comprising a single asset whose value
^{1}*S*_{1}represents the accumulated value of the stock.

The first two representations are financially equivalent. One approach (probably the first) will be computationally easier to work with, but both will result in the same value-at-risk. The third representation is different. It excludes financing from the portfolio. With the third approach, the random variable represents something different than it does with the first two approaches.

As we shall see, every value-at-risk measure must directly characterize a conditional probability distribution for some vector of risk factors, such as prices, interest rates, spreads, or implied volatilities. Those risk factors are called **key factors**. They are the components of the **key vector** . Occasionally, we use asset values as key factors. This was the case in our examples of Leavens’ PMMR and the value-at-risk measure for industrial metals. We explore the role of key factors in more detail shortly.