### Chapter 11

#### Historical Simulation

# 11.1 Motivation

One of the three “methods” early authors identified for calculating value-at-risk was called **historical simulation** or **historicalvalue-at-risk**. A contemporaneous description of historical simulation is provided by Linsmeier and Pearson (1996). Updated to reflect our terminology and notation, it reads:

The distribution of profits and losses is constructed by taking the

currentportfolio, and subjecting it to theactualchanges in the key factors experienced during each of the last α periods … Once the hypothetical mark-to-market profit or loss for each of the last α periods have been calculated, the distribution of profits and losses and the value-at-risk can then be determined.

Stated more formally, historical simulation employs the Monte Carlo method to calculate value-at-risk. But rather than construct a pseudorandom realization {^{1}*r*^{[1]}, ^{1}*r*^{[2]}, … , ^{1}*r*^{[m]}} of a sample for ^{1}** R**, it constructs {

^{1}

*r*^{[1]},

^{1}

*r*^{[2]}, … ,

^{1}

*r*^{[m]}} directly from historical data for

^{1}

**.**

*R*In Section 10.4, we discussed transformation procedures that employ the Monte Carlo method with pseudorandom realizations. If a transformation procedure employs the Monte Carlo method with historical realizations, we call it an **historical transformation procedure**. Historical simulation is then use of an historical transformation procedure to calculate value-at-risk.

Historical simulation is controversial because it is *ad hoc*. In any situation where it might be applied, a better result can be obtained using a pseudorandom realization {^{1}*r*^{[1]}, ^{1}*r*^{[2]}, … , ^{1}*r*^{[m]}}, especially if one employs variance reduction or selective valuation of realizations (both discussed in Section 10.5).

So why mention historical simulation? The unfortunate truth is that historical simulation is popular, at least among banks. Pérignon and Smith (2010) report that, of banks that disclosed their methodology for calculating value-at-risk in 2005, 73% used historical simulation. Most of the rest—14%—used value-at-risk measures with Monte Carlo transformation procedures.

In this chapter, we describe how to construct a realization {^{1}*r*^{[1]}, ^{1}*r*^{[2]}, … , ^{1}*r*^{[m]}} from historical data—and how to use it to calculate value-at-risk. We then provide context with a brief history of historical simulation. We review arguments that have been made to support the methodology, and we explain why, not surprisingly, historical simulation is inferior to more standard approaches for calculating value-at-risk.