Value at Risk and Bank Capital Management

CHAPTER

♦

3

Market Risk

Chapter 1 defi ned value at risk (VaR) as the maximum potential loss that a business unit

or a position can generate in a given time horizon within a defi ned percentage of potential

scenarios (the so-called confi dence level), where extremely adverse scenarios are excluded.

In formal terms, if V is the value of a given portfolio, V

0

is its initial value, and 1 − a is

the desired confi dence level (e.g., 99%), then VaR

1−a

(i.e., the amount of loss that can be

exceeded only in a percentage of potential cases equal to a) is the amount such that

Pr(V − V

0

< −VaR

1−a

) = a

or alternatively, since V

0

− V represents a loss whenever V < V

0

,

Pr(V

0

− V > VaR

1−a

) = a

How can the risk manager estimate VaR in practice? Let us consider VaR for a trader

dealing in UK stocks and calculated on a daily horizon at a 99% confi dence level. If the

trader has invested in a basket of stocks that closely resembles the FTSE100, a simple

way would be to consider FTSE100 daily returns in the last 200 business days, sort them

in ascending order, and then identify the percentage VaR with the third-lowest return in

the sample. If, for instance, the three worst losses were −4.1%, −4.7% and −6.95%, the

maximum loss that can be excedeed only in 1% of cases is 4.1%. In fact, according to

historical experience, the index (and therefore the trader) may lose more than VaR only

in 1% of cases (i.e., two days out of a 200-day sample). This method is known as histori-

cal simulation, and the risk manager would implicitly be assuming that the distribution

of tomorrow’s returns could be reasonably approximated by the return distribution of the

last 200 days. Of course, it is crucial to decide how many days should be included in the

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