Value at Risk: A Worked Example
At the recent PAC hearing on the sale of part
of the UK gold reserves, the PAC Chairman, David Davies, asked
for a note on value at risk. The following describes the value
at risk concept and provides a worked example for two different
levels of gold holding within the UK net reserves.
Linsmeier and Pearson (1996) describe the concept
of value-at-risk as, "Value at risk is a single, summary
statistical measure of possible portfolio losses. Specifically,
value at risk is a measure of losses due to `normal' market movements.
Losses greater than the value at risk are suffered only with a
specified small probability. Subject to the simplifying assumptions
used in its calculation, value at risk aggregates all of the risks
in a portfolio into a single number suitable for use in the boardroom,
reporting to regulators, or disclosure in an annual report. Once
one crosses the hurdle into using a statistical measure, the concept
of value at risk is straightforward to understand. It is simply
a way to describe the magnitude of the likely losses on the portfolio".
Kevin Dowd in his book "Beyond value at
risk" states that, "in its most literal sense, VaR refers
to a particular amount of money, the maximum amount we are likely
to lose over some period, at some specific confidence level".
In other words, since risk reflects uncertain
outcomes, it can usefully be described in terms of probabilities.
Value-at-risk uses this approachit is an estimate, with
a given confidence level (ie some statistical probability of changes
in price), of the loss that might be expected in a portfolio if
positions are held over some set time horizon. It answers the
question "how much can I lose with x per cent probabillity
over a given time horizon".
There are three methods of calculating value
(1) Numerical value at riskwhereby
the historical movements of the portfolio as a whole are used
to calculate the value at risk,
(2) Parametric value at riskwhereby
the relationships (variance and covariances) between the assets
held in the portfolio are estimated and the value at risk is calculated
using this information, and
(3) Monte-Carlo simulationwhereby
the value at risk is derived by simulating the relationships between
the assets held in the portfolio.
The numerical method is the more intuitive of
the three approaches. The following outlines how the value at
risk might be calculated for a hypothetical portfolio using the
numerical example. We then go on to describe, using the Parametric
approach (the method most commonly used in the financial markets),
how the value at risk would be calculated for the net reserves.
Daily returns on a portfolio over time, say
a year, are likely to be distributed broadly in line with those
shown in figure 1. There may be a number of occasions when the
portfolio made large positive returns and a number of occasions
when the portfolio made large negative returns but the majority
of observations will tend to lie towards the middle of the distribution,
around or just above zero.
Given that we are talking about risk (Value
at risk) we are only concerned with losses and, given that we
are considering maximum expected losses subject to a high confidence
limit, essentially with large losses, ie the observations in the
left hand tail. From the hypothetical data shown in figure 1 we
can see that losses of US$30 million, US$26 million and US$24
million were made on one occasion each.
In determining the value at risk it is first
necessary to decide on a cut off point for the confidence interval,
for example 99 per cent. If we choose a 99 per cent confidence
interval then we are implicitly saying that we would not expect
to lose more than the value at risk figure on all but three (1
per cent of trading days) trading days per year.
Value at risk is the maximum expected loss over
a given horizon period at a given level of confidence. In the
above example, the maximum expected daily loss, with a 99 per
cent confidence interval, is US$24 million.
A WORKED EXAMPLE
UK NET RESERVES
Consider the case of a portfolio worth $14 billion
where gold makes up 50 per cent of the portfolio (ie $7 billion)
and the remainder is split 40:40:20 into $, Euro and Yen (ie $2.8
billion is held in dollars, $2.8 billion is held in euro and $1.4
billion is held in Yen).
If the portfolio is managed in $, there are
three currency risksthe risk arising from movements in
the dollar-gold exchange rate, the risk arising from movements
in the dollar-euro exchange rate and the risk arising from movements
in the dollar-yen exchange rate.
The variance-covariance matrix for a ten day
holding period is:
|Yen||(0.053 per cent||0.004 per cent
||0.017 per cent)|
|Euro||(0.004 per cent
||0.042 per cent||0.012 per cent)
|Gold||(0.017 per cent
||0.012 per cent||0.054 per cent)
(Data on variances and covariances are based on historical
movements in asset prices and are published by a number of firms,
for instance by JP Morgan as part of its RiskMetrics service.)
So this portfolio's value-at-risk is $459 million, ie this
is the maximum amount we could expect to lose (with a 99 per cent
confidence interval) over the next 10 days.
Value-at-risk for a single position is calculated as:
VaR = sensitivity of position to change in market prices * estimated
change in price
The VaR figure, in this example, reflects the maximum amount
we could expect to lose over the next 10 days. However, the variance
and covariance matrix is estimated from a long run of data (eg
one to two years of data). Back