We investigate the behavior of a fictive zero-bond of
a given credit quality with principal 1 USD,
which matures after years. In all simulations
denotes the beginning
and
the end of the lifetime of the zero-bond. The starting point of the simulation is denoted
by
, the end by
. The observation period, i.e., the time window investigated, consists of
trading days and the holding period of
trading days. The confidence level for the VaR
is
. At each point in time
the risky
yields
(full yield curve) and the riskless treasury yields
(benchmark curve)
for any time to maturity
are contained in our data set for
, where
is the number of different
maturities.
The corresponding spreads are defined by
for
.
In the following subsections 3.4.1 to 3.4.5 we specify different variants of the historical simulation method which we use for estimating the distribution of losses from the zero-bond position. The estimate for the distribution of losses can then be used to calculate the quantile-based risk measure Value-at-Risk. The variants differ in the choice of risk factors, i.e., in our case the components of the historical yield time series. In Section 3.6 we describe how the VaR estimation is carried out with XploRe commands provided that the loss distribution has been estimated by means of one of the methods introduced and can be used as an input variable.
1. Basic Historical Simulation:
We consider a historical simulation, where the risk factors are given by the full yield curve,
for
.
The yield
at time
for the remaining time to maturity
is determined by means of linear interpolation
from the adjacent values
and
with
(for
reasons of simplicity we do not
consider remaining times to maturity
and
):
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(3.5) |
In analogy to (3.1) for time and remaining
time to maturity
one obtains
Now the fictive present value
is
compared with the present value for unchanged yield
for each scenario
(here the remaining time to maturity is not changed, either).
2. Mean Adjustment:
A refined historical simulation
includes an adjustment for
the average of those relative changes in the observation period
which are used for generating the scenarios according to
(3.3).
If for fixed
the average of relative changes
is different from 0, a trend is projected from the past to the future
in the generation
of fictive yields in (3.4).
Thus the relative changes are corrected
for the mean by replacing
the relative change
with
for
in (3.4):
3. Volatility Updating:
An important variant of historical simulation uses volatility
updating Hull (1998). At each point in time the exponentially
weighted volatility of
relative historical changes is estimated for
by
In a situation, where risk factor volatility is heteroscedastic
and, thus, the process of risk factor changes is not stationary,
volatility updating cures this violation of the assumptions made
in basic historical simulation, because the process of re-scaled
risk factor changes
is stationary.
For each
these renormalized relative changes are used in analogy to
(3.4) for the determination of fictive scenarios:
The other considerations concerning the VaR calculation in historical simulation remain unchanged.
4. Volatility Updating and Mean Adjustment:
Within the volatility updating framework, we can also apply a correction for the average change according to 3.4.1(2). For this purpose, we calculate the average
In this subsection the risk factors are
relative changes of the benchmark curve instead of the
full yield curve. This restriction
is adequate for quantifying
general market risk, when there is no need to include spread risk.
The risk factors are the yields
for
. The yield
at time
for
remaining time to maturity
is calculated similarly to
(3.1) from adjacent values by
linear interpolation,
Linear interpolation yields
In the determination of the fictive full yield we now assume that the spread remains unchanged within the holding period. Thus, for the
When we take the view that risk is only caused by spread changes
but not by changes of the benchmark curve, we investigate
the behavior of the spread risk factors for
.
The spread
at time
for time to maturity
is again obtained by linear interpolation. We now use
In the conservative approach we assume full correlation between risk from the benchmark curve and risk from the spread changes. In this worst case scenario we add (ordered) losses, which are calculated as in the two preceding sections from each scenario. From this loss distribution the VaR is determined.
Finally, we consider simultaneous relative changes of the benchmark curve and the spreads. For this purpose (3.18) and (3.21) are replaced with