3.4 Historical Simulation and Value at Risk

We investigate the behavior of a fictive zero-bond of a given credit quality with principal 1 USD, which matures after $ T$ years. In all simulations $ t=0$ denotes the beginning and $ t=T$ the end of the lifetime of the zero-bond. The starting point of the simulation is denoted by $ t_0$, the end by $ t_1$. The observation period, i.e., the time window investigated, consists of $ N\ge1$ trading days and the holding period of $ h\ge1$ trading days. The confidence level for the VaR is $ \alpha\in[0,1]$. At each point in time $ 0\le t\le t_1$ the risky yields $ R_i(t)$ (full yield curve) and the riskless treasury yields $ B_i(t)$ (benchmark curve) for any time to maturity $ 0<T_1<\dots<T_n$ are contained in our data set for $ 1\le i\le n$, where $ n$ is the number of different maturities. The corresponding spreads are defined by $ S_i(t)=R_i(t)-B_i(t)$ for $ 1\le i\le n$.

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.


3.4.1 Risk Factor: Full Yield

1. Basic Historical Simulation:

We consider a historical simulation, where the risk factors are given by the full yield curve, $ R_i(t)$ for $ i=1,\dots,n$. The yield $ R(t,T-t)$ at time $ t_0\le t\le t_1$ for the remaining time to maturity $ T-t$ is determined by means of linear interpolation from the adjacent values $ R_i(t)=R(t,T_i)$ and $ R_{i+1}(t)=R(t,T_{i+1})$ with $ T_i\le T-t<T_{i+1}$ (for reasons of simplicity we do not consider remaining times to maturity $ T-t<T_1$ and $ T-t>T_n$):

$\displaystyle R(t,T-t)={[T_{i+1}-(T-t)]R_i(t) +[(T-t)-T_i]R_{i+1}(t)\over T_{i+1}-T_i}\,.$ (3.1)

The present value of the bond $ PV(t)$ at time $ t$ can be obtained by discounting,

$\displaystyle PV(t)={1\over\big[{1+R(t,T-t)}\big]^{T-t}},\qquad t_0\le t\le t_1.$ (3.2)

In the historical simulation the relative risk factor changes

$\displaystyle \Delta_i^{(k)}(t)={R_i\big(t-k/N\big) -R_i\big(t-(k+h)/N\big) \over R_i\big(t-(k+h)/N\big)}, \quad 0\le k\le N-1,$ (3.3)

are calculated for $ t_0\le t\le t_1$ and each $ 1\le i\le n$. Thus, for each scenario $ k$ we obtain a new fictive yield curve at time $ t+h$, which can be determined from the observed yields and the risk factor changes,

$\displaystyle R_i^{(k)}(t+h)=R_i(t)\big[1+\Delta_i^{(k)}(t)\big], \qquad 1\le i\le n,$ (3.4)

by means of linear interpolation. This procedure implies that the distribution of risk factor changes is stationary between $ t-(N-1+h)/N$ and $ t$. Each scenario corresponds to a drawing from an identical and independent distribution, which can be related to an i.i.d. random variable $ \varepsilon_i(t)$ with variance one via

$\displaystyle \Delta_i(t)=\sigma_i \varepsilon_i(t).$ (3.5)

This assumption implies homoscedasticity of the volatility of the risk factors, i.e., a constant volatility level within the observation period. If this were not the case, different drawings would originate from different underlying distributions. Consequently, a sequence of historically observed risk factor changes could not be used for estimating the future loss distribution.

In analogy to (3.1) for time $ t+h$ and remaining time to maturity $ T-t$ one obtains

$\displaystyle R^{(k)}(t+h,T-t)={[T_{i+1}-(T-t)]R_i^{(k)}(t)
+[(T-t)-T_i]R^{(k)}_{i+1}(t)\over T_{i+1}-T_i}
$

for the yield. With (3.2) we obtain a new fictive present value at time $ t+h$:

$\displaystyle PV^{(k)}(t+h)={1\over\big[{1+R^{(k)}(t+h,T-t)}\big]^{T-t}}.$ (3.6)

In this equation we neglected the effect of the shortening of the time to maturity in the transition from $ t$ to $ t+h$ on the present value. Such an approximation should be refined for financial instruments whose time to maturity/time to expiration is of the order of $ h$, which is not relevant for the constellations investigated in the following.

Now the fictive present value $ PV^{(k)}(t+h)$ is compared with the present value for unchanged yield $ R(t+h,T-t)=R(t,T-t)$ for each scenario $ k$ (here the remaining time to maturity is not changed, either).

$\displaystyle PV(t+h)={1\over\big\{{1+R(t+h,T-t)}\big\}^{T-t}}\,.$ (3.7)

The loss occurring is

$\displaystyle L^{(k)}(t+h)=PV(t+h)-PV^{(k)}(t+h) \qquad 0\le k\le N-1,$ (3.8)

i.e., losses in the economic sense are positive while profits are negative. The VaR is the loss which is not exceeded with a probability $ \alpha$ and is estimated as the $ [(1-\alpha)N+1]$-th-largest value in the set

$\displaystyle \{L^{(k)}(t+h)\mid 0\le k\le N-1\}.
$

This is the $ (1-\alpha)$-quantile of the corresponding empirical distribution.

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 $ 1\le i\le n$ the average of relative changes $ \Delta_i^{(k)}(t)$ 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 $ \Delta_i^{(k)}(t)$ with $ \Delta_i^{(k)}(t)-\overline
\Delta_i(t)$ for $ 1\le i\le n$ in (3.4):

$\displaystyle \overline \Delta_i(t)={1\over N}\sum_{k=0}^{N-1}\Delta_i^{(k)}(t),$ (3.9)

This mean correction is presented in Hull (1998).

3. Volatility Updating:

An important variant of historical simulation uses volatility updating Hull (1998). At each point in time $ t$ the exponentially weighted volatility of relative historical changes is estimated for $ t_0\le t\le t_1$ by

$\displaystyle \sigma_i^2(t)=(1-\gamma) \sum_{k=0}^{N-1}\gamma^k\big\{\Delta_i^{(k)}(t)\big\}^2, \qquad 1\le i\le n.$ (3.10)

The parameter $ \gamma\in[0,1]$ is a decay factor, which must be calibrated to generate a best fit to empirical data. The recursion formula

$\displaystyle \sigma_i^2(t)=(1-\gamma) \sigma_i^2(t-1/N)+\gamma\big\{\Delta_i^{(0)}(t)\big\}^2, \qquad 1\le i\le n,$ (3.11)

is valid for $ t_0\le t\le t_1$. The idea of volatility updating consists in adjusting the historical risk factor changes to the present volatility level. This is achieved by a renormalization of the relative risk factor changes from (3.3) with the corresponding estimation of volatility for the observation day and a multiplication with the estimate for the volatility valid at time $ t$. Thus, we calculate the quantity

$\displaystyle \delta_i^{(k)}(t)=\sigma_i(t)\cdot {\Delta_i^{(k)}(t)\over \sigma_i(t-(k+h)/N)}, \qquad 0\le k\le N-1.$ (3.12)

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 $ \Delta_i(t)/\sigma_i(t))$ is stationary. For each $ k$ these renormalized relative changes are used in analogy to (3.4) for the determination of fictive scenarios:

$\displaystyle R_i^{(k)}(t+h)=R_i(t)\big\{1+\delta_i^{(k)}(t)\big\}, \qquad 1\le i\le n,$ (3.13)

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

$\displaystyle \overline \delta_i(t)={1\over N}\sum_{k=0}^{N-1}\delta_i^{(k)}(t),$ (3.14)

and use the adjusted relative risk factor change $ \delta_i^{(k)}(t)-\overline \delta_i(t)$ instead of  $ \delta_i^{(k)}(t)$ in (3.13).


3.4.2 Risk Factor: Benchmark

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 $ B_i(t)$ for $ i=1,\dots,n$. The yield $ B(t,T-t)$ at time $ t$ for remaining time to maturity $ T-t$ is calculated similarly to (3.1) from adjacent values by linear interpolation,

$\displaystyle B(t,T-t)=\frac{\{T_{i+1}-(T-t)\}B_i(t) +\{(T-t)-T_i\}B_{i+1}(t)} {T_{i+1}-T_i }\,.$ (3.15)

The generation of scenarios and the interpolation of the fictive benchmark curve is carried out in analogy to the procedure for the full yield curve. We use

$\displaystyle \Delta_i^{(k)}(t)={B_i\big(t-k/N\big) -B_i\big(t-(k+h)/N\big) \over B_i\big(t-(k+h)/N\big)}, \quad 0\le k\le N-1,$ (3.16)

and

$\displaystyle B_i^{(k)}(t+h)=B_i(t)\big[1+\Delta_i^{(k)}(t)\big], \qquad 1\le i\le n.$ (3.17)

Linear interpolation yields

$\displaystyle B^{(k)}(t+h,T-t)={\{T_{i+1}-(T-t)\}B_i^{(k)}(t)
+\{(T-t)-T_i\}B^{(k)}_{i+1}(t)\over T_{i+1}-T_i}\,.$

In the determination of the fictive full yield we now assume that the spread remains unchanged within the holding period. Thus, for the $ k\hbox{-th}$ scenario we obtain the representation

$\displaystyle R^{(k)}(t+h,T-t)=B^{(k)}(t+h,T-t)+S(t,T-t),$ (3.18)

which is used for the calculation of a new fictive present value and the corresponding loss. With this choice of risk factors we can introduce an adjustment for the average relative changes or/and volatility updating in complete analogy to the four variants described in the preceding subsection.


3.4.3 Risk Factor: Spread over Benchmark Yield

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 $ S_i(t)$ for $ i=1,\dots,n$. The spread $ S(t,T-t)$ at time $ t$ for time to maturity $ T-t$ is again obtained by linear interpolation. We now use

$\displaystyle \Delta_i^{(k)}(t)={S_i\big(t-k/N\big) -S_i\big(t-(k+h)/N\big) \over S_i\big(t-(k+h)/N\big)}, \quad 0\le k\le N-1,$ (3.19)

and

$\displaystyle S_i^{(k)}(t+h)=S_i(t)\big\{1+\Delta_i^{(k)}(t)\big\}, \qquad 1\le i\le n.$ (3.20)

Here, linear interpolation yields

$\displaystyle S^{(k)}(t+h,T-t)={\{T_{i+1}-(T-t)\}S_i^{(k)}(t)
+\{(T-t)-T_i\}S^{(k)}_{i+1}(t)\over T_{i+1}-T_i}\,.
$

Thus, in the determination of the fictive full yield the benchmark curve is considered deterministic and the spread stochastic. This constellation is the opposite of the constellation in the preceding subsection. For the $ k\hbox{-th}$ scenario one obtains

$\displaystyle R^{(k)}(t+h,T-t)=B(t,T-t)+S^{(k)}(t+h,T-t).$ (3.21)

In this context we can also work with adjustment for average relative spread changes and volatility updating.


3.4.4 Conservative Approach

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.


3.4.5 Simultaneous Simulation

Finally, we consider simultaneous relative changes of the benchmark curve and the spreads. For this purpose (3.18) and (3.21) are replaced with

$\displaystyle R^{(k)}(t+h,T-t)=B^{(k)}(t+h,T-t)+S^{(k)}(t,T-t),$ (3.22)

where, again, corrections for average risk factor changes or/and volatility updating can be added. We note that the use of relative risk factor changes is the reason for different results of the variants in subsection 3.4.1 and this subsection.