We illustrate the power of Monte Carlo simulation (MCS) through a simple example. Consider our GARCH(1,1)-normal model of returns, where

and

As mentioned earlier, at the end of day t we obtain R_t and we can calculate , which is tomorrow's variance in the GARCH model.

Using random number generators, which are standard in most quantitative software packages, we can generate a set of artificial (or pseudo) random numbers

drawn from the standard normal distribution, N(0, 1). MC denotes the number of draws, which should be large, for example, 10,000. To confirm that the random numbers do indeed conform to the standard normal distribution, a QQ plot of the random numbers can be constructed.

From these random numbers we can calculate a set of hypothetical returns for tomorrow as

Given these hypothetical returns, we can update the variance to get a set of hypothetical variances for the day after tomorrow, t + 2, as follows:

Given a new set of random numbers drawn from the N(0, 1) distribution,

we can calculate the hypothetical return on day t + 2 as

and the variance is now updated using

Graphically, we can illustrate the simulation of hypothetical daily returns from day t + 1 to day t + K as

Each row corresponds to a so-called Monte Carlo simulation path, which branches out from on the first day, but which does not branch out after that. On each day a given branch gets updated with a new random number, which is different from the one used any of the days before. We end up with MC sequences of hypothetical daily returns for day t + 1 through day t + K. From these hypothetical future daily returns, we can easily calculate the hypothetical K-day return from each Monte Carlo path as

If we collect these MC hypothetical K-day returns in a set then we can calculate the K-day value at risk simply by calculating the 100pth percentile as in

We can also use Monte Carlo to compute the expected shortfall at different horizons

where takes the value 1 if the argument is true and zero otherwise.

Notice that in contrast to the HS and WHS techniques introduced in Chapter 2, the GARCH-MCS method outlined here is truly conditional in nature as it builds on today's estimate of tomorrow's variance,

A key advantage of the MCS technique is its flexibility. We can use MCS for any assumed distribution of standardized returns—normality is not required. If we think the standardized t(d) distribution with d = 12 for example describes the data better, then we simply draw from this distribution. Commercial software packages typically contain the regular t(d) distribution, but we can standardize these draws by multiplying by as we saw in Chapter 6. Furthermore, the MCS technique can be used for any fully specified dynamic variance model.

In Figure 8.1 we apply the NGARCH model for the S&P 500 from Chapter 4 along with a normal distribution assumption. We use Monte Carlo simulation to construct and plot VaR per day, as a function of horizon K for two different values of . In the top panel the initial volatility is one-half the unconditional level and in the bottom panel is three times the unconditional level. The horizon goes from 1 to 500 trading days, corresponding roughly to two calendar years.

The VaR coverage level p is set to 1%. Figure 8.1 gives a VaR-based picture of the term structure of risk. Perhaps surprisingly the term structure of VaR is initially upward sloping both when volatility is low and when it is high. The VaR term structure is driven partly by the variance term structure, which is upward sloping when current volatility is low and downward sloping when current volatility is high as we saw in Chapter 4. But the VaR term structure is also driven by the term structure of skewness and kurtosis and other moments. Kurtosis is strongly increasing at short horizons and then decreasing for longer horizons. This hump-shape in the term structure of kurtosis creates the hump in the VaR that we see in the bottom panel of Figure 8.1 when the initial volatility is high.

In Figure 8.2 we plot the per day, , against horizon K.

The coverage level p is again set to 1% and the horizon goes from 1 to 500 trading days. Figure 8.2 gives an ES-based picture of the term structure of risk, which is clearly qualitatively similar to the term structure of VaR in Figure 8.1. Note however, that the slope of the ES term structure in the upper panel of Figure 8.2 is steeper than the corresponding VaR term structure in the upper panel of Figure 8.2. Note also that the hump in the ES term structure in the bottom panel of Figure 8.2 is more pronounced than the hump in the VaR term structure in the upper panel of Figure 8.1.

In the book so far, we have discussed methods that take very different approaches: Historical Simulation (HS) in Chapter 2 is a completely model-free approach, which imposes virtually no structure on the distribution of returns: the historical returns calculated with today's weights are used directly to calculate a percentile. The GARCH Monte Carlo simulation (MCS) approach in this chapter takes the opposite view and assumes parametric models for variance, correlation (if a disaggregate model is estimated), and the distribution of standardized returns. Random numbers are then drawn from this distribution to calculate the desired risk measure.

Both of these extremes in the model-free/model-based spectrum have pros and cons. Taking a model-based approach (MCS, for example) is good if the model is a fairly accurate description of reality. Taking a model-free approach (HS, for example) is sensible in that the observed data may capture features of the returns distribution that are not captured by any standard parametric model.

The Filtered Historical Simulation (FHS) approach, which we introduced in Chapter 6, attempts to combine the best of the model-based with the best of the model-free approaches in a very intuitive fashion. FHS combines model-based methods of variance with model-free methods of the distribution of shocks.

Assume we have estimated a GARCH-type model of our portfolio variance. Although we are comfortable with our variance model, we are not comfortable making a specific distributional assumption about the standardized returns, such as a normal or a distribution. Instead, we would like the past returns data to tell us about the distribution directly without making further assumptions.

To fix ideas, consider again the simple example of a GARCH(1,1) model:

where

Given a sequence of past returns, , we can estimate the GARCH model and calculate past standardized returns from the observed returns and from the estimated standard deviations as

We will refer to the set of standardized returns as . The number of historical observations, m, should be as large as possible.

Moving forward now, at the end of day t we obtain R_t and we can calculate , which is day t + 1's variance in the GARCH model. Instead of drawing random s from a random number generator, which relies on a specific distribution, we can draw with replacement from our own database of past standardized residuals, The random drawing can be operationalized by generating a discrete uniform random variable distributed from 1 to m. Each draw from the discrete distribution then tells us which τ and thus which to pick from the set .

We again build up a distribution of hypothetical future returns as

where FH is the number of times we draw from the standardized residuals on each future date, for example 10,000, and where K is the horizon of interest measured in number of days.

We end up with FH sequences of hypothetical daily returns for day t + 1 through day t + K. From these hypothetical daily returns, we calculate the hypothetical K-day returns as

If we collect the FH hypothetical K-day returns in a set then we can calculate the K-day Value-at-Risk simply by calculating the 100pth percentile as in

The ES measure can again be calculated from the simulated returns by simply taking the average of all the s that fall below the number; that is,

where as before the indicator function returns a 1 if the argument is true and zero if not.

An interesting and useful feature of FHS as compared with simple HS is that it can generate large losses in the forecast period, even without having observed a large loss in the recorded past returns. Consider the case where we have a relatively large negative z in our database, which occurred on a relatively low variance day. If this z gets combined with a high variance day in the simulation period then the resulting hypothetical loss will be large.

In Figure 8.3 we use the FHS approach based on the NGARCH model for the S&P 500 returns. We use the NGARCH-FHS model to construct and plot the per day as a function of horizon K for two different values of . In the top panel the initial volatility is one-half the unconditional level and in the bottom panel is three times the unconditional level. The horizons goes from 1 to 500 trading days, corresponding roughly to two calendar years.

The VaR coverage level p is set to 1% again. Comparing Figure 8.3 with Figure 8.1 we see that for this S&P 500 portfolio the Monte Carlo and FHS simulation methods give roughly equal VaR term structures when the initial volatility is the same.

In Figure 8.4 we plot the per day against horizon K.

The coverage level p is again set to 1% and the horizon goes from 1 to 500 trading days. The FHS-based ES term structure in Figure 8.4 closely resembles the NGARCH Monte Carlo-based ES term structure in Figure 8.2.

We close this section by reemphasizing that the FHS method suggested here combines a conditional model for variance with a Historical Simulation method for the standardized returns. FHS thus captures the current level of market volatility via but we do not need to make assumptions about the tail distribution. The FHS method has been found to perform very well in several studies and it should be given serious consideration by any risk management team.

In order to simulate the model forward in time using Monte Carlo we need to assume a multivariate distribution of the vector of shocks, . In this chapter we will rely on the multivariate standard normal distribution because it is convenient and so allows us to focus on the issues involved in simulation. In Chapter 9 we will look at more complicated multivariate t distributions.

The algorithm for multivariate Monte Carlo simulation is as follows:

Loop through these four steps from day t + 1 until day t + K. Now we can compute the portfolio return using the known portfolio weights and the vector of simulated returns on each day.

Repeating these steps times gives a Monte Carlo distribution of portfolio returns. From these MC portfolio returns we can compute VaR and ES from the simulated portfolio returns as in Section 2.

Multivariate Filtered Historical Simulation can be done easily when we assume constant correlations.

Loop through these steps from day t + 1 until day t + K. Now we can compute the portfolio return using the known portfolio weights and the vector of simulated returns on each day as before.

Repeating these steps times gives a simulated distribution of portfolio returns. From these FH portfolio returns we can compute VaR and ES from the simulated portfolio returns as in Section 2.

As mentioned before, random number generators typically provide us with uncorrelated random standard normal variables, , and we must correlate them before simulating returns forward.

At the end of day t the GARCH and DCC models provide us with and without having to do simulation. We can therefore compute a random return for day t + 1 as

where .

Using the new simulated shock vector, , we can update the volatilities and correlations using the GARCH models and the DCC model. We thus obtain simulated and . Drawing a new vector of uncorrelated shocks, , enables us to simulate the return for the second day ahead as

where .

We continue this simulation from day t + 1 through day t + K, and repeat it for vectors of simulated shocks on each day. As before we can compute the portfolio return using the known portfolio weights and the vector of simulated returns on each day. From these MC portfolio returns we can compute VaR and ES from the simulated portfolio returns as in Section 2.

In Figure 8.5 we use the DCC model for S&P 500 returns and the 10-year treasury bond index from Chapter 7 to plot the expected future correlations. We have assumed four different values of the current correlation, ranging from −0.5 in the blue line to +0.5 in the purple line. Note that over the 60-day horizon considered, the correlations converge toward the long-run correlation value but significant differences remain even after 60 days.

In Chapter 4 we saw how the expected future variance can be computed analytically from current variance using the GARCH model dynamics. The key equations were repeated in Section 2. Unfortunately for dynamic correlation models, such exact analytical formulas for expected future correlation do not exist. We need to rely on the simulation methods developed here in order to construct correlation forecasts for more than one day ahead. If we for example want to construct a forecast for the correlation matrix two days ahead we can use

where the Monte Carlo average is done element by element for each of the correlations in the matrix.

When correlations across assets are assumed to be constant then FHS is relatively easy because we can draw from historical asset shocks, using the entire vector (across assets) of historical shocks. The (constant) historical correlation will be preserved in the simulated shocks. When correlations are dynamic then we need to ensure that the correlation dynamics are simulated forward but in FHS we still want to use the historical shocks.

In this case we must first create a database of historical dynamically uncorrelated shocks from which we can resample. We create the dynamically uncorrelated historical shock as

where is the vector of standardized shocks on day and where is the inverse of the matrix square-root of the conditional correlation matrix .

When calculating the multiday conditional VaR and ES from the model, we again need to simulate daily returns forward from today's (day t) forecast of tomorrow's matrix of volatilities, and correlations, .

From the database of uncorrelated shocks , we can draw a random vector of historical uncorrelated shocks, called . It is important to note that in order to preserve asset-specific characteristics and potential nonlinear dependence in the shocks, we draw an entire vector representing the same day for all the assets.

From this draw, we can compute a random return for day t + 1 as

where .

Using the new simulated shock vector, , we can update the volatilities and correlations using the GARCH models and the DCC model. We thus obtain simulated and . Drawing a new vector of uncorrelated shocks, , enables us to simulate the return for the second day as

where . We continue this simulation for K days, and repeat it for FH vectors of simulated shocks on each day. As before we can compute the portfolio return using the known portfolio weights and the vector of simulated returns on each day. From these FH portfolio returns we can compute VaR and ES from the simulated portfolio returns as in Section 2.

The advantages of the multivariate FHS approach tally with those of the univariate case: It captures current market conditions by means of dynamic variance and correlation models. It makes no assumption on the conditional multivariate shock distributions. And, it allows for the computation of any risk measure for any investment horizon of interest.