Multi-asset portfolio: Bootstrapping Value at Risk

• Monte Carlo VaR using historical probability distribution curve
• Correlations naturally embedded in the process

Hello all! Hope everyone is well and safe.

As discussed a couple of posts ago (here), Value at risk can be obtained by at least three main methods: Monte Carlo, Historical, and variance-covariance. There are particular assumptions, uses  and variations for each one of them that I will discuss in a future post. In this post my goal is to briefly talk about calculation of Monte Carlo Value at Risk based on the probability density obtained from the historical distribution. This process is called bootstrapping.

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The first step in the VaR calculation is the identification of the risk drivers that affect the prices of the assets in the portfolio. Next step is to calculate returns from the time series produced by these drivers. Let's consider a concrete example. A portfolio $\Pi (t) = \beta_S S + \beta_B B$ contain a position of size $\beta_S$ of a stock $S$ and a position of size $\beta_B$ of a bond $B$. For the stock, the driver is simply its own price. For the bond, there are at least two underlyings that one would take into account, named interest rate and credit spread. Such choice of drivers would lack of the additional spread priced by the market that we will generally call OAS (Option Adjustment Spread), for bonds that do not have optionality embedded it is called Z-spread. A simpler driver choice would be to take directly the yield to maturity from the quoted values of $B$. Notice however that the use of the yield as the risk driver requires a bond with issue date longer than the period used for the historical analysis, although some data proxies might be used to replace missing data if necessary.

Once we have defined the risk drivers (stock value and yield, for our example) we obtain the log-returns for the specified period (question for an interview: could we use arithmetic returns instead?). We build a two-column matrix in which each row denotes a scenario that is comprised of returns of stock and yield. Once we have such a matrix, the process is quite straightforward. The simulation process is done by drawing numbers from a uniform distribution ranging from one to the number of rows in the scenario matrix. Each sorted number relates directly to a row in such a matrix and the value of the returned portfolio can be calculated from the picked scenario. 

Using log-returns, we define $\delta_S = log(\frac{S}{S_0})$, therefore $S=S_0 exp(\delta_S)$ and the variation of stock value is $\Delta S = S_0(exp(\delta_S) - 1)$. Similarly, for the yield one can write $\Delta y = y_0(exp(\delta_y) - 1)$. Therefore, variations of the portfolio value $\Pi (t)$ can be calculated by means of:

$\Delta \Pi (t) = \beta_S \Delta S + \beta_B B \{ D (t) \Delta y + \Gamma (t) (\Delta y)^2 \}$

where $D(t)$ and $\Gamma(t)$ denote the duration and convexity of the bond, and $y$ is the yield. Notice that some variations of this formula are possible depending on the used definition of duration and convexity. Nevertheless the concept is pretty clear, that is, one relies on the derivatives over the risk drivers to compute the return of an asset. A linear relation works well for linear assets such as stocks and contracts for difference. Conversely, options and swaptions are clearly non-linear, then the second order derivative is crucial to keep accuracy of the calculation. Despite the fact bonds are non-linear, some managers assume it linear for sake of computational cost (consider a portfolio with hundreds of assets, for instance). By the way, another good interview question would be: what improvements can be done in this process when we have a portfolio with hundreds of assets?

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One of the advantages of this method is the fact that correlations are naturally embedded in the resulting probability distribution and no further assumption is required on this matter. However, a significant drawback is the fact that the use of historical returns requires data of a long period in addition to assume that past history is a good proxy for future returns. In addition, fast-paced scenarios cannot not be captured by such approach. For instance, a pure historical approach could not reproduce or indicate the losses due to the pandemic, mainly because we were living in a significant bull market before that.
At last, this post is just a scratch on the surface of bootstrapping VaR. One could consider for instance the use of a term structure interest rate as part of the risk drivers. Another interesting possibility is the fitting of a kernel to the historical distribution that is embedded into a copula. This method would allow for dimension reduction but requires further considerations on the historical correlation. This is an interesting topic, I shall come back to it at some future post.

I hope you enjoyed the post, leave your comments and share!

May the Force be with you!

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Diogo Pedroso

LinkedIn: www.linkedin.com/in/diogomourapedroso