In previous posts, I covered the calculation of portfolio Value at Risk (VaR) and Expected Shortfall using the historical simulation and linear modeling methods. Now, in order to close the topic, I will briefly touch on several additional interesting aspects.
Bond Portfolio – Interest Rate Risk
In the previous notes, VaR calculations referred to an equity portfolio. What happens in the case of bonds?
For bonds, we can use the same linear model, but it requires modification. Here, we must account for interest rate risk. Since a portfolio may contain bonds with different maturities and terms, their cash flows are decomposed, and zero-coupon bond yields are used for modeling purposes.
After that, we use the historical return statistics of the zero-coupon bonds and construct the variance–covariance matrix of returns, just as we do in the case of equities. Finally, the VaR measure is calculated.
The model is built in this Excel file:
Options Portfolio – Quadratic Model
For an options portfolio, calculations can also be performed using a linear model, but the results do not remain within reasonable economic boundaries, so additional complexity becomes necessary.
The issue is that if we use a linear model, we rely only on delta (i.e., sensitivity to the underlying asset price). However, to obtain correct results, we must also account for gamma (the change in delta).
Gamma affects the probability distribution of the portfolio value — it is no longer normal; it becomes skewed.
The diagrams below illustrate why this happens. A Long Call has positive gamma, while a Short Call has negative gamma.
I have not developed this model — hopefully, I won’t need to measure the risk of an options portfolio…
Monte Carlo
As mentioned, VaR can be calculated using either historical simulation or modeling methods. The Monte Carlo method is essentially a third approach.
In historical simulation, we take historically observed scenarios and call it a simulation because we attempt to explain the future using the past.
In the modeling approach, we also use historical data, but not the actual realized scenario vectors. Instead, we calculate individual asset variances and covariances and derive results from them.
The Monte Carlo method requires somewhat more advanced mathematics. Based on the correlations between assets, we generate possible future scenarios. This requires generating correlated random variables from a multivariate normal distribution. One way to do this is by using the Cholesky decomposition matrix.
It sounds complicated, but in Excel it is relatively straightforward.
Ultimately, we obtain simulated scenarios and compute the VaR measure.
Stressed VaR
Stress testing means evaluating what happens if events develop in an extremely adverse manner. For example, one may use the historical statistics from 2007–2008, when markets were under severe stress.
This is important because during stress periods, normal correlation structures tend to break down.
In simulations, practitioners sometimes use 5 standard deviations, which theoretically implies an event occurring once every 7,000 years*. In practice, however, it is not uncommon to observe assets whose price volatility reaches 5 sigma within 10–20 years.
Principal Components Analysis
Finally, there is the Principal Components Analysis (PCA) method.
This method is used for highly correlated assets. A matrix of assets and risk factors is constructed, and the impact of each factor on portfolio value is calculated.
For example, what happens to a bond portfolio with various maturities when interest rates shift in parallel? (Excel-Principal Component Analysis)
Adapted from:
Options, Futures & Other Derivatives, John C. Hull