The interest rate risk is a particular type of market risk and can be measured with the same criteria as the general market risk and, in particular, equity risk. Specifically, by applying the definition of value at risk (VaR) to a fixed-income portfolio it can be stated that the VaR is a measure of the maximum potential change that the value of a portfolio can suffer, in this case afixed-income portfolio, at a time horizon and at a given probability level, due to changes in interest rates.
The difficulty with using the VaR in afixed-income portfolio is not so much in its definition, which is a direct application of the general definition of value at risk, but rather in the fact that characterising the dynamics of interest rates is very complex, for two main reasons: the wide range of products and the close relationship that exists between them.
In this sense, even though characterising this dynamic is complex and exceeds the scope of this book, some general principles will be established.
Thefirst is that by studying the interest rate at a particular term, as seen in the graph in Sect. 5.1.3., (Fig. 5.1) dynamics of mean reversion are evident, that is, to characterise the dynamics of the interest rate at a given term the best model of those presented in this book is the autoregressive process, thus ruling out processes such as random walk because, as shown in thefigure, interest rates, whatever the maturity, are in a stable interval most of the time. Fig.5.1shows that the government
3As will be discussed in later chapters, such hedging strategies are known as dynamic hedging.
bond over ten years is always between 1 % and 6 % throughout the period.
The main reason for the interest rate dynamics presenting mean reversion is that what is bought or sold is not the interest rate but the bond, and when this bond matures, unless there are credit risk issues, it pays for the nominal—nothing more and nothing less. This is why its price before maturityfluctuates around this value, leading to afluctuation in the interest rate which in turn involves the movement of mean reversion.
However, although there are stochastic processes that characterise interest rate dynamics at a given maturity reasonably well, it is difficult to coherently characterise the joint dynamics of interest rates at different maturities. In these cases, it is normally assumed that all TSIR variability is given by a certain number of stochastic factors. Thus, the dynamics of each of the interest rates at each term is defined using these stochastic factors, establishing the relationship between the different dynamics.
Once this definition has been obtained, the parameters of these stochastic factors are set so that the dynamics of the theoretical model are as close as possible to the actual dynamics observed using statistical techniques such as maximum likelihood.
There are many simple approaches in this regard. One of them is based on the work of Black et al. (1990) in which it is assumed that the implicit interest rates between periodtand periodt+ 1, whereΔtis the time between period t and periodt+ 1, have the following dynamics: ln (rt,t+ 1τ+ 1)ẳ ln (rt,t+ 1τ) +θ(t)Δτ+σετ, being τ today, Δτ the time betweenτand τ+ 1, andετa random variable that is distributed by a normal with mean zero and variance one, independent ofετ* for all τ andτ*. As demonstrated, this dynamic depends on the moment in time (τ) that occurs and the time until maturity (t). The value of these implicit interest rates varies as time (τ) passes and also according to maturity (t), because at any given time the implicit rate between one year and one year and one month and between ten years and ten years and one month is not the same. Thus, as in the other stochastic
processes for market variables, over time the value of the implicit rate, which is the market variable in this case, varies in a non-perfectly predictable way mainly due to ετ, but also in this case the dynamics of the implicit rate are different depending on the term and are not the same at different maturities.
Thus, from the current TSIR and if possible those from the past, the parametersσ and θ(t) are estimated and the theoretical model is adapted as much as possible to the actual data. Once the parameters have been estimated, by numerical simulation, in the same way as in the examples presented previously, the VaR can be calculated.
While there are many simple approaches, the one presented here is based on a single factor because, as can be seen, at each point in timeτ there is only one source of uncertainty which isετ. However, although it is beyond the scope of the book, there are more sophisticated approaches based on several factors.
Given the complexity of this dynamic, as established in previous chapters, a particularly interesting option would be to estimate the VaR using historical simulation. In this case, historical simulation is especially useful since it does not require any probabilistic model, which greatly simplifies the calculation.
At the same time, it must also be highlighted that for the interest rate dynamics of a TSIR of risky bonds, it is necessary to simulate the risk-free interest rate and the risk premium, which also varies over time, together.
The risk-free rate and the premium could be calculated separately, which introduces more complexity into the already complex problem of coher- ently characterising interest rate dynamics.
Finally, it should be mentioned that when the net present value at risk (NPVaR) calculation was discussed, a constant interest rate was assumed for simplicity; however, as it is being considered in this chapter, although with difficulty, it is possible to characterise the interest rate dynamics and incorporate them when estimating the NPVaR, which will provide more accurate estimates although they are also more complex.