Overview
The historical simulation works best if an asset can be found such that price returns can be computed and applied to the asset. Unfortunately, fixed income assets are of relatively short duration, and cannot not be effectively used in a historical simuation. Rather it is interest rates that are simulated.
When interest rates are simulated, the challenge is to decide how to difference the historical data. Does the anlyst compute arithmetic returns on the interest rate, or does she compute a straight difference (subtraction). Usingn differences can lead to negative interest rates, but arithmetic returns are generally not thought to apply to interest rates.
Zero Coupon Bond Simulation
In the case of interest rates, each rate can be converted into the price of an asset, namely a zero coupon bond of the given maturity. This makes it possible for the analyst to compute zero coupon bond prices at each day, and then to compute an arithmetic return for each zero coupon bond from which the new set of interest rates can be computed.
Rate Curve Model Simulations
In a model based simulation, a model is chosen to fit represent the yield (or discount curve) at point in the history. A common model that is used is the Nelson Siegel for example.
Then, for each point in the history, the model is fit to that data point, returning a set of parameters (in the case of the Nelson Siegel model, four parameters) that represent the curve at that point.
Then a set of differences are compouted between each data point and the next point in the history. Next, the model is fit to todays day. Lastly, the set of parameter differences are sampled and then applied to todays model.
Care must be taken with approach that the parameters never take on disallowed values. For example, the {% \tau %} parameter cannot be less than or equal to zero in the Nelson Siegel model. (Often times, the {% \tau %} parameter is replace by {% \lambda = \frac{1}{\tau} %} in order to deal with this situation)