# Do stochastic interest rate models forecast future interest rate?

And if so, can they be used to estimate the future price of bonds?

## 2 Answers

Yes, every stochastic interest rate model makes a forecast for the future interest rate. For example, the random walk model

$$dr_t = \sigma dW_t$$

predicts all future interest rates to be equal to the current rate. The Vasicek model,

$$dr_t = a(\bar{r} - r_t)dt + \sigma dW_t$$

predicts that the future interest rate reverts to the long-term rate $\bar{r}$, i.e.

$$E(r_t) = \bar{r} + (r_0 - \bar{r})e^{-at}$$

More complex models might make forecasts for the entire maturity structure of interest rates, rather than just for the short rate.

Every interest rate forecast can be converted into a forecast for bond prices, since bond prices are (more or less) a function of interest rates. However, you should note that

1. You can't be sure that you have the right interest rate model.
2. Even if you have an accurate model, most interest rate models are risk-neutral, i.e. they don't take the risk premium associated with duration into account. You can't expect them to give accurate forecasts of real-world interest rates.
3. Stochastic models generally don't take external information into account (e.g. government monetary policy, risk sentiment, business cycle, bond market microstructure). They are potentially discarding valuable information.

For these reasons I would be very careful when using a pure stochastic interest rate model to make forecasts for bond prices.

• I am slightly confused about the model $dr_{t} = \sigma dW_{t}$. Presumably $dW_{t}$ is a step in the Wiener process, which is sampled from a standard normal distribution with domain {-3, ..., 3}. It would seem more intuitive to have a Bernoulli sampling with domain {-1, 1} and probabilities {0.5, 0.5}. That way the interest rate either goes up or down by the standard deviation $\sigma$. – A.L. Verminburger Oct 3 '17 at 14:05
• Why is that more intuitive? Interest rates don't necessarily move in discrete steps (well, target rates do. But not effective rates) – Chris Taylor Oct 5 '17 at 10:41

Yes, they do forecast the interest rate. But that is not really what they are used for. They are used to forecast the variability of interest rates (i.e. various outcomes for interest rates that we could have) and therefore allow the pricing of interest rate derivatives (not of the bonds themselves).

In other words these models generate a set of interest rate paths (thousands of them). The mean (expected) interest rate path from the model is not particularly interesting or useful for trading bonds. The model is generally calibrated so the mean outcome is derived from the forward structure in the market, or from the forecast of the person running the model. So you "learn nothing new" by averaging the model's predictions, you are just getting back what you put in.

So a stochastic interest rate model is a machine for automatically generating plausible i.r. scenarios, not a crystal ball that tells you what interest rates will be.