1. What are forward default free interest rates and what contributes to their unconditional volatility?

The instantaneous nominal forward interest rate contains information about the expectation held by market participants of future nominal default-free interest rates. Default-free means the equivalent of the interest rate on a short-term government bond with no default risk. An n-year forward rate contains information about expected short-term interest rates, n-years ahead; but there may be other determinants of the n-year nominal forward rate; firstly, forward interest rates contain a technical convexity effect that reflects the volatility of the short rate, such a convexity effect will drive a wedge between the expected nominal short rate and the forward rate even if investors are risk neutral. Secondly, if investors are averse to risk they may require a holding period risk (or term) premium for acquiring a longer maturity bond to compensate them for unexpected changes in bond prices and their own risk aversion. Expectations and premia are not observable and re-assessment of each of these components may contribute to the total volatility of the n-period nominal forward rate. Quantifying the contribution of each of these components to total volatility of forward interest rates is challenging!

2.How effective is it to use historical data to measure unconditional volatility? Does this pose a challenge for pension funds?

We may use historical data to assess an appropriate level of total unconditional volatility out to maturities of 20 to 25 years. There are, however, two reasons why we may want to be cautious using historical data to assess the level of future unconditional volatility; firstly, monetary policy changes over time and so does the structure of an economy. The switch to a specified target for inflation by many central banks may reduce the total volatility of n-year forward rates. Whilst it could still be prudent to use historical data to assess future unconditional volatility, we may wish to attach slightly more weight to recent observations when carrying out our analysis. Secondly, there may be a number of non-fundamental factors that distort the information we get from our data, particularly from, longer dated forward interest rates. There are two key issues here: a) Yield curve construction methods used by many practitioners and central banks will introduce spurious variation in longer maturity nominal forward rates. b) Small measurement/pricing errors (for long maturity financial instruments) can induce large spurious variation in the long end of the forward rate curve. It might be prudent to ignore information in historical nominal forward rate data of a maturity beyond 10-15 years.

This poses a challenge for a pension fund with a long term planning horizon as they will have to make an assumption about unconditional volatility of interest rates in their interest rate projections which could easily extend to between 50 and 80 years into the future. Unconditional volatility would tell us something about the distribution of possible interest rate outcomes this far into the future and thus the potential loss if unexpected changes in interest rates occur. Ideally we would also need to be able to make an assumption about the fraction of volatility explained by stochastic interest rates, term premia and convexity effects. By decomposing total volatility between each of these components, the pension fund could obtain the appropriate risk return relation of longer maturity default-free bonds.

For a pension fund which typically has a planning horizon beyond 10-15 years, the task is even more complicated as there is little reliable data available to give some guidance for an appropriate assumption on unconditional volatility. But we know that many theoretical term-structure models imply that the unconditional volatility of the change in the log nominal forward rate must be zero as the maturity goes to infinity. The only requirement for this to hold is that the expected path of the short rate and term premia (if present) are stationary. Both seem to be plausible assumptions. Using such theoretical restrictions and the information we get about the decay of the volatility (with maturity) from historical data of maturities up to 10 to 15 years, it is possible to extrapolate and get an appropriate unconditional volatility term-structure for the change in the nominal log forward rates of maturities ranging from a year to infinity.

3. Can expected inflation and/or real interest rates contribute to the variation of forward rates at such long maturities?

Using such an approach to construct an unconditional volatility term-structure of changes in the log nominal forward rate often shows that the n-year nominal forward rates can be quite volatile even when n is as large as 30 years. This level of volatility can be induced by changing expectations about future inflation, changing expected real interest rates, term-premia and technical convexity effects. The latter two seem to be the best candidates to explain volatility of forward interest rates with such long maturities as it is hard to rationalise large changes in long-term expected inflation or expected real rates over a short period of time. But in economies without an explicit target for inflation, or an explicit target for inflation that is not credible, it is more likely that changing expectations of long-term inflation can contribute to the unconditional volatility of nominal n-year forward rates.