Longstaff and Schwartz have developed a two-factor general equilibrium model of the term structure, hereby referred to as the LS model. The two factors used in the LS model are short-term interest rate and the volatility on the short-term interest rate.
They suggest using the two-factor model instead of a single-factor model when looking at the term structure, this is because of the single-factor model showing a perfect correlation between the returns on bond of all maturities, which doesn't reflect how it works in reality. Similar two-factor models have been developed before, like the CIR model, which is the closest to the LS model. However the advantage of the LS model persists in the dynamics and risk premium in the volatility factor, which are endogenously determined. Also using the interest rate volatility as a second variable is useful, because it is a key variable when pricing interest rate options and bonds.
In the LS model they make assumptions regarding the production and preferences. They assume that all physical investment is performed by a single stochastic constant-returns-to-scale technology which produces a good that is either consumed or reinvested in production. With the use of different formulas and statistical tests they come up with that the results are appealing and that the cross sectional restrictions imposed by the model on the evolution of the term structure seems consistent with the data.
The LS model also seems to work appealing in practical situations, and might therefore become a useful tool when valuing and hedging of interest rate-contingent-claim. The dynamics in the factors are also very appealing. But because of lack of use, their will have to be more testing before being the preferred model when valuing interest.
Bond price Volatility and Term to Maturity: A generalized respecification
Michael H. Hopewell and George G. Kaufman challenge the common proposition about when all factors being constant, the longer the maturity, the greater becomes the price volatility of bonds. They say that the price volatility is proportionally related to the duration of the bond and that it becomes greater as the duration becomes higher.
Duration
Duration is the weighted average maturity of a bond (average life) where the weights are the relative discounted cash flows in each period, which means that it identifies the time (from the present) at which the bond generates the average present value dollar.
The difference between maturity and duration is small for small maturities but increases with time. It's also stated that there is an inverse relationship between duration and coupon rates (higher coupon rates giving lower duration everything else being equal and vice versa)
Approximating the percentage price change
With some basic mathematical manipulation it is shown that for a given change in interest rates, percentage changes in bond prices vary proportionately with duration (dP/P=-Ddr). Hopewell and Kaufman then state that the relationship between changes in bond prices and duration can be used in a practical way.
The duration is shorter the higher the interest rates are which means that the biggest differences in duration between short-term and long-term bonds occur when interest rates are low. As a result, long-term bonds provide more yield than short-term bonds in an economy with a symmetrical business cycle. This reflects the failure to maintain constant in the difference in duration.
Expectations, Bond Prices, and the Term Structure of Interest Rates, by Burton G. Malkiel
In this article they showed that an explicit examination of theoretical bond price movements shines light upon understanding the observable structure of market interest rates. That the yield curve has a tendency to flatten out as term to maturity extends was both explained and demonstrated. A model was used to derive an ascending and descending yield curve with behavioural assumptions which were not as demanding as the ones in the Lutz theory as expectations were introduced to the extent that future interest-rate fluctuations were anticipated to be contained within previously existing ranges. When the level of interest rates were high in the historical range, a descending yield curve could be explained even where investors had formed no expectations as to the probable direction of interest-rate changes.
They then introduced stronger elements of foresight, where their model had an advantage of being in closer conformity with the practices of bond investors who had always considered the Lutz theory unrealistic. The Keynes-Hicks modifications were also assimilated into the analytical framework. Modifications could also be done to the model to take account of differences in yield which result from features as; coupon differences, call features, tax advantages of discount bonds and institutional rigidities as they might apply to a particular maturity area. This way, the traditional exceptional theory has been yet polished further to provide a more accurate picture of how the actual investing practices of bond investors make the theory both simpler in its assumptions, and modifications that are easier to alter with. This gave some insight to how the yield of securities with different terms to maturity behaves.
