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5.1.1 Term structure models for dummies

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Imagine a portfolio with two identical discount bonds, except that the first $ P(t,T_1)$ expires some time before the second $ P(t,T_2)$ . What is the effect of a market fluctuation, which suddenly rises the spot rate at a time $ t<T_1<T_2$ before the first bond reaches maturity? The bonds are correlated and both will loose some of their original value; since there is more time left for another fluctuation to step back in the opposite direction, it is reasonable to assume that the second bond with a longer time to maturity will be less affected.

Taking advantage of this correlation, Vasicek creates a portfolio with a positive holding in the first bond and a negative holding in the second. By choosing exactly the right balance, this delta-hedging cancels out the uncertain effect from fluctuations and leaves only a deterministic change in the portfolio value. This is then used to calculate the fair price of a bond. The normalized value of the discount function is of course known at the maturity $ P(T,T)=1$ and the calculation is carried out with a forecast of the interest rates backward in time to predict the fair value $ P(T-t,T)$ for an increasing lifetime $ T-t$ .

The VMARKET applet below illustrates the procedure for a bond lifetime with up to RunTime=10 years.

VMARKET applet:  press Start/Stop to simulate the price of a zero-coupon bond backward in time, for a market with a volatile spot rate paying a reward for the associated risk. The plots show the value of the discount function as a function of the spot rate (P[r] in black) for an increasing time to maturity t (Time on the top of the window, in years). Directly derived from that using (2.2.2#eq.1), two plots show the evolution of the yield curve (Y[r] in blue, for a fixed Time) and the term structure of the interest rates (Y[t] in grey, for a fixed SpotRate). The latter acquires a finite value and sweeps across the plot window over the time span of one simulation [0; RunTime] and is best viewed after rescaling with Display.

For a given value of the spot rate $ r$ (horizontal axis, chosen to reflect the current market conditions), the discount function $ P(t,T)$ is decreasing backward in time $ t$ . Indeed, investors expect a return from their investment, which shows up as a growth of the discount function when the time runs forward so as to reach exactly one at maturity. The reward can be measured using (2.2.2#eq.1) as a yield $ Y(t,T)=-\log(P)/(T-t)$ and differs from the spot rate $ r$ because of the uncertain evolution of the future rates.

Virtual market experiments: evolving the yield curve
  1. Press Display to study the evolution of the yield curve $ Y(r)$ for a fixed lifetime of the bond (specified under Time) and the term structure of the interest rates $ Y(t)$ that is plotted for the specified SpotRate.
  2. Set Volatility=0 and compare the output obtained for a constant interest rate with the simple discounting previously used in (1.3#eq.6).

Due to the cyclic nature of the economy and the changes in the central bank interest rates, economists generally forecast what may be the future evolution of spot rates $ r(t^\prime)$ with $ t^\prime\in[t,T]$ . This opinion consists of a drift (``the spot rate will fall'') and a volatility (``the spot rate will fluctuate'') that can be estimated from historical values (exercise 1.05).