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Imagine a portfolio with two identical discount bonds, except that the first expires some time before the second . What is the effect of a market fluctuation, which suddenly rises the spot rate at a time 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 and the calculation is carried out with a forecast of the interest rates backward in time to predict the fair value for an increasing lifetime .
For a given value of the spot rate (horizontal axis, chosen to reflect the current market conditions), the discount function is decreasing backward in time . 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 and differs from the spot rate because of the uncertain evolution of the future rates.
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 with . 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).