Interest rates futures

There is a clear correlation between quality or cost of materials and the durability/life expectancy of buildings. Greater resistance of better materials to wear and tear can be assumed, with obvious implications on future maintenance. Striking the best balance between initial capital outlay and maintenance cost requires complex calculations that take into account such intangibles as future interest rates and taxation of building operations. 

In this section, we describe the relationship between the price of a zero-coupon bond and spot and forward rates. We assume a risk-free zero-coupon bond of nominal value 1, priced at time t and maturing at time T. We also assume a money market bank account of initial value P t, T) invested at time t. The money market account is denoted M. The price of the bond at time t is denoted P t, T) and if today is time 0 (so that t > 0), then the bmid price today is unknown and a random factor (similar to a future interest rate). The bond price can be related to the spot rate or forward rate that is in force at time t. 

The most straightforward models to implement are normal models, followed by square root models and then lognormal models. The process that is used will have an impact on the distribution of future interest rates predicted by the model. A generalised distribution is given in Figure 3.2. 

FIGURE 3.2 Distribution of future interest rates implied by different processes. 

The average future interest rate over the time period (f, T) is given by Equation (3.37) ... 

Implied forward rates indicate the expected short-term (one-period) future interest rate for a specific point along the term structure they reflect the spread on the marginal rate of return that the market requires if it is investing in debt instruments of longer and longer maturities. 

Extremely long-dated zero coupon and forward rates can never decline, even when expected long-term future interest rates fall therefore, this limits the extent to which very long-dated bond yields are affected by a change in the current interest-rate environment (Dybvig et al., 1996). 

In a conventional positive yield curve environment, it is common for the 30-year government bond to yield say 10-20 basis points above the tlO-year bond. This might indicate to investors that a 100-year bond should yield approximately 20-25 basis points more than the 30-year bond. Is this accurate As we noted in the previous section, such an assumption would not be theoretically valid. Marshall and Dybvig have shown that such a yield spread would indicate an undervaluation of the very long-dated bond and that should such yields be available an investor, unless he or she has extreme views on future interest rates, should hold the 100-year bond. 

The change in the short rate will result in a 50-basis point decline in all the expected future interest rates. However, this will not result in a uniform fall in all bond yields. The impact on the zero-coupon curve and the forward rate curve is shown in Figure 7.4. 

With respect to the first assumption, the risk that an investor faces is that future interest rates will be less than the yield to maturity at the time the bond is purchased. This risk is called reinvestment risk. As for the second assumption, if the bond is not held to maturity, it may have to be sold for less than its purchase price, resulting in a return that is less than the yield to maturity. This risk is called interest rate risk. 

The adjustment required to convert a futures interest rate to a forward interest rate. 

FutUPes Prices Futures prices are quoted as (100 - Futures interest rate x 100). The quarterly compounded futures interest rates adjusted for convexity are converted to continuously compounded zero rates, as follows. 

From (3.36) and (3.37), it is clear that these two versions of the expectations hypothesis are incompatible unless no correlation exists between future interest rates. Ingersoll (1987) notes that although such an economic environment would be both possible and interesting to model, it is not related to reality, since interest rates are in fact highly correlated. Given a positive correlation between rates over a period of time, bonds with terms longer than two periods will have higher prices under the unbiased version than under the return-to-maturity version. Bonds with maturities of exactly two periods will have the same price under both versions. 

Any models using implied forward rates to generate future prices for options underlying bonds would be assuming that the future interest rates implied by the current yield curve will actually occur. An analysis built on this assumption would, like yield-to-worst analysis, be inaccurate, because the yield curve does not remain static and neither do the rates implied by it therefore future rates can never be known with certainty. To avoid this inaccuracy, a binomial tree model assumes that interest rates fluctuate over time. These models... 

A rise in volatility generates a range of possible future paths around the expected path. The actual expected path that corresponds to a zero-coupon bond price incorporating zero OAS is a function of the dispersion of the rai e of alternative paths around it. This dispersion is the result of the dynamics of the interest-rate process, so this process must be specified for the current term structure. We can illustrate this with a simple binomial model example. Consider again the spot rate structure in Table 12.1. Assume that there are only two possible future interest rate scenarios, outcome 1 and outcome 2, both of equal probability. The dynamics of the short-term interest rate are described by a constant drift rate a, together with a volatility rate a. These two parameters describe the evolution of the short-term interest rate. If outcome 1 occurs, the one-period interest rate one period from now will be... 

At 0 percent volatility, the prices generated by the up and down moves are equal, as the future interest rates are equal. Hence, the forward rate is the same... 

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