In this part, we learn how to estimate the term structure from a given set of market quotes on instruments including coupon bonds, LIBOR, forward rates, and swap rates. We will distinguish between exact methods and smoothing methods. Exact methods will give us discount curves that exactly reproduce the market data. But this comes at the cost of somewhat irregular forward curves. If we insist on the regular forward curves we use a smoothing method which gives us smooth yield curves. But at the cost of not exactly matching the market data. We will eventually see what PCA, Principal Component Analysis tells us about the basic shapes of the yield curve. We now start with the bootstrapping method, which is an exact method. It is a very simple method and it is most used among trading desks. The idea is to build up the term structure from shorter maturities to longer maturities. And it is most instructive to learn about the Bootstraping method from an example. We look at this following data from the U.S market on October 1st, 2012. We are given LIBOR rates, futures rates, and swap rates. The day count convention is actual by 360. The aim is to find the discount curve for all the reset cash flow dates that exactly matches these quotes. Let's look at the first block of market data, the LIBOR rates. They're given for overnight, 1 month, and 3 months maturities. We denote these maturity dates by S-1, S-2, and S-3. In terms of the day count convention, the year fractions are given here. The corresponding zero coupon bond prices are given in this 1 to 1 relation. The next block contains quotes on futures prices. Recall that the futures price for a maturity date Ti+1 is quoted as: 100 x (1 - the futures rate) which is the futures rate for the period with reset date Ti and maturity date Ti+1. We have 5 futures prices. The reset date for the first futures is T-1 is equal to the 19th of December, 2012. We use futures rates as proxies for forward rates as shown here. When looking at the maturity spectrum, notice that we have a maturity overlap as shown here. The first reset date of the first futures contract lies between S-2 and S-3 the maturities of the LIBOR rates. From this, we get The LIBOR rate for T-1 by linearly interpolating The S-2 LIBOR, and the S-3 LIBOR rates. From the T-1 LIBOR rate, we then get the T-1 bond and then we derive the T-2 bond using this formula which relates forward rates to ratios of zero coupon bonds. By iterative usage of this formula, We eventually get the T-3, T-4, T-5 and T-6 zero coupon bond prices. The third block contains swap rates for 9 different lengths. These swaps pay annual coupons at the following 30 dates. The dates marked in red, are the dates on which the 9 swaps mature. For these 9 dates, we are given swap rates. What we will do next is we will use linear interpolation to get swap rates for the black dates U-6, U-8, U-9, and so on and so forth. We will get the swap rate for U-1 as shown on the following page. Again we note that there is a maturity overlap and U-1 falls between T-4 and T-5 which are maturity dates of the futures. From the futures quotes, we derived the corresponding zero coupon bond prices. So we are given the T-4 bond and the T-5 bond prices. From these, we infer the T-4 and the T-5 simple spot rates and then we use linear interpolation of these 2 simple spot rates to get the simple spot rate with maturity U-1. The simple spot rate with maturity U-1 gives us the U-1 bond price, and then we can use the swap rate formula which relates the swap rate for maturity U-2 to the U-1 and U-2 bond prices. We can solve this equation for the U-2 bond price as shown here. For the remaining cash flow dates we use linear interpolation to obtain the corresponding swap rates. We can then invert the swap rate formula to derive iteratively the U-n bond price in terms of the previous bond prices and the corresponding swap rates as shown here. We eventually obtain the U-n bond prices for all the remaining maturities n from 3 to 30. Setting the initial value of the discount curve equal to 1 we thus have constructed the discount curve for 40 maturity time points as shown here. Recall that the blue maturities S-1, S-2, S-3 correspond to the LIBOR rates. The red maturities T-1 up to T-6 correspond to the futures rate, and the green maturities U-1 up to U-30 correspond to the swap rates. We also see again the overlap of these maturity spectra. The T maturities overlap with the S maturities and the green swap maturities overlap with the red T maturities. The figure on the left hand side shows us the discount curve as a function of time to maturity. We see that the bulk of the maturity dates correspond to the swap contracts. If we zoom in to the first 3 years this is what we see on the right hand side. We see the short end of the discount curve. So here we can nicely distinguish between the LIBOR maturity dates, the futures reset and maturity dates, and then the first cash flow dates of the swap contracts. The figure on the left hand side shows 3 curves. The first curve is the yield curve, that corresponds to the discount curve. The second curve is the curve of simple spot rates both of which look decently smooth. But if we look at the forward curve, this is the middle curve here, we see some saw tooth. If we again zoom in to the first 3 years, this is what is shown on the right hand side, we see irregularities also showing up here. So in sum the bootstrapping gives us an entire term structure from relatively few instruments. We had 3 LIBOR, 5 futures contracts, and 9 swap contracts. The method exactly reconstructs these market prices and this is desirable for interest rate risk management and trading desks because they have to marked to market their books. But the implied forward curve is irregular and sensitive to bond price variations. Bootstrapping is an example of an exact estimation method.
