# Step 3: Calibrate to Obtain Zero Rates for First Two Years

To value a 2-year swap, one has to resort to using the formula given in equation (2.6). To do this, one would need the discount factors at times 6, 12, 18, and 24 months (or equivalently the zero rates *r*0,0.5, *r*0,1, *r*0,1.5, and *r*0,2). Since one can easily read off the zero rates *r*0,0.5 and г0д from Table 2.10, it remains to only find the zero rates *r*0,1.5 and *r*0,2•

Given that the maturities in Table 2.10 go out to only 1.25 years, one has to first guess the zero rate r0**,**2 and then linearly interpolate the value of *r*0,1.5 (using the values *r*0,1.25 and *r*0,2 so as to ensure that the 2-year swap rate is matched.

To do this, first observe that the discount factors associated for times 1.5 years and 2 years are given by the expressions and respectively. Putting all these together into equation (2.6) to value a 2-year, fixed-floating interest rate swap, one has

**TABLE 2.11** Zero Rates for First Two Years

where and refer to the discount factors for times 0.5 years and 1 year, respectively. Using the linear interpolation of the zero-rate curve in the time interval (1.25,2) yields,

Putting all these together, one can arrive at Table 2.11.

# Step 4: Calibrate to Obtain Zero Rates for First Five Years

Applying a similar idea to step 3, one can arrive at Table 2.12.

The information in Table 2.12 can be alternatively represented as in Figure 2.1.

Given the constructed zero-rate curve in Figure 2.1, one can easily read off zero rates for any maturity. Although the methodology discussed provides a reader with a good idea on how linear interpolation can be used to bootstrap a zero curve using market information, in practice the reader has to bear in mind some of the following subtleties associated with this method:

■ Instead of linearly interpolating and bootstrapping zero rates, one can alternatively linearly interpolate and bootstrap discount factors or natural logarithm of discount factors or forward rates. Doing this can possibly yield a completely different zero curve – in the process arriving at different values when discounting cash flows. As a consequence, for the purposes of consistency, it is imperative for one to do what market

**TABLE 2.12** Zero Rate Term Structure

practitioners generally do as opposed to doing something that is theoretically correct. The reader is referred to Van Deventer, Imai, and Mesler (2005) for further details.

■ In the discussion presented, I used only the overnight rate, Eurodollar futures, and the swap rates to construct the zero curve. The reason for

**FIGURE 2.1** Term Structure of Zero Rates for the First Five Years

this stems from the fact that I wanted to construct a zero rate curve using the LIBOR (or rates at which investment banks can borrow and lend from each other). To similarly build a zero-government curve (or any other curve of a different credit rating), one has to use the appropriate liquid financial instruments with the appropriate credit ratings.

■ In my discussion thus far, I did not pay any heed to the settlement process associated with the underlying financial instruments (i.e., take into consideration the time lag between transaction and settlement). In practice, this needs to be factored so as to ensure that instruments priced using the constructed zero-rate curve reproduce the market (or traded) prices of the instruments with appropriate settlement conventions.

■ To construct a zero-rate curve to value instruments using LIBOR, I used instruments directly linked to LIBOR. In the beginning of this chapter, I made reference to the fact that in constructing a zero-rate curve, it is imperative for one to be able to use liquid instruments that sometimes are the derivatives of the underlying instruments that would have been, theoretically, the correct choice of instruments to use. In such an instance, market makers would use related instruments including the appropriate spreads to ensure that appropriate liquid-market instruments are used.^{[1]}

- [1] Fueled by the market crash of 2008 and the Dodd-Frank Act, the market practice has been trending toward using a floating rate that is supported by more collateralization, such as the Overnight Indexed Swap (OIS) rate. As a consequence, interest rate swaps whose floating rates are indexed to the OIS rates instead of LIBOR are becoming more popular. See, for example, Smith (2012).