# INFERRING THE FORWARD CURVE

Suppose that you do not have access to a term structure model or the requisite adjustment to the futures rates. You still can infer the LIBOR forward curve if you observe the fixed rates on plain vanilla interest rate swaps. For example, suppose that you know the current level of 3-month LIBOR is 0.50% and that the fixed rates on 1-year and 2-year swaps are 2.12% and 3.40%. These swaps are for quarterly settlements and 30/360 day-counts. Suppose further that you observe the full range for the intermediate-maturity swaps: The 0.50-year fixed rate is 1.04%, the 0.75-year is 1.58%, the 1.25-year is 2.44%, the 1.50-year is 2.76%, and the 1.75-year is 3.08%.

Okay, it is incredibly unrealistic that you could observe all these swap fixed rates. In practice, commercial banks making markets in swaps quote fixed rates for standard time frames, such as for the 1-year and 2-year contracts. Typically, you will need to * interpolate* to get the other rates. Suppose you use simple straight-line interpolation, adding 54 basis points to the observed level of 3-month LIBOR to get 1.04% and 1.58% and then 32 basis points for each quarter in the second year to go from 2.12% to 3.40%. This is clearly arbitrary and adds

*to the analysis.*

**model risk**Given these observed or interpolated swap fixed rates, we can dip into our bond math toolkit to infer the LIBOR forward curve. The trick is to transform these swaps into bonds by adding 100 in par value to date 0 and to the maturity date and then use the bootstrapping technique from Chapter 5 to get the implied spot rates. Here are the equations to solve for the series of spot * (Spot)* rates, quoted in terms of months.

*0×6 is the 6-month rate as of today (day 0);*

**Spot***0×9 is the 9-month rate, and so forth.*

**Spot**Can you imagine doing these types of repetitive calculations in the olden days before spreadsheets and being able to do the bootstrapping with discount factors? Anyway, now we have the implied spots, all annualized for the same periodicity. We can use equation 5.4 from Chapter 5 to get the sequence of implied forward rates.

You'll notice that this forward curve on 3-month LIBOR turns out to be the same as in Table 8.1 – this is how I put together the example on my spreadsheet. There are two points to this exercise. First, implied spot rates can be bootstrapped by working either * down* from the forward curve or

*from the cash market for fixed-coupon securities (I'm envisioning upwardly sloped curves). Second, and more important, this demonstrates how we can infer the forward curve that is consistent with observed (and, likely, interpolated) swap fixed rates. That provides the inputs needed to price nonvanilla swaps.*

**up**Suppose a corporation plans to issue at par value a 1-year, fixed-rate bond in three months to refinance some maturing debt. This is a classic * pre-issuance* interest rate risk management problem. The risk is that market rates jump up unexpectedly prior to issuance, raising the coupon rate on the new debt. “Unexpectedly” here is important; nothing can be done about widely anticipated rate changes because they already are priced into the derivatives that might be used to hedge the risk. The corporation could sell some interest rate futures contracts but a commonly used strategy is to enter a 3 x 15

*as the fixed-rate payer. This is a 1-year swap that starts in three months. In practice, this might be referred to as the*

**forward-starting swap***forward swap.*

**3mly**A concern for the corporate treasurer is that this is not a plain vanilla swap having a widely quoted fixed rate. Suppose a commercial bank offers to “sell” the swap to the corporation for a fixed rate of 2.98%. Is that reasonable pricing? You only know for sure that 3-month LIBOR is 0.50% and that 1-year and 2-year vanilla swaps are at 2.12% and 3.40% but you assume it is reasonable to use straight-line interpolation between those observed rates. Given your now fully loaded bond math toolkit, you can set up and solve this equation.

This solves for the “average” of the relevant 3 x 15 segment of the LIBOR forward curve. The answer is * SFR* = 2.93%. This neglects credit risk and transactions costs. Once again, you can think of it as the mid-market rate around which the swap market maker builds the bid-ask spread as compensation for the costs and risks of entering the derivative. The corporation now has a basis to negotiate, perhaps arguing that 2.98% is too high a fixed rate.