A Teaching Note on Pricing and Valuing Interest Rate Swaps Part 1

A Teaching Note on Pricing and Valuing Interest Rate Swaps Using LIBOR and OIS Discounting The intent of this note is to extend the discussion of pricing and valuing interest rate swaps that appears in chapter eight of my book, Bond Math: The Theory Behind the Formulas (Wiley Finance, 2011), to include recent developments in the use of OIS (Overnight Indexed Swap) discounting. In Bond Math, I use the traditional method of bootstrapping implied spot (i.e., zero-coupon) swap rates, using either the LIBOR forward curve or fixed rates on a series of “at-market” interest rate swaps that have a market value of zero. In the last few years, swap dealers have started to use implied spot rates and corresponding discount factors that have been bootstrapped from fixed rates observed in the OIS market. This note explains why and how this is being done using simplified numerical examples based on the Bond Math chapter. Three important calculations for interest rate swaps to be covered are: (1) pricing an at-market (or par) swap, (2) valuing an off-market swap, and (3) inferring the forward curve that is consistent with a sequence of at-market swaps. I use “pricing” to mean determining the fixed rate at inception and “valuing” to mean determining the market value thereafter. To keep the examples simple, I neglect transactions costs, use a 30/360 day-count convention, and assume that the pricing and valuation are on a settlement date. However, the same techniques can be used to deal with bid-ask spreads, more common day-count conventions such as actual/360 and actual/365, and between-settlement-date calculations. Starting with the LIBOR forward curve, pricing an at-market swap entails “monetizing” each forward rate by multiplying by the notional principal and day-count 1 fraction and then discounting the cash flows. The idea is to derive a uniform fixed rate such that when it is monetized using the same notional principal and day-count fractions, the same present value is obtained. Therefore, an at-market swap, which is also called a “par” swap, has an initial value of zero by construction. The swap fixed rate is an “average” of the LIBOR forward curve, not a simple arithmetic or geometric average, but an average in the time-value-of-money sense in that the two legs of the swap have the same present value. An obvious challenge is to obtain the LIBOR forward curve needed to price the swap because it generally is not observed. It helps if there are actively traded futures contracts on the reference rate, such as the Eurodollar contract at the CME Group in Chicago. Its trading results provide data on 3-month LIBOR for quarterly delivery dates out to ten years. However, a convexity adjustment to the observed futures rate is needed to get the forward rate. That requires an interest rate term structure model and assumptions about the future rate volatility and correlations across points along the yield curve. The need for this adjustment, which is discussed in greater detail in Bond Math, arises because of daily mark-to-market and settlement practices on exchange-traded futures contracts. Swap valuation involves: (1) comparing the contractual fixed rate to that on an atmarket swap having otherwise matching terms, (2) getting an annuity for the difference in the fixed rates, and (3) calculating the present value of the annuity using a sequence of discount factors corresponding to the settlement dates. An alternative approach is to interpret the interest rate swap as a long/short combination of a bond paying the fixed rate on the swap and a floating-rate bond paying the money market reference rate, e.g., 3- 2 month LIBOR. In the traditional methodology for swap valuation, the implicit floater maintains its par value on rate-reset dates while the fixed-rate bond can be valued at a premium or discount. The difference in the prices of these two bonds is the value of the interest rate swap. With OIS discounting, the result that the implicit floating-rate bond paying LIBOR is priced at par value no longer holds. It is useful to infer the LIBOR forward curve from observed fixed rates on atmarket swaps. This implied forward curve, also called the projected curve, is used to price and value non-standard contracts. For example, a “vanilla” interest rate swap has a constant notional principal and an immediate start date. Non-vanilla varieties can have varying notional principals and deferred start dates. Pricing such swaps entails getting the “average”, perhaps a “weighted average” if the notional principal varies, of the relevant segment of the forward curve. As shown later in the paper, the implied LIBOR forward curve calculated for OIS discounting is needed to value collateralized interest rate swaps using the combination-of-bonds approach. The first section of the note repeats the examples of pricing and valuing interest rate swaps in chapter eight of Bond Math and uses the LIBOR swap curve for discounting. In addition to showing the equations that produce the implied spot rates, the corresponding discount factors are calculated. The second section explains OIS discounting and identifies its impact on the swap values and the implied forward curve. The third section generalizes the numerical examples for practical applications in pricing and valuing swaps using spreadsheet analysis. 3 The Traditional Method to Price and Value Interest Rate Swaps Suppose the sequence of fixed rates on at-market interest rate swaps is: 1.04% for 6 months, 1.58% for 9 months, 2.12% for 12 months, 2.44% for 15 months, 2.76% for 18 months, 3.08% for 21 months and 3.40% for 24 months. These swaps are for quarterly settlements tied to 3-month LIBOR and assume the 30/360 day-count convention. Current 3-month LIBOR is 0.50%. The assumed 30/360 day-counts for all rates is to simplify the exposition—in practice, actual/360 and actual/365 are common. The implied spot rates given this sequence of observed (or, perhaps, interpolated) fixed rates are determined via the standard bootstrapping technique using the combination-of-bonds interpretation of a swap. That is, the implicit fixed-rate bond is priced at par value (100) and redeems that amount at maturity. Each quarterly coupon payment on this bond is the annual swap fixed rate times 100, and then divided by four because there are assumed to be 90 days in each quarter of the 360-day year. The 6month implied spot rate is the solution for Spot0x6 in this expression: 100 = 1.04 / 4 1.04 / 4 + 100 + , Spot 0 x 6 = 0.010407  0.0050   Spot  2 + 1   0 x6  4  1 +   4   The first cash flow is discounted using current 3-month LIBOR. The result for Spot0x6 becomes an input in the solution for the “0x9” implied spot rate. 100 = 1.58 / 4 1.58 / 4 1.58 / 4 + 100 + + , Spot 0 x9 = 0.015829  0.0050   0.010407  2  Spot  3 1 +  0 x9   1 + 4  1 +  4    4  These are the remaining equations for the bootstrapped implied spot curve. 4 100 = 2.12 / 4 2.12 / 4 2.12 / 4 2.12 / 4 + 100 + + + , 4  0.0050   0.010407  2  0.015829  3  Spot  1 +  0 x12   1 +  1 + 4  1 +   4 4     4   Spot 0 x12 = 0.021272 2.44 / 4 2.44 / 4 2.44 / 4 2.44 / 4 + + +  0.0050   0.010407  2  0.015829  3  0.021272  4 1 +   1 +  1 +  4  1 +  4 4 4       2.44 / 4 + 100 + , Spot 0 x15 = 0.024506 5  Spot 0 x15  1 +  4   100 = 2.76 / 4 2.76 / 4 2.76 / 4 2.76 / 4 + + +  0.0050   0.010407  2  0.015829 3  0.021272  4 1 +   1 +  1 +  4  1 +  4 4 4       2.76 / 4 2.76 / 4 + 100 + + , Spot 0 x18 = 0.027756 5 6  0.024506   Spot 0 x18  1 +  1 +  4   4   100 = 3.08 / 4 3.08 / 4 3.08 / 4 3.08 / 4 + + +  0.0050   0.010407  2  0.015829  3  0.021272  4  1 +   1 + 1 +  4  1 +  4 4 4       3.08 / 4 + 100 3.08 / 4 3.08 / 4 , Spot 0 x 21 = 0.031025 + + + 5 6 7  Spot 0 x 21   0.027756   0.024506   1 +  1 +  1 +   4 4     4   100 = 3.40 / 4 3.40 / 4 3.40 / 4 3.40 / 4 + + + 2 3 4  0.0050   0.010407   0.015829   0.021272  1 +  1 +  1 +  1 +  4   4 4 4       3.40 / 4 3.40 / 4 3.40 / 4 3.40 / 4 + 100 + + + + , 5 6 7 8  0.024506   0.027756   0.031025   Spot 0 x 24  1 +  1 +  1 +  1 +  4 4 4       4   100 = Spot 0 x 24 = 0.034316 All of the reported results for these and the following equations are calculated on a spreadsheet to preserve accuracy in the bootstrapping process, which is very sensitive to 5 rounding. However, the rounded results are shown in the equations for the purpose of consistent exposition. Each implied spot rate converts to a discount factor (DF), starting with current 3month LIBOR. 1 = 0.998752 0.0050    1 + 4   DF 0 x3 = DF 0 x 6 = DF 0 x 9 = 1  0.010407  1 +  4   1 = 0.988222  0.015829  3  1+   4 DF 0 x12 = DF 0 x15 = DF 0 x18 = DF 0 x 21 = DF 0 x 24 = = 0.994817 2 1  0.021272  1 +  4   4 1  0.024506  5  1+   4 1  0.027756  1 +  4   6 1  0.031025  7  1+   4 1  0.034316 8  1+   4 = 0.979008 = 0.969923 = 0.959358 = 0.947352 = 0.933943 Discount factors are particularly useful in spreadsheet analysis because a future cash flow needs only to be multiplied by the factor to obtain its present value. 6 The forward LIBOR curve that is consistent with these at-market swap fixed rates can be calculated from either the implied spot rates or the discount factors. The implied, or projected, 3-month forward rate between months 3 and 6 is denoted Rate3x6; the implied rate between months 12 and 15 is Rate12x15. These are the results using the bootstrapped spot rates: Rate3x 6 2     1 + 0.010407     4  − 1 * 4 = 0.015821 =     0.005000 1    1 + 4    Rate6x9 3     1 + 0.015829     4  − 1 * 4 = 0.026694 =     0.010407  2    1 + 4    Rate9x12 4     1 + 0.021272     4  − 1 * 4 = 0.037647 =     0.015829 3    1 + 4    Rate12x15 5     1 + 0.024506      4  = − 1 * 4 = 0.037468   0.021272  4     1 + 4    Rate15x18 6     1 + 0.027756     4  − 1 * 4 = 0.044047 =    0.024506  5     1 + 4    7 Rate18x 21    0.031025  7   1+     4  = − 1 * 4 = 0.050696    0.027756  6    1 + 4    Rate21x 24 8     1 + 0.034316     4  − 1 * 4 = 0.057427 =    0.031025  7     1 + 4    The same series of implied forward rates can be obtained using the discount factors.  0.998752  − 1 * 4 = 0.015821 Rate3x 6 =   0.994817   0.994817  − 1 * 4 = 0.026694 Rate6x9 =   0.988222   0.988222  Rate9x12 =  − 1 * 4 = 0.037647  0.979008   0.979008  − 1 * 4 = 0.037468 Rate12x15 =   0.969923   0.969923  − 1 * 4 = 0.044047 Rate15x18 =   0.959358   0.959358  Rate18x 21 =  − 1 * 4 = 0.050696  0.947352   0.947352  Rate21x 24 =  − 1 * 4 = 0.057427  0.933943  Suppose this sequence of implied forward rates is in fact the observed LIBOR forward curve and each is the fixed rate on a forward rate agreement (FRA) on 3-month LIBOR. An FRA is just a one-period interest rate swap; alternatively, a swap is a series 8 of FRAs. For example, a 3x6 FRA can be used to lock in 1.5821% for an exposure to 3month LIBOR, three months into the future. A company that issues a floating-rate note tied to LIBOR can create a “synthetic” fixed-rate liability by entering a series of payfixed FRAs or a pay-fixed interest rate swap. The difference is that the FRAs would fix rates that vary according to the shape of the LIBOR forward curve whereas the swap establishes the same fixed rate for each period. The implied spot curve, and the corresponding discount factors, can be computed as the geometric average of the LIBOR forward rates in addition to being bootstrapped from the swap fixed rates. A few examples illustrate the process. 2 Spot 0 x 6  0.005000   0.015821     , 1 +  * 1 +  = 1 + 4 4      4  Spot 0 x 6 = 0.010407 3 2  0.010407  * 1 + 0.026694  =  Spot 0 x9  ,   1 +   1+ 4   4 4     Spot 0 x9 = 0.015829 4 3  0.015829   0.037647   Spot 0 x12   ,  = 1 +  * 1 + 1 + 4   4 4     Spot 0 x12 = 0.021272 Notice again that the result from one calculation is used as an input in the next—that is the hallmark of the bootstrapping technique. An at-market swap fixed rate is the “average” of the relevant segment of the LIBOR forward curve. Each forward rate is multiplied by the notional principal (assume it is 100) and by the day-count factor (which is 0.25 because of the assumed 30/360 convention). These cash flows are discounted using the implied spot rates corresponding to each time period. The 24-month swap fixed rate (SFR) is the uniform rate such that when it is 9 multiplied by 100 and by 0.25 and discounted using the same spot rates, its present value is equal to that of the forward curve. In sum, SFR is the solution to this expression: 0.5000% *100 * 0.25 1 0.005000   1 +  4   + + 1.5821% *100 * 0.25 + 4.4047% *100 * 0.25 3.7468% *100 * 0.25 5 0.024506    1 + 4   0.010407   1 +  4   + 2 6 0.027756    1 + 4   2.6694% *100 * 0.25 3 0.015829   1 +  4   + 3.7647% *100 * 0.25 + 4 0.021272   1 +  4   5.0696% *100 * 0.25 0.031025    1 + 4   7 + 5.7427% *100 * 0.25 8 0.034316    1 + 4   = SFR *100 * 0.25 1 0.005000    1 + 4   + SFR *100 * 0.25 5 0.024506    1 + 4   + SFR *100 * 0.25 + 2 0.010407    1 + 4   + SFR *100 * 0.25 6 0.027756    1 + 4   SFR *100 * 0.25 3 0.015829    1 + 4   + + SFR *100 * 0.25 0.031025    1 + 4   7 SFR *100 * 0.25 4 0.021272    1 + 4   + SFR *100 * 0.25 8 0.034316    1 + 4   The result for the 24-month swap on 3-month LIBOR is that SFR = 3.400%. It is no surprise that the same swap fixed rate of 3.400% arises via the discount factors in a more compact equation. (0.5000% *100 * 0.25 * 0.998752) + (1.5821% *100 * 0.25 * 0.994817) + (2.6694% *100 * 0.25 * 0.988222) + (3.7647% *100 * 0.25 * 0.979008) + (3.7468% *100 * 0.25 * 0.969923) + (4.4047% *100 * 0.25 * 0.959358) + (5.0696% *100 * 0.25 * 0.947352) + (5.7427% *100 * 0.25 * 0.933943) = (SFR *100 * 0.25 * 0.998752) + (SFR *100 * 0.25 * 0.994817) + ( SFR *100 * 0.25 * 0.988222) + (SFR *100 * 0.25 * 0.979008) + (SFR *100 * 0.25 * 0.969923) + (SFR *100 * 0.25 * 0.959358) + (SFR *100 * 0.25 * 0.947352) + (SFR *100 * 0.25 * 0.933943) 10