Pricing a swap is the determination of the fixed rate at origination; valuing the swap is determining its fair value thereafter. A plain vanilla swap starts with an initial value of zero because by construction the present values of the fixed-rate leg and the floating-rate leg are equal. As time passes and as interest rates change, the swap takes on positive or negative value. That's important because accounting rules for derivatives require that the fair value of the swap be recognized on the balance sheet as an asset or liability. Moreover, depending on the applicability of hedge accounting treatment, the change in fair value from period to period might have to flow through the income statement. That can impact the closely watched earnings per share numbers.

Tet's go back to the 2-year, 3.40% fixed versus 3-month LIBOR, quarterly settlement in arrears, $60 million notional principal, plain vanilla interest rate swap between Party A and B depicted in Figure 8.1. Suppose that three months go by. Party A, the fixed-rate payer, makes a net settlement payment of $435,000 to its counterparty because the initial observation for 3-month LIBOR is 0.50% in Table 8.1.

The fair value of the swap now depends on the current market and, in particular, on the relevant segment of the new LIBOR forward curve. This

TABLE 8.2 3-Month LIBOR Forward Curve and Implied Spot Rates and Discount Factors Three Months Later

Time Period

Forward Rates

Time Period

Implied Spot Rates

Discount Factors

0×3

0.7500%

0×3

0.7500%

0.998129

3×6

1.6210%

0×6

1.1853%

0.994100

6×9

2.6780%

0×9

1.6822%

0.987489

9×12

3.5460%

0×12

2.1474%

0.978812

12×15

3.5980%

0×15

2.4371%

0.970086

15×18

4.2520%

0×18

2.7390%

0.959882

18×21

4.7090%

0×21

3.0198%

0.948713

forward curve and the implied spot rates are shown in Table 8.2. Notice that the forward curve has twisted and flattened – the near-term rates have gone up a bit and the longer-term rates have come down.

Now we can solve for the fixed rate on a 1.75-year swap using the forward rates and the discount factors in Table 8.2.

The result is that SFR = 3.00%.

Therefore, after three months, the swap has become a liability to Party A because it has an obligation to pay 3.40% fixed for the next seven quarterly periods when the going market rate is only 3.00%. Likewise, the swap has become an asset to Party B. Notice that these changes in market value since origination have occurred with 3-month LIBOR going up from 0.50% to 0.75%. A common mistake based on looking at Figure 8.1 is to conclude that Party A gains when LIBOR goes up and Party B gains when LIBOR goes down. The current observation on LIBOR is important – it determines who pays whom on the next settlement date. In this case, Party A now owes Party B $397,500 at the end of the period.

However, it is the entire forward curve that determines the market value of the swap. Current 3-month LIBOR is merely the first observation on that forward curve.

The fair value of the swap is the present value of the annuity representing the difference between the contractual fixed rate of 3.40% and the mark-to-market rate of 3.00%. The amount of the annuity is $60,000 per period for the remaining seven quarterly periods, given the $60 million notional principal and the 30/360 day-count.

This is the unambiguous part of swap valuation. The idea is that the two counterparties could enter a “mirror swap” at 3.00% to offset 3-month LIBOR for the remaining 1.75 years. In principle, Party A owes B a total of $420,000 to be paid in seven installments of $60,000.

The ambiguous part of swap valuation is in calculating the present value of the $60,000 per period annuity. Suppose this is a voluntary unwind of the derivative contact. Party A or B for some reason wants to exit the deal and asks for a settlement payment. If this is an unsecured swap, Party A might argue that the appropriate discount rate is its 1.75-year cost of funds on fully amortizing debt because it is extinguishing its liability. However, if this swap is collateralized in some manner, Party B might argue for a lower discount rate (and higher settlement payment from Party A to close out the contract). For routine accounting valuations, the traditional method has been to get the discount rate (or rates) from the current swap market.

What do you suggest – discount the $60,000 annuity at the current swap fixed rate of 3.00% or the sequence of implied spot rates (or discount factors)? Are you anticipating the same result, differing perhaps only by rounding? It's an interesting bond math problem. First use 3.00% to get $407,678 for the value of the swap. Here's the full equation, but it's easily obtained on a calculator.

Now use the implied spot rates to get $410,233. The discount factors would obtain the same value with any difference due only to rounding.

Party A should recognize the interest rate swap on its balance sheet as a liability, as Party B books an asset, but for how much: $407,678 or $410,233 ?

There is not a big difference between the two values given the $60 million notional principal; the issue is theoretical correctness. Here's a hint: The new swap fixed rate of 3.00%, as an “average” of the forward curve, is also the 1.75-year par yield as described in Chapter 5. It's the solution for PMT in the next expression.

Given this implied spot curve, a 3.00%, quarterly payment, 1.75-year bond is priced at par value. Its yield to maturity of 3.00% is the weighted average of the spot rates, with most of the weight on the last rate that has largest cash flow. But our swap valuation problem entails the present value of an annuity, not a fixed-rate bond that redeems principal at maturity. The theoretically correct value is $410,233, obtained using the sequence of implied spot rates (or discount factors).

The key point here is that the fixed rate on a swap is the initial “average” of the relevant segment of the forward curve for the money market reference rate. Later, the value of the swap depends on the new “average” of the remaining segment of the new forward curve. In our example, the 3.40% swap is marked to market using a 3.00% fixed rate. But notice that any number of shifts and twists to the forward curve after three months could have resulted in a new “average” of 3.00%. The annuity component of value is still $60,000 per period. However, the present value of the annuity depends on the particular shift and twist – they determine the new implied spot curve and the fair value for the swap.

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