GF561 Unit 7 Interest Rate Swap Valuation Problem Example

Two Bonds, Then Three Forwards: A Unit 7 Swap Valuation Problem for a Paving Contractor's Loan

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Purdue University Global

GF561: Derivatives and Hedging

Unit 7 Swap Valuation Problem

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[Date]

The paving contractor, its loan and every rate are composite and illustrative; the article and textbooks cited are real.

The Swap

The contractor borrowed $40 million at a floating rate to buy pavers, milling machines and trucks, and entered a swap to fix its interest cost. It pays a fixed 4.10 percent a year on $40 million and receives a floating rate set at the start of each annual period. The floating rate was set at 3.70 percent at the last reset, nine months ago. Three annual payments remain, due in 0.25, 1.25 and 2.25 years.

The Curve

Continuously compounded zero rates for the three dates are 4.30, 4.05 and 3.90 percent, giving discount factors of:

1. 0.25 years: e raised to negative 0.043 times 0.25, about 0.989308.

2. 1.25 years: e raised to negative 0.0405 times 1.25, about 0.950635.

3. 2.25 years: e raised to negative 0.039 times 2.25, about 0.915990.

Part 1: The Swap as Two Bonds

The fixed leg. It behaves like a bond paying $1,640,000 each year, 4.10 percent of $40 million, with $40 million at the end. Its value is $1,640,000 times the sum of the three discount factors, 2.855933, which is about $4,683,730, plus $40 million times 0.915990, which is $36,639,594. Total: about $41,323,323.

The floating leg. Right after each reset, a floating-rate bond is worth its face value. So the floating leg is worth the next known payment, $1,480,000, 3.70 percent of $40 million, plus $40 million of notional, all discounted for 0.25 years: $41,480,000 times 0.989308, about $41,036,478 (Hull, 2022).

Value to the contractor. The contractor receives floating and pays fixed, so the swap is worth $41,036,478 less $41,323,323, about negative $286,845.

Part 2: The Swap as Forward Rate Agreements

Each remaining exchange can be valued as a separate forward agreement, using forward rates implied by the curve.

First payment, in 0.25 years. The floating rate is already known at 3.70 percent. The net payment is 3.70 less 4.10 percent of $40 million, negative $160,000, worth about negative $158,289 today.

Second payment, in 1.25 years. The forward rate from 0.25 to 1.25 years, from the zero rates, is 4.05 times 1.25 less 4.30 times 0.25, about 3.988 percent continuously compounded, or about 4.068 percent with annual compounding. The net payment is 4.068 less 4.10 percent of $40 million, about negative $12,773, worth about negative $12,142 today.

Third payment, in 2.25 years. The forward rate from 1.25 to 2.25 years is about 3.712 percent continuously compounded, or about 3.782 percent annually. The net payment is about negative $127,090, worth about negative $116,414 today.

Total. About negative $286,845, matching Part 1.

Both methods agree because they price the same promises: the bond view nets two present values at the end, while the forward view nets each year first, and the swap is worth about negative $287,000 to the contractor because the fixed 4.10 percent now sits above every floating rate the curve expects.

What the Negative Value Means

The swap has lost value for the contractor since it was signed, because expected short-term rates have fallen below the 4.10 percent it agreed to pay. It is not a loss on the loan: the contractor's floating-rate interest has fallen by the same amount. The swap did what it was meant to do, which is fix the total cost of borrowing at about 4.10 percent plus the loan's credit spread, whatever happens to rates.

Sensitivity to Rates

If every zero rate fell by one basis point, the swap's value to the contractor would fall by about $7,792. Most of that sensitivity comes from the fixed leg's final $40 million, which rises in value as rates fall. A fall of 100 basis points would make the swap worth about negative $1.08 million, while lowering the loan's interest by a similar present value.

A Third Check: Valuing at Each Reset

The same answer can be reached a third way. Standard textbook treatments show that a swap's value at any date equals the present value of the floating leg less the present value of the fixed leg, whichever method produces those present values (McDonald, 2013). Because the floating leg resets to par at each payment date, its value never drifts far from the notional; nearly all of the swap's value, and its sensitivity to rates, sits in the fixed leg. That is why the swap behaves much like a short position in a two-and-a-quarter-year fixed-rate bond.

Credit Considerations

The negative value means the contractor owes the bank on a mark-to-market basis. If the swap agreement requires collateral above a threshold, the contractor may have to post cash as rates fall, even though its overall interest cost is unchanged. Research on swap spreads finds that the credit quality of the parties affects swap pricing, which is one reason banks require collateral or charge more to riskier customers (Duffie & Huang, 1996).

Conventions

The floating payment uses the rate set at the last reset. Forward rates are converted from continuous to annual compounding to match the annual payment schedule.

References

Duffie, D., & Huang, M. (1996). Swap rates and credit quality. The Journal of Finance, 51(3), 921-949. https://doi.org/10.1111/j.1540-6261.1996.tb02712.x

Hull, J. C. (2022). Options, futures, and other derivatives (11th ed.). Pearson.

McDonald, R. L. (2013). Derivatives markets (3rd ed.). Pearson.