Tell apart what a swap is worth and what it is priced at
The idea
A swap's price and its value are different things. The price is the fixed rate, the swap rate, agreed at initiation. It is set so that the present value of the fixed payments equals the present value of the expected floating payments, giving the swap zero value at the start. Using discount factors for each payment date, the swap rate is (1 - final discount factor) divided by the sum of the discount factors. Once the swap is under way the fixed rate stays the same, but market rates move, so the swap acquires value. The value to the fixed-rate payer is roughly the difference between the current market swap rate and the original fixed rate, times notional, times the sum of the discount factors for the remaining payments. If rates rise, the pay-fixed party gains because it is locked into paying a below-market rate; if rates fall, the receive-fixed party gains. Value is zero-sum between the two counterparties.
A practice question
At initiation, the value of a plain vanilla interest rate swap to each party is:
- the full notional principalWhy not: Notional is not exchanged in an interest rate swap and is not its value.
- zero to both parties — correct
- one fixed payment's worthWhy not: That is one fixed payment, not the swap's value at the start.
The swap rate is chosen so the present values of fixed and floating legs are equal, so neither side has value at initiation.
A harder one
Two years after entering a pay-fixed interest rate swap, market swap rates have risen well above the original fixed rate. The swap's value to the fixed-rate payer is:
- positive — correct
- negativeWhy not: Holds for the receive-fixed party, who now receives a below-market rate.
- zeroWhy not: A swap has zero value only at initiation, or if rates return to the original swap rate.
The payer is locked into paying less than the current market rate while receiving a floating rate that has risen, so the swap is an asset to it.
The hardest tier
Discount factors for years one, two and three are 0.970, 0.935 and 0.895. What fixed rate (price) gives a new three-year annual-pay swap zero value?
- 3.50%Why not: Divides by the number of payments, 3, rather than by the sum of the discount factors.
- 10.50%Why not: Stops at 1 minus the final discount factor and never divides by the annuity factor.
- 3.75% — correct
Swap rate = (1 - 0.895) / (0.970 + 0.935 + 0.895) = 0.105 / 2.800 = 3.75%.
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