Swap value after inception: time and rate changes

Once a swap is in place, what makes it gain or lose value?

The swap rate is the swap’s price, and it never changes after inception. What changes is the swap value : what the contract would be worth if it were marked to market today. The mark-to-market (MTM) value starts at zero and then moves with interest rates and with the passing of time. This is the price versus value of a swap distinction the exam tests.

Price versus valuethe swap rate is a price; what the contract is worth is its value

• Price: the swap rate Memorize set at inception so that value is zero; fixed for the life of the contract
• Value: mark-to-market Memorize zero at inception; later PV(received) − PV(paid)
  • Moves with rates rates up, fixed payer gains
  • Moves with time on a sloped curve, even if forwards do not change

How much does a swap pay each period?

Each period the fixed-rate payer receives the market reference rate (MRR) that has just set and pays the fixed swap rate. The MRR is like the “spot” price on that date and the swap rate is the agreed “forward” price, so the periodic settlement value to the fixed-rate payer has the same form as the payoff of any long forward.

$$P = (MRR - s_N) \times N \times t$$

Here t is the fraction of a year in the period (0.25 for quarterly, 0.5 for semiannual). A positive result is received by the fixed-rate payer; a negative result is paid. The fixed-rate receiver’s settlement is the same number with the opposite sign.

What happens to the value when rates change?

The value to the fixed-rate payer is what it expects to receive minus what it expects to pay, in present value terms.

$$V = \sum PV(\text{floating received}) - \sum PV(\text{fixed paid})$$

If forward rates rise after inception, the present value of the floating payments rises while the fixed payments stay put, so the fixed-rate payer gains and the receiver loses. A fall in rates does the reverse. Solving for a new swap rate on the new curve gives a number above the old one when rates have risen: the payer locked in a cheaper rate than the market now offers.

A quick way to size the gain is to price the difference. A new swap for the remaining term would carry the new swap rate, while this one carries the old rate, so the payer is ahead by the rate gap on every remaining period.

$$V_{payer} = (s_{new} - s_{old}) \times N \times t \times \sum DF$$

The sum of the discount factors covers the remaining payment dates only, on the new curve. The receiver’s value is the negative of the payer’s.

Fixed-rate payer versus fixed-rate receiverthe two sides move in opposite directions

• Rates rise (forward curve shifts up) Memorize PV(floating received) > PV(fixed paid)
  • Fixed-rate payer gains
  • Fixed-rate receiver loses
• Rates fall (forward curve shifts down) Memorize PV(floating) < PV(fixed)
  • Fixed-rate payer loses
  • Fixed-rate receiver gains
• Time passes, upward-sloping curve, forwards unchanged Use it early net payments are made by the payer
  • Fixed-rate payer value turns positive
  • Fixed-rate receiver value turns negative

Slide the forward curve up and down. The swap rate you started with stays on the chart, and the shaded gap between it and the new curve is the value that moves from one side to the other.

Exam tip Rates up, fixed payer up. The swap rate never changes; its value does.

What happens as time passes?

Even if no rate changes, value moves, because the early periods settle and drop out. On an upward-sloping curve the fixed payer pays net in the early periods and expects to receive net later (the previous lessons showed why). After the early settlements, the periods that remain are the ones where it receives, so its swap value turns positive and the receiver’s turns negative.

Why is a receive-fixed swap like a bond?

The fixed-rate receiver is long a fixed-rate bond and short a floating-rate note. A floating-rate note stays near par, so the swap’s value moves like the fixed-rate bond’s. If rates fall, the value of a receive-fixed swap as a long bond rises. If the curve slopes upward and rates follow their forwards, a par fixed-rate bond drifts to a discount as time passes, which is the receiver’s loss in the example above.

Go deeper Duration of a swap

Because a receive-fixed swap is a long fixed-rate bond plus a short floating-rate note, its interest rate sensitivity is roughly the duration of the fixed-rate bond minus the small duration of the FRN. Managers use receive-fixed swaps to add duration without buying bonds. Duration itself is covered in Fixed Income.

Go deeper Other swaps follow the same logic

Currency and equity swaps also start at zero value and move with the underlying variables. The price is whatever fixed terms make them fair at inception, and the value afterward is the present value of what you receive minus what you pay. For the Level I exam, recognition is enough.