A swap as a series of forwards: what is the same and what is different

If a swap is just several forwards, why does it have one fixed rate?

A swap is an agreement to exchange a series of future cash flows. An interest rate swap is, economically, a series of forward contracts on the market reference rate (MRR): on each payment date the fixed-rate payer settles the gap between the MRR and the fixed rate, on a notional that never changes hands. A single period of that is a forward rate agreement, and a swap is a strip of them.

The twist is the rate. Interest rates have a term structure, so a series of FRAs would each carry a different fixed rate, the implied forward rate for that period. A standard swap carries one single fixed rate , the swap rate, for every period. That one difference explains almost everything on the exam.

What is the same?

Both are forward commitments, so they share the same traits.

• Firm commitment, symmetric payoff. Neither side can walk away, and one side’s gain is the other’s loss.
• No cash at the start. The fixed rate is set so that the contract begins at zero value.
• Net cash settlement. The payment is based on the difference between the MRR and the fixed rate, times the notional and the period length.
• Credit exposure. Both are over-the-counter contracts, so each side carries counterparty risk.

What is different?

A swap has periodic settlement , one payment at the end of each period under a single contract. An FRA has one settlement, made at the start of its interest period and discounted. And because the swap rate is one number laid over a whole curve, the contract is a set of off-market forwards : forwards whose fixed rate differs from the market’s forward rate for that period.

Swap versus a series of forwardssame building block, different fixed rate

• What is the same Memorize both are forward commitments on the reference rate
  • Firm commitment symmetric payoff, no option to walk away
  • No cash at the start value is zero at inception
  • Net cash settlement (MRR − fixed rate) × notional × period
  • Counterparty credit exposure both are over-the-counter contracts
• What is different Memorize the fixed rate and the timing
  • Fixed rate one swap rate for all periods, not each period's own forward rate
  • Value of each period at the start non-zero (off-market); only the total is zero
  • Settlement swap pays at the end of each period; an FRA settles at the start
  • Contract one agreement, not a separate contract per period

For the fixed-rate payer, the value of each period at inception is the gap between that period’s forward rate and the swap rate, times the notional, discounted to today.

$$V_i = (f_i - s_N) \times N \times DF_i$$

Add all the periods and the total must be zero, otherwise the swap would not be fair to start with.

$$\sum_{i=1}^{N} (f_i - s_N) \times N \times DF_i = 0$$

Change the spot rates and watch the bars. Each period’s gap between the forward bar and the swap-rate line is that period’s off-market value, and the running total always lands on zero.

Exam tip If a statement says every forward inside a new swap is worth zero, it is false. If it says the whole swap is worth zero at the start, it is true.

How is a swap like two bonds?

The swap as a bond portfolio view is the second way to see the same contract. The fixed-rate payer makes fixed payments, like the issuer of a fixed-rate bond , and receives floating payments, like the holder of a floating-rate note (FRN) that pays the MRR. At the start the purchase and sale prices of the two bonds offset, and at maturity the two principals offset. Only the difference between the coupons is left, which is exactly the swap’s net payment.

Interest rate swapprincipals cancel, so only the coupon difference remains

• Pay fixed, receive floating Memorize long FRN + short fixed-rate bond
  • Gains when rates rise
• Receive fixed, pay floating Memorize long fixed-rate bond + short FRN
  • Gains when rates fall

$$\text{Pay fixed} = \text{long FRN} + \text{short fixed-rate bond}$$

Go deeper Why an off-market forward has a value

An FRA is fair when its fixed rate equals the implied forward rate. If you enter a forward at a different fixed rate, you lock in a rate better or worse than the market’s, and the difference, discounted, is the contract’s value. Each period of a swap is exactly such a forward, with the swap rate in place of the forward rate.

Go deeper Where the credit exposure sits

On an upward-sloping curve the early periods have negative value for the fixed payer and the late periods positive value. As the early periods settle, the remaining value is positive and sits in the later periods. So the credit exposure that builds up on the fixed payer’s side over time is a direct result of the off-market structure.