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Show how an interest rate swap works as a string of forward commitments

CFA® Level I · Derivatives and Risk Management · Pricing and Valuation of Interest Rate and Other Swaps · DE.1.c

The idea

A plain vanilla interest rate swap exchanges a fixed rate for a floating rate on a notional that is never itself exchanged. Each payment date can be read as a separate forward contract on the reference rate, and the swap's fixed rate is the single rate that makes the whole strip worth zero at inception. That is why a swap has no value when it is struck and acquires value afterwards as rates move: the package of implied forwards is repriced while the contracted fixed rate stays where it was.

Formulas

Swap rate from forward rates

DF_t = 1 ÷ [(1 + f₁)(1 + f₂)…(1 + f_t)] and R = (1 − DF_n) ÷ Σ DF

DF_t
the discount factor for date t
f₁
the forward rate for period 1
f₂
the forward rate for period 2
f_t
the forward rate for period t
R
the swap rate, the fixed rate that makes the swap worth zero
DF_n
the discount factor for the last date
DF
the discount factors for every payment date

The fixed rate is a weighted average of the forward rates, weighted by the discount factors.

A practice question

From the question bank, with the reason each wrong option is wrong.

What is exchanged on each payment date of a plain vanilla interest rate swap?

  1. The notional principalWhy not: The notional is a reference amount for calculating the payments and never changes hands.
  2. The net of the fixed and floating interest payments - correct
  3. Both parties' full interest paymentsWhy not: In a standard swap the two payments are netted, so only the difference moves between the parties.

In an interest rate swap, one side pays a fixed rate and the other pays a floating rate (one that moves with the market). The notional is just the amount used to calculate the interest and is never handed over. On each payment date the two interest amounts are netted, so only the difference is paid. Net payment = Notional × (Fixed rate − Floating rate). Example: 10,000,000 × (5% − 4%) = 100,000, paid by the fixed payer to the floating payer, not the two full amounts.

A harder one

A plain vanilla interest rate swap can be broken down into simpler contracts. How can the exchange on each payment date be viewed?

  1. As a separate option on the reference rate for each periodWhy not: No optionality exists: each payment is owed whatever the rate turns out to be.
  2. As a separate loan from one party to the other each periodWhy not: No principal is lent or repaid, which is what distinguishes a swap from borrowing.
  3. As a separate forward contract on the reference rate - correct

A swap can be seen as a series of forward contracts on the floating interest rate, one for each payment date. A forward is a promise to settle a price fixed today. The fixed rate is chosen so that all these forwards together are worth zero at the start. Example: in a three-year swap paying 4% fixed and receiving floating, the second payment is like a forward to pay 4% and receive the year 2 floating rate. It is not an option, because each payment is owed whatever happens.

The hardest tier

Market interest rates for all maturities rise sharply after a plain vanilla swap is struck. What happens to the value of the pay-fixed (receive-floating) position?

  1. It becomes positive - correct
  2. It becomes negative, because it must pay moreWhy not: The fixed payment does not change when rates rise; that is what makes it fixed. It is the floating receipt that rises.
  3. It stays zero for the life of the contractWhy not: Zero value holds only at inception, when the fixed rate was set against the then-current curve.

The pay-fixed party is locked into paying the old, lower fixed rate. Meanwhile the floating payments it receives reset to the new, higher rates, so it receives more than it pays. Net gain per period = Notional × (Floating rate − Fixed rate). Example: on 10,000,000 with fixed 4% and floating now 5%, the gain is 10,000,000 × 1% = 100,000 a year. The present value (today's worth) of these expected gains is the positive value of the position.

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