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Show how a plain vanilla interest rate swap converts an asset or liability into a different one, and work out the cash flows that result.

FRM® Part I · Financial Markets and Products · Swaps · FRM1.FM.20.b

The idea

A swap changes the character of an existing asset or liability without touching the original contract. Take a borrower paying a floating rate plus a spread who enters a swap paying fixed and receiving the same floating rate. The floating legs cancel and the borrower is left with a fixed-rate liability: All-in cost = swap fixed rate + loan spread. Now take an investor holding a fixed-rate bond who pays fixed and receives floating on a swap. The bond becomes a floating-rate asset earning the floating rate plus the gap between its coupon and the swap rate: Asset return = floating rate + (bond coupon − swap fixed rate). Always net each leg period by period, keep the spread in the position it came from, and remember the notional and payment dates of the swap should match the underlying loan or bond. Exam questions ask for the resulting all-in rate or the net cash flow in a period.

Formulas

All-in cost of a floating loan swapped to fixed

all-in cost = swap fixed rate + loan spread

swap fixed rate
the fixed rate paid on the swap
loan spread
the spread over the reference rate on the loan

Holds when the swap receives the same reference rate flat on the loan's notional and dates. Any spread on the swap's floating leg must be netted too.

Return on a fixed-rate bond swapped to floating

asset return = floating rate + (bond coupon − swap fixed rate)

floating rate
the reference rate received on the swap
bond coupon
the coupon rate on the bond held
swap fixed rate
the fixed rate paid on the swap

The investor pays fixed and receives floating on the bond's notional. Net each leg in the same period.

A practice question

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

A company has a floating-rate loan and wants certainty about its interest cost. Which interest rate swap position achieves this?

  1. Receive fixed and pay floating, on the loan's notional and payment dates.Why not: Doubles the floating exposure instead of removing it; the company would pay floating on both legs.
  2. Pay fixed and receive floating, on the loan's notional and payment dates. - correct
  3. Receive fixed and pay fixed in two different swaps.Why not: Two fixed legs offset each other and leave the floating loan cost unchanged.
  4. Pay floating and receive floating on a different reference rate.Why not: A basis swap changes the floating index but leaves the company exposed to floating rates.

Paying fixed and receiving floating means the floating receipt offsets the loan's floating payment, leaving the swap fixed rate (plus any loan spread) as the net cost.

A harder one

An investor holds USD 30 million of a bond paying a 5.5% annual coupon, paid semi-annually. She enters a semi-annual swap on USD 30 million in which she pays 4.9% fixed and receives the floating rate. For one period, the floating rate is 4.0%. What is her net cash inflow for the period across the bond and the swap?

  1. USD 1.380 millionWhy not: Uses the full-year amount without the 0.5 accrual fraction.
  2. USD 0.510 millionWhy not: Subtracts the 0.6% spread instead of adding it to the floating rate.
  3. USD 0.690 million - correct
  4. USD 0.825 millionWhy not: Counts only the bond coupon and ignores the swap's net payments.

Bond coupon 30 x 5.5%/2 = 0.825; swap pays 30 x 4.9%/2 = 0.735 and receives 30 x 4.0%/2 = 0.600. Net = 0.825 - 0.735 + 0.600 = 0.690, equal to floating plus 0.6% on the notional for half a year.

The hardest tier

An investor holds a EUR-denominated floating-rate note paying the reference rate plus 0.5% and wants a fixed-rate asset. She enters a swap in which she receives 3.6% fixed and pays the same reference rate. What is her net return per year?

  1. 4.1% - correct
  2. 3.1%Why not: Subtracts the note spread instead of adding it to the swap fixed rate.
  3. Reference rate + 4.1%Why not: Leaves a floating component, but the swap's floating payment offsets the note's floating receipt.
  4. 3.6%Why not: Ignores the 0.5% spread on the note, which she keeps.

The floating payment on the swap offsets the floating coupon of the note, leaving 3.6% swap fixed + 0.5% spread = 4.1% fixed.

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