Case 010Interest rate derivativesCore

A company pays fixed 7.5% semi-annually on a Rs 100 crore swap with two years left. Given the four discount factors, find today's par swap rate and the swap's value to the company.

The situation

Kodachadri Cables entered a five-year interest rate swap three years ago, paying a fixed 7.5% semi-annually and receiving the floating six-month benchmark on Rs 100 crore of notional. Two years remain, with four semi-annual payments left, and today is a reset date so the next floating coupon has just been fixed at the current six-month rate.

The desk's discount factors for 0.5, 1.0, 1.5 and 2.0 years are 0.9662, 0.9335, 0.9019 and 0.8714. Ignore day counts and credit adjustments.

Your task

What fixed rate would a new two-year swap carry today, what is the existing swap worth to Kodachadri, and which side would have to pay the other to tear it up?

Quick check

Before any arithmetic: the discount factors imply six-month rates near 7%. Is Kodachadri's swap an asset or a liability to it?

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

Today's par rate is about 7.00%, and the swap is worth about minus Rs 0.91 crore to Kodachadri. The four discount factors sum to 3.6730; the par rate is one less the last factor, 0.1286, divided by half that annuity. The fixed leg at 7.5% is worth 100.91 crore and the floating leg, at a reset date, is worth exactly 100, so the payer of fixed is behind by 0.91 crore. Kodachadri would have to pay that to tear the swap up.

Step 1What is the par rate, and why does it come straight from the discount factors?

A fair swap is one where the fixed leg and the floating leg are worth the same today. The floating leg plus notional is worth exactly the notional at a reset date, so the par rate is whatever fixed coupon makes the fixed leg plus notional also worth 100: one less the final discount factor, divided by the discounted annuity for a half-year coupon. The discount factorWhat one rupee due at a future date is worth today, read off the swap curve.s sum to 3.6730, so the par rate is 0.1286 over 0.5 times 3.6730, about 7.00%. Think of it as the rent that makes a two-year lease fair given what money is worth on each payment date.

The relationship

s = \frac{1 - d_4}{0.5 \sum_{i=1}^{4} d_i} = \frac{1 - 0.8714}{0.5 \times 3.6730} \approx 7.00\%

s	the par swap rate, annualised, paid semi-annually
d_i	the discount factors for 0.5, 1.0, 1.5 and 2.0 years
0.5	the half-year coupon period

What it says in wordsThe par rate is the fixed coupon whose discounted payments plus discounted notional equal the notional today, which is what the floating leg is worth at a reset.

Step 2What is the existing swap worth?

Value each leg with the same discount factors. The fixed leg pays 3.75 crore four times, worth 3.75 times 3.6730, 13.774 crore, plus notional at 0.8714, 87.14, a total of 100.914 crore. The floating leg is worth 100 at a reset date, which you can check by discounting the forward coupons implied by the same factors, about 3.50 each, and adding the notional. Kodachadri receives the leg worth 100 and pays the leg worth 100.91, so the swap is worth about minus Rs 0.91 crore to it. The shortcut gives the same answer: it overpays by 7.5% less 7.00%, half a point a year, 0.25 crore a half-year, times the annuity of 3.6730.

Discounted with the same four factors, the fixed leg at 7.5% plus notional is worth 100.91 crore while the floating leg plus notional is worth 100.00, so Kodachadri, which pays fixed, is behind by about Rs 0.91 crore, and the par rate that would make a new swap fair is 7.00%.

Date	Discount factor	Fixed coupon, Rs crore	PV fixed	Forward rate	PV floating
0.5 y	0.9662	3.750	3.623	7.00%	3.380
1.0 y	0.9335	3.750	3.501	7.01%	3.270
1.5 y	0.9019	3.750	3.382	7.01%	3.160
2.0 y	0.8714	3.750	3.268	7.00%	3.050
Notional	0.8714	100	87.14		87.14
Leg value			100.914		100.000

The forward six-month rates implied by the discount factors are all close to 7.0%, which is why the floating leg discounts back to exactly 100 and the par rate is 7.00%; the fixed leg at 7.5% is worth 100.914.

Step 3Who pays whom to tear it up, and what would you add?

Kodachadri would pay its counterparty about Rs 0.91 crore to cancel, or it could enter an offsetting swap receiving 7.00% and lock in a net payment of half a point a year for two years, which is the same thing spread out. The valuation is the mark-to-market a bank would show on its statement, and it is also the amount of collateral the swap would need if it were under a margining agreement. Say the limits: real swaps have day-count conventions and a stub that change the coupons slightly; the discount curve and the forward curve are no longer the same curve on most desks, so the floating leg is projected off one and discounted on the other; and a credit adjustment reduces what a bank would actually pay to a corporate to unwind.

Where candidates lose it

Candidates compute the par rate by averaging the forward rates or by reading the two-year point of the curve. The par rate is a weighted average of the forwards with discount-factor weights, which here lands at 7.00% because the forwards are flat; on a steep curve the shortcut is wrong.

The second loss is valuing the floating leg by guessing future rates. At a reset date it is worth par, full stop, and saying so in one line is what separates a candidate who understands swaps from one who has memorised a formula.

What the interviewer asks next

Three months after this reset, with the next floating coupon already fixed at 7.02%, how do you value the floating leg?
The counterparty's credit has weakened. Does the swap's value to Kodachadri go up or down?
What is the DV01 of this swap, and which way does Kodachadri benefit from a rate rise?
Why might a treasurer keep paying 7.5% rather than unwind at a loss?

Company names and figures are illustrative.