Goldman Sachs and Google entered into an interest rate swap on September 25th, 2

Question

Goldman Sachs and Google entered into an interest rate swap on September 25th, 2015 on a principal notional of $50 million dollars. Goldman Sachs will make annual floating payments according to 1-year T-Bill rates plus 50 basis points. Google in return will make fixed payments on annual basis. The first cash flow exchange will occur on September 25th, 2016. The contract will last for a period of 10 years, e.g., there will be a total of 10 payments for each company. On September 25th, 2015, the following T-Bill rates are observed.

Maturity T-Bill Rate (%) 1 year 6.03 2 years 6.21 3 years 6.35 4 years 6.47 5 years 6.65 6 years 6.83 7 years 6.95 8 years 7.17 9 years 7.35 10 years 7.83

a) If there is no cash settlement at the initiation of the contract, what should be the fair fixed rate that Google must pay annually?

b) If Google prefers a fixed rate of 7.0% annually, what amount of cash settlement is needed between the parties on September 25th, 2015 in order to have a fair contract?

Explanation / Answer

a) To find fixed rate we need to equate the PV of fixed rate payment and present value of floating rate payment.

First we need to calculate the PV factors for all the different maturities T-Bill rates. It can be calculated using the following formula:

PVF (Present Value Factor) = 1/ (1+ Rate * (n/360)) where Rate is the corresponding T- Bill rate and n is the time until maturity for that T-Bill. We use 360 days as convention. So for example for a 5 year T-Bill n = 5 x 360

Using the above formula we arrive at the following present value factors:

Using the above PVF we get fixed rate in case of no notional exchange as follows

Fixed Rate = (1- last period PVF)/Sum of all PVF = (1- 0.560852)/(0.943129 + 0.889521 +.....+0.560852) = 5.94%

b) Let us assume that the principal amount for Google is $ 50 million and for Goldman it is $ Y million

Now we have to find Y and the cash settlement would be 50-Y

We need to equate the total fixed payment and total floating payment.

Periodic Interest Payment by fixed side = 50 * 0.07 = 3.5

We need to find the total present value of all the interest payments by fixed. This can be done by multiplying Periodic interest payment of 3.5 with PVF found in part (a) . We also need to find the present value of $ 50 million which will be settled at end of 10 yr. This can be done by multiplying 50 with PVF of 10 Yr. T-Bill.

Fixed side = 3.5 * sum of PVF + 50 * PVF-10 Yr. T-Bill = 3.5* 7.3977 + 50 * 0.560852 = 53.9347

This should also be the amount that is paid by Goldman Sachs for the contract it to be fair.

So if the notional value was $50 million and Google pays 7% fixed then Goldman has to pay the difference to Google i.e.

Cash Amount = 53.9347 - 50 = $ 3.9347 million (paid by Goldman to Google)

Time to Maturity 1 2 3 4 5 6 7 8 9 10 Rate (Given) 6.03% 6.21% 6.35% 6.47% 6.65% 6.83% 6.95% 7.17% 7.35% 7.83% PVF 0.9431293 0.8895214 0.8399832 0.7944074 0.7504690 0.7093205 0.6727212 0.6354855 0.6018658 0.5608525

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