Determining interest rate forwards and their application to swap valuation

Supposing the company wants to borrow a sum of money for three years on

the basis of the above rates:

i. it could pay annual interest at a rate of 5.40% in each of the three

years, or

ii. it could pay interest at a rate 3.50% in the first year, 5.71% in the

second year and 7.02% in the third year, or

iii. it could pay annual interest at a rate of 4.60% in each of the first two

years and 7.02% in the third year.

USING INTEREST RATE FORWARDS TO VALUE A SIMPLE INTEREST RATE SWAP CONTRACT

Supposing the above company has $100m borrowings in the form of variable

interest rate loans repayable in five years and pays interest based on the above yield curve. It expects interest rates to increase in the future and is therefore keen to fix its interest rate payments.

The bank offers to swap the variable interest rate payments for a fixed rate,

such that the company pays a fixed rate of interest to the bank in exchange for

receiving a variable rate of return from the bank based on the above yield rates

less 50 basis points. The variable rate receipts from the bank will then be used

to pay the interest on the loan.

The fixed equivalent rate of interest the company will pay the bank for the swap can be calculated as follows:

The current expected amounts of interest the company expects to receive from

the bank, based on year 1 spot rate and years 2, 3, 4 and 5 forward rates are:

Year 1       0.0300 x $100m = $3.00m

Year 2       0.0521 x $100m = $5.21m

Year 3       0.0652 x $100m = $6.52m

Year 4       0.0773 x $100m = $7.73m

Year 5       0.0660 x $100m= $6.60m

Note: The rates used to calculate the annual amounts are reduced by 50 basis

points or 0.5%.

At the start of the swap, the net present value of the swap receipts based on

the variable rates from the bank will be the same as the costs based on the

fixed amount paid to the bank.

Let’s say R is the fixed amount of interest the company will pay the bank, then

($3.00m – R) x 1.035-1 +

($5.21m – R) x 1.046-2 +

($6.52m – R) x 1.054-3 +

($7.73m – R) x 1.061-4 +

($6.60m – R) x 1.063-5

= 0

Removing the brackets, the above expands to:

2.90m – 0.966R + 4.76m – 0.914R + 5.57m – 0.854R +

6.10m – 0.789R + 4.86m – 0.737R= 0

Simplifying this, adding all the $ flows together and R-flows together, gives:

24.19m – 4.26R = 0

$5.68m = R

In percentage = $5.68m/$100m = 5.68%

In practice the receipts and payments of the swap would be netted off such

that the company will expect to pay $2.68m ($5.68m – $3.00m) to the bank in

year one, and expect to receive $0.84m ($6.52m – $5.68m) from the bank in

year three, and so on for the other years. The present values of these net

annual flows, discounted at the yield curve rates, will be zero. The fixed rate of

5.68% is lower than the five-year spot rate of 6.30% because some of the

receipts and payments related to the swap contract occur in earlier years when

the spot yield curve rate is lower.

Although at the commencement of the contract, the present value of the swap

is zero, as interest rates fluctuate, the value of the swap will change. For

example, if interest rates increase and the company pays interest at a fixed

rate, then the swap’s value to the company will increase. The value of the swap contract will also change as the swap approaches maturity, and the number of receipts and payments reduce.

CONCLUSION

The previous article and this article considered the relationship between

bonds, interest rates, spot and forward yield curves, culminating with how the

forwards rates can be used to determine the equivalent fixed rate of a simple

swap contract. The examples used were simplified into annual cash flows and

rates, and students undertaking Paper P4 should be able to demonstrate their

knowledge and understanding of these areas to this extent.

In practice, valuation of bonds and related products is more complicated

because of factors such as: cash flows occurring more frequently than once a

year, early redemption of products, change in product values, and so on.

However, these aspects are beyond the scope of the Paper P4 syllabus.

Shishir Malde is examiner for Paper P4