That iiwhere the swaps expected net cash flows occurring

MACQUARIE UNIVERSITY

APPLIED FINANCE CENTRE

More specifically, a swap is a contract calling for an exchange of cash flows on predetermined dates determined by the difference in two prices. One party makes a payment to the other, depending on whether a price turns out to be greater or less than a reference price. A swap, therefore, provides a way to hedge a stream of risky cash flows. This can be more convenient, and involve lower transaction costs than hedging with a series of forward contracts.

In this topic, we will consider commodity, interest rate, currency, and total return swaps. We will discuss the motives for using these instruments, how they are priced and valued, and consider how a market maker can hedge the risks in these swaps.

In this section we will follow the commodity swap example in McDonald. The ideas developed with this example will help you understand the structure of a swap, how the swap is priced and valued, and how it can be hedged by a market maker.

6.1.1 Pricing the swap

The present value of the two forward prices is $37.383. Thus, IP could pay an oil supplier $37.383 and the supplier would commit to delivering 1 barrel in each of the next 2 years – this transaction is known as a prepaid swap.

With a prepaid swap, the buyer might worry about the resulting credit risk. Therefore, a better solution is to defer payments until the oil is delivered, while still fixing the total price. Any payment stream with a present value of $37.383 is acceptable. Typically though, a swap will call for equal payments. For example, the payment per year per barrel, x, will have to be $20.483 to satisfy the following equation

Notice that because the swap is financially settled, the principal (in this case, barrels of oil) is notional. That is, the principal is not physically delivered but rather is used to determine the cash flows on the swap.

6.1.3 Relationship to forwards

The market maker is receiving the fixed swap price of $20.483/barrel. The market maker therefore gains as the spot price of oil falls and loses as the spot price rises.

To hedge the swap, the market maker could enter into two forward contracts to buy oil – the first expires in 1 year at $20/barrel and the second expires in 2 years at $21/barrel.

6.1.5 Valuing the swap

As with all financial instruments, the value of a swap is the present value of its expected future cash flows. That is,

We can estimate the floating cash flows on this swap using the forward prices since these are the prices that IP would receive if it were to close out the swap with forwards.

Thus, from IP’s perspective, the value of the swap at its origination is

After the swap has been dealt, the value will generally move away from zero. This is because forward oil prices and zero coupon yields will change over time. In addition, even if oil prices and interest rates do not change, the implicit borrowing/lending in the swap means that the value of the swap at the next settlement will not be zero.

For example, assume that the 1 and 2 year forward prices rise to $22/barrel and $23/barrel immediately after IP deals the swap. The new value is

which is what IP would receive if it wanted to unwind the swap with its original counterparty.

An interest rate swap is a series of cash flows between two parties. Each set of cash flows is based on either a fixed or floating interest rate.

The most common type of interest rate swap is a plain vanilla swap. With this swap, a firm agrees to pay cash flows based off a fixed interest rate and receives cash flows based off a floating interest rate. The floating rate is a reference rate such as LIBOR.

•XYZ Corp pays 1 year LIBOR on its borrowings;
•receives 1 year LIBOR on the swap; and
•pays the fixed rate of 6.9548% per annum on the swap.

Let’s now broaden our minds and think of some other motivations to use an interest rate swap.

XYZ Corp would have dealt the interest rate swap with a market maker. Suppose the market maker dealt a back-to-back swap with another customer, ABC Corp.

In other words, the fixed rate is determined such that the present value of the swap’s fixed cash flows equals the present value of the swap’s expected floating cash flows.

We will illustrate these ideas with the example in McDonald. Recall XYZ Corp is receiving 1 year LIBOR and paying the fixed rate on a notional principal of $200 million.

•Discount the expected net cash flows of the swap.

That is, we solve for R in the following equation:

This means that in general, that the per period fixed rate, R, for an interest rate swap of any maturity n can be computed using the formula for the par coupon rate presented on page 9 of the Topic 5 lecture notes. That is,

If the swap settles more than once per year, the per annum fixed swap rate is computed by multiplying the per period fixed swap rate by the number of settlements per year.

This methodology shows that R, the par fixed swap rate (the one which yields a value of zero at the swap’s origination), is totally dependent on the level/slope/shape of the relevant spot zero coupon yield curve.

Here is a summary of the approach we have used so far to value an interest rate swap:

We can replicate the cash flows on this 3 year interest rate swap to receive fixed and pay floating by borrowing at a floating rate of 1 year LIBOR and lending at a fixed rate. That is, by issuing a floating rate note and buying a coupon bond.

The value of this interest rate swap is thus the difference between the present value of a coupon bond and the present value of a floating rate note (FRN). That is,

For our example, the value of the coupon bond is

and the value of the floating rate note is

Once a swap has been transacted its value will, in general, start to drift away from zero as time passes. Revaluing the swap on an ongoing basis requires continual recalculation of the zero coupon yield curve. This yield curve moves around and steepens/flattens as the general level of interest rates moves with the markets’ reaction(s) to new information. The swap’s fixed rate remains just that: fixed, and essentially a reflection of history. When the zero coupon yield curve changes, the expected cash flows and the discount factors will also change.

The following questions ask you to revalue the 3 year swap we have just looked at. We will also consider the interest rate risks in the swap and think about the ways in which a market maker could hedge these risks.

3. Describe the movements in the spot zero coupon yields that will

•Amortising swap: This is a swap with a decreasing notional principal. Amortising swaps can be used to hedge financial instruments with decreasing principals such as mortgages.

•Accreting swaps: In this swap, the notional principal increases over time. The increase in the notional principal might be designed to match drawdowns in a loan agreement.

Read McDonald pages 264-268

6.3.1 Structure of a currency swap

For example, consider an Australian firm that wishes to borrow Australian dollars (AUD) for 5 years at a fixed rate. Rather than issuing 5 year fixed rate AUD debt, the firm could issue 5 year fixed rate debt in the United Sates capital market and convert the U.S. dollar proceeds into AUD by transacting a currency swap to receive the 5 year USD fixed swap rate and pay the AUD fixed swap rate. Firms typically do this because there is more liquidity in offshore capital markets or because they want to diversify their borrowings geographically.

Firms that are starting out in other countries are also attracted to currency swaps. For example, consider an Australian firm that is expanding its business in Europe. The firm has established domestic bank relationships but is relatively unknown in Europe. This means that if the firm borrows euros in the European capital market it is unlikely to obtain terms as favourable as it would by borrowing in the Australian capital market. The firm can fund its European operations by issuing AUD fixed rate debt and converting the proceeds into euros by

McDonald explains how a currency swap can be replicated by a portfolio of currency forward contracts. He then proceeds to value the currency swap as a portfolio of currency forwards and presents formulae to compute the fixed swap rate for one given the fixed rate of the other currency.

We can also replicate a currency swap by borrowing and lending in each currency.

The value of the swap in the terms currency for a swap where the commodity currency’s interest rate is received and the terms currency’s interest rate is paid is

Swap value = S BComm−B Terms

Similarly, the value of the swap in the terms currency for a swap where the terms currency’s interest rate is received and the commodity currency’s interest rate is paid is

You are a market maker in currency and interest rate swaps at MAF Bank, where you manage the risks in a portfolio of currency and interest rate swaps.

Available to you on the trading floor are the related markets: spot and forward foreign exchange, borrowing and lending cash of any maturity, forward rate agreements, currency futures, and interest rate futures.