Valuation of IRS using zero /discount curve.

What affects Swap Value?

From a valuation perspective swaps are not much different from customized notes and bonds. The fixed cash flows in a swap are akin to the interest payments on a high quality fixed rate note, while swap floating cash flows are like the interest payments on a floating rate note. Market variables that affect swap pricing include changes in the level of interest rates, changes in swap spreads, changes in the shape of the yield curve, and FX rates (for currency swaps). Key transaction-specific variables that affect swap valuation include notional principal amount and amortization, time to maturity, swap payment frequency, and floating rate reference index.

Mechanics of Swap Pricing

Measuring the current market value of an interest rate swap involves four distinct elements:

1.	Constructing a zero coupon* yield curve
2.	Extrapolating a forecast of future interest rates to establish the amount of each future floating rate cash flow
3.	Deriving discount factors to value each swap fixed and floating rate cash flow
4.	Discounting and present valuing all known (fixed) and forecasted (floating) swap cash flows.

While this may sound complicated, the curve building and discounting techniques are the same techniques used to establish the theoretical market value of any interest bearing security.

Constructing a Yield Curve

The first step in swap valuation is to build a yield curve from current cash deposit rates, eurodollar futures prices, treasury yields, and interest rate swap spreads. These known market rates are “hooked” together to form today’s coupon yield curve. The coupon curve is the raw material from which a zero coupon yield curve is constructed, usually using a method called “bootstrapping”. This involves deriving each new point on the curve from previously determined zero coupon points (hence the phrase, “bootstrapping”).

Zero rates are higher than coupon rates when the yield curve is positively sloped and lower when the curve is inverted. The gap is widest at the far end of the yield curve. When rates are low and the yield curve flat the difference between coupon and zero rates will be minimal, but when rates are high and the curve steep, the difference is significant.

Because the cash flow dates of the swap to be valued rarely exactly match the dates for which zero curve points have been developed, interpolation between data points is needed to solve the problem. While this sounds simple, some extremely complicated algorithms have been developed to minimize the errors that can arise from interpolation.

Forecasting Future Short-Term Rates

One half of the cash flows in a simple swap are floating rate. What makes the floating leg of the swap hard to price is the uncertainty of the forward rates—only today’s floating rate is known for certain. A forecast of future floating rates — a forward yield curve of short term interest rates—is needed before prospective floating rate cash flows can be generated. In fact, the forward curve is just an extension of the zero coupon yield curve; once the zero curve has been developed, it easily transforms into the forward curve needed to generate the swap’s floating rate cash flows.

Deriving Discount Factors

Discount factors, used to present value each swap cash flow, are developed as part of the process of bootstrapping the zero coupon yield curve. Like forward interest rates, discount factors are just a transformation of zero coupon rates. In fact, there is a simple formula for converting one to the other**.

Valuing the Swap

With all the calculations concluded, the only step remaining is to apply the discount factors to find the present value of fixed and floating swap cash flows. These values are then netted to determine the swap’s current market value. This value can be positive, zero, or negative, depending on how market interest rates have changed since the swap was created. For a floating to fixed swap, higher market rates will create a gain for the hedger, lower rates a loss. In the example below, a swap with a remaining term of 2 years is valued at a point when market interest rates have risen 1% across the yield curve from the time the swap was put into place.

Simple Rule of Thumb for estimating the sentivity of a Swap's Value to changes in Market interest rates results in Change in Value

While understanding the basics of swap valuation should make senior financial officers more comfortable with the balance sheet implications of the firm’s hedging activity, a quick and dirty way to estimate a hedge’s value and rate sensitivity can also prove useful.

A Simple Rule of Thumb for Estimating the Sensitivity of a Swap’s Value to Changes in Market Interest Rates While understanding the basics of swap valuation should make senior financial officers more comfortable with the balance sheet implications of the firm’s hedging activity, a quick and dirty way to estimate a hedge’s value and rate sensitivity can also prove useful. DV ’01—the change in dollar value of a swap for a one basis point change in market interest rates— is a simple measure for benchmarking how the value of an interest rate swap changes as interest rates change. However, because the DV ’01 changes as market rates go up and down and the shape of the yield curve changes, it can only be used to estimate the change in a swap’s value for small shifts in rates.

The chart below describes the DV ’01 for a $25 million interest rate swap with maturity between 2 and 10 years. These values are based on today’s yield curve (May, 1999). The values needed to estimate the potential gain or loss on a hedge include the actual fixed rate on the swap, the market fixed rate for a swap of equal remaining life, and the DV ’01 of the swap.

For example, ***Company ABC has a two month window in which to execute a five-year swap program covering $25 million of its bank debt. The company could hedge the debt today at a fixed swap rate of 5.90%, but it hopes that the market will improve over the next few weeks. Using the DV ’01 value, the company can estimate how much it stands to gain or lose from delaying the hedge. From the chart, the DV ’01 of a 5 year $25 million swap is $9,375. A 25 basis point change in rates (a not uncommon occurrence over a two month period), would trigger an opportunity gain or loss around $234,375 ($9,375 X 25 basis points). With an available estimate now of how much is at risk, the company can decide how confident it is in its interest rate forecast.

Summing Up

Hedgers have typically been concerned with the actual market value of interest rate swaps only at financial reporting dates or when considering the early termination of a hedge. However, as active hedge management increases and accounting ground rules change, understanding the basic concepts underlying swap valuation takes on more importance.

* Zero coupon rates are associated with instruments that pay no interest until maturity. The zero coupon curve is used in securities valuation because it requires no assumptions about reinvestment rates on intermediate cash flows.

** Discount factors can be derived from zero coupon rates using the formula 1/(1+r)^n where r is the periodic rate and n is the number of periods. Where zero rates are derived on a continuously compounded basis (the usual method), the formula becomes 1/e rt where e is the natural log, r is the zero rate and t is the term in years.