## Valuation of Interest Rate Swaps

To price a swap, we need to determine the present value of cash flows for each leg of the transaction. In an interest rate swap, the fixed leg is fairly straightforward because the cash flows are specified by the coupon rate set at the time of the agreement. Pricing the floating leg is more complex because, by definition, the cash flows change with future changes in interest rates. The forward rate for each floating payment date is calculated by using the forward curves.

## Yield to Maturity

Yield to maturity (YTM) is the expected rate of return for a bond that is held until its maturity, assuming that all coupon and principal payments are made in full when due and that coupons are reinvested at the original YTM.

YTM is perhaps the most familiar pricing concept in bond markets. To understand this concept, it is important to clarify how YTM is related to spot rates and a bond’s expected and realized returns.

How is the yield to maturity related to spot rates?

In bond markets, most bonds outstanding have coupon payments and many have various options, such as a call provision. The YTM of these bonds with maturity T would not be the same as the spot rate at T. But, the YTM should be mathematically related to the spot curve. Because the principle of no arbitrage shows that a bond’s value is the sum of the present values of payments discounted by their corresponding spot rates, the YTM of the bond should be some weighted average of spot rates used in the valuation of the bond.

The following example addresses the relationship between spot rates and yield to maturity.

Is the yield to maturity the expected return on a bond?

In general, it is not, except under extremely restrictive assumptions. The expected rate of return is the return one anticipates earning on an investment. The YTM is the expected rate of return for a bond that is held until its maturity, assuming that all coupon and principal payments are made in full when due and that coupons are reinvested at the original YTM. However, the assumption regarding reinvestment of coupons at the original yield to maturity typically does not hold. The YTM can provide a poor estimate of expected return if (1) interest rates are volatile; (2) the yield curve is steeply sloped, either upward or downward; (3) there is significant risk of default; or (4) the bond has one or more embedded options (e.g., put, call, or conversion).

If either (1) or (2) is the case, reinvestment of coupons would not be expected to be at the assumed rate (YTM). Case (3) implies that actual cash flows may differ from those assumed in the YTM calculation, and in case (4), the exercise of an embedded option would, in general, result in a holding period that is shorter than the bond’s original maturity.

The realized return is the actual return on the bond during the time an investor holds the bond. It is based on actual reinvestment rates and the yield curve at the end of the holding period. With perfect foresight, the expected bond return would equal the realized bond return.

## Yield Curve

The yield curve is a graphical representation of the different interest rates at various maturities.

There are different types of yield curves: spot, forward, par etc. No matter what type of yield curve, it represents different interest rates (at different maturities) at a single point of time.

## Slope of the Yield Curves

In developed markets, yield curves are most commonly upward sloping with diminishing marginal increases in yield for identical changes in maturity; that is, the yield curve “flattens” at longer maturities. Because nominal yields incorporate a premium for expected inflation, an upward-sloping yield curve is generally interpreted as reflecting a market expectation of increasing or at least level future inflation (associated with relatively strong economic growth). The existence of risk premiums (e.g., for the greater interest rate risk of longer-maturity bonds) also contributes to a positive slope.

An inverted yield curve is somewhat uncommon. Such a term structure may reflect a market expectation of declining future inflation rates (because a nominal yield incorporates a premium for expected inflation) from a relatively high current level. Expectations of declining economic activity may be one reason that inflation might be anticipated to decline, and a downward-sloping yield curve has frequently been observed before recessions.

A flat yield curve typically occurs briefly in the transition from an upward-sloping to a downward-sloping yield curve, or vice versa.

A humped yield curve, which is relatively rare, occurs when intermediate-term interest rates are higher than short- and long-term rates.

Daily Treasury Yield Curve

## Par Curve

The par curve is a hypotenetical yield curve for coupon-paying Treasury securities that assumes all securities are priced at par. It represents the yields to maturity on coupon-paying government bonds, priced at par, over a range of maturities. In practice, recently issued (“on the run”) bonds are typically used to create the par curve because new issues are typically priced at or close to par.

The par curve is important for valuation in that it can be used to construct a zero-coupon yield curve. The process makes use of the fact that a coupon-paying bond can be viewed as a portfolio of zero-coupon bonds. The zero-coupon rates are determined by using the par yields and solving for the zero-coupon rates one by one, in order from earliest to latest maturities, via a process of forward substitution known as bootstrapping (a statistical method for estimating a sample distribution based on the properties of an approximating distribution).

## No-arbitrage Principle

The no-arbitrage principle says that tradable securities with identical cash flow payments must have the same price.

## Forward Rate

A forward rate is an interest rate that is determined today for a loan that will be initiated in a future time period. It can be think of as the spot rate in the future.

The term structure of forward rates for a loan made on a specific initiation date is called the forward curve.

Forward rates and forward curves can be mathematically derived from the current spot curve.

## Spot Rate

A spot rate, or a spot interest rate, is a rate of interest on a security that makes a single payment at a future point in time.

## Term Structure of Interest Rates

The term structure of interest rates is a mathematical description of the relationship among market interest rates at various maturities. The graphical representation of the term structure is the yield curve.