# Yield curves

Yield curves form a critical part of how FIA works.

## Zero curves

A zero curve is the most basic type of yield curve available. Zero yields may be used directly to discount future cash flows and for pricing of securities.

## Converting a par coupon curve to a zero curve

### Rationale

While a zero coupon curve represents the purest form of the state of interest rate expectations, many curves are instead quoted as par curves. For instance, Bloomberg's yield curve pages quote a mixture of zero coupon (spot rate) sovereign curves and par credit curves.

The par curve is a graph of yields to maturity (YTM) against maturity for bonds that are trading at par, or at face value. A bond trading at par has its coupon equal to its YTM.

### Why use par curves?

In other words, why not try to use all bond data available in the marketplace to calculate the zero curve?

The reason is that not all bond prices are equally valuable for spot rate calculation. For instance, bonds selling at a high premium or discount may be subject to various price distortions, as they may be in high demand when interest rates are low and are about to rise, so their yields will be lower than usual.

The subset of bonds selling at or near par are therefore likely to be more representative of the underlying term discounting rates implicit in the market, as they are generally not subject to these distorting forces.

Par curves are also used to select an appropriate fixed rate for interest-rate swaps, and to choose the coupon for new bonds that are issued at par.

### Converting from par curves to zero curves

Zero curves may be calculated from related par curves using a standard bootstrap approach.

Suppose we are given a par curve for a market in which bonds pay coupons annually. The curve is quoted at 0.25 years, 0.5 years, then 1 to 10 years.

Recall that the par yield is the yield to maturity of a bond than is worth 100 (face value) at the given maturity. At the 0.25, 0.5 and 1 year points, there is at most one cash flow remaining in the bond's lifetime. The bond can therefore be treated as a bill or a zero coupon bond, for which the yield to maturity is the zero (spot) yield. The 0.25, 0.5 and 1.0 year points on the zero curve are therefore the same as on the par curve.

Suppose now that we have a set of zero coupon yields

<math>\left \{ z_1, z_2\dots,z_n \right \}</math>

at maturities

<math>\left \{ m_1, m_2\dots,m_n \right \}</math>

where all yields <math>z_i\</math> at maturities <math>m_i\</math> are known except the last, using arguments such as that in the preceding paragraph. We know the par yield <math>C_n\</math> at maturity <math>m_n\,</math>. What is the corresponding zero yield <math>z_n\</math>for that maturity?

The key relationship here is that the price of the bond is its face value. Setting the coupon of the bond to its par yield (which is the defining characteristic of a par bond) and assuming constant compounding gives

<math>P=C_n \sum_{i=1}^{N-1} e^{-z_i m_i} + (100+C_n) e^{-z_n m_n} = 100</math>

All quantities here are known except for <math>z_n\,</math>, which is given by

<math>z_n = \frac{log(100+C_n) - log(100-S)}{m_n}</math>

where

<math>S=C\sum_{i=1}^{N-1} e^{-z_i m_i}</math>

The case for bonds that pay coupons twice a year is similar, but the expression <math>C_n</math> should be replaced by <math>C_n/2\,</math>.

### Verification

A useful spot check on the correctness of the zero rates calculated from a bootstrap calculation is to reprice all the bonds used in the calculation. In every case, setting the coupon of the bond to the par yield for that bond's maturity should give a price of 100 (par value).

### Implementation

Flametree Technologies have a conversion utility available to our customers that reads par curves from Excel spreadsheets, calculates par curves and stores the results in a format usable by FIA.

## Constant maturity curves

Some data providers (notably, the US Department of the Treasury Resource Center, www.treasury.gov) make sovereign curve data available in the form of constant maturity treasury (CMT) rates. These are described as follows:

*These rates are commonly referred to as "Constant Maturity Treasury" rates, or CMTs. Yields are interpolated by the Treasury from the daily yield curve. This curve, which relates the yield on a security to its time to maturity is based on the closing market bid yields on actively traded Treasury securities in the over-the-counter market. These market yields are calculated from composites of quotations obtained by the Federal Reserve Bank of New York. The yield values are read from the yield curve at fixed maturities, currently 1, 3 and 6 months and 1, 2, 3, 5, 7, 10, 20, and 30 years. This method provides a yield for a 10 year maturity, for example, even if no outstanding security has exactly 10 years remaining to maturity.*

Although not clearly stated on the Federal Reserve's website, CMT yields are yields to maturity, the coupons for the bonds to which the yields refer are the constant maturity yields, and the prices are par (Whaley, Robert E.; *Derivatives markets, valuation, and risk management*, Wiley, 2006). For instance, the coupon of a 10 year constant maturity bond is its quoted CMT yield, and its price is 100. This allows the CMT curve to be treated as a par curve, and a conversion routine such as that shown in the previous section will allow extraction of zero coupon yields from CMT yields.