The Treasury High Quality Market Corporate Bond Yield Curve 

For the Pension Protection Act of 2006 

October 9, 2007 

 This note describes the methodology used to construct the Treasury High Quality Market 

(HQM) Corporate Bond Yield Curve, which provides discount rates for the Pension Protection 

Act of 2006 (PPA). The description applies to the methodology in use at present; the 

methodology may be changed in the future because of changes in market conditions or for other 

reasons. Any significant changes in this methodology will be announced by notice. 

 The yield curve shows yields at different maturities from a particular category of fixedincome securities. The HQM yield curve corresponds to U.S corporate bonds whose credit 

quality is a market-weighted average of the top three qualities – AAA, AA, and A. 

 There are different types of yields and yield curves. For the purposes of the PPA, the 

relevant type of yield is the spot yield, which is the yield on a bond with a single payment at a 

future date, that is, a zero-coupon bond. 

 The yield curve pertains to a specific point in time, usually late in a business day, and is 

computed from the prices at that time on a set of bonds. The first step in constructing the curve 

is to assemble an appropriate set of bonds. 

 The HQM methodology currently uses a set of U.S. corporate bonds that covers the 

AAA, AA, and A markets. The maximum maturity of the bonds is 30 years, and currently, the 

par amount outstanding of each bond is at least $250 million. The part of the curve at lower 

maturities is filled in with AA commercial paper rates from the Board of Governors of the 

Federal Reserve System. The bond prices are bid prices. 

 The bonds currently chosen for the bond set pay fixed nominal semiannual coupons and 

the principal amount at maturity. Bonds with other characteristics, such as convertible bonds or 

bonds with floating coupons, are generally omitted. Bonds with embedded options are also 

excluded at present, although bonds with make-whole call provisions are included. All bonds are 

issued by U.S. corporations; asset-backed bonds are excluded. 

 With the set of bonds selected, the next step is to determine the prices of the bonds, 

including accrued interest, and the amounts and timings of the cash flows from the bonds. For 

precision, the HQM methodology moves forward the dates of cash flows that are scheduled to be 

received on weekends or holidays to the next business day. 

 The yield curve construction proceeds by fitting a statistical model that relates the bond 

prices to their cash flows, and computing the spot yield curve rates from this model. The model 

has two parts. The first part is based on the concept of a discount function, labeled d(t), which 

gives the present value of $1 to be received at maturity t years in the future with credit quality 

equal to the market-weighted average of the top three qualities. The present value of the cash 

flows from a bond is obtained by applying d(t) to the flows. 
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However, the price of the bond can deviate from the present value because the credit 

quality of the bond may not be average high quality. Therefore, in the second part of the model, 

adjustment factors are added to the present value to compensate. The adjustment factors are 

specified later in this note. 

Writing the bond price model symbolically, the price p of a bond is given as: 

∑ ∑ = =

= + +

n

i

m

j

p d ti ci bj xj e

1 1

( )

where the ci for i=1,...,n are the n cash flows from the bond to be received at times ti, the bjxj for 

j=1,...,m are the adjustment factors, made up of the m coefficients bj on the m adjustment 

variables xj, and e is the random disturbance for this bond. The bj are fixed across the bonds, 

while the values of the xj variables change with the bonds. 

The computation of the yield curve is done by estimating the d(t) function and the bj

coefficients from the bond data set of prices and cash flows. Given d(t), the spot rate y(t) at 

maturity t in percent (for maturities of 1⁄2 year or greater) is directly calculated by the following 

formula, which follows market convention including semiannual compounding: 

( 1) ( )

1 ( ) 200

2 = 1 − d t t y t

In order to estimate the bond price model, a specific functional form must be chosen for 

d(t). This is done by writing d(t) in terms of the forward rate f(t), which is the short-term interest

rate on a forward contract t years in the future, and applying the functional form to f(t). The 

following formula writes d(t) in terms of f(t) as the exponential of the negative of the integral of 

f(t): 

( )

0

( ) exp ( ) ∫ = −

t

d t f z dz

To determine the functional form for f(t), the entire maturity span of the bonds from zero 

to 30 years maturity is divided into maturity ranges. Currently, the HQM methodology uses five 

ranges, which are delineated by the maturity points 0, 1.5, 3, 7, 15, and 30 years. 

Within each of these five ranges, f(t) is given by a cubic equation, which is an equation 

that can have terms raised to the third power, and the five cubic equations are smoothly joined 

together across the ranges, with continuous first and second derivatives. A set of cubic equations 

connected like this is called a spline, so f(t) is represented by a cubic spline. 
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 Three constraints are placed on the cubic spline for f(t). First, for smoothness, f(t) is 

constrained to be approximately a line at short maturities; specifically, the second derivative of 

f(t) is set to zero at maturity zero. 

 Second, the value of f(t) at the furthest maturity – 30 years – is constrained to equal its 

average value from 15 years to 30 years, and f(t) is kept constant at this average for all maturities 

beyond 30 years. Third, f(t) is made to approach this constant value smoothly; specifically, the 

derivative of f(t) is set to zero at maturity 30 years. The constant value of f(t) beyond 30 years is 

used to project d(t) and the spot yields beyond 30 years maturity. 

 It can be demonstrated that the cubic spline for f(t) as set out here is determined by five 

parameters. These parameters are estimated from the bond data. 

 In addition, the adjustment factors in the second part of the bond price model must also 

be specified and estimated. The HQM methodology currently includes two adjustment variables 

x1 and x2, which adjust the bond prices so that d(t) and the spot rates represent market-weighted 

average credit quality of the top three qualities – AAA, AA, and A. 

 To define the variables, let w1 be the total par amount outstanding of the AA bonds in the 

bond set relative to total amount outstanding of AAA and AA bonds. Then the first adjustment 

variable x1 is defined as w1 times the maturity for AAA bonds, w1-1 times the maturity for AA 

bonds, and zero for A bonds. Larger values of w1 adjust AAA bonds more and move d(t) nearer 

to a AA discount function relative to AAA. 

 Analogously, let w2 be the total par amount outstanding for A bonds relative to the total 

for all bonds in the bond set, which are rated AAA, AA, or A. Then the second adjustment 

variable x2 is defined as w2 times the maturity for AAA and AA bonds, and w2-1 times the 

maturity for A bonds. Larger values of w2 adjust AAA and AA bonds more, and move d(t)

nearer to a discount function at the A level. 

 These variables imply that the amount of the adjustment to each bond’s price relative to 

the present value of the bond’s cash flows given by d(t) is greater when the quality class of the 

bond is a smaller proportion of the high-quality market. Consequently, the resulting estimate of 

d(t) is more representative of the quality classes with larger shares and pertains to the marketweighted average of the top three quality classes. 

 In sum, the seven parameters, comprising five parameters in the cubic spline for f(t) and 

the two adjustment coefficients bj on the adjustment variables, are estimated from the bond price 

data. The estimation is done by nonlinear least squares, that is, the seven parameter estimates are 

chosen to minimize the sum of the squared differences between the actual bond prices and the 

prices given by the bond price model. In the estimation, the adjustment factors can be regarded 

as linear regression terms added to the present value part of the model. 

 Before the estimation is carried out, the bond data are weighted. The weighting consists 

of two stages. 
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 In the first stage, equal weights are assigned to the commercial paper rates at the short 

end of the curve, and the par amounts outstanding of all the bonds are rescaled so that their sum 

equals the sum of the weights for commercial paper. Then the squared price difference for each 

bond is multiplied by the bond’s rescaled par amount outstanding, and the squared difference for 

each commercial paper rate is multiplied by the commercial paper weight. 

 In the second stage, for bonds with duration greater than unity, the weighted squared 

price difference for each bond from the first stage is divided by duration. Commercial paper 

durations are less than unity. 

 The nonlinear least squares estimation is carried out after weighting. The estimation 

produces values for the five cubic spline parameters and the two adjustment variable coefficients. 

The spline parameters are used to generate f(t), which, in turn, produces d(t) and the spot rates. 

 The HQM methodology is applied to the bond data for each business day, and spot rates 

are computed for maturities at half-year intervals from 1⁄2 year through 100 years. The monthly 

average spot rates are the arithmetic means of the spot rates at these maturities for all the 

business days of the month.