Introduction

This presentation describes the current Treasmethodology for constructing the corporate borequired by the Pension Protection Act of 200Previous versions of this methodology are set Treasury White Paper (February 7, 2005) fouhttp://www.treas.gov/offices/economic-policy/reports/pension_yand the White Paper Update (January 24, 200http://www.treas.gov/offices/economic-policy/reports/wp_up_x.p

The methodology may be changed in the futurbecause market conditions change, or for othe
The Pension Protection Act

 The PPA mandates that Treasury publish a 

yield curve for calculating the present valuliabilities and lump sum distributions.

 The methodology chosen for this yield curv

a curve that satisfies the general requiremreliable yield curve that successfully capturbehavior, as well as the specific requiremen

 The following discussion sets out these requ
PPA Requirements

 The PPA requires that the yield curve repr

corporate bond market rather than the U.S

market, as in the typical yield curve. Corpmuch more heterogeneous than Treasuriesspecial features, and criteria must be develdeciding upon the bond set for the curve.

 The yield curve must be a single blended cu

high quality corporate bonds, i.e., bonds ra

A. Typical yield curves do not combine difffrom disparate markets, and so an approacdeveloped for the combination.
Yield Curve Requirements: Projectio

 An important requirement for any yield cu

yield curve must be projected for indefinite

maturities beyond 30 years maturity. This 

because the yield curve may be used to disc

well beyond 30 years into the future. The mmust be developed for doing this; the usual stops at 30 years and contains no provision The projected discount rates must be reliabreflect the behavior of long-term interest ra

 In the case of pension liabilities, the yield c

projected out through 100 years maturity. 
Yield Curve Requirements: Product

 Since the yield curve must be estimated da

production environment, the yield curve m

provide estimates that are robust and stabl

to perturbations in the bond set while captumovements in the market. Consequently, tshould evolve smoothly over time.

 While the methodology can be changed as w

market conditions or for other reasons, sucshould be infrequent, and the curve should tuned every day. 
Additional Yield Curve Requiremen

 In addition to the reasoning behind the yie

methodology, the results produced by the mmust be good indicators of market condition

 In particular, the yield curve should not be 

criticism that it is arbitrary, in the sense thequally valid methodologies or equally valiimplementing a given methodology that prresults from the same data. Therefore, therationale for setting every parameter contamethodology, and no parameter should be l
Additional Requirements, continued

 Yield curves can exhibit arbitrary behavior 

derived from overly complicated models thastrong assumptions.

 However, stringent assumptions usually do 

bond markets, and so such assumptions caresults. Also, bond data do not show clear to distinguish among competing complicateagain the models give different results. 

 To avoid problems of arbitrariness, the cho

methodology should aim toward robust and straightforward market averages that provindicators of central market tendencies.
The White Paper Methodology

This presentation focuses on the current versTreasury White Paper methodology, which prcorporate bond yield curves for the PPA repremarket-weighted average quality of the combquality bond market. These yield curves are Quality Market-Weighted (HQM) Corporate BCurves.
 The White Paper methodology computes th

a point in time from data for a selected set bonds. The bond set covers the range of ma30 years and includes the three high qualitCriteria for bond selection are discussed lat

 The curve is directly fitted to the bond set w

of a separate yield curve for Treasury securon a Treasury curve is unnecessary and necomplex.

 The yield curve employs a bond pricing mo

discount function, and extends the model wterms for factors not contained in the disco

The White Paper Methodology, cont
The Discount Function

 The starting point for the White Paper met

discount function.

 The discount function 

δ

(

τ) gives for each ma

years) the amount that must be invested to the future at that maturity. The discount ffor the point in time for which the yield curcomputed.

 The discount function as used here pertain

weighted average credit quality in the high and gives the present prices of future paymquality. The interpretation of the discount made more precise later in the discussion oterms.
The Forward Rate

The forward rate curve 

ξ(

τ) is defined as the r

curvature of the discount function:

Forward rates are instantaneous future interon the discount function. The discount functiin terms of the forward rate:

))(exp()( ∫

=

0

−=

τ

τδ

ξ

z

dzz

)(

1)( )( τδτ

δ

τ τξ

d

d

−=
The Spot Rate

The relevant interest rate concept for discounflows, such as pension benefits, is the spot rat

yield on a bond with a single payment at matucoupon bond). Therefore, the spot yield curve spot yields for all maturities, is the central yieconcept.With semiannual compounding, the spot rate related to the discount function as follows:

/1)( )200/)(21( , 0 0 5./1)( )200/)(1( , 5.0 2

= + < <

= + ≥

τδ ττ τ

τδ τ τ τ

y

y
The Spot Rate, continued

In terms of the discount function, the spot rat

0),2/()1)(/1(200)(

0 5.

200)( )(/1(

2 )1 , 0 5. 1

−×= <<

×=

−

≥

τ τττδ

τ τδ

τ

τ

yy

It can be shown by continuity that 

y(0) = 100·

ξ

If the forward rate eventually settles down to 

the spot rate will converge to:

y

=

×

ξ

− )1)2/*((exp200*

which is not the same as 

ξ

* because the spot r

semiannually compounded.
The Par Rate

 Another yield curve concept is the par yield 

gives for each maturity the yield on a bond maturity and paying semiannual coupons, trading at par, that is, whose flat price (excinterest) is 100.

 Par yields are also computed from the disco

The coupon rate at par is determined with function, and the par yield is the yield for t

 Par yields are indicators of bond market co

therefore they can be used to compare the cmarket with the Treasury market.
Bond Price Equation

Using the discount function and forward rate for the 

kth bond out of a set of 

P bonds can be w

where 

p

k is the price of the 

kth bond (including 

interest), the 

cki for i = 1,...,

n

k are the 

n

k cash flo

bond to be received at maturities 

τki, the 

ζj for j 

m coefficients for the 

m regression variable va

the random disturbance.

ξ

ζ

δ

τ

ζ

ε

τ

ki

j kj

k kiki

j kj

k

dzz xc

p xc

m

j

n

k

i

ki

z

m

j

n

k

i

∑ −=

∫

+

∑

+

=

∑

+

∑

+

=

=

=

=

=

1 1 01 1

exp( )(

)

(

)
Functional Form of the Discount Fu

 In order to estimate the bond price equatio

form must be determined for the discount f

 It is more useful to work with the forward r

because forward rates all have the same inmaturity and can be compared across matu

 Moreover, to make economic sense it is nec

δ

(

τ) > 0 and 

δ(0) = 1, and any functional form 

rate curve satisfies these conditions.

 It is also generally reasonable to impose th

dδ

/

dτ < 0, which says that $1 received later 

lower present price. If the functional form 

chosen to be positive, this condition is also 
Maturity Ranges

 In determining the functional form, it is us

that corporate bond maturities tend to diviThe ranges reflect the fact that many tradehave preferred maturity habitats. The forweach range are related.

 The maturity ranges can change with signi

in bond market conditions. For recent yearPaper methodology has delineated the rang

bonds by the maturity points 0, 1.5, 3, 7, 15

 The range 0 to 3 years reflects short-term t

there is enough activity that it should be futwo with a point at 1.5 years.
Maturity Ranges, continued

 The range 3 to 7 years contains bonds of so

term centering around 5 years. And the rayears contains the bulge of bonds frequentl10 years.

 Finally, the last range 15 to 30 years reflec

bonds. There is not enough information in term bonds to determine whether to break 

further, so all the longer bonds are kept togrange.
Averages in the Ranges

 Given the ranges, the forward rate in each 

approximated by its average. However, it iallow some movement of the forward rate wrange, especially in the last range, and also forward curve that is smooth over all matu

 The way to do this that does not depart too 

averages is to use a cubic polynomial for ea

string the polynomials together smoothly awith knots at the 6 maturity points. Smootthat at each of the interior knots 1.5, 3, 7, aspline is made to be continuous with continsecond derivatives.
Averages in the Ranges, continued

 This discussion shows that the use of a cub

forward rate arises naturally from averaginmaturity ranges together with smoothness

 The choice of fixed knots for the spline base

ranges significantly increases the stability curve over time.
Constraint in the Nearest Range

 Because there are no bonds with zero matu

forward curve has no anchor at the zero pomaturity range. 

 For this reason it makes sense to impose an 

linear constraint on the movement of the fothe earliest maturities. This is accomplishconstraining the second derivative to zero a

0 )0(

2

2

=

d

τ

d

ξ
The Long-Term Forward Rate

 Special constraints are needed for the last 

15 to 30 years, because this range has to leprojected forward rate beyond 30 years thaprojections of the yield curve.

 Because it is usually not possible to estima

movements in the forward rate beyond matthe projected forward rate is fixed at a consits long-term average.

 Moreover, the forward curve should flatten 

to connect smoothly with this long-term conleads to the first constraint for the last mat

0 )30(

=

dτ

dξ
 One approach to estimating the forward ra

years maturity focuses on the typical drop-occurs a few years before 30 years, and prodecline. The forward rate given by this appfalls to a very low level at long maturities.

 This approach is dubious because it allows 

behavior for a few years before 30 to determrates for many decades beyond 30 years.

 And as a practical matter, the low forward 

projects would imply relatively low spot ratconsequently relatively high discounted prepension obligations beyond 30 years.

The Long-Term Forward Rate, conti
The Long-Term Forward Rate, conti

 More significantly, this approach places ins

importance on risk and term premiums at f

 In contrast, the White Paper methodology e

fact that the forward rate is the marginal rlengthening the maturity of a loan, and as influenced by term premiums, which refleccannot be hedged.

 Accordingly, aversion to risk has the effect 

forward rates, and this suggests that the lorate may remain at a relatively high level.
The Long-Term Forward Rate, conti

 Based on these risk considerations, the forw

maturity 30 years and beyond can be assumdetermined on average by the factors that aforward rate in the 15- to 30-year maturity that range is sufficiently distant to reflect lattitudes toward risk.

 Therefore, the constant long-term forward 

maturity 30 forward is assumed to be the arate in the 15- to 30-year range. This impladditional constraint on the forward rate cu

)30( )(15 3015

ξ

ξ dzz

z

=

∫

=
 The effect of the last 

constraint on spot yield curves is shown in the side picture, which depicts a typical situation. The constraint averages out the drop-off in yields frequently seen in a small maturity range right before year 30.

The Long-Term Projection

 In sum, the constraint fixes the forward ra

its average, with the result that the yield cmaturities around 30 years and beyond refmarket yields.

0 10 20 30 40 

W
The Spline for Estimation

For the purposes of estimation, the cubic splinforward rate curve 

ξ(

τ) can be written as a lin

of 8 B-splines, 

b1(

τ),...,

b

8

(

τ).

The B-splines are derived by expanding the kvector (0,0,0,0,1.5,3,7,15,30,30,30,30); each of the derived from a sequence of 5 of these knots: 

b

(0,0,0,0,1.5), 

b

2 is derived from (0,0,0,1.5,3), and 

derived from (15,30,30,30,30).

The chart on the next page depicts the 8 B-sp
0 5 10 15 20 25 

0.00.20.40.60.81.0

b1b2b3b4

b5b6b7b8

B-SPLINES

Knots 0,1.5,3,7,15,30
The Spline for Estimation, continue

The next step is to impose the 3 constraints on splines. The result is a set of 5 nonnegative cμ

α

(

τ), for 

α = 1,...,5. The constraints can be rep

following matrix (written to 2 decimal places)

×

⎟⎟⎟⎟⎟⎟⎠⎞

⎜⎜⎜⎜⎜⎜⎝⎛

=Μ

76.076.010000024.024.0010000

0000100000000133.0000000067.01

where 

Μ is the column vector 

(μ1(

τ),...,μ

5

(

τ))’, a

column vector 

(

b1(

τ),...,

b

8

(

τ))’. The cubic spline

plotted in the next chart.
0 5 10 15 20 2

0.00.20.40.60.81.0

BASIS SPLINES WITH CONSTR

Knots 0,1.5,3,7,15,30

μ1

μ2

μ3 μ4

μ5
The Spline for Estimation, continue

Therefore, the forward rate curve 

ξ(

τ) includin

constraints can be written as a linear combina

with 5 coefficients 

β

α to be estimated:

)( )(

5

1

τξ

β

μ α

τ

α

α

=

∑

=

Defining

dzz

z )( )(

=

∫

=

0

τ

ν

α

τ

μα

the discount function is written as

τδ

β ανα

τ

α

)( (exp ))(

5

1

=

−

∑

=
The Spline for Estimation, continue

If the estimates of the 5 parameters 

β

α are str

they have been in recent years, the resulting function fulfills the conditions discussed earlipositive and declining with a value of 1 at 

τ = 

The next chart plots the 5 integrated function
0 5 10 15 20 2

0369

1215

INTEGRATED BASIS SPLINE

WITH CONSTRAINTS

Knots 0,1.5,3,7,15,30

ν1

ν2

ν3

ν4
Regression Variables

 In addition to the 5 parameters 

β

α, the coef

regression variables 

xj must be estimated. 

Paper methodology currently includes 2 regvariables. These variables could change in 

 The 2 variables are used to combine the thr

qualities into a single pricing model; therefimplement the PPA requirement that a sinbe produced for the combined high quality market.
Regression Variables, continued

 The variables are chosen so that the discou

pertains to the quality level that is the maraverage of the three qualities, with weights par amounts outstanding of the respective 

 The greater the effect of the variables in ad

prices of bonds in a particular quality classdiscount function is representative of that qget a market-weighted discount function, thadjustment should vary inversely with the quality class.

 As a result, the spot curve derived from the 

function also pertains to market-weighted 
Definition of the Regression Variabl

 Let 

ΓAAA, 

ΓAA, and 

ΓA be the total par amoun

in the bond set for AAA, AA, and A bonds, rDefine:

AAA AA

A

A

AAA AA

AA

Γ+Γ+Γ

Γ

=

Γ+Γ

Γ

=

2

1

ω

ω
Definition of the First Regression V

 The first variable 

x1 is defined as follows: fo

bond, 

x1 is 

ω1τAAA, where 

τAAA is the bond’s m

AA bond, 

x1 is 

(

ω1-1)

τAA, where 

τAA is the bon

and for each A bond, 

x1 is zero.

 The larger the weight of AA bonds 

ω1, the g

price adjustment through 

x1 for AAA bonds 

bonds, and so the closer is the discount fundiscount function relative to AAA.
Definition of the Second Regression 

 The second variable 

x

2 is defined analogous

bond, 

x

2 is 

ω

2

τAAA, where 

τAAA is the bond’s m

AA bond, 

x

2 is 

ω

2

τAA, where 

τAA is the bond’s 

for each A bond, 

x

2 is 

(

ω

2-1)

τA, where 

τA is th

maturity.

 The larger the weight of A bonds 

ω

2, the gr

adjustment for AAA and AA bonds relative the closer is the discount function to a A direlative to AAA and AA.
Market Weighting

 These 2 variables together cause the discou

be set in between the AAA and AA markets market sizes, and then to be moved further up in between the AAA-AA market and the again based on market weights. The resultfunction therefore represents market-weighquality across the three quality levels AAA
Regression Coefficients

 The coefficients 

ζ1 and 

ζ2 show the price dif

AAA and AA bonds, and AAA-AA and A borespectively.

 Specifically, since 

x1 is scaled by maturity, 

average amount per year of maturity that iwilling to pay for the insurance against risk the AAA rating relative to AA. Analogousl

average amount for AAA-AA bonds relative 

 Although more complicated functions of ma

defined for these variables, straightforward by maturity helps capture average market stable fashion over time.
Estimation

The final price equation for estimation is writfor the kth bond:

with notation as before. In the current White approach, there are 7 coefficients to be estimacoefficients, the βα, and 2 regression coefficien

Estimation is by nonlinear least squares, whiusual covariance matrix to be computed for thvariables. The next sections discuss the bond weighting scheme.

∑ −= ∑ + ∑ 

= = =

kn

i

m

j

k kiki 

j p xc 1 1

5

1

τνβ ))((exp 

ζ

α

αα
Data

 The PPA requires that the corporate bond y

derived from a set of high quality corporate must accurately represent the high quality market.

 In contrast to Treasury coupon issues, corp

come in many varieties, not all of which proinformation for the curve. So it is necessarselection from the universe of all high qualupon several types of characteristics.

 Since the curve shows rates of return over 

spans, each selected bond must generally punambiguous stream of fixed future cash flfor an explicit price.
Data, continued

 Therefore, the basic type of selected bond is 

conventional Treasury coupon issues: a bonfixed semiannual nominal coupon denomindollars until maturity, when the principal i

 Bonds that differ from the basic type are ge

excluded. Bonds with floating coupons proinformation about future rates of return, siinterest payments are changing over time. bonds do not have a clear price for cash flowprice depends on equities.

 The chosen bonds must be issued by corpor

backed bonds are excluded, as well as bondsponsored agencies.
Data, continued

 The bond set covers maturities up through 

Currently, bid prices are used for the bondsbelow 1 year can be filled in by Federal Respaper rates.

 To ensure sufficient liquidity, each bond m

minimum size threshold in terms of par amoutstanding. The current minimum is $25

 At present, callable bonds are excluded, un

feature is make whole. Putable bonds and sinking funds are also excluded.

 The number of corporate bonds that fulfill 

specifications is sufficiently large that therin computing the yield curve.
Weighting

 Before estimation, the White Paper method

weights to the bond data in two stages.

 In the first stage, equal weights are assign

commercial paper rates in the data set, and amounts outstanding of the bonds are rescasum equals the sum of the commercial pap

 Then the data are multiplied by the square 

commercial paper weights and the rescaled which implies that the squared residuals of the least squares fit are weighted by par amoutstanding.
 The purpose of the first stage is to give gre

larger bonds because they are more liquid aimportant in the market.

 Commercial paper rates are assigned a hig

they can anchor the curve at the short end.

 In the second stage of weighting, for bonds 

greater than unity, the weighted bond data the square root of (Macaulay) duration.

 The second stage corrects for heteroscedast

higher duration are more volatile, and the terms of such bonds have higher variance.

Weighting, continued
Software The model derived from the White Paper m

estimated using software developed at the Department.

 The software provides the capability to esti

yield curves by splining the forward rate cu

 As discussed, the spline on the forward cur

from B-splines, and arbitrary linear constrimposed on the B-spline coefficients. The 3 used here are examples of linear constraint
Software, continued A special feature of the software is that the 

the (possibly constrained) B-splines can alsto be nonnegative, thus ensuring that the rdiscount function is nonincreasing. This febeen necessary in recent data for the corpocurve, but it can be useful with particularly data sets.

 The software also enables arbitrary linear 

variables to be added to the yield curve, as corporate curve.

 The software takes account of coupon and p

payment dates that fall on weekends or holmoves them forward to the next business d
Results The following charts present sample result

Quality Market-Weighted (HQM) CorporatCurve computed by the White Paper methobusiness day Wednesday, June 20, 2007.

 In the bond set for this date, there are 81 A

417 AA bonds, and 953 A bonds, for a total There are 12 commercial paper rates.

 The numbers of bonds whose maturities fal

ranges 0-1.5, 1.5-3, 3-7, 7-15, and 15-30, ar99, 237, 523, 315, and 277.
 The 5 βk spline coefficients estimated for th

percentage terms are:

β1 5.396

β2 5.404β3 5.973β4 6.666β5 6.769

Results, continued
Results, continued The estimates of the regression coefficients 

in basis points are as follows:

Variable Coefficient T-

x1 6.8 

x

2 9.7 1

 These coefficients say that the price of a AA

basis points higher per year of maturity thAA bond with the same characteristics, and AAA-AA bond was 9.7 basis points higher pmaturity than an A bond.
Results, continued As a measure of fit, the mean absolute erro

actual and fitted prices for this date was 0.point.

 The mean absolute price error relative to a

(including accrued interest) was 0.8 percen

 This is a close fit in view of the diverse cha

corporate bonds.
Results, continued

 The next two charts depict the spot and pa

curves for this day.

 The spot curve is projected out 100 years, a

upward.

 The par chart includes the yields to maturi

bonds in the bond set, and shows that the Hfalls within the set of yields.
0 10 20 30 40 50 60 70 80 

012345678

HQM CORPORATE SPOT YIELD C

6/20/2007, Percent

Maturity, Years
AAA

AAA

0 5 10 15 20 2

012345678

Maturity, Years

HQM CORPORATE PAR YIELD C

6/20/2007, Percent
Results, continued

 The next chart shows the forward rate curv

extended out to 40 years.

 The chart illustrates the cubic equations w

maturity ranges indicated by the vertical liis fixed at 30 years maturity and beyond at forward rate average of 6.75 percent.
0 5 10 15 20 25 30 

012345678

Maturity, Years

Projected Beyond Average 15 to 30 Value at 30 of 6.7

Zero

Derivative

Zero SecondDerivative

Knots: 0,1.5,3,7,15,3

HQM CORPORATE

FORWARD RATE CURVE

6/20/2007, Percent
Results, continued

 And the following chart plots the discount f

date, corresponding to the forward curve in chart.

 The discount function equals unity at matu

positive and declining, eventually convergi
0 10 20 30 40 50 60 70 80 

0.00.20.40.60.81.0

Maturity, Years

HQM CORPORATE DISCOUNT FUN

6/20/2007
Results, continued

 The next chart pulls together and compare

and forward HQM curves for June 20, 2007

 The chart illustrates the smooth projection 

curves out 100 years.
0 10 20 30 40 50 60 70 80 

012345678

SpotParForward

Maturity, Years

HQM CORPORATE YIELD CURV

6/20/2007, Percent
 The next charts show select features of the 

methodology over time. The data in the chfrom monthly averages of results for the Hestimated for the 683 business days from Jthrough September 2007.

 The first chart depicts the βα coefficients ov

span, where β1 is the coefficient on the lowe

basis spline, up through β5, which is the coe

last basis spline.

 The chart shows that β1 and β2 generally ro

period.

Results over Time
J M M J S N J M M J S N J M M 

012345678

BETA COEFFICIENTS

HQM CORPORATE BOND YIELD C

1/2005 - 9/2007, Percent, Monthly Ave

β1

β2

β3

β4

2005

β5

2006 
 The next chart depicts the 2 regression coe

for quality levels over this time period.

 The chart shows that ζ1 rose a bit over the 

spiked up recently because of financial marThe coefficient ζ2 also increased on balance

Results over Time, continued
J M M J S N J M M J S N J M M 

048

121620

REGRESSION COEFFICIENTS

HQM CORPORATE BOND YIELD C

1/2005 - 9/2007, Basis Points, Monthly A

ζ2

2005 2006 

ζ1
 And the following chart depicts the t-ratios 

regression coefficients over this time span.

 The chart shows that the coefficients were 

significant throughout the period.

Results over Time, continued
J M M J S N J M M J S N J M M 

0369

121518

REGRESSION T-RATIOS

HQM CORPORATE BOND YIELD C

1/2005 - 9/2007, Monthly Average

ζ2

2005 2006 

ζ1
Results over Time, continued

 The next chart plots monthly average par y

same period, at the four maturities of 2 yeayears, and 30 years.

 The chart indicates some flattening of the p

this period.
J M M J S N J M M J S N J M M 

012345678

Maturity

2 Years5 Years10 Year30 Year

1/2005 - 9/2007, Percent, Monthly Aver

2005 2006 

HQM CORPORATE PAR YIEL
Results over Time, continued

 All these charts show that the 7 estimated 

the White Paper HQM methodology have eover this period.

 The final chart is a three-dimensional char

monthly averages of the HQM spot yield cuyears October 2003 through September 200average spot curve at each month is shown back slice with the spot rate on the vertical maturity on the axis extending into the slidviewer.
Results over Time, continued

 The chart shows that these curves have evo

smoothly throughout the period while pickimovements. At 100 years maturity, the spover 7-1/2 percent in late 2003 and end aroin September 2007. At 30 years maturity tabout 6-1/2 percent.

 At the short end, the spot rates rise over tim

1-1/4 percent to about 5-1/2 percent and threcently, reflecting the Federal Reserve tigbegan in the middle of 2004 and the easing 2007. These movements mirror the behaviterm 

β coefficients.

Conclusion

This presentation describes the Treasury Whimethodology for constructing the High QualitWeighted (HQM) Corporate Bond Yield CurveThe White Paper methodology meets the yield requirements, and provides smooth and robusresults over time that capture market behavio