Introduction

● The High Quality Market (HQM) Corporate Bond Yie

the Pension Protection Act ( ) PPA uses a methodology 

at Treasury to construct corporate bond yield curves bextended reg yg ressions on maturity ranges.

● This presentation describes the conceptual basis of th

the methodology for its construction. (For links to mo

d i b h h l lid ) documentation about the curve, see the last slide.)

● The HQM methodology is general. It has been applie

Treasury inflation Treasury inflation-indexed securities (TIPS) and has 

indexed securities (TIPS) and has 

construct other corporate bond yield curves; a samplicurves appears at the end of this presentation.
HQM Methodology Characteristics

The HQM methodology contains features and capathat do not appear in other yield curve approaches● It uses regression variables It uses regression variables.● It j t i ld b d th l t t it d 

It pro

jec

ts y

i

eld curves 

beyon

d the 

longes

t ma

turity 

d

● It makes use of established bond market characterist

generate a stable curve.
HQM Methodology: Regression Variable

● The HQM methodology has the special capability o

regressi i bl ith th i ld Th 

ion var

i

ables with the y

i

eld curve. The reg

terms adjust for the particular attributes of individ

● The HQM yield curve represents the high quality c

The HQM yield curve represents the high quality c

bond market, i.e., corporate bonds rated AAA, AA, HQM curve contains two regression terms. These 

adj f h bl d AAA AA d A b 

djustment 

factors t

hat blen

d AAA, AA, an

d

A

bon

single HQM yield curve that is the market-weighte(MWA) 

q y gq y ualit

y of hi

gh 

qualit

y bonds. 

● Other yield curve approaches were typically develo

Treasury securities, and they do not take into acco

distinctive characteristics of corporate bonds. Othapproaches do not have regression variables and cbonds of different qualities.bonds of different qualities.
HQM Methodology: Projections

● The HQM methodology projects yields beyond 30 

The HQM methodology projects yields beyond 30 

maturity (out to 100 years maturity to get discoufor long-dated pension liabilities). 

● The methodology ensures that the projections ar

consistent with yields before 30 years maturity a

lo gn

g

-te i e t e t et a ailable i the 

term 

inv

e

s

t

m

e

n

t returns 

a

vailable

in the m

● Other yield curve approaches generally stop at 3

maturity and contain no provision for projection maturity and contain no provision for projection
HQM Methodology: Established Market 

● Th HQM h d l i d i d f b i h 

The HQM met

h

o

d

o

logy 

is 

der

ive

d

from 

bas

ic 

hyp

about corporate bond markets. This informs how 

parameters r

a

mete

r

s

that

underpin nde

r

pin 

the

analysis n

a

lys

i

s

are set.

● Other yield curve approaches can be arbitrary in

the

y y yp are not informed b

y basic market hypothes

makes it difficult to choose among them.
Developing the HQM Yield Curve: Summ

● The HQM yield curve is a spot yield curve The HQM yield curve is

a spot yield curve. 

● The HQM yield curve is calculated from an estim

discount function discount function.

● The discount function is estimated using bond p

other data and constraints consistent with relatiother data and constraints consistent with relati

that hold in forward rates. 

● Spot rates from the curve are used to value pens

Spot rates from the curve are used to value pens

liabilities.
The Spot Yield Curve

● The spot rate for any maturity is the yield on a b

provides a single payment at that maturity. Thicoupon bond. Because each spot rate pertains to cash flow it is the relevant interest rate concept cash flow, it is the relevant interest rate concept 

discounting a pension liability at the same matu

● The HQM Corporate Bond Yield Curve is a spot 

The HQM Corporate Bond Yield Curve is

a spot 

curve: it gives the spot rate for each maturity at year maturities 1⁄2 year up through 100 years, fo

200 t t 200 spo

t ra

tes.

● Zero coupon corporate bonds may not be general

avail bl i th k t b t t t till b il

able 

in the mar

k

e

t, 

b

u

t spo

t ra

tes can still

b

computed, because they can be inferred from mabehavior.
The Spot Yield Curve, continued

● Each H

Qy p p p M 

yield curve 

pertains to a s

pecific 

poin

The curve is calculated late in the day for each bday, and the monthly HQM spot rates are averafor all the business days of the month Segment for all the business days of the month. Segment 

required by the PPA are derived from the monthaverages.

● The quality of the HQM curve is the market-wei

average (MWA) quality of the AAA, AA, and A bo

to compute it.

● The next chart plots the monthly average spot r

through 100 years maturity from the HQM yield January and February of this year. The Februaabove January beyond the shortest maturitiesabove January beyond the shortest maturities.
9 

HQM SPOT YIELD CURVES

Monthly Averages, Percent

789567345

February 2010January 2010

12

Maturity

10 20 30 40 50 60 70 80 90 

0 

Maturity
The Discount Function

● The HQM methodology uses the discount functio

known concept in finance known concept in finance.

● The discount function 

δ

(

τ) gives for each maturit

the present value of $1 to be received at that mathe present value of $1 to be received at that ma

is, the amount that must be invested now to get in the future.

● Markets do not provide an explicit discount func

it can be extracted from market behavior throug

curve, as i h f E i i h 

in t

he case o

f spot rates. 

Est

imat

ing t

h

function is a key goal of the HQM methodology.

● Th di t f ti f t th i t i 

The discoun

t

function re

fers 

to the same po

i

n

t

in 

the yield curve. It represents future payments wcredit 

qualit

y is e

qual to the market-wei

ghted av

q yq g 

credit quality of high quality bonds.
The Discount Function, continued

● The values of the discount function depend upon 

f i l di h f hi h k factors, 

inc

l

uding t

he purposes 

for whi

c

h mar

ket 

participants borrow and lend, their expectations 

future, and their attitudes toward risk. future, and their attitudes toward risk.

● The next chart illustrates the typical shape of re

discount functions. The discount function is uni

maturity zero, because the present value of $1 rimmediately is $1, and normally decreases, becapayment i d l t i ti i th l t d 

t rece

ive

d

l

a

ter 

in time 

is worth

less 

t

o

da

received earlier.

● The discount function says that The discount function says that

δ(10) is the amou

is the amou

must be invested now to get $1 after 10 years. Fchart, this is about 58 cents. The value 

δ(20) fal

cents, and 

δ(30) is 14 cents.
1.0 

HQM DISCOUNT FUNCTION

0.8 0.6 0.4 0.2 

Maturity, Years

0 10 20 30 40 50 60 70 80 90 

0.0 
The Spot Rate

● Once the discount function is known, the spot ra

readil t d S ifi ll th di t f 

dily computed. Specifically, the discount fun

implies that an investment of $δ(τ)x gives a singl

of $x at maturity τ,j p 

ust like a zero coupon bond. 

rate at maturity τ is the yield of such a bond.

● The formula for g g py etting the spot yield curve y(τ) 

as follows for maturities of at least 1⁄2 year:

( ) 200 (1/ ( )2 1) 1 y τ = × δ τ τ −

● For example, at 10 years maturity, using the dis

y(τ ) = 200× (1/δ (τ )2τ 1)

function from the last chart:

(10) 200 (1/ 0 5758 1) 5 60 20

1

y = × − =y(10) 200×(1/ 0.5758 1) 5.60
The Price of a Bond

● To estimate the discount function, the price of a 

be linked to the discount function.

● The price of a bond is the sum of three componen

present values of the bond’s future cash flows diby the discount function, plus any adjustment fathe bond the bond s’ specific characteristics plus a random 

specific characteristics, plus a random 

characteristics of the bond that are not adjusted

● In the case of the HQM curve there are two adju

In the case of the HQM curve, there are two adju

factors, described later, that adjust each bond fothat the discount function pertains to market-we

average (MWA) li hil h i di id l b 

(MWA) quality, while each individual bo

either AAA, AA, or A quality.
The Price of a Bond, continued

The formula for the price 

p of a bond is:

p

=

∑

δ

τ

c

+

∑

ζ

x

+

ε

j

m

j

j

i

n

i

i

1

1

(

)

=

=

j

j

j

i

1

1

where p is the price of the bond (including accrued interesci for i = 1,...,

n are the 

n cash flows from the bond to 

i

at maturities 

τi, 

ζj for j = 1,...,

m are the 

m coefficients for the 

m regre

j

variables 

xj, giving 

m adjustment factors 

ζj

xj, and 

ε is the random error.
The Price of a Bond, continued

● With this price formula, the yield curve is estim

choosing an appropriate set of bonds and calculadiscount function and adjustment factors which best fit of the formula to all the bonds. The spot best fit of the formula to all the bonds. The spot 

curve is computed from the resulting discount fu

● The description of the discount function so far is 

The description of the discount function so far is 

specific enough for estimation. To estimate, a prof the discount function must be chosen.

● The main source of differences among yield curv

approaches is their different choices of discount 

f Th HQM th d l b th di forms. The HQM meth

o

d

o

logy 

bases the discoun

form on market maturity ranges.
The Forward Rate

● The concep p

t of the forward rate is useful for pick

for the discount function. The forward rate and discount function are equivalent in that one can from the other and the forward rate is easier to from the other, and the forward rate is easier to 

● The forward rate is straightforward: for each ma

the forward rate is the interest rate obtained by the forward rate is the interest rate obtained by 

an investment maturing at τ by a small amount 

● The extended investment is in essence made up 

The extended investment is in essence made up 

investments: the initial investment that yields a through maturity τ, and the reinvestment of this 

the extension period. The forward rate is the fuinterest rate on the reinvestment.
The Forward Rate, continued

● An example of the forward rate is the calculation 

yi ld ft d f T iti Th F 

i

elds often 

done 

for 

Treasury securities. The 

F

e

average 1-year yield in the Treasury market wa

p , ercent, and the 2-

yy p ear 

yield was 0.86 

percent. 

yields imply that the 1-year forward yield 1 year future was about 1.38 percent. That is, an invesbe indifferent between buying a 2 be indifferent between buying

a

2

-year note and 

year note and 

1-year note plus a second 1-year note at this yielfrom now.

● The forward rate for the HQM curve is analogou

example, although it is defined as a short-term y

applies to corporate bonds. In general, similar tdiscount function, the market does not provide frates

.
The Forward Rate, continued

● The forward rate curve 

ξ(

τ) gives the forward rat

maturity Relative to the discount function the maturity. Relative to the discount function, the 

rate is given by the following formula:

(

)

(

)

1

(

)

τ

δ

τ

δ

τ

ξ

τ

d

d

=

−

● The forward rate is higher at a given maturity w

investors who are trading at that maturity are linvestors who are trading at that maturity are l

lend based on their views of uncertainty and theexpectations about the market, while borrowers 

eager t b b d th i ti Th 

to 

borrow 

base

d on th

e

ir perceptions. The 

rate summarizes market views at each maturity number.
The Forward Rate, continued

● The forward rate is especially useful because it c

compared across maturities, and so it can be usebasis for developing a functional form for estimadiscount function discount function.

● The next chart plots the forward rate curve up t

years maturity from the discount function in slidyears maturity from the discount function in slid

averages of the forward rate in maturity ranges 3, 3-7, 7-15, and 15-30 years are included, as dis

l ) h h h h hf d later.

) T

he c

hart s

hows t

hat t

he 

forwar

d rate cu

a hump around 10 years maturity, then flattene

was constant beyond 30 years maturity. was constant beyond 30 years maturity.
9 

HQM FORWARD RATE CURVE

78563412

Maturity, Years

0 5 10 15 20 25 30 35 

0 

y
Maturity Ranges

● Corporate bond trades tend to divide into matur

Corporate bond trades tend to divide into matur

Prices of bonds in each range are related, becaustraders who have chosen to trade in that range t

h hd f d i have shared preferences and expectations.

● As a consequence, the forward rates in each mat

are rel d b h fl h i f 

lated too, because they reflect the views of 

● The HQM methodology uses five maturity range

d li t d b th t it i t 0 1 5 3 7 15 delineated by the maturity points 0, 1.5, 3, 7, 15

years maturity.
Maturity Ranges, continued

● The range 0 to 3 years reflects short-term tradin

th i h ti it th t it h ld b f th there 

is enoug

h acti

vity th

a

t it s

hould

be 

further 

two with a point at 1.5 years.

● Th 3 t 7 t i b d f h 

The range 

3

to 

7 years con

t

a

ins 

bon

ds o

f somew

h

term centering around 5 years maturity. The ra

y g qy ears contains the bul

ge of bonds fre

quentl

y see

10 years maturity. The last range 15 to 30 years bonds with the longest maturities. 

● For maturities beyond 30 years, the forward rat

projected from the five ranges, as described later

● The five ranges are depicted in the previous cha

22 as the maturities corresponding to the five fo

averages

.
Averages in the Ranges

● Given this choice of ranges, the forward rate in e

as a first approximation could be simply taken tconstant, reflecting its average in the range.

● The previous chart on slide 22 shows what these 

would look like.

● H h h i di h h i i 

However, t

he c

hart 

i

ndicates t

hat t

here 

is a s

ign

amount of movement in the forward rate curve t

captured by the averages and that needs to be incaptured by the averages and that needs to be in

the curve to be accurate.

● Moreover,g , the avera

ges are not connected, and in 

get a well-behaved forward rate curve, the valuecurve in the ranges must join together smoothly
Averages in the Ranges, continued

● A smooth forward rate curve can be obtained wit

departing too far from averages, and without intexcessive volatility, by using a cubic polynomial the curve in each range instead of a constant, anthe curve in each range instead of

a constant, an

smoothly joining the polynomials together at the each range.

● The forward curve specified in this manner is a 

with knots at the six maturity points that deline

ranges.

● The use of a cubic spline arises naturally as a pa

way t fit th t f th thl 

to fit the componen

ts o

f the curve smoothly 

ranges. In particular, the spline is not imposed forward curve without 

justification.
Averages in the Ranges, continued

● Moreover, p the choice of fixed knots for the s

pline 

maturity ranges increases significantly the stabyield curve estimates over time.

● The next step is to estimate the spline and get th

rate curve, the discount function, and finally the 

curve However before doing this it is necessarcurve. However, before doing this, it is necessar

three constraints upon the forward curve, includconstraint at the short end and two at the long e
Constraint in the Nearest Range

● Because there are no bonds with zero maturity, 

rate curve has no anchor at the zero point in the maturity range.

● For this reason, an approximate linear constrain

imposed on the movement of the forward rate at 

earliest maturities This is the constraint on the earliest maturities. This is the constraint on the 

rate curve at the short end.

● This constraint sets the second derivative of the 

This constraint sets the second derivative of the 

curve to zero at the beginning:

(

0

)

2

d

ξ

0

(

0

)

2

=

d

τ

d

ξ
The Long-Term Forward Rate

● Forward rates can be estimated u

p gy throu

gh 30 

y

maturity, the maximum typical maturity of corpbonds. However, for pension liability discountinnecessary to derive spot rates up through 100 yenecessary to derive spot rates up through 100 ye

maturity. Consequently, the forward rate curve projected out through 100 years maturity.

● Because there are usually not enough data to es

accurately the movements in the forward rate be

years maturity, the projected forward rate is set constant, which is an estimate of the average forbeyond 30 years maturity This constant is callebeyond 30 years maturity. This constant is calleterm forward rate.
The Long-Term Forward Rate, continue

● Two constraints are needed for the last maturity 

to 30 years to derive the constant long-term forwand to ensure well-behaved projections beyond 3maturity.

● The first constraint makes the forward rate curv

out at maturity 30 years and thereby connect smout at maturity 30 years, and thereby connect sm

with the constant long-term forward rate at 30 y

● As a formula this constraint sets the first deriva

As

a formula, this constraint sets the first deriva

forward curve to zero at 30 years maturity:

dξ

(30

)

0

(30

)

=

dτ

dξ
The Long-Term Forward Rate, continue

● In addition, q there is the critical 

question of how 

determine the constant long-term forward rate.

● The H

Q gy p M methodology

postulates that forward 

largely influenced by market aversion to risks thbe hedged and that generally grow over time. 

● Based on these considerations, the HQM method

assumes that the long-term forward rate is dete

the same factors that affect forward rates in the the same factors that affect forward rates in the year maturity range, since that range is likely sudistant in time to reveal underlying long-term atoward risk.
The Long-Term Forward Rate, continue

● Therefore, the constant long-term forward rate f

years maturity up through 100 years maturity iconstrained to be the average forward rate in thyear maturity range This is the third and final year maturity range. This is the third and final 

on the forward rate curve.

● As a formula this constraint is: As

a formula, this constraint is:

(

)

(

) 30

ξ

ξ

z

dz

∫

(30

)

(

)

15

15

ξ

ξ

z

=

∫

=
The Adjustment Factors

● The adj g ustment factors are estimated alon

g with 

discount function. This is a unique aspect of the methodology. The HQM yield curve has two adjufactors.

● The HQM curve needs the adjustment factors be

discount function pertains to market discount function pertains to market

-weighted a

weighted a

AAA, AA, and A quality (MWA quality).

● Therefore the price of each bond must be adjust

Therefore, the price of each bond must be adjust

reflect what it would be if the bond were of MWA The adjustment makes the price consistent with 

quality of the discount function.
The Adjustment Factors, continued

● The adjustment factors enable the entire set of A

and A bonds to be blended in the estimation ratand

A bonds to be blended in the estimation, rat

estimating separate curves for each quality ratinblending produces a more stable result, because 

individual markets may be small and variable (ethe AAA market).

● Blending the bonds is also useful because yield c

the three qualities tend to be related. Blending 

of the common relationships among the individuof the common relationships among the individu

to derive a more robust overall average yield cur

● The first adjustment factor adjusts AAA and AA 

The first adjustment factor adjusts AAA and AA 

be market-weighted average AAA-AA prices. Thfactor adjusts average AAA-AA prices and A pric

MWA i f ll h li iMWA pr

ices 

for all t

hree quali

t

ies.
The First Adjustment Factor

● This factor assumes that for a bond of maturity 

difference between the price of the bond at AAA the price at AA quality is 

ζ1

τ, that is, a constant a

multiplied by maturity multiplied by maturity

τ.

● The reason for multiplying by maturity is that t

rating is like insurance against risk, and the lonrating is like insurance against risk, and the lon

maturity, the greater the amount of the insurancoefficient 

ζ1 has been around 50 basis points rec

● The first factor also uses the value 

ω

1, which is t

par amount outstanding of AA bonds as a fractio

pl AA t t t di Thi l h 

lus AA par amoun

ts ou

t

s

tanding. This va

lue 

h

around 59 percent recently.
The First Adjustment Factor, continued

● The value 

ω

1 can be interpreted as the distance from A

to market to market

-weighted AAA weighted AAA

-AA quality as a fraction of th

AA quality as

a fraction of th

from AAA to AA quality.

● Each AAA bond price is adjusted to market Each AAA bond price is adjusted to market

-weig

weig

AA quality by subtracting 

ζ1

ω

1

τ from the price. Sub

full amount 

ζ1

τ would adjust the AAA price all the way 

To adjust the AAA price down to AAA-AA quality, the is multiplied by the fraction 

ω

1.

● Similarly, each AA bond price is adjusted to mar

weighted AAA-AA quality by adding 

ζ1(1-

ω

1

)

τ to t

● In the estimation, the regression variable 

x

1 is d

ω

1

τ for AAA bonds, 

(

ω

1-1)

τ for AA bonds, and zero 

bonds. The value

ζ1 is the regression coefficient 

bonds. The value

ζ1 is the regression coefficient 

estimated.
The Second Adjustment Factor

● The second adjustment factor transforms the pri

bonds of market-weighted AAA-AA quality and b

quality into prices of MWA quality. It is entirely 

to the first adjustment factor to the first adjustment factor.

● For the second factor, the difference between the 

bond at market bond at market

-weighted AAA weighted AAA

-AA quality and th

AA quality and th

A quality is 

ζ2

τ; again, the longer the maturity, th

the amount of insurance against risk. The coeffi

h b d 31 b i i t tl has 

been aroun

d 31

bas

is po

i

n

ts recently.

● This factor also uses the value 

ω

2, which is the t

amount t t di f A b d f ti f A

t ou

t

s

tanding o

f

A

bon

ds as a 

fraction o

f A

and A par amounts outstanding. This value has around 67 

percent recentl

y.

p y
The Second Adjustment Factor, continue

● As before, the value 

ω

2 , is the distance from AAA-AA 2 

MWA quality as a fraction of the distance from AAA-Aquality.

● Therefore, a AAA-AA bond price is adjusted to M

quality by subtracting 

ζ2

ω

2

τ from the price. Simila

bond price bond price is adjusted to MWA quality by adding is adjusted to MWA quality by adding

the price.

● In the estimation the regression variable In the estimation, the regression variable

x is d

2 is d

ω

2

τ for AAA and AA bonds, and 

(

ω

2-1)

τ for A bond

value 

ζ2 is the regression coefficient to be estima
Estimation

● The cubic spline with its three constraints and t

The cubic spline with its three constraints and t

adjustment factors altogether contain seven parbe estimated.

● The estimation is done by using least squares to 

price formula on slide 16 to the bond data.

● The next sections discuss the data and the weigh

scheme that is applied to the data before estima
Data

● The HQM yield curve uses data from a set of hig

The HQM yield curve uses data from a set of hig

corporate bonds, rated AAA, AA, or A, that accurrepresent the high quality corporate bond marke

● The bonds are selected from the universe of all h

bonds based upon several types of characteristic

● The HQM curve is meant to represent all bonds 

market. So a bond that is priced and available f

i l d d if it i ld t b tli included even if its yield appears to be an outlier 

other bonds. Such a bond is excluded only if thereason other than unusual yield for the exclusio
Data, continued

● The basic type of selected bond is analogous to 

conventi l T i : b d th 

tional Treasury coupon issues: a bond tha

fixed semiannual nominal coupon denominated dollars until maturity, p p when the principal is retu

● Bonds that differ from the basic type are genera

excluded. Bonds with floatingpp 

 coupons provide l

information about future rates of return, since tinterest payments are changing over time. Convbonds do not have a clear price for cash flows bebonds do not have a clear price for cash flows be

price depends on equity values.

● Bonds must be issued by corporations Asset Bonds must be issued by corporations. Asset-ba

ba

are excluded, as well as bonds issued by U.S. spoagencies.
Data, continued

● The bond set covers maturities up through 30 ye

prices are used. Maturities below 1 year are fillFederal Reserve commercial paper rates.

● E h b d i i i h h ld 

Each bond must meet a minimum size threshold 

of par amount outstanding. The current minimu

million.

● Callable bonds are excluded at present, unless th

feature is make whole. Putable bonds and bond

sinking funds are also excluded.

● The number of corporate bonds that fulfill these 

specifications is sufficiently large at all maturitithere is no problem in computing the yield curve
Weighting

● Before estimation, Q gy pp 

 the HQM methodology applie

to the bond data in two stages.

● The first stag g yp e weights the bonds by par amount

outstanding and weights the commercial paper requally with the bonds. This is done because bolarger par amounts outstanding are more imporlarger par amounts outstanding are more impor

because the commercial paper rates must anchoend of the curve.

● In the second stage, for bonds with duration gre

unity, the bond data are divided by the square ro

(Macaulay) duration. The second stage corrects greater volatility of bonds with higher duration.
Results

● The following charts present sample results for t

The following charts present sample results for t

yield curve for a single day, the last business day December 31, 2009. As noted earlier, the publis

curve f b i hl for December is a monthly average.

Characteristics of the bonds included in the Decem

Maturity Range

(years)

Rating Number of 

Bonds

0 – 1.5 1.5 – 3 

3 7 

AAA 101

AA 253

A 1 211 3 – 7 

7 – 15 

15 – 30 

A 1,211

Commercial paper 7

Total 1 572Total 1,572 15 – 30 
Results, continued

● The first chart shows the forward rate curve for 

31, 2009 plotted through 30 years maturity, and projected out through 40 years maturity.

● The chart depicts the five cubics for the five mat

ranges, delineated by the vertical lines at the m

1 5 3 7 15 and 30 years The cubics are knotte1.5, 3, 7, 15, and 30 years. The cubics are knotte

to get a smooth curve. The near constraint damcurve at maturity zero.

● The forward curve is fixed at 30 years maturity 

at the long-term forward rate of 6.57 percent, wh

gi b h f h f d i h 1

iven by the average of the forward rate in the 1

maturity range. The second constraint flattens 

curve at 30 years maturity.c ve a 30 yea s a y.
9 

HQM FORWARD RATE CURVE

12/31/2009, Percent

789

Knots: 0,1.5,3,7,15,30 Projected Beyond 30

567

Flattens Out

345

Average 15 to 30 =

123

Maturity Years

Average 15 to 30 = Value at 30 = 6.57 Becomes Linear

0 5 10 15 20 25 30 35 

01

Maturity 

, Years
Results, continued

● The following chart plots the discount function f

December 31, 2009.

● Th h h h h di f i 

The chart shows that the discount function equa

maturity zero, because the present value of $1 r

immediately is $1, and it declines, because the pimmediately is $1, and it declines, because the p

value of a payment received later in time is sma
1.0 

HQM DISCOUNT FUNCTION

12/31/2009

0.8 0.6 0.4 0.2 

Maturity, Years

0 10 20 30 40 50 60 70 80 90 

0.0 
Results, continued

● The third chart shows the spot yield curve for D

31, 2009, derived from the discount function in t

previous chart previous chart.

● The spot curve is projected out through 100 year

gently upward from about 15 years maturity onwgently upward from about 15 years maturity onw

reaches 6.42 percent at 30 years maturity, and 6at 100 years maturity. It is near zero at the earl

maturities, reflecting the low short-term rates o
9 

HQM SPOT YIELD CURVE

12/31/2009, Percent

789563412

Maturity, Years

0 10 20 30 40 50 60 70 80 90 

0 
Results, continued

● The estimates of the two adjustment factor coeff

f D b 31 2009 i b i i t f ll for December 31, 2009 in basis points are as foll

Variable Coefficient T-Ratio

x1 51 8.46

x2 32 10.91

● These coefficients say that on this date, the pric

bond was 51 basis p g py 

oints higher per year of mat

the price of a AA bond, and the price of a AAA-A32 basis points higher per year of maturity than 

● So at 10 years maturity, the prices of AAA and A

bonds were 510 and 320 basis points, respective

the prices of AA and A bondsthe prices of AA and A bonds.
Sample Yield Curves

● The last two charts contain sample yield curves 

The last two charts contain sample yield curves 

December 31, 2009 at the four qualities AAA, AABBB. Please note that these yield curves are pr

and h ffi i l i ld d t

hey are not offi

c

i

a

l Treasury y

i

eld curves.

● The sample curves were constructed using the H

meth d l difi d h diff l

h

o

d

o

logy modifi

e

d to get t

hese different qual

instead of the market-weighted average quality H

Q cu ve. M cu

rve.

● The curves illustrate the fact that the HQM met

a 

generalp p y gy ur

pose 

yield curve methodology that 

applied to different kinds of yield curves.
Sample Yield Curves, continued

● The fi t cha t di la the ot ield c e fo 

The firs

t cha

r

t displays the spot yield

curv

es for 

qualities, with the HQM curve included for compOn December 31, 2009, the AA and A curves are 

close, with the AAA curve somewhat lower. The curve is near the AA curve. The BBB curve is hi

● The second chart displays the par yield curves fo

quality levels. These curves are shown for illust

because par yield curves are standard in financibecause par yield curves are standard in financi
9 

SPOT YIELD CURVES

12/31/2009, Percent

7856

HQM

34

AAA AAA

12

BBB 

Maturity, Years

0 10 20 30 40 50 60 70 80 90 

0 
9 

PAR YIELD CURVES

12/31/2009, Percent

7895634

AAA AAA

12

BBB 

Maturity, Years

0 5 10 15 20 25 

0 
For More Information

● The High Quality Market (HQM) Corporate Bond Yie

the Pension Protection Act ( ) PPA is published by the 

month and is available on the IRS website. 

● For more details on the mathematics behind the curv

For more details on the mathematics behind the curv

more documentation in general, visit the Office of EcoPolicy website. Go to www.treas.gov, under “Offices” 

“E i P li ” th h “Th C t B d “Econom

ic 

P

olicy,

” then c

hoose “The 

Corpora

te 

Bon

d 

Curve.”

● The HQM methodology has been applied to Treasury 

The HQM methodology has been applied to Treasury indexed securities (TIPS). See papers 0601 and 0501 Economic Policy Research Paper Series on the Office Polic ebsite Policy website