Introduction

● The High Quality Market (HQM) Corporate Bond Yield 

the Pension Protection Act (PPA) uses a methodology 

 d

at Treasury to construct yield curves from extended regon maturity ranges.

● This presentation discusses basic concepts for understa

This presentation discusses basic concepts for understa

using the HQM yield curve, and presents an overview ocurve methodology. For links to more detailed documenand technical descrip , tions of the curve, see the last slid

● The methodology for the HQM curve is general. It has 

applied to Treasury inflation-indexed securities (TIPS) 

been used to construct other corporate bond yield curvebeen used to construct other corporate bond yield curve

example of a AA curve appears at the end of this presen
HQM Yield Curve Summary

● A yield curve provides information about a sector of the 

market at a point in time. The information includes yimarket at

a point in time. The information includes yi

different types of bonds in this sector at various maturi

● The HQM yield curve pertains to the high quality corpo

market i e corporate bonds rated AAA AA or A The market, i.e., corporate bonds rated AAA, AA, or

A. The

regression methodology of the HQM curve blends AAA, bonds into a single yield curve that represents the mark

weighted average quality of high quality bonds weighted average quality of high quality bonds.

● The HQM methodology also projects yields beyond 30 y

maturity out through 100 years maturity. The methodo

ensures that the projections are consistent with yields bensures that the projections are consistent with yields b

years maturity and with long-term investment returns in the market.
Spot Rates

● The yields provided by the HQM Corporate Bond Y

C t t Th HQM id Curve are spot rates. The HQM curve provides spo

for each maturity at half-year intervals for maturiyear up through 100 years, for a total of 200 spot r

● The spot rate for any maturity is defined as the yi

bond that gives a single payment at that maturity

called a zero coupon bond. called a zero coupon bond.

● Because high quality zero coupon bonds are not ge

available, the HQM methodology computes the spo

so as to make them consistent with the yields on oquality bonds.
Spot Rates, continued

● The HQM yield curve is calculated late in the day 

b i d d th bli h d thl HQM bus

iness 

day, an

d the publi

s

h

e

d monthly HQM sp

are averages of rates for all the business days of th

● Segment rates required by the PPA are derived fro

Segment rates required by the PPA are derived fro

monthly average spot rates.

● The next chart plots the 200 monthly average spot 

out through 100 years maturity from the HQM yiefor July and August of this year. The August curvJuly beyond the shortest maturities.
9 

HQM SPOT YIELD CURVES

Monthly Averages, Percent

678456

August 2011

23

July 2011

10 20 30 40 50 60 70 80 90 10

01

Maturity
Discounting Liabilities

● The HQM spot rates are used to discount future p

li biliti t id th t f di l l li

abilities 

to provide the presen

t

funding 

leve

l nee

the liabilities. Therefore, regardless of the assets do the funding, the HQM spot rates give the fundi

at kt a mar

k

e

t-wei ht d hi h dit lit i

ght

e

d hi

g

h credit quality.

● A different quality yield curve would give a differe

funding level funding level: the higher the credit quality of the b

the higher the credit quality of the b

the yield curve used for discounting, the higher thlevel and the greater the funding burden on the pe

plan but also the lower the risk that the plan will plan, but also the lower the risk that the plan will 

underfunded.
Discounting Liabilities, continued

● Therefore, the choice of the credit quality for the y

curve implies a decision about the tradeoff betweefunding levels and less pension underfunding risklower funding levels and more risk.

● Consequently, a different yield curve from the HQ

curve may change funding requirements substant

E g the use of a Treasury curve would increase E.g., the use of

a Treasury curve would increase

significantly the funding burden relative to the HQcurve.
Blended High Quality Bonds

● Although no individual bond has the credit quality 

represented by the HQM yield curve the curve carepresented by the HQM yield curve, the curve ca

for an appropriate blended set of high quality bond

● For example, if 5 percent of the high quality mark

rated, 25 percent is AA rated, and 70 percent is A rblended high quality bond of 10 years maturity coof 5 percent 10-year AAA, 25 percent 10-year AA, percent 10-year A can be priced by applying the Hrates to the cash flows from these bonds.

● Th f i i i l i li biliti ld b 

There

fore, 

in pr

inc

i

p

le, pens

ion li

abilities could

be 

by a portfolio of blended high quality bonds, and thof this portfolio would track the HQM yield curve.
Blended High Quality Bonds, continued

● Of course, this approach is difficult for a variety of 

including ratings changes within the portfolio whiincluding ratings changes within the portfolio whi

require rebalancing, and lack of bonds beyond 30 ymaturity.

● Moreover, changes in the relative sizes of the thre

quality ratings categories would require changes iblended bonds to remain consistent with the H

Q 

M 

curve.

● Therefore, it’s no surprise that typical bond portfo

d t t f th diffi lti d ’t t k th do no

t accoun

t

for these difficulties 

don’t

trac

k the 

curve exactly.
HQM Risk Aspects

● Nevertheless, it’s not clear that any other yield cu

be easier to hedge than the HQM yield curve. EveTreasury coupon issues stop at 30 years maturity, special inconsistencies such as on-the-run issues.

● In analyzing hedging, it must also be kept in mind 

HQM yield curve provides a suitable level of pensi

funding that is both accurate and stable over time funding that is both accurate and stable over time

Therefore, the HQM curve can be relied upon for pand in that sense reduces risk.
HQM Risk Aspects, continued

● Also, the monthly average feature of the HQM yie

may further reduce certain types of risk.

● This is because no portfolio tracks the HQM curve 

perfectly and therefore it perfectly, and therefore, it s’ important that the cur

important that the cur

stable for planning purposes. The monthly averagreduces the possible volatility in the curve that cou

f i l b h i t th from spec

i

a

l

b

e

hav

ior a

t month-en

d.
Yield Curve Methodology

● The following slides contain an overview of the HQ

curve meth d l o

d

o

logy.

● Different yield curve methodologies can produce v

different results, which affect pension liabilities andifferent results, which affect pension liabilities an

everything else for which the curve is used. Thereessential to develop the methodology that accurate

captures market behavior captures market behavior.

● Moreover, an understanding of the HQM methodo

greatly in the appropriate use of the HQM curve.
HQM Methodology Features

The HQM yield curve methodology contains features capabilities that do not appear in other yield curve capabilities that do not appear in other yield curve 

approaches:

● This methodology combines regression variables w

This methodology combines regression variables w

yield curve.

● The methodology projects yields beyond 30 years.

● The methodology makes use of established bond m

characteristics to help generate a stable curve tha

captures market movements captures market movements. 
HQM Methodology: Regression Variables

● The HQM methodology has the special capability of 

regression variables with the yield curve. The regreregression variables with the yield curve. The regre

terms adjust for the particular attributes of individu

● The HQM yield curve contains two regression terms

t dj t t f t th t bl d AAA AA terms are adjus

tmen

t

fac

tors th

a

t blen

d AAA, AA, an

bonds into a single HQM yield curve that pertains to market-weighted average quality of high quality bon

● Other yield curve approaches were typically develop

Treasury securities, and they do not take into accoundistinctive characteristics of cor

porate bonds. Other 

approaches do not have regression variables and canbonds of different qualities.
HQM Methodology: Projections

● The HQM methodology projects yields beyond 30 y

maturity out to 100 years maturity to get discount maturity out to 100 years maturity to get discount 

long-dated pension liabilities. 

● The methodology p j ensures that the 

pro

jections are 

consistent with yields before 30 years maturity anlong-term investment returns available in the mar

● Oth i ld h ll t t 30 Other y

i

eld curve approac

hes generally s

top a

t 30 

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

● The HQM methodology is derived from basic hypo

about corporate bond markets. The hypotheses insetting of the parameters that underpin the HQM curve.

● Other yield curve approaches typically use mathem

functions that don’t have any foundation in marke

hypotheses.

● Consequently, differences among yield curves give

other app 

roaches with different mathematics are a

in that there is no way to choose among them.
Estimation of HQM Spot Rates

● Estimation of the spot rates in the HQM yield cur

b h i i t t f b d (d ib d by c

hoos

ing an appropr

i

a

te se

t o

f

bon

ds (describ

e

d

● The regression terms are computed so that the pri

each bond adjusted by the regression terms equals each bond adjusted by the regression terms equals 

would have been if the bond were of market averaquality. (Technical discussion of the regression ter

found in the references ) found in the references.)

● The spot rates are chosen to be the rates that give 

statistical fit to all the prices in the bond set. This 

that the estimated prices of the bonds, given as thflows discounted by the spot rates, come closest on to their actual prices.
Estimation of HQM Spot Rates, continued

● Note that the HQM yield curve estimation employ

of bond prices The estimation does not directly uof bond prices. The estimation does not directly u

yields or averages of yields.

● However, the description so far of the estimation i

general actually to get spot rates statistically. To cthe estimation, the spot rates must be assumed to mathematical pattern or functional form.

● The main source of differences among yield curve 

approaches is their different choices of functional 

and as noted functional forms in other approacheand as noted, functional forms in other approache

typically are not grounded in market hypotheses. HQM methodology bases the functional form on h

b t t it i th k t

a

bou

t ma

turity ranges 

in the mar

k

e

t.
The Forward Rate

● The concept of the forward rate is useful for pickin

The concept of the forward rate is useful for pickin

functional form for estimating the spot rates.

● The forward rate is straightforward: for each matu

consider entering into a contract to invest an amouthe time of that maturity for a small amount of timbeyond that maturity. The forward rate at that mthe future interest rate on this investment.
The Forward Rate, continued

● In 

g,g p , eneral, analo

gous to s

pot rates, the market do

provide explicit forward rates, and they must be infrom other prices in the market.

● Th f d t i hi h t i t it h The 

forwar

d ra

te 

is hi

g

her a

t a g

iven ma

turity w

h

investors who are trading at that maturity are leslend based on their views of uncertainty and their 

expectati b h k hil b ions a

bout t

he mar

ket, while 

borrowers ar

eager to borrow based on their perceptions. The forate summarizes market views at each maturity in 

number.
Maturity Ranges

● Trading in bonds tends to divide into maturity ran

trades that take place in each maturity range on areflect similar preferences and expectations of tradthat range.

● As a consequence, the forward rates in each matur

are related, because they reflect the views of trade

● Therefore, the HQM methodology models the forw

in each maturity range as a smooth (cubic) functio

j g y( 

oins the functions together smoothly (as a cubic s
Maturity Ranges, continued

● The HQM methodology uses five maturity ranges The HQM methodology uses five maturity ranges, 

delineated by the maturity points 0, 1.5, 3, 7, 15, ayears maturity.

● The choice of fixed maturity ranges increases sign

the stability of the HQM yield curve estimates ove
The Long-Term Forward Rate

● Assum

p pj tions must be made in order to 

pro

ject forw

rates beyond 30 years maturity so as to obtain spoout through 100 years maturity.

● Fi t f ll th l Firs

t o

f all, the 

long-t f d t b d 30 term 

forwar

d ra

te 

beyon

d 30 

maturity is set to a constant, because there are usenough data to estimate accurately movements in 

f d t b d 30 t it forwar

d ra

te 

beyon

d 30 years ma

turity.
The Long-Term Forward Rate, continued

● Next the HQM methodology postulates that the lo

forward rate is determined by the same factors thaforward rates in the longest 15- to 30-year maturitsince that range is likely sufficiently distant in timreveal underlying long-term attitudes toward risk

● Therefore, the constant long-term forward rate fro

years maturity up through 100 years maturity is tyears maturity up through 100 years maturity is t

be the average forward rate in the 15- to 30-year mrange.
Estimation

● The forward rates in the maturity ranges are estim

least squares, and the spot rates are derived from estimated forward rate curve.

● The regression terms and adjustment factors are The regression terms and adjustment factors are

simultaneously estimated. Before estimation, the data are weighted. (The adjustment factors and w

are described in the references ) are described in the references.)
Data

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

corporate bonds rated AAA AA or A that accuratcorporate bonds, rated AAA, AA, or A, that accurat

represent the high quality corporate bond market.

● The bonds are selected from the universe of all hig

bonds based upon several types of characteristics.

● The HQM curve is meant to represent all bonds in 

mark t S b d th t i i d d il bl f 

ket. So a bond that is priced and available for 

included even if its yield appears to be an outlier ryields on other bonds. Such a bond is excluded on

i h h l i ld f h is some reason other than unusual yield for the ex
Data, continued

● The basic type of selected bond is analogous to 

conventional Treasury p 

 coupon issues: a bond that 

fixed semiannual nominal coupon denominated in dollars until maturity, when the principal is return

● Th b d i i h h 30 

The bond set covers maturities up through 30 year

maturities below 1 year filled in by Federal Reservcommercial paper rates. Bid prices are used. Eac

must meet a minimum size threshold in terms of pamount outstanding, currently $250 million.

● The number of corporate bonds that fulfill these The number of corporate bonds that fulfill these

specifications is sufficiently large at all maturities there is no problem in computing the HQM yield c
Results

● The following charts present sample results for th

yi ld f i l d A t 31 2011 th l 

ield curve for a single day, August 31, 2011, the la

August. As noted earlier, the published HQM curAugust is the monthly average.

Characteristics of the bonds included in the August 3

Rating Number of Maturity Range N

(years) 

0 – 1.5 

1.5 – 3 

Bonds

AAA 42

AA 299 1.5 3 

3 – 7 

7 – 15 

AA 299

A 1,345

Commercial paper 12

Total 1,698 15 – 30 
Results, continued

● The first chart shows the estimated forward rate c

A t 31 2011 l tt d th h 30 t it August 31, 2011 plotted through 30 years maturit

projected out through 40 years maturity.

● The chart depicts the forward rate in the five matu

The chart depicts the forward rate in the five matu

ranges, delineated by the vertical lines at the mat1.5, 3, 7, 15, and 30 years.

● The forward curve is fixed at 30 years maturity an

at the long-term forward rate of 6.76 percent, whicgiven by the average of the forward rate in the 15–

maturity range.
HQM FORWARD RATE CURVE

9

Knots: 0 1 5 3 7 15 30

HQM FORWARD RATE CURVE

8/31/2011, Percent

Projected Beyond 30

678

Knots:

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

456

Flattens Out

234

Average 15 to 30 =

Becomes Value at 30 = 6.76Linear

0

5 10 15 20 25 30 35 

01

Maturity, Years

Linear
Results, continued

● The second chart shows the spot yield curve for Au

The second chart shows the spot yield curve for Au

2011.

● The spot curve is projected out through 100 years. 

gently upward from about 30 years maturity onwareaches 6.00 percent at 30 years maturity, and 6.6at 100 years maturity. It is near zero at the earliematurities, reflecting the low short-term rates on 
9 

HQM SPOT YIELD CURVE

8/31/2011, Percent

678456123

Maturity Years

0 10 20 30 40 50 60 70 80 90 10

01

Maturity, Years 
Sample AA Yield Curve

● The last chart adds a sample spot yield curve for A

31 2011 t AA lit t th HQM t 31, 2011 a

t AA quality 

to the HQM spo

t curve. Pl 

e

that this yield curve is a prototype, and it is not an Treasury yield curve.

● The sample AA curve was constructed using the H

methodology modified for AA quality instead of the 

weighted average high quality of the HQM curve weighted average high quality of the HQM curve.

● The AA curve illustrates the fact that the HQM 

methodology is a general-purpose yield curve meth

that can be applied to different kinds of yield curv
9 

SPOT YIELD CURVES

8/31/2011, Percent

678456

HQMAA

123

0 10 20 30 40 50 60 70 80 90 10

01

Maturity, Years 
For More Information

● The High Quality Market (HQM) Corporate Bond Yield 

the Pension Protection Act (PPA) is published by the IR

month and is available on the IRS website. 

● For more details on the mathematics behind the curve 

more documentation in general visit the Office of Econmore documentation in general, visit the Office of Econ

Policy website. Go to www.treasury.gov, under “Resour

choose “Economic Policy,” then choose “Corporate Bond Curve Pa

pers.”

● The HQM methodology has been applied to Treasury in

indexed securities (TIPS). See papers 0601 and 0501 in 

Economic Policy Research Paper Series on the Office of Economic Policy Research Paper Series on the Office of 

Policy website.