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Introduction 

● This presentation discusses three yield curves that provide 

discount rates for calculating present values of future cash flows:

● The first curve is the Treasury Nominal Coupon-Issue (TNC) 

Yield Curve, which pertains to Treasury nominal coupon issues. 

● The second is the Treasury Real Coupon-Issue (TRC) Yield Curve 

for Treasury Inflation-Protected Securities (also known as TIPS).

● And the third is the High Quality Market (HQM) Corporate Bond 

Yield Curve, which pertains to U.S. high quality corporate bonds.

● The presentation summarizes information about the three yield 

curves with emphasis on the Treasury curves. More information 

can be found in the references in the last slide.
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Yield Curve Definition 

● A yield curve provides information about a sector of the 

bond market at a point in time. The information includes 

yields on different types of bonds in this sector at various 

maturities.

● For the TNC and TRC curves, the sectors are Treasury 

nominal coupon issues and TIPS, respectively. For the 

HQM curve, the sector is U.S. high quality corporate bonds.
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The Par Yield Curve 

● One set of information provided by yield curve analysis is 

the par yield curve.

● The par yield curve shows for each maturity the yield on a 

security of that maturity that is selling at par (price 

excluding accrued interest equals 100).

● The par yield curve provides a picture of the respective 

market sector for securities with coupons and is used for 

market analysis. Time series of par yields show market 

movements over time.
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The Spot Yield Curve 

● However, for the purpose of discounting future cash flows, 

another set of information provided by yield curve analysis 

is more useful, and that is the spot yield curve.

● The spot yield curve shows for each maturity the yield on a 

security without coupons that provides a single payment at 

that maturity. Such a security can be called a zero coupon 

bond. The yields are called spot rates.

● All the yield curves discussed here are estimated from 

coupon securities; there are no actual zero coupon 

securities in the estimation. Therefore, the spot rates are 

calculated so that they are consistent with the yields on the 

coupon securities, and they can be obtained approximately 

in the market by a portfolio of coupon securities.
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The Spot Yield Curve, continued 

● The spot rates are the appropriate discount rates to be 

used for discounting future cash flows.

● Each future cash flow is discounted by the spot rate whose 

maturity is the same as the future point in time when the 

cash flow occurs to get the present value of the cash flow. 

The present values of a series of cash flows are then added 

to get the total present value of the series.

● Sometimes par yields are used to discount cash flows. This 

approach is flawed because par yields include the effects of 

coupon payments and do not match the cash flows in time.
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The HQM Yield Curve 

● The HQM corporate bond yield curve is produced as mandated by 

the Pension Protection Act of 2006 (PPA). This curve pertains to 

high quality corporate bonds, that is, bonds in the top three 

qualities AAA, AA, and A.

● The curve data are disseminated by IRS and by the Treasury 

Office of Economic Policy, and include spot rates and segment 

rates derived from the spot rates that are used by singleemployer pension plans to discount future liabilities.

● To meet the requirements of the PPA, it was necessary to invent 

a new yield curve methodology at Treasury for the HQM curve. 

The methodology is described below.

● The HQM yield curve is available back through 1984.
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The TNC Yield Curve 

● The HQM methodology was subsequently applied to produce the 

TNC yield curve, which pertains to Treasury nominal coupon 

issues, both notes and bonds.

● The TNC curve includes both on-the-run issues (securities most 

recently issued of each maturity) and older off-the-run issues. 

However, in this presentation the focus is on spot rates from offthe-run issues, which can be derived for all maturities without 

on-the-run distortions and are more suitable for discount rates.

● The initial application of the TNC yield curve was to discount 

future liabilities of various federal agencies, such as pension and 

other postemployment liabilities, for the agencies’ audited 

financial statements and for the annual Financial Report of the 

U.S. Government.

● The TNC curve is available back through 2003, and is being 

extended back another 25 years to the mid 1970s.
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The TRC Yield Curve 

● The HQM and TNC methodology was subsequently used to 

produce the TRC yield curve. The TRC yield curve is the real 

yield curve derived from Treasury Inflation-Protected Securities 

or TIPS, and includes both real notes and real bonds.

● The nominal TNC and real TRC yield curves can be combined to 

derive the Treasury Breakeven Inflation (TBI) Curve, where the 

breakeven inflation rate for any maturity is the inflation rate 

that equates nominal and real Treasury spot rates at that 

maturity in dollar terms. The TBI curve can be used to estimate 

inflationary expectations.

● The TRC yield curve is available back through 2003.
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Yield Curve Data 

● These yield curves each provide several sets of data, 

including spot rates, selected par yields, and forward rates. 

The TNC curve provides off-the-run and on-the-run data.

● The curves are calculated late in the day on each business 

day, and the results are disseminated each month. The 

spot rate data include monthly averages of daily spot rates, 

as well as end of month spot rates from the TNC and TRC 

curves.

● Spot rates are available for each maturity at half-year 

intervals starting at 1⁄2 year up through 100 years, for a 

total of 200 maturities. Rates beyond 30 years maturity 

are projected as described below.
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Yield Curve Data, continued 

● All spot rates are semiannually compounded, following 

market convention. However, some applications for 

discounting may choose to ignore the semiannual 

compounding and apply the rates as if they were annual.

● Off-the-run TNC and TRC spot rates can be interpreted as 

the risk-free social rates of time preference. This means 

that social preference is indifferent between future cash 

flows and their present values discounted by the TNC spot 

rates in the case of nominal flows or TRC spot rates for 

flows in real terms.
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Yield Curves for December 31, 2015 

● The next chart contains the par yield curve for this day.

● There are: 278 Treasury nominal coupon issues in the TNC 

dataset for this day including 6 on-the-run issues; 37 TIPS in the 

TRC dataset; and 1,600 AAA, AA, or A rated securities plus 9 AA 

commercial paper rates in the HQM yield curve dataset.

● For this day, all yields are close to the respective curves.

● TIPS yields are negative at the earliest maturities.
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TNC Yield Curve

TRC Yield Curve

TNC and TRC PAR YIELD CURVES

12/31/2015, Percent

Maturity

TNC On-The-Run Yields

TNC Off-The-Run Yields

TRC Yields
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Yield Curves for December 31, 2015, continued 

● Slide 16 contains TNC, TRC, and HQM spot yield curves for this 

day plus the TBI breakeven curve which is derived from the TNC 

and TRC spot curves. All curves are projected out to 100 years 

maturity. 

● The HQM spot rate is 5.00 percent at 30 years maturity, 

compared with 3.25 percent and 1.35 percent for the TNC and 

TRC spot rates, respectively. The TBI rate is 1.89 percent at 30 

years maturity.

● The HQM rate exceeds the TNC rate largely because of corporate 

bond default risk, and the TNC rate exceeds the TRC rate with 

positive inflation.
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● The spot yield curves rise gradually throughout the projection 

range from 30 years maturity through 100 years maturity, which 

is typical. At 100 years maturity, the HQM spot rate is 5.25 

percent, the TNC rate is 3.60 percent, the TRC rate is 1.55 

percent, and the TBI rate is 2.04 percent.

● The real TRC spot yield curve is negative at the earliest 

maturities. Discounting with negative spot rates actually 

increases present values.

Yield Curves for December 31, 2015, continued 
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SPOT YIELD CURVES

12/31/2015, Percent

Maturity

HQM

TNC

TRC

TBI
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The methodology used for the yield curves contains features 

and capabilities that do not appear in other yield curve 

approaches: 

Methodology Features 

● The methodology makes use of established bond market 

characteristics to generate a stable yield curve that 

captures market movements. 

● The methodology projects yields beyond 30 years.

● The methodology combines regression variables with the 

yield curve.

The following slides discuss details of these features. 
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Methodology Features, continued 

● The requirements of the PPA made necessary the invention 

of this methodology, which is new and more extensive and 

more powerful.

● Because different yield curve methodologies can produce 

significantly different results which affect discounted cash 

flows and everything else for which yield curves are used, 

it was essential that Treasury develop and implement the 

methodology that most accurately captures market 

behavior.
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● In particular, the mathematical form of the yield curve 

must be selected in order to compute the curve.

● Other yield curve approaches typically use mathematical 

forms that don’t have any foundation in bond market 

characteristics. Consequently, differences among yield 

curves given by other approaches with different 

mathematics are arbitrary, in that there is no way to 

choose among them.

● In contrast, the methodology used here is grounded in 

established bond market characteristics, which generate 

the mathematical form of the yield curve and ensure that 

the form is not arbitrary.

Methodology Features, continued 
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The Forward Rate 

● The concept of the forward rate is useful for constructing a 

mathematical form for the yield curve that accords with 

established market characteristics.

● The forward rate is easy to define: for each maturity, consider 

entering into a contract to invest some money at the time of that 

maturity for a small amount of time beyond that maturity. The 

forward rate at that maturity is the future interest rate on this 

investment.

● The forward rate is higher at a given maturity when investors 

who are trading at that maturity are less eager to lend based on 

their assessment of uncertainty and their expectations and 

purposes, while borrowers are more eager to borrow based on 

their perceptions. The forward rate summarizes market views 

for each maturity in a single number.
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Maturity Ranges 

● Moreover, trading in securities tends to divide into maturity 

ranges, such that the trading activity in each range on average 

reflects similar purposes, similar views of risk, and similar 

expectations about securities in that range.

● Because market views can be considered similar for securities in 

the same range, the forward rates in each maturity range can be 

assumed to be related to each other in a simple fashion.

● Consequently, this methodology models the forward rates in each 

maturity range as a smooth (cubic) function, and joins the 

functions together smoothly across ranges (as a cubic spline).
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Maturity Ranges, continued 

● This methodology at the present time uses five maturity ranges, 

delineated by the maturity points 0, 1.5, 3, 7, 15, and 30 years 

maturity. These points provide separate ranges for the critical 

maturities of 2 years, 5 years, the benchmark 10 years, and 30 

years.

● The choice of fixed maturity ranges increases significantly the 

stability of the yield curve estimates over time.

● In addition, the methodology is statistically straightforward to 

estimate. In contrast, certain other yield curve approaches 

generate statistical models that are ill-conditioned and unstable.
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The Long-Term Forward Rate 

● The methodology must include the capability of projecting 

forward rates beyond 30 years maturity so as to obtain yields out 

through 100 years maturity. This is necessary because cash 

flows to be discounted frequently extend beyond 30 years.

● First of all, the long-term forward rate at and beyond 30 years 

maturity is set to a constant, because there are not enough data 

to estimate accurately movements in the forward rate beyond 30 

years maturity.
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The Long-Term Forward Rate, continued 

● Next, this methodology postulates that the long-term forward 

rate is determined by the same factors that affect forward rates 

in the farthest 15- to 30-year maturity range, since that range is 

sufficiently distant in time to reveal underlying long-term 

attitudes toward risk and return.

● Therefore, the constant long-term forward rate from 30 years 

maturity up through 100 years maturity is taken to be the 

average forward rate in the 15- to 30-year maturity range.
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Methodology: Projections 

● Using the long-term forward rate, spot rates can be projected 

beyond 30 years maturity out to 100 years maturity. The 

projected spot rates provide discount rates for long-dated cash 

flows. 

● Moreover, the method of construction of the long-term forward 

rate ensures that yields around 30 years maturity are consistent 

with yields before 30 years maturity and with long-term 

investment returns available in the market. 

● Other yield curve approaches generally stop at 30 years maturity 

and contain no provision for projection and no mechanism to 

ensure that yields around 30 years maturity are consistent with 

earlier yields. Building the projection methodology into the yield 

curve solves these problems.
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Methodology: Regression Variables 

● This methodology also has the special capability of combining 

regression variables with the yield curve. This capability is not 

available in other approaches. The regression variables adjust 

for various features of the market and particular attributes of 

individual securities.

● The TNC and TRC yield curves use regression to measure the 

hump in yields that is sometimes seen around 20 years maturity.

● The TNC curve also uses regression to measure price effects of 

on-the-run securities. As opposed to other approaches, the TNC 

yield curve includes both on- and off-the-run securities in the 

same yield curve and measures on-the-run effects.
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Data 

● The TNC yield curve represents all nominal coupon issues 

and the TRC yield curve represents all TIPS. The dataset 

for each business day is derived from securities that are 

priced on that day. Bid prices are used.

● The HQM yield curve represents all high quality corporate 

bonds that have similar features to Treasury coupon issues.

● The dataset for each day excludes securities with fewer 

than two coupon payments remaining.
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Estimation 

● The forward rates in the maturity ranges are chosen to be 

those rates that give the best statistical fit (by least 

squares) to all the prices in the dataset for that day. The 

par and spot yields are derived from the estimated forward 

rates.

● Regression terms are simultaneously estimated with the 

forward rates.

● Before estimation, the data are weighted by the (square 

root of the) inverse of duration.
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Results for the TNC Yield Curve 

● The following charts show results for the TNC yield curve 

over the period January 2003 through December 2015, for 

a total of 3,247 business days and 156 months.

● The first chart in slide 30 shows the monthly average par 

yield curve over the entire period.

● The chart shows that the par yield curve flattened out 

before the 2007-2009 recession, and that yield spreads 

were large in the financial crisis of 2009 forward. After 

that, spreads decreased a bit.
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Maturity

TNC PAR YIELD CURVE

Monthly, Percent

2003 2005 200720092011

2015

2013
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Results for the TNC Yield Curve, continued 

● The second chart in slide 32 shows monthly average spot 

rates over the period at maturities of 2 years, 5 years, 10 

years, 30 years, and 100 years.

● Again there is a significant bunching of the spot rates 

before the 2007-2009 recession.
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2003200420052006200720082009201020112012201320142015

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5 Years

10 Years

30 Years

100 Years

TNC SPOT RATE AT SELECTED MATURITIES

Monthly, Percent
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Results for the TRC Yield Curve 

● The following chart in slide 34 shows the analogous spot 

rates for the TRC yield curve over the same period January 

2003 through December 2015.

● Real TRC spot rates spiked at the end of 2008, due in part 

to reduced liquidity in the financial crisis.

● Similar to the nominal spot rates, the chart shows that the 

real rates bunched together before the 2007-2009 recession.

● Over the last few years, there have been episodes of 

negative spot rates at lower maturities.
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30 Years

100 Years

TRC SPOT RATE AT SELECTED MATURITIES

Monthly, Percent
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For More Information 

● Data and documentation for the TNC and TRC yield curves are 

available on the Treasury Economic Policy website at:

https://www.treasury.gov/resource-center/economic-policy/corp-bondyield/Pages/TNC-YC.aspx

● The HQM corporate bond yield curve for the PPA is published by 

the IRS each month. Data and documentation are also available 

on the Treasury Office of Economic Policy website at:

https://www.treasury.gov/resource-center/economic-policy/corp-bondyield/Pages/Corp-Yield-Bond-Curve-Papers.aspx

● Previous applications of the methodology presented here to TIPS 

yield curves can be found in papers 0601 and 0501 in the 

Economic Policy Research Paper Series on the Office of Economic 

Policy website.