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Introduction 

● This presentation introduces the Treasury Real CouponIssue (TRC) Yield Curve. The TRC yield curve is derived 

from Treasury Inflation-Protected Securities, which are 

called TIPS.

● The TRC yield curve is a companion to the Treasury 

Nominal Coupon-Issue (TNC) Yield Curve, and uses the 

same methodology with a few variations. This 

methodology is also used for the High Quality Market 

(HQM) Corporate Bond Yield Curve. The TNC and TRC 

curves are combined to derive breakeven inflation rates. 

● This presentation assumes basic familiarity with the TNC 

yield curve; see the last slide for references for the TNC 

and HQM curves.
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TRC Yield Curve Summary 

● The TRC yield curve provides information including yields 

about Treasury real coupon issues. Real coupon issues are 

called Treasury Inflation-Protected Securities, or TIPS, and 

are made up of both real notes and real bonds.

● TIPS are analogous to Treasury nominal coupon issues in 

that they pay semiannual fixed coupons and the principal 

at maturity. The difference is that the coupon and 

principal payments from TIPS are in real terms.

● Payments in real terms are converted to dollars by 

adjusting the payments up or down by changes in the (not 

seasonally adjusted) Consumer Price Index for All Urban 

Consumers (CPI-U). (Except that when the CPI-U falls, 

the principal is not adjusted downward.)
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TRC Yield Curve Summary, continued 

● The nominal TNC yield curve estimates separately the 

yields of on-the-run issues (securities most recently issued 

of each maturity) and older off-the-run issues. In contrast, 

TIPS appear to have minimal on-the-run effects, so no 

distinction is made in the real TRC yield curve between onthe-run and off-the-run.

● Similar to the TNC yield curve, the TRC yield curve 

projects real yields beyond 30 years maturity out through 

100 years maturity. The methodology ensures that the 

projections are consistent with real yields before 30 years 

maturity and with long-term investment returns available 

in the market.
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The Real Par Yield Curve 

● Analogous to the TNC yield curve, the real TRC yield curve 

provides the Treasury par yield curve and spot yield curve 

in real terms.

● The real par yield curve shows for each maturity the real 

yield on a TIPS of that maturity that is selling at par (price 

excluding accrued interest equals 100).

● Analogous to the TNC curve, the TRC curve generates real 

par yields for each maturity at half-year intervals for 

maturities of 1⁄2 year up through 100 years, for a total of 

200 yields. Following market convention, yields are 

semiannually compounded.
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The Real Spot Yield Curve 

● The TRC yield curve also provides the Treasury real spot 

yield curve, which shows for each maturity the yield on a 

Treasury security with a single real payment at that 

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

Treasury bond.

● The real spot rates are inferred from the TRC par yield 

curve. 

● TRC real spot rates can be interpreted as risk-free real 

social rates of time preference. This is analogous to the 

interpretation of off-the-run TNC spot rates as nominal 

social rates of time preference with no default risk but with 

uncertain inflation.
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Yield Curves for End of June 

● The next two charts show the TRC par and spot yield 

curves for June 30, 2015, the last business day in June. 

There are 38 TIPS in the dataset for this day, compared 

with 280 Treasury nominal securities in the TNC curve.

● The par yield curve in slide 8 includes a scatter diagram of 

individual TIPS yields, and shows that the yields are close 

to the curve on this day. The par curve reaches 1.12 

percent near 30 years maturity and has no hump.

● The yields are negative at low maturities. This occurs 

when the price of the TIPS is greater than the sum of real 

coupon and principal payments. Of course, the nominal 

return on the TIPS can still be positive once real payments 

are converted to dollars.
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TRC PAR YIELD CURVE

6/30/2015, Percent

Maturity
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Yield Curves for End of June, continued 

● The real spot yield curve in slide 10 is derived from the real 

par curve and is projected out to 100 years maturity. The 

slide also contains the nominal TNC spot yield curve, 

which is above the real curve. Both curves are very smooth 

on this day.

● The real spot rate rises to 1.17 percent at 30 years 

maturity, and continues to rise throughout the projection 

range beyond 30 years maturity, reaching 1.42 percent at 

100 years maturity.
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TNC AND TRC SPOT YIELD CURVES

6/30/2015, Percent

Maturity

Nominal TNC

Real TRC
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Breakeven Inflation 

● The nominal TNC and real TRC spot yield curves can be 

combined to derive the Treasury Breakeven Inflation (TBI) 

Curve, which provides breakeven inflation rates.

● The breakeven inflation rate (sometimes called inflation 

compensation) for any maturity is the inflation rate that 

equates nominal and real Treasury spot rates at that 

maturity in dollar terms.

● Therefore, the breakeven inflation curve shows possible 

future inflation rates that, if realized, would equate the 

nominal and real spot yield curves in dollars.

● As opposed to yields, breakeven inflation rates are 

calculated as annual rates, which is the usual convention 

for inflation data.
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Breakeven Inflation, continued 

● Breakeven inflation rates are often used as measures of 

inflationary expectations.

● However, breakeven rates may be biased as expectations 

indicators. The most important possible bias stems from 

the fact that nominal yields embody a risk factor stemming 

from the uncertainty of future inflation that real yields do 

not have. This factor acts as a term premium, pushing up 

breakeven rates above expected inflation.

● The chart in slide 13 shows the breakeven inflation curve 

calculated from the spot curves in the previous chart, 

projected out through 100 years. The breakeven inflation 

rate at 30 years maturity is 2.19 percent.
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BREAKEVEN INFLATION RATE CURVE

6/30/2015, Percent

Maturity
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TRC Yield Curve Methodology 

● The methodology for the TRC yield curve is the same as the 

methodology for the TNC yield curve with some variations.

● The following slides summarize this methodology, and 

point out differences from the TNC curve.

● Further information on the TNC methodology is in the 

references at the back.
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TRC Methodology: Regression Variable 

● As in the TNC curve, the TRC yield curve contains the 

regression variable to measure the hump in yields that is 

sometimes seen around 20 years maturity.

● However, in contrast to the TNC curve, the TRC yield 

curve makes no distinctions between on-the-run and offthe-run securities. This is because no noticeable on-therun effects appear in the TIPS market. Moreover, the 

number of TIPS is so small that it would not be possible to 

sort out statistically any on-the-run effects.

● Therefore, the TNC regression variables for on-the-run and 

first off-the-run are omitted from the TRC yield curve.
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TRC Methodology: Projections 

● The TRC methodology projects real par and spot yields 

beyond 30 years maturity out through100 years maturity. 

● The projection methodology is the same as the TNC 

methodology cast in real terms, and ensures that the 

projections are consistent with real yields before 30 years 

maturity and with long-term real returns for maturities 

over 30 years.
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● The TRC yield curve methodology is derived from the same 

basic hypotheses about bond markets that underlie the 

TNC yield curve, translated into real terms.

● As summarized in the next slides, these hypotheses imply 

forward rate functions over selected maturity ranges that 

provide the functional form for estimating the TRC curve.

TRC Methodology: Established Market Views 
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The Real Forward Rate 

● Analogous to the TNC yield curve, the concept of the real 

forward rate is used for setting up the functional form for 

estimating the TRC yield curve.

● The definition of the real forward rate is the same as the 

nominal forward rate in real terms: 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 real forward rate at that 

maturity is the real interest rate on this investment.
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Maturity Ranges 

● Furthermore, similar to nominal securities, trading in 

TIPS 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 TIPS 

in the same range, the real forward rates in each maturity 

range can be assumed to be related to each other.

● Therefore, the TRC methodology models the real forward 

rates in each maturity range as a cubic equation, and joins 

the equations together smoothly across ranges as a cubic 

spline.
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Maturity Ranges, continued 

● The current TRC methodology uses the same five maturity 

ranges as the TNC yield curve, delineated by the maturity 

points 0, 1.5, 3, 7, 15, and 30 years maturity. 

● However, earlier in the sample period there were 

insufficient numbers of TIPS at low maturities, requiring 

that the earlier ranges be combined. All five ranges were 

used since mid 2008. 

● The choice of fixed maturity ranges ensures the stability of 

the TRC yield curve estimates over time, even with small 

samples of TIPS.
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The Long-Term Real Forward Rate 

● Analogous to the TNC yield curve, the long-term real 

forward rate beyond 30 years maturity through 100 years 

maturity is set to a constant, which is taken to be the 

average real forward rate in the 15- to 30-year maturity 

range.

● This approach enables real yields and breakeven inflation 

rates to be projected beyond 30 years maturity even when 

there are few long-term TIPS in the market. The longterm inflation projections provide a sense of market 

expectations for future inflation.
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Estimation 

● The real forward rates in the maturity ranges are 

estimated by least squares, and the real par and spot yields 

are derived from the estimated forward rate curve. The 

regression term is simultaneously estimated with the 

forward rates.

● Before estimation, the TIPS data are weighted by the 

(square root of the) inverse of duration.

● However, in contrast to the TNC yield curve, the TRC 

estimates of the real forward rates are allowed to be 

negative, thereby giving negative yields.

● As in the TNC curve, the TRC methodology is statistically 

straightforward and is well-conditioned and stable.
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Data 

● The TRC yield curve represents all extant TIPS on each 

business day.

● Therefore, each day’s dataset includes all TIPS existing on 

that day, with the exception that TIPS with fewer than two 

coupon payments remaining are excluded.

● The daily TIPS pricing data are bid prices.

● Because TIPS do not have call features, no provision must 

be made for call options.

● The sample period in this presentation covers the period 

January 2003 through June 2015, for a total of 3,121 

business days and 150 months. The TNC yield curve also 

starts in 2003.
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Real Par Yields over Time 

● The chart in slide 25 shows TRC real par yields from 2003 

forward at 2 years, 5 years, 10 years, and 30 years 

maturities.

● Real yields generally fell over this period, and recently 

yields were frequently negative at lower maturities.

● Yields spiked in the financial crisis at the end of 2008, 

reflecting to some extent lower liquidity of TIPS relative to 

nominal Treasuries. Also, at that time there may have 

been the expectation of short-term deflation, which would 

have made short-term nominal Treasuries more desirable.
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TRC PAR YIELDS AT SELECTED MATURITIES

Monthly, Percent
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Breakeven Inflation over Time 

● The following chart in slide 28 shows TBI breakeven 

inflation at selected maturities. The chart includes the 5-

year CPI-U inflation rate for comparison, smoothed by 

logarithmic trend line least squares.

● The chart shows that 5-year inflation started around 2-1/2 

percent or above early in the period and has decelerated to 

around 2 percent recently.

● The lagged breakeven rates as forecasts are reasonably 

close to actual inflation, even picking up the recent 

deceleration. This suggests that the breakeven rates were 

not significantly biased upward and did not contain much 

of a term premium.
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Breakeven Inflation over Time, continued 

● However, the term premium may be larger in a period 

when inflation is not so stable and harder to forecast.

● Also, the breakeven rates in the chart are somewhat higher 

at higher maturities: in the recovery since mid 2009, the 

average difference between the 10-year and 5-year rates is 

41 basis points, and the average difference between the 30-

year and 10-year rates is 23 basis points. This may 

indicate that inflation expectations rose with maturity, but 

these differences may also reflect some term premium.

● The chart in slide 28 shows sharp declines in breakeven 

inflation in the crisis at the end of 2008, corresponding to 

the spikes in real yields seen in the previous chart.
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5-Year CPI-U

BREAKEVEN INFLATION RATES

 AT SELECTED MATURITIES

Monthly, Percent
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Breakeven Inflation over Time, continued 

● The chart in slide 30 shows the monthly TBI breakeven 

inflation over the entire period.

● The chart indicates that breakeven inflation was relatively 

flat and stable over this period and across all maturities.

● The chart depicts the sharp declines in breakeven inflation 

in the crisis at the end of 2008.
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BREAKEVEN INFLATION RATES

Monthly, Percent

2003 2005200720092011

2013

2015
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Forward Breakeven Inflation 

● The chart in slide 32 shows selected forward breakeven 

inflation rates.

● Forward breakeven inflation is the inflation rate over a 

given horizon at a future time.

● The series included in the chart are common measures of 

future inflation expectations.

● All the breakeven rates converge around 2 percent at the 

end, similar to the breakeven rates on slide 28. This may 

suggest that inflation is expected to remain near 2 percent 

for some time.
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30-Year Forward 1-Year

SELECTED FORWARD

BREAKEVEN INFLATION RATES

Monthly, Percent
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10-Year Breakeven Inflation 

● The chart in slide 34 plots two measures of breakeven 

inflation at 10 years maturity.

● The first measure is the TBI rate already shown. The 

second measure is the 10-year Treasury constant maturity 

nominal yield minus real yield. The second measure is 

often used for the 10-year breakeven rate.

● The second measure is less than the TBI rate in each 

month. The average difference between the two is 18 basis 

points throughout the period and 14 basis points since mid 

2009.
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TBI Rate

10-Year Nominal Minus Real Yield

10-YEAR BREAKEVEN INFLATION

Monthly, Percent
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Breakeven Inflation and Forecasts 

● The chart in slide 36 compares breakeven inflation with 

CPI forecasts from the Survey of Professional Forecasters 

at the Federal Reserve Bank of Philadelphia.

● The breakeven rates are close to the forecasts, and there 

seems to be little evidence of a term premium.

● The 10-year breakeven and forecast inflation are 

particularly close: the average difference between 10-year 

breakeven and forecast rates is -6 basis points over the 

entire period and -3 basis points since mid 2009.
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10-Year Breakeven

1-Year Forecast

10-Year Forecast

BREAKEVEN AND FORECAST 

INFLATION RATES

Quarterly, Percent

Forecasts from Survey of Professional Forecasters
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Breakeven Inflation and Forecasts, continued 

● The chart in slide 39 compares breakeven inflation with 

CPI forecasts from the long-range consensus in Blue Chip 

Economic Indicators.

● The chart plots the Blue Chip 5-year and 10-year forecasts, 

with the latter derived from the first and second 5-year 

average forecasts.

● Because these forecasts begin a year ahead, the breakeven 

rates for comparison are extended a year to 6 years and 11 

years.
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Breakeven Inflation and Forecasts, continued 

● The breakeven rates are generally close to the forecasts. 

As in the previous chart, there does not seem to be much 

evidence of a term premium.

● For the 5-year, the breakeven rates fall below the forecast 

rates toward the end of the period. The 10-year breakeven 

rates are a bit above the forecast earlier in the period.

● The 10-year breakeven and forecast inflation are again 

especially close: the average difference between 10-year 

breakeven and forecast rates is 1 basis point over the 

entire period and -3 basis points since mid 2009.
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11-Year Breakeven

5-Year Forecast

10-Year Forecast

BREAKEVEN AND FORECAST 

INFLATION RATES

Semiannual, Percent

Forecasts from Blue Chip Long-Range
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More Information and References 

● The High Quality Market (HQM) Corporate Bond Yield Curve for 

the Pension Protection Act (PPA) is published by the IRS each 

month. Data are also available on the Economic Policy website.

● Data for the Treasury Nominal Coupon-Issue (TNC) Yield Curve 

are published monthly on the Economic Policy website. 

● For more details on the methodology for the TNC and HQM yield 

curves and for the data, visit the Office of Economic Policy 

website. Go to www.treasury.gov, under “Resource Center” 

choose “Economic Policy,” then choose “Treasury Yield Curve” and 

“Corporate Bond Yield Curve Papers.”

● Previous applications of this methodology 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.