Technical Paper Series

Congressional Budget Office

Washington, D.C.

Using the Treasury Nominal and Inflation-Indexed

Spread to Estimate Expected Long-Run Changes in

the CPI-U under Inflation Uncertainty, Risk

Aversion, and the Probability of Deflation

Angelo Mascaro

Congressional Budget Office

Washington, D.C.

E-mail: angelo@cbo.gov

November 2003

Technical papers in this series are preliminary and are circulated to stimulate

discussion and critical comments. These papers are not subject to CBO’s formal

review and editing processes. The analysis and conclusions expressed in them are

those of the author and should not be interpreted as those of the Congressional

Budget Office. References in publications should be cleared with the author. Papers

in this series can be obtained at www.cbo.gov (select Publications and then

Technical Papers).
Abstract

Yield spreads between rates on Treasury nominal and inflation-indexed securities are

thought to be distorted measures of expected inflation because of various biases.

Three sources of bias are investigated in this paper: risk aversion, inflation

uncertainty, and failure to account for the explicit option on the redemption value of

the Treasury’s inflation-indexed security when there is a probability of deflation.

The analysis produces estimates of all three biases. It finds that significant bias could

arise from the second and third sources, while any bias from the first source appears

negligible.
1 The theoretical basis for the roles of inflation variability and risk aversion were first

demonstrated by Stanley Fischer (1975). The application of options pricing analysis to the

Treasury Inflation Indexed bond’s terminal option is new.

2 See Scott Hein and Jeffery Mercer (2003).

1

Introduction

Yields calculated from prices of Treasury’s inflation-indexed securities provide a

market-based measure of inflation-adjusted, or real, interest rates. That information,

together with yields on Treasury’s conventional, or nominal securities, can be used

to provide prompt information on the bond markets’ consensus of medium-term to

long-term expectations of inflation. In addition, such information can help to

determine how unfolding events are influencing changes in those expectations. In

contrast, surveys of professional forecasters, which are often used to measure

inflation expectations, are available much less promptly and may be less useful for

analyzing the effects on inflation expectations of specific events.

But as a measure of expected inflation, the spread between nominal and indexed

Treasuries could be distorted by inflation uncertainty, risk aversion, and the

probability of deflation.1

 Inflation uncertainty imparts a downward bias to the

measure of expected inflation that varies directly with the degree of inflation

uncertainty. Risk aversion imparts an upward bias that varies directly with the

degree of risk aversion. The probability of deflation matters because of the option

on the redemption value of Treasury’s indexed bond as explained later. That

probability imparts an upward bias to the measure of the real yield on indexed bonds

that varies directly with the probability of deflation; thus, this source imparts a

downward bias to the spread-estimate of expected inflation. A fourth distortion, not

examined here, could arise from the different tax treatment of indexed and nominal

Treasuries. The inflation accrual is taxable when earned so that the indexed-bond

investor must finance tax payments from other sources of income, whereas the

nominal-bond investor can finance the payments directly from interest income. This

distortion has been examined elsewhere, however, and does not appear to be

significant.2

This study develops and applies an easily understandable framework for using

market and econometric information to extract a measure of long-term expected

inflation adjusted for the three potential distortions just described. Briefly, the

adjustments produce an estimate of long-term inflation expectations that is higher

than the unadjusted spread, but, for ten-year Treasuries, still somewhat lower than

that reported in a survey of professional economists. However, longer-term

Treasuries yield expected inflation measures that are very close to the survey

measures, possibly suggesting that survey respondents also might have in mind a

longer horizon than ten years.
3 These estimates of volume and trading are from the Treasury Department (2003).

4 This lag feature of the inflation adjustment stems from the timing of the CPI release and

the decision by the Treasury Department that an investor would always want to be able to

ascertain the value of the inflation adjusted par value at the time of sale of purchase.

5 See Brian Sack and Robert Elsasser (2002). Their estimate, obtained from an informal

survey in early 2002, found, for maturities greater than five years, TIIS spreads of 2/ 32nds

to 6/32nds compared to off-the-run nominal Treasury spreads of .25/32nds to 1/32nd. In

contrast, quotes published daily, for example, in the Wall Street Journal (which gives

eSpeed/Cantor Fitzgerald as source), tend to show TIIS spreads of 1/32nd.

2

B

cV

R

cV

R

cV

R

E Max V

R t

t

t

t

t

t M

t

M

t tM

t

M = +

+

+

+ +

+

+

+

+ ++

+

+ +

+

+ +

+

δ

δ

δ

δ

δ

δ

δ

δ ( )( ) ............... ( )

{, }

11 1 ( )

1

1

1

1 (1)

Treasury’s Inflation-indexed Securities

Treasury inflation-indexed securities (TIIS) have been issued regularly since January

1997. Outstanding volume has risen from about $15 billion in 1997 to about $155

billion by mid-2003 and trading among primary securities dealers has risen from a

daily average of $1 billion in early 1998 to about $3 billion by late 2002.3 Interest

payments are calculated as the product of the semiannual coupon rate that is set at

origination and the inflation adjusted par value. The inflation measure is the

seasonally unadjusted CPI-U (consumer price index, all urban consumers). The

inflation adjustment for any given month augments par value cumulatively as the

ratio of the CPI three months earlier to the reference CPI (the value of the CPI-U

three months prior to the auction, where the indexed value for days within a month

is obtained through daily, linear interpolation between the first and last day of the

month).4 The value for redemption at maturity is the larger of either the original

issue par value or the cumulative, inflation-adjusted par value, a feature that will be

shown to have prominence when investors give more than negligible probability to

the chance of deflation. TIIS are sold at auction in maturities thus far of 10 and 30

years, in multiples of $1,000 original issue par value, with interest payable semiannually. Informal estimates of bid-ask spreads, a measure of trading liquidity, have

tended to be higher for TIIS than for nominal Treasuries.5

 

Basic mechanics of price and yield

The yield-to-maturity calculation for an indexed bond follows from a present value

equation in much the same way as that for a conventional bond with the additions of

mechanisms for inflation adjustment and redemption valuation at maturity. A

present value equation for an indexed security is shown here:
6 The presentation in the text simplifies notation. Since bond interest intervals are

semiannual, the rates of interest and inflation should be understood as semiannual, or as

annual with implicit division by two, and the period numbers, 1,...,M, as semiannual.

3

B

cV

r

cV

r

cV

r

E Max V

R t

t

t

t

t

t

t

M

t tt

M

t

M = +

+

+

+ +

+

+

+

+ + +

+

+ ( )( ) .......... ( )

{, ( ) }

11 1 ( )

1 1

1 δδ δ 1

δ

δ

π

V Vk M tk t

k

+ + t

+

δ =+ = δ ( ) , , ,....., 1 01 π

where: Bt = current (settlement date) price; c=coupon rate at origination; *=fractional

time from settlement date to next coupon payment period; Et = current expectation

of maximum function, Vt+*+k=cumulative inflation adjustment from the settlement

date to the k-th coupon date; Rt

=nominal interest rate, or yield to maturity;

M=maturity period.6

The index bond’s cumulative inflation adjustment from the settlement date to any

future coupon payment period can be written as:

(2)

where Bt is average inflation from settlement date, “t,” to future date, “t+*+k,” and

Vt is the value of the adjustment factor from the original issue date to the current

settlement date. The inflation term for future periods should also be thought of as

the inflation prediction embedded in a nominal bond of the same maturity, under the

assumption that the TIIS and nominal Treasury markets are sufficiently integrated.

Using the inflation adjustment (Equation 2) in the bond price (Equation 1) results in

a price equation involving both an inflation-adjusted rate and a nominal interest rate,

where “rt” is the inflation-adjusted interest rate, or real yield, to remaining maturity:

(3)

This last equation indicates three features in computing the yield to maturity of

inflation-indexed bonds. First, it shows how inflation indexation converts an original

coupon rate into an inflation-adjusted rate through the product term, cVt; this term

scales the original coupon rate “c,” by the cumulative adjustment for inflation.

Second, except for the final term, it shows how indexation converts nominal yield to

maturity Rt into an inflation-adjusted yield to maturity rt. Third, it shows that

computing the inflation-adjusted yield requires solving the maximum function. If

deflation has not occurred and is not expected, the function would be evaluated as its

rightmost term which will exceed unity. In that case, the standard solution based on

the market price, Bt, is the inflation-adjusted yield to maturity, rt. But if a deflation

of sufficient magnitude is expected, the rightmost term could become less than unity

and the function would evaluate as unity. In that case, the terminal value would be

discounted at a nominal rate before solving for rt

. The true value of the maximum
4

function will be shown later to be the value of an option on the inflation-indexed

bond’s value at maturity.

Some Properties of Calculated Yields to Maturity

The six-year history of TIIS has revealed several patterns (see Figures 1 and 2):

• First, the calculated yields to maturity have been lower for TIIS than for nominal

Treasuries of comparable maturities, presumably because medium- to long-term

expected inflation has been positive.

• Second, yields on TIIS display a term-structure similar to that for nominal

Treasuries, suggesting that both types of Treasuries may have comparable

cyclical behavior.

• Third, while yields on TIIS moved in a narrower range than yields on nominal

Treasuries during the financial turmoil of 1997-1998, they moved more in

tandem during the turmoil that set in after early 2000. This could imply close

alignment between the yield on TIIS and the real component of the yield on

nominal Treasuries for the latter period.

• Fourth, as a measure of expected inflation, the yield-spread between nominal

Treasuries and TIIS has tended to be lower than the survey measure reported by

the Federal Reserve Bank of Philadelphia (see Figure 2). That difference

between the survey and bond measures has been much greater for the 10-year

bond measure than for the 30-year bond measure, except for the period near mid2000 when nominal interest rates were at a peak. This closer correspondence of

the survey to the TIIS 30-year measure reflects the fact that the rate on the 30-

year bond has varied much less than that on the 10-year bond over the recent

economic cycle.
5

Figure 1. Treasury Nominal and Inflation-Indexed Rates, February 1997 to August

2003.

Source: Haver Analytics.

Note to Figure 1: The 10-year inflation-indexed Treasury rate consists of the rates for

on-the-run issues (i.e., the most recent issue for each month in the chart).

Figure 2. Yield Spread between Nominal and TIIS vs the Philadelphia Federal Reserve

(quarterly) Survey of 10-year Inflation Forecasts, 1997Q1 to 2003Q3.

Sources: Haver Analytics, Federal Reserve Bank of Philadelphia.

1997 1998 1999 2000 2001 2002 2003

8.0

7.0

6.0

5.0

4.0

3.0

2.0

1.0

0.0

8.0

7.0

6.0

5.0

4.0

3.0

2.0

1.0

0.0

Yield Yield

N om inal Treasury Rates:

30-year

10-year

Inflatio n-Indexed Treasu ry R ates:

30-year

10-year

1997 1998 1999 2000 2001 2002 2003

4.0

3.0

2.0

1.0

0.0

4.0

3.0

2.0

1.0

0.0

10-ye ar CPI-U Fore cas t

(Philade lphia FR B Surve y)

Implie d Fore cas ts from nominal-TIIS s pre ads : 10-ye ar, 30-ye ar
6

Distortions to the Bond-Based Measure of Inflation

The fact that the survey measure of long-term inflation expectations has almost

always been higher than that computed from the spread between nominal and TIIS

yields suggests that simply using the nominal-TIIS spread may give a distorted

estimate of inflation expectations. The presence of distortions due to inflation

uncertainty, risk aversion, and possibly mis-measurement because of the maximum

function explain a portion of the difference between the survey and bond measures

of expected inflation.

The Distortions Due to Inflation Uncertainty and Risk Aversion

These two distortions are examined separately. The distortion due to inflation

uncertainty is examined first. When future inflation is uncertain and investors are risk

neutral, they will tend to pay more for a nominal bond and less for an indexed bond

than if they were risk averse; this, in turn, will cause the yield spread between

nominal and inflation-indexed Treasuries to understate expected inflation. A simple

example illustrates this result.

Consider two investment possibilities for an investor. First, suppose the investor

could sell one unit of purchasing power today, invest the dollar proceeds of P(0) in

a conventional bond that returns nominal purchasing power of P(0)[1+R] tomorrow

and real purchasing power of P(0)[1+R]/P(1), where “R” is the nominal rate, P(0) and

P(1) are today’s and tomorrow’s respective price levels, and the expression effectively

is purchasing-power yield or real return. Alternatively, suppose the investor could

directly invest one unit of purchasing power today in an inflation-indexed bond that

returns inflation protected purchasing power of [1+r] tomorrow. Given the two

possibilities, a risk neutral investor could demand a real rate of “r” that equates

purchasing power from the inflation-protected bond to the expected purchasing power

of the conventional bond. If so, then,

(4)

where EP(1)=P(0)(1+EB), defines expected inflation. The inequality follows from

Jensen’s inequality, which implies that E{1/(1+B)}>{1/(1+EB)} (see Figure 3).

As a result, (1+R)/(1+ r) <(1+ EB), which implies that uncertainty about inflation

causes the spread between the nominal and inflation-indexed rates to be less than

expected inflation.

( ){ ( )}( ) { }( ) ( { }) ( )

( 1 )

1

1

1

1

1 1

1

0

1

0

1

0

1

+ =

+

=+ >+ =

+

+

r E

P R

P

R PE

P

R P

E P

R

Eπ
7

EU

R

U E

R

( ) ( { })

1

1

1

1

+

+

<

+

π π +

 

Figure 3. Jensen’s Inequality for E(1/(1+B)) > 1/(1+EB)

To examine the distortion due to risk aversion, I first exploit the definition of risk

aversion. This leads directly to an equation with an adjustment factor that is needed

to correct the nominal-TIIS spread for this distortion. It is then shown that the

specification of the adjustment factor in that equation will differ according to

assumptions about investor behavior. 

When investors are risk averse instead of risk neutral, they exhibit diminishing,

incremental satisfaction from each increment to resources available for consumption

and are willing to pay to avoid risky outcomes. As before, they can hold a nominal

bond offering risky purchasing power in the future or an index bond offering certain

purchasing power in the future. (This is a simplification. As seen later, it does not

mean investors will hold only one type of bond.) To avoid risk, they will pay an

additional amount for the inflation-indexed bond compared to the risk-neutral case,

and receive a smaller, safe return. The difference in the return on the inflationprotected bond under risk aversion versus risk neutrality is accounted for by the risk

premium. With risk aversion, the expected satisfaction, or utility, from the real return

on the risky bond is less than the satisfaction from the expected real return on the

risky bond as shown here (see Figure 4).

(5)

One way to eliminate risk could be for the investor to give up an amount such that the

sure satisfaction from the safe return is the same as the expected satisfaction from the

risky return. The risk premium, denoted by D, would then be as defined in

proportional form in Equation 6:
7 That is, if inflation is normally distributed with dispersion, FB, then:

Ee-B

.(1+EB)

-1(1+FB

2

). For additional discussion, see A.G. Malliaris and W.A. Brock (1982).

8

1

1

1

+ = 1

+

+ r E( − R

π )( ) ρ

(6)

A graphical representation of the risk premium is shown in Figure 4:

Figure 4. Risk Aversion and the Risk Premium 

With risk aversion and the risk premium, the real return on the indexed bond can then

be viewed as equivalent to the safe return, or:

(7)

The interpretation of Equation 7 is that under risk aversion the indexed bond will

command a higher price and lower real yield than under risk neutrality, and this

makes the estimate of expected inflation based on the spread between TIIS and

nominal Treasury rates higher than otherwise. Thus, in contrast to the downwardly

biased estimate for expected inflation because of inflation uncertainty, risk aversion

creates an upward bias.

The two sources of bias, inflation uncertainty and risk aversion, can be written out

directly from Equation 7 if inflation follows a normal process.7

 In that case,

U r UE

R

EU

R

( ) ( { }{ }) ( ) 1

1

1

1

1

1

+ =

+

+ − = +

π + ρπ
9

()( )

1

1

1

1

1 2

+

+

= +

+

+

















R

r

Eπ

ρ

σ π

 

( )() ( )

( ) 1

1

1

1

1

2

+ =

+

+

+

+

E

R

r

π

σ

ρ

π 

(8)

where, FB, is the standard deviation of the inflation process.

If Equation 8 is solved for the spread, using the approximation that (1-D) . 1/(1+D),

then the net bias will be seen to depend on the relative strengths of the biases from

inflation uncertainty and risk aversion as shown here:

. (9a)

Rewriting Equation 9a to solve for expected inflation shows that expected inflation

will exceed, equal, or fall short of the spread depending on the relative strengths of

the inflation-uncertainty and risk-aversion biases. 

(9b)

Obviously, based on Equation 9b, a time-series analysis using observations on the

spread between the nominal Treasury and TIIS yields has two unobserved (though

not unobservable) components to reckon with before extracting an estimate of

expected inflation, EB. Estimates of risk aversion and inflation uncertainty are

therefore required. Estimates of the inflation uncertainty parameter are described in

the next section since they also play a key role in the option on the terminal value of

TIIS.

 

Having exploited the definition of risk aversion to derive an equation for adjusting the

nominal-TIIS spread still leaves open the specification of the adjustment factor. Any

specification of the premium for risk aversion, D, will depend on the utility function

(as suggested by Equation 6) and the assumptions about investor behavior. For

utility, it is convenient to assume constant relative risk aversion. For investor

behavior, assuming the investor only attempts to equalize the level of satisfaction

between the risky and inflation-protected bond leads to an expeditious specification

that follows directly from the definition of risk aversion and requires only the

variance of the TIIS rate of interest (as suggested by Equation 7). If the utility

function has constant relative risk aversion, the risk premium has the form (see

Appendix 1):

1

1

1

1 1 2 + =

+

+

r + −

R

E ( )( )( )

π

σ ρ π
8 I am indebted to Albert Metz of CBO for suggesting this alternative expression for the

risk premium and for providing key insights for the derivation of equation 10b.

10

ρ σγ γ

γ = = −

1 −

2

2 1

1

1 r , () when U C C (10a)

Alternatively, we could adopt the assumption that investors behave according to the

consumption-capital asset pricing model. In that framework, a risk-averse investor

takes into account the correlation between inflation and consumption, choosing the

portfolio of nominal and inflation-indexed bonds that provides the maximum amount

of current and future satisfaction from consumption. Taking this approach results in

an alternative expression for the risk premium as shown here (see Appendix 1)8:

ρ (10b)

π

γ

∗ + +

−

=− +

+

> ==





 

 ( ) (, ) ,  '( )

'( ) 1

1

1

0 1 1 R Cov m wherem

U C

U C

C

C

t

t

t

t

Estimates of the utility parameter “(” are available from the finance literature and, for

the Equation 10a form of risk aversion, estimates of the variance of the real rate can

readily be obtained from data on TIIS yields. For example, using on-the run TIIS

yields, we could select a 12- or 24-month moving-standard deviation of monthly yield

levels (see Figure 5). Various estimates of “(” are found in the literature. Those

reported in Prescott (1996) range between 1 and 2, but other estimates have ranged

even higher such as 50 or beyond. For the alternative form of risk aversion shown in

Equation 10b, estimates of the covariance expression are similarly computable.

Figure 5. Level and Moving-Standard Deviation of on-the-run, 10-year TIIS Yields,

January 2000 to August 2003.

For both measures of the risk aversion factor, however, the computed estimates turn

out to be quite small. For example, using the Equation 10a definition and an average

2000 2001 2002 2003

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.65

0.55

0.45

0.35

0.25

0.15

0.05

Yield

Standard

Deviation

TIIS Y ield

Standard Deviation: 24-month

12-month
9 It is well known from the literature on the equity premium puzzle that abnormally high

values of ( are needed to explain the spread between returns on equity and Treasury bills.

On this see, Rajnish Mehra and Edward C. Prescott (February 2003).

11

estimate of the standard deviation of about 0.4 percentage points, the risk-aversion

factor ranges from 1.2x10-5 for (=1.5, to 40x10-5 for (=50. For Equation 10b, the

same range of values of (, and the estimate of the covariance term over the period

since the early 1980s, the estimated range for the risk-aversion factor again is quite

small, from 5.6 x 10-6 to about 2.0 x 10-4. Such small factors suggest that risk aversion

plays little role in distorting the nominal TIIS spread.9

Deflation and Distortion of the Indexed-Yield to Maturity

Finally, the third source of bias to be examined is the upward bias to the calculated

yield on the indexed bond in the presence of a positive probability of deflation. As

already indicated, published yields on TIIS impose the assumption that the value of

the maximum function in Equation 3 is its last argument which, in turn, implies an

assumption of no probability of deflation over the remaining maturity of the bond.

The expected value of the maximum is examined to determine the consequence of this

assumption.

The expected value of the maximum function is the value of an option on the indexed

bond’s principal at maturity. In real terms, the option’s benefit to holders is the

promise of an increase in real purchasing power when there is deflation, the same

benefit received by holders of conventional Treasury securities. In contrast, with

inflation, the purchasing power of principal at maturity is preserved for TIIS while it

is eroded for conventional Treasuries. In nominal terms, as a result of the option,

deflation is not allowed to erode nominal principal for TIIS below original par value,

thus promising holders at maturity no less than they would receive from a

conventional Treasury at maturity.

By itself , the maximum function in Equation 3 is the option’s “intrinsic value.” That

value will depend on the accumulated inflation adjustment that already has occurred

since the bond was issued, Vt

, and the critical value of deflation, B*, that would erode

those accumulated adjustments over the period remaining to maturity, (* + M).

Taking these factors together into the composite term, Vt

(1+ B*)* + M, determines the

“strike price” of the option. Because the strike price has a value of unity, the critical

rate of deflation can be determined immediately for any TIIS, as Vt

-1/(* + M) -1. For

example, greater accumulated adjustment from past inflation requires even greater

deflation before the guaranteed minimum becomes effective. A shorter remaining

maturity has the same interpretation (see Figure 6). Critical values for currently

outstanding TIIS with original maturities of 10 years are shown in Table 1.
12

Figure 6. Option-Value of the Indexed Bond’s Principal at Maturity

Table 1. Inflation Indexed Bonds: Accumulated Inflation Adjustments, Critical

Deflation Rates, and Spreads to Nominal Treasuries.

Bond’s 

Maturity 

15-Jan

2007

15-Jan

2008

15-Jan

2009

15-Jan

2010

15-Jan

2011

15-Jan

2012

15-Jul

2012

InflationAdjustment

Factor, Vt

1.16 1.14 1.12 1.09 1.06 1.04 1.02

Critical 

Deflation 

Factor, B*

-3.59 -2.52 -1.87 -1.25 -0.68 -0.39 -0.24

Spread to 

Nominal

Treasury

1.20 1.26 1.36 1.41 1.48 1.58 1.71

Note: Author’s calculations using bond ask prices for May 27, 2003 as published in The Wall

Street Journal, May 28, 2003.

While the option’s intrinsic value can be determined without reference to uncertainty,

finding the expected value of the option requires additional information about the

stochastic process governing inflation as measured by the consumer price index.

Using the same assumption as before, that the inflation rate follows a normal process

(which is equivalent to assuming that the price level, and thus the index adjustment,

Vt, follows a log-normal process), the option’s expected value, or the expected value

of the maximum function, will additionally depend on the volatility of inflation

forecasts over the time remaining until maturity. The greater the volatility and the

longer is remaining maturity, the greater will be the option’s expected value. An

illustration of the option’s value is shown by the curved line in Figure 6.
10 Congressional Budget Office (July 2000).

13

B

cV

r

V

r

N

L

R

N

L t

t

t

j

j

j M

t

t

M w

v

v

t

M y

v

= ++

+

+ +

+ − +

=

=

∑ + + () () ( ) ( ) [ ( )] , 1 1

1

1

1 0

δδ δ σ

σ

σ

Using standard methods in pricing options, the resulting equation for the inflationindexed bond, inclusive of the expected value of the embedded option is shown here

(see Appendix 2 for derivation):

(11)

where:

Nw(..), is the cumulative normal distribution with mean, -Fv, and unit variance;

Ny(..), is cumulative standard normal (zero mean and unit variance);

Fv

2

 = FB

2

M;

L = Etln(VT) = ln(Vt) + [EtB - 1⁄2(FB)

2

]M = ln(Vt) + (Rt - rt)M - 1⁄2(FB)

2

M.

According to Equation 11, given price Bt

, the calculated yield to maturity involves

additional terms in rt

 that appear in the two cumulative normal distributions. The first

of these two distributions, Nw(..), gives the probability that inflation will be higher

than the critical value of deflation, B*, shown in Figure 5; the second distribution, 1-

Ny(..), gives the probability that deflation outcomes will be at or below the critical

value, B*. As deflation becomes less and less probable the weights of the two

distributions shift increasingly in favor of the term, Vt

/(1+rt)

*+M.

Because investors are aware of the contingent nature of payment at maturity, they

presumably price the securities and produce some consensus likelihood for each of

the two states of inflation and deflation. The existence of a positive deflation

probability implies, according to Figure 5 and Equation 11, that the real rate could be

much lower than that reported in published calculations.

Estimates of the bias due to ignoring the option on the bond’s redemption value at

maturity require an estimate of the standard deviation, or volatility, of inflation. One

way to estimate volatility is to use the standard deviation of semiannual inflation, the

same frequency at which interest payments are made until the bond reaches maturity.

Using this measure, the standard deviation is estimated to be about 0.016. A second

way, using monthly inflation in the CPIU, where inflation is calculated as 12 times

the logarithmic first difference, finds the estimated standard deviation over the period

since 1951 (annualized), is higher at about 0.04. Over 60-month spans, however, the

annualized, monthly standard deviation has varied from just below 0.02 to just above

0.05 (see Figure 7). A third way is to use the errors in CBO’s 2-year ahead forecast

of the CPI which have an annualized standard deviation of 0.008.10 A fourth way, not
14

explored here, would be to base volatility on the estimate of standard deviations taken

from econometric price equations.

Figure 7. CPIU(NS), 60-Month Moving Standard Deviation of Annualized Monthly

Logarithmic Change, February 1956 to August 2003.

Sources: Haver Analytics and Author’s calculations.

An illustration of the potential bias to the real rate has been constructed, based on

Equation 11 and using, as alternative values of volatility, multiples of a standard

deviation equal to 0.016. Table 2 shows the results using market data in late May 2003

when deflation concerns were prevalent. For a given maturity, the bias is measured as

the difference between the yield ignoring volatility (volatility = 0 x standard deviation)

and that including volatility (volatility = (1 or 2 ) x standard deviation)

Table 2. Illustrative Estimates of TIIS Rate Adjusted for the Option at Maturity

volatility:

multiples

 of

Std. Dev.

Inflation-adjusted yields per maturity using multiples of standard deviation in eq. 11

(Calculations use ask prices as of May 27, 2003)

------------------------------------Maturity of TIIS Issue----------------------------------------

15-Jan-07 15-Jan-08 15-Jan-09 15-Jan-10 15-Jan-11 15-Jan-12 15-Jul-12

0 0.698 0.984 1.254 1.445 1.597 1.674 1.611

1 0.698 0.984 1.253 1.439 1.575 1.648 1.589

2 0.530 0.741 0.991 1.125 1.202 1.294 1.259

Source: Author’s calculations using bond price data from The Wall Street Journal,

Wednesday, May 28, 2003 (p. B4).

It appears from Table 2 that a greater distortion to real-rate estimates occurs for TIIS

with longer remaining maturities and for larger estimates of the volatility of inflation.

Both results are consistent with the theory of options pricing. For bonds with longer

maturities, their options have longer lives and offer greater expected value to the

1956 1961 1966 1971 1976 1981 1986 1991 1996 2001

0 .0 6

0 .0 5

0 .0 4

0 .0 3

0 .0 2

0 .0 1

0 .0 0

0.06

0.05

0.04

0.03

0.02

0.01

0.00

S tandard D eviation S tandard D eviation
11 Arguments for treating the parameters as variable can be found in Robert C. Merton

(1990). 

15

holder for any given level of volatility; with greater volatility, a wider range of

outcomes is possible, which offers another source of value to the holder. Bonds with

longer maturities also have less accumulation from past inflation and therefore require

less deflation in the future as indicated in Table 1.

The results shown in Table 2 offer a range of estimates for adjusting published TIIS

yields under the current low inflation environment. They suggest that the distortion

could range from being negligibly small, as in the case on the TIIS maturing in January

2007 using estimated volatility = .016, to being possibly as large as 35 basis points, as

in the case of the TIIS maturing in July 2012 and using estimated volatility = .032.

Combined estimate of bias to expected inflation

A framework that combines the three separate sources of bias to the nominal-TIIS

spread estimate of long-term inflation expectations is Equation 12. The adjusted

estimate of expected inflation consists of three adjustments to the spread between the

nominal rate and the observed rate on TIIS: an upward adjustment to offset the effect

of ignoring the option on terminal value; an upward adjustment to offset the effect of

inflation uncertainty; and a downward adjustment to offset the effect of risk aversion.

( $) , (12) ( )

( )

( )

( )

1

1

1

1

1

2

1

2

2 + =

+

+ ∗

+

+

π

σ

γσ

R π

r r

 

where r* represents the option-adjusted yield on TIIS.

Supplying values to the parameters needed for computing the adjusted measure of

expected inflation is likely to be as much art as science. It is unlikely the parameters

are constant.11 An example compares the unadjusted spread between nominal

Treasuries and TIIS, the adjusted spread, and the ten-year ahead forecast of the

Consumer price index that is published quarterly by the Federal Reserve Bank of

Philadelphia. For the example, we use the matched sequence of on-the-run nominal

Treasury and TIIS yields with original maturities of ten years, a 15 basis point

adjustment for the terminal option, a five-year moving variance of CPIU inflation, a

24-month moving variance of the observed yield on TIIS, and a utility parameter of 1.5

(see Figure 8).
16

Figure 8. Comparing Measures of 10-year Expected Inflation: Survey average

from Philadelphia Federal Reserve vs. Spread Between Nominal and

Inflation-Index Treasury, and Spread Adjusted for Distortions.

Sources: Philadelphia Federal Reserve, Haver Analytics, Author’s calculations.

According to the example of Figure 8, the adjustments imply a higher level of expected

inflation than is otherwise found in the spread between yields on nominal and inflationindexed Treasury securities. The adjustments for inflation uncertainty and the terminal

option account for virtually all of the increase, as the adjustment for risk aversion is

negligible. Nevertheless, the adjusted measure of expected inflation is still

significantly below the survey measure of long-term inflation. It may be that bondmarket participants, who implicitly determine the expected inflation in the spread, have

taken a greater hold on the possibility of substantially low inflation than have the

economists participating in surveys on the long-term inflation outlook. However, if

the survey measure is compared to the longer-term spread, the difference is greatly

diminished (see Figure 9):

2000 2001 2002 2003

3.3

3.0

2.7

2.4

2.1

1.8

1.5

1.2

3.3

3.0

2.7

2.4

2.1

1.8

1.5

1.2

Expected Inflation Expected Inflation

Philadelphia FRB Survey Spread, Adjusted Unadjusted
17

Figure 9. Comparing Measures: Implied Inflation from 30-year, Original Maturity

Spread Between Nominal and Inflation-Index Treasury, Spread Adjusted

for Distortions, and Survey Average of 10-year Expected Inflation from

Philadelphia Federal Reserve.

Sources: Philadelphia Federal Reserve, Haver Analytics, Author’s calculations.

This last figure reinforces the earlier suggestion that the survey measure may

encompass a longer horizon, one that also may be more consistent with expectations

embedded on the longer-maturity Treasuries. Without the adjustments, however, even

the longer-term spread has been everywhere lower than the survey estimate. But the

adjustments for distortions also tend to reduce the downward bias in the spread

between nominal and inflation-indexed bonds.

2000 2001 2002 2003

3.0

2.8

2.6

2.4

2.2

2.0

1.8

1.6

3.0

2.8

2.6

2.4

2.2

2.0

1.8

1.6

Expected Inflation Expected Inflation

Philadelphia FRB Survey Spread, Adjusted Unadjusted
12 This derivation follows that in Laffont (1989).

U r U Er

U

r r Er

U

r ( r Er ~)( ) ( ) {( ~) ( )} ( ) {( ~ 1 1 ) ( )} 1

1 1

1

2 1

1 1

2

2

2 += + +

+

+ −+ +

+

+ −+

∂

∂

∂

∂

EU r U Er

U

r r ( ~)( ) ( )

1 1

1

1

2

2

2 1

2 += + +

+ +

∂

∂ σ

U Er U Er

U

Er

Er

U

Er ({ }{ }) ( ) Er ( ) ( ) ( )

11 1 ( ) 11

1

1 1

2

2

2

2 2 + −= + −

+

+ +

+ ρ +

∂

∂ ρ

∂

∂ ρ

U Er U Er

U

Er ({ }{ }) ( ) Er( )

11 1 ( ) 1+ −= +− 1

+ ρ +

∂

∂ ρ

Appendix 1. Derivation of Equations 10a and 10b.

This appendix expands upon Equation 6 to show the derivation of the risk premium as

defined in Equations 10a and 10b. First, in the case of Equation 10a, the following

simplified notation is used:12

, with mean, 1+Er, and variance about the mean, . (A1.1)

1

1

1

+

+

= +

R

r

π

~ σ r

2

Using A1.1, rewrite the second and third terms of Equation 6 as shown here:

U E r EU r ( { (A1.2) ~}{ }) ( ~ 11 1 + −= ρ + ).

The derivation proceeds by taking a second order Taylor’s expansion of the terms on

each side of the equality in A1.2 around (1+Er).

Expanding the right-hand side of the equality in A1.2 yields:

(A1.3)

Then, taking the expected value yields:

. (A1.4)

Taking the Taylor expansion of the left-hand side of the equality in A1.2 yields:

(A1.5)

If the risk premium is not too large, this can be simplified further as:

(A1.6)

Finally, equating the right-hand sides of Equations A1.4 and A1.6, eliminating terms

common to both sides of the resulting equality, and solving for the risk premium

yields:
13 See, for example, Sumru Altu— and Pamela Labadie (1994).

19

ρ

∂

∂

∂

∂

σ = − +

+

+

1

2

2

2 2 1

1

1

U

Er

U

Er

Er

r ( )

( )

( ) . (A1.7)

Using the Utility function in the text, the definition of the coefficient of relative risk

aversion as,

γ , (A1.8)

∂

∂

∂

∂

= − +

+

+

2

2 1

1

1

U

Er

U

Er

Er ( )

( )

 ( )

and assuming that Er is sufficiently small gives the approximation that is shown in

Equation 10a of the text.

For Equation 10b, the investor’s decision problem is the following:

( )

max{ ( ) ( )}

, ( )

( )

Uc Uc

subject to c d b a c

R

d rb

t t

t t t tt t t

+

+ += =

+

+

+ +

+

+

β

π

1

1

1

1

1

where “d, b, and a” respectively represent the real values of nominal bonds, inflationindexed bonds, and initial endowment, and $ is time preference. This problem is the

basis for the “consumption capital-asset pricing model,” that has become standard in

the financial economics literature.13 The solution to the problem results in,

E (A1.9) r

R

R U c

U c

t

t

1

1

1

1

1 1

1

1

1

+





 



 = +

+





 



 − +

+





 



 +

π π

( ) cov( , )

'

'

( )

( )

Finally, using the expression,

(A1.10) E

E

( )

( )

1

1

1

1

2

+ =

+

+

π

σ

π

π

 

and substituting for A1.9 yields Equation 10b in the text..
14 This derivation follows that in Stephen Figlewski, William L. Silber, and Marti G.

Subrahmanyam (1990).

20

E V

R R

E V

R

t t

M

t

M

t

M

t t

M

t

M

max{ , ( ) }

() ()

max{ , ( ) }

( )

1 1

1

1

1

01 1

1

+

+ = +

+

+ −

+

+

+ +

+

+

π π δ

δ δ

δ

δ

f V dL e dL e

dV

V T

L

L L

L

L L

T

T

L L ( )

() ()

= = − − − −

1

2

1

2

2

2

2

2 2 2

σ σ

σ σ

Π Π 

C e E V e V f V dL R M

t T

R M

T T

VT

= −= − −+ −+

=

+ ∞

∫ () () max{ , } ( ) ( ) δ δ 01 1

1

Appendix 2. Deriving the Value of the Option on the Redemption Value of TIIS.

This appendix shows how the expected terminal value expression in Equation 3 of the

text is equal to the option on terminal value that appears in Equation 11 of the text.14

It is easiest to begin by first noting the following equivalence from adding and

subtracting one in the numerator of the expression to the left of the equality:

(A2.1)

In this way the maximum function is scaled to zero, instead of one, when B is at or

below the critical value, B*, as pictured in Figure 6 of the text.

We can now use standard methods to evaluate the discounted expected maximum

function as written on the right-hand side of Equation A2.1. In doing so, it is

convenient to proceed in continuous time and afterwards convert back into discrete

time. The first step is to assume the dynamics governing the behavior of the inflation

index, Vt, are log normal and define the following quantities:

L V LEL EL L M (A2.2) ≡ ≡ ≡ −= + T t Lt

− −

ln( ); ( ); ( ) ( ) σ σδ π

2 22

Log normality implies that the probability density of VT has the following form:

(A2.3)

Now we can define the value of the call option on terminal value as:

(A2.4)

The integral begins at one, instead of at -4, because the maximum function has a zero

value up to the value VT = 1 (see Figure 6).
21

y

L L

dy dL M

dV

L L V

T

T

= − = = σ σ σ δ + π;

1 1

e f V dL e e dy e N R M L

T

RM y

LV L

R M

y

L T

L

−+ −+ −

= −

+ ∞

− + = =− − ∫ ∫ () ()

{ }

( ) ( ) [ ( )] δ δ

σ

δ

σ

1

2

1

2

1 Π

Ve N

L V

r

N

L t

RE M

w

L

L

t

M w

L

L

−− +

+ + ≈

+

+ ( )( ) ( ) ( ) ( ) π δ

δ σ

σ

σ

σ

1 

We can now value the two parts of the integral separately, taking first that part

involving the strike price of one. The procedure involves the change-of-variable

method to standardize the normal distribution, using the following definitions:

(A2.5)

Using the definitions from Equation A2.5, the following results obtain for the part of

the integral in Equation A2.4 that involves only the strike price “1”:

(A2.6)

Using the equivalences, N(-Z) = 1 - N(Z), results in this portion of the call having the

value:

e N . (A2.7)

L

R

N R M L

y

L

M y

L

− +

+ ≈ +

( ) ( ) ( ) ( ) δ

δ σ σ

1

1

Because a nearly identical procedure applies to that part of the integral in Equation

A2.4 involving VT, we will not show all the steps here. (They are identical to that

developed in Figlewski, et. al (1990), pp.130-132.) They result in the remaining part

of the call option in Equation A2.4 having the value,

(A2.8)

Substituting terms from Equations A2.7 and A2.8 into Equation A2.4 and then into

Equation A2.1 results in Equation 11 of the text. 
22

References

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NY: Academic Press, 1994).

Bitsberger, Timothy S., (presentation of U.S. Treasury Deputy Assistant Secretary for

Federal Finance at the Bond Market Association’s Inflation-Linked Securities

Conference, New York, NY, June 26, 2003). Available at:

http://www.treasury.gov/press/releases/js505.htm.

Congressional Budget Office, The Budget and Economic Outlook: An Update (July

2000).

Figlewski, Stephen, William L. Silber, and Marti G. Subrahmanyam, (eds.), Financial

Options: From Theory to Practice (Homewood, Ill.: Business One Irwin,

1990). 

Fischer, Stanley (1975). “The Demand for Indexed Bonds,” Journal of Political

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Hein, Scott, and Jeffery Mercer (2003). “Are TIPS Really Disadvantaged?

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2003). 

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Mass: The MIT Press, 1989).

Malliaris, A.G., and W.A. Brock, Stochastic Methods in Economics and Finance (New

York, NY: Elsevier/North-Holland, Inc., 1982).

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Working Paper 9525 (Cambridge, Mass: National Bureau of Economic

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Merton, Robert C., “On the Mathematic and Economics Assumptions of ContinuousTime Models,” in Merton, Continuous-Time Finance (Cambridge, Mass:

Blackwell Publishers, 1990), Chapter 3, pp. 57-93.

Prescott, Edward C., “Theory Ahead of Business Cycle Measurement,” in Preston J.

Miller (ed.), The Rational Expectations Revolution: Readings from the Front

Line (Cambridge, Mass.: The MIT Press, 1996).

Sack, Brian, and Robert Elsasser, “Treasury Inflation-Indexed Debt: A Review of the

U.S. Experience,” Finance and Economics Discussion Paper 2003-32

(Washington, D.C.; Federal Reserve Board of Governors, June 2002).