Diversification Benefits of TIPS

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Diversification Benefits of Treasury Inflation

Protected Securities: An Empirical Puzzle##

Abdullah Mamun

Department of Finance and Management Science

University of Saskatchewan, Saskatoon

25 Campus Drive Saskatoon, Canada SK S7N 5A7.

Email:[email protected]

Phone: (306) 966-1862

Fax: (306) 966-2515

Nuttawat Visaltanachoti*

Department of Commerce

Massey University

Private Bag 102 904, NSMC,

Auckland, New Zealand.E-mail: [email protected]

Phone: +64-9-414 0800 ext. 9460

Fax: +64-9-441-8177

Abstract

This paper empirically tests the benefits of Treasury inflation protected securities

(TIPS) for investors. This study examines whether TIPS enhance the risk return

characteristics of an investors portfolio. The results of conditional spanning tests

show that adding TIPS to any combined portfolio of stocks, Treasury bonds,

Treasury bills, corporate bonds, and real estate provides investors with

diversification benefits. This paper also shows that United Kingdom (UK) inflation-

linked gilts (ILGs) enhance the risk return characteristics of an investors portfolio.

These findings hold in different economic and inflationary environments, and they

confirm the prediction of economic theory that indexed bonds are important for

investors who are vulnerable to inflation.

Key words: Treasury inflation protected securities (TIPS), portfolio diversification

JEL Classification: G11, G12

## Acknowledgements: We are grateful to Charles Corrado, Lawrence Rose, Henk Berkman, Ben

Jacobsen, and Greg Bauer for their valuable comments. We especially wish to thank Raymond Kan for

his helpful comments about spanning test methods. The article has also benefited from participants in

workshops at University of Saskatchewan, Massey University, University of Auckland, Thammasart

University, Bank of Canada, and Reserve Bank of New Zealand and at the 2005 Financial Management

International Annual Meeting.*Corresponding author: Nuttawat Visaltanachoti. Private Bag 102904, Department of Commerce,

Massey University, Auckland, New Zealand. Phone: +64 9 4140800 ext 9460. Fax: +64 9 4418177.

Email address: [email protected]

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mailto:[email protected]:[email protected]


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Diversification Benefits of Treasury Inflation Protected Securities:

An Empirical Puzzle

Abstract

This paper empirically tests the benefits of Treasury inflation protected

securities (TIPS) for investors. This study examines whether TIPS enhance

the risk return characteristics of an investors portfolio. The results of

conditional spanning tests show that adding TIPS to any combined portfolio

of stocks, Treasury bonds, Treasury bills, corporate bonds, and real estate

provides investors with diversification benefits. This paper also shows that

United Kingdom (UK) inflation-linked gilts (ILGs) enhance the risk return

characteristics of an investors portfolio. These findings hold in different

economic and inflationary environments, and they confirm the prediction of

economic theory that indexed bonds are important for investors who are

vulnerable to inflation.

Key words: Treasury inflation protected securities (TIPS), portfolio

diversification

JEL Classification: G11, G12

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1. Introduction

Theoretically, inflation-indexed bonds are important for long-term

investors who are vulnerable to inflation. Fischer (1975) shows that the

introduction of an indexed bond enables investors to guard against the risk of

inflation, which previously could not be fully hedged. Roll (1996) claims that

Treasury inflation protected securities (TIPS) are the least risky of all assets

because they are immune to both default and inflation risks. Campbell and

Viceira (2001) show that in the presence of a significant inflation risk

nominal bonds are risky and that they are not good substitutes for indexed

bonds. However, empirically indexed bonds (e.g., TIPS) may not deliver the

benefits predicted by theory (Hunter and Simon, 2005).

This study makes two contributions to the literature. First, the results

show that the theoretical prediction about the benefits of indexed bonds to

investors empirically holds for TIPS. In addition, the results show that for

any combination of assets, including stocks, nominal Treasury bills, nominal

Treasury bonds, corporate bonds, and real estate, in a benchmark portfolio

TIPS provide statistically significant diversification benefits. Second, the

findings based on the United Kingdom (UK) inflation-linked gilts (ILGs) data

from January 1981 to August 2005 show that the diversification benefits hold

for other indexed bonds over different sample periods in different inflationary

and economic environments.

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Hunter and Simon (2005) show that TIPS do not enhance the mean-

variance efficiency of a diversified portfolio. This contradiction between the

theory and empirical evidence warrants attention because equity, nominal

bonds, and bills do not provide a hedge against inflation. Therefore, indexed

bonds should enhance the risk return characteristics of a long-term investors

portfolio. The results of the present study suggest that Hunter and Simons

conclusion that TIPS do not enhance the mean-variance efficiency of a

diversified portfolio may be premature.

The results of this study are in line with the findings of Kothari and

Shanken (2004) and Roll (2004), who investigate asset allocation among

stocks, indexed bonds, Treasury bonds, and a riskless asset using a mean-

variance framework. In addition to historical TIPS data, Kothari and

Shanken (2004) also construct a hypothetical index bond from historical

yields on conventional Treasury bonds and an inflation-forecasting model.

Using the hypothetical indexed bond, they investigate asset allocation among

stocks, indexed bonds, Treasury bonds, and a riskless asset. Both studies

conclude that an economically significant weight should be assigned to TIPS

in an optimal portfolio. It is well known that an optimal portfolio choice

under mean-variance analysis is highly sensitive to the expected mean value

of asset returns1; According to Roll (2004), historical mean returns,

particularly ones computed over such short sample periods such as seven

years, are completely unreliable as estimates of future expected returns (p.

1See Chopra and Ziemba (1993).

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49). Therefore, Kothari and Shankens (2004) and Rolls (2004) findings could

be sensitive to their sample period.

2. Data2.1 Data Source

Six classes of assets are used in the present study: equity, Treasury

bills, Treasury bonds, corporate bonds, real estate, and indexed bonds.2 The

return of the S&P 500 composite index is used as a proxy for equity returns.

The return from the Merrill Lynch U.S. Treasury bond index is used as a

proxy for nominal Treasury bonds returns. The return from the Merrill Lynch

3-month Treasury bill index is used as a proxy for Treasury bill returns. The

return from the Merrill Lynch corporate master bond index is used as a proxy

for corporate bonds returns, and the return from the National Association of

Real Estate Investment Trust index is used as a proxy for real estate

returns.3 The TIPS return is calculated from an index of all maturity TIPS

available from Barclay Capital.4 The sample period starts in February 1997

(when the data are first available5) and ends in August 2005.

2On December 31, 2004, the relative weight of the six classes of assets were as follows: real estate was

1.8%, TIPS were 1.7%, Treasury bonds were 14%, Treasury bills were 4.4%, corporate bonds were

11.6%, and equity was 66.6%.

3Although real estate is the smallest in terms of market value compared to other asset classes, it is

included in the present study because of its inflation-hedging capability. Two main sources of historical

real estate data used in the empirical literature are CREFs and REITs. Ibbotson and Siegel (1984)

argue that as a result of appraisal smoothing and imperfect marketability it is necessary to be careful

about directly comparing measured real estate return (i.e., CREFs return) with those of other assets.

Therefore, REITs are used to represent real estate in the present analysis.4See http://www.barcap.com5TIPS are designed to offer investors protection against inflation. TIPS pay a semi-annual fixed real

coupon rate on an inflation-adjusted principal. The principal is adjusted on a daily basis using the U.S.

consumer price index of all urban customers (i.e., CPI-U, set at a lag of 3 months). TIPS were first

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2.2 Real Monthly Return and Its Characteristics

Table 1 presents the descriptive statistics and assets return correlation

for real monthly returns for all assets used in this study. Panel A, Table 1

presents descriptive statistics from March 1997 to August 2005. Monthly

inflation is calculated from CPI-U data available from the Federal Reserve

website.6 TIPS returns are exceptionally large during this period compared to

the returns of other asset classes, except for the REIT returns. This

exceptionally high return for TIPS is also noticed by Roll (2004). TIPS have a

higher volatility than Treasury bills, Treasury bonds, and corporate bonds,

but they have a lower volatility than REIT and equity.

Panel B, Table 1 shows the correlation among the real monthly returns

of TIPS, Treasury bills, Treasury bonds, corporate bonds, real estate, and the

S&P 500 equity index. The correlation between TIPS and all other asset

classes, except equity, is positive. Roll (2004) reports a negative correlation

between TIPS and equity indices and a positive correlation between TIPS and

Treasury bills and Treasury bonds in daily returns. As in Rolls (2004) study,

the results of the present study show a high correlation between TIPS and

Treasury bonds.

issued in 1997. Since then, the U.S. Treasurys inflation-linked bond market has grown to more than

$281.04 billion as of December 2004. Market capitalization for TIPS is still small compared to nominal

Treasury bonds. The U.S. Treasury has, till the end of December 2004, issued 16 inflation-linked bonds

with different issues sizes. The Treasury issues TIPS with three different maturities (i.e., 5 years, 10

years, and 30 years).6See http://www.federalreserve.gov

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3. Method

3.1 Unconditional Spanning Test

Spanning tests are used to investigate the diversification benefits of

TIPS for investors. Spanning tests investigate whether a set ofKassets (i.e.,

benchmark assets) span the mean-variance efficient frontier of the Nassets

(i.e., test assets) plus the Kbenchmark assets. Let R1t be a K-vector of the

real returns on the Kbenchmark assets and R2t be an N-vector of the real

returns on the Ntest assets. When Rt is the N+Krisky asset returns at time

t, the expected returns () and the covariance matrix (V) of the N+Krisky

assets are

[ ]1 2t tR R R= t

[ ] 12

tE R

= =

[ ] 11 1221 22

tV VVar R V V V

= =

t

(1)

The projection ofNtest assets on the space spanned by the Kassets is

2 1t tR R = + +

0

(2)

Huberman and Kandel (1987) provide the necessary, sufficient conditions for

spanning in terms of a restriction on the coefficient of regression (2):

0 1 1: 0 ,Nx N Nx N H = = (3)

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where 1 1N N K K N = . Therefore, a spanning test is a joint test of a constant

that is equal to 0 and the sum of the coefficients (i.e., sum of s) that is equal

to 1. The null hypothesis determines whether the benchmark assets span the

return space of the benchmark assets and test assets. The rejection of the

null hypothesis indicates that the inclusion of the test assets can shift the

efficient frontier of the benchmark assets upward.

Equation (2) is estimated using a traditional regression method with

the heteroskedasticity and autocorrelation consistent (HAC) covariance

matrix suggested by Newey and West (1987).7 The null hypothesis is tested

in equation (3) using the Wald coefficient restriction test8:

( )' 1

' RW R q R R q

= 2(2) (4)

where ; +

=

2 1

1 0 ... 0

1 1 ... 1K

R

1 1

K

+

=

;

2 1

0

1q

=

; 1 1 K K+ + is the covariance

matrix of and .

In order to improve the finite sample properties, the following F-test is

used:

1,2~2 = KTFWF (5)

7According to Kan and Zhous (1999) simulation, tests based on the regression method have better size

and power properties in small sample tests than tests based on the stochastic discount factor (SDF)

approach. Jaganathan and Wang (2002) argue that Kan and Zhou implicitly assume that the expected

mean of factors is known, and they show that the two approaches deliver similar results when they use

the same moments.8Kan and Zhou (2001) show a closed form of this test statistics.

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3.2 Conditional Spanning Test

Minimum-variance efficiency can be examined using conditioning

information. Hansen and Richard (1987) show that an unconditionally

efficient strategy with respect to information must be conditionally efficient,

but the reverse is not true. Dybvig and Ross (1985) point out that superior

performance arises because of superior information. For an uninformed

observer, an investment strategy with superior information can appear to be

either inside or outside of the mean-variance efficient frontier.

Ferson and Siegel (2001) explain the relationship between conditional

and unconditional efficiency with respect to conditioning information in

terms of optimal utility functions. They show that an unconditionally efficient

portfolio maximizes the conditional expectation of a quadratic utility

function. As a result, an unconditionally efficient portfolio must be a

conditionally mean-variance-efficient portfolio because investors with a

quadratic utility function choose a mean-variance-efficient portfolio.

However, investors with a different utility function (e.g., exponential

function) choose a conditional mean-variance-efficient portfolio that is not

unconditionally efficient with respect to information. Therefore, conditional

efficiency does not imply unconditional efficiency with respect to information.

As a result, the conditional spanning approach is likely to be more

appropriate.

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There are several ways to incorporate conditioning information into a

spanning test. Bekaert and Urias (1996) and Cochrane (1996) incorporate

conditioning information as scaled payoffs and estimate the model as an

unconditional model. This is an intuitive way of incorporating conditional

information; however, as indicated by DeRoon and Nijman (2001), the scaled

return rapidly increases the dimension of the estimation and testing problem.

Alternatively, conditional information can be specified in the form of

predictability in a regression framework. In these forms of conditional

testing, it is assumed that the expected returns are linear in the conditional

variables and conditionally homoskedastic. Campbell and Viceira (1996) and

Harvey (1989), among others, have used this approach to incorporate

conditional information. A later development of the conditional spanning test

(see Ferson and Schadt, 1996; Shanken, 1990) considers the regression

coefficients that are linear in the conditional variables. DeRoon and Nijman

(2001) show that incorporating conditional information by assuming the

expected returns are linear in the conditional variables is a special case of the

approach assuming regression coefficients are linear in the conditional

variables.

Following Shanken (1990) and Ferson and Schadt (1996), it is assumed

that and of equation (2) are linear functions of the instruments:

0

0 1

'

'

i i t i

i i t i

a z a

b z b

1

= +

= + (6)

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Four instruments are used in the conditional model presented in this article.

They are the lagged value of yield curve slope between a Treasury bond and

Treasury bill, the lagged yield spread between a Treasury bond and TIPS,

and the lagged return of TIPS, and the lagged return of the S&P 500

composite. As mentioned by Hunter and Simon (2005), the yield curve slope

captures the business cycle and monetary policy expectation, and the yield

spread allows for a risk-adjusted expected inflation rate and periodic trade-off

of returns, inflation risk, and liquidity. In addition, lagged TIPS and S&P 500

composite returns capture a lead-lag relationship between stocks and TIPS

returns.

4. Portfolio Diversification with an Inflation-Indexed Bond

4.1 The Case of TIPS

In the present study, investors pursue a buy-and-hold strategy with a

monthly holding period. Their benchmark portfolios contain any combination

of two to five asset classes (i.e., stocks, nominal Treasury bills, nominal

Treasury bonds, corporate bonds, and real estate). The results of the

spanning tests are presented in four groups. In panels A, B, and C, Table 2,

investors hold a portfolio with any two, three, or four assets, respectively. In

panel D, Table 2, it is assumed that investors hold a diversified portfolio that

includes all five assets classes.

The results of the unconditional spanning tests are presented in

column 2, Table 2. The results show that for several benchmark portfolios the

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spanning hypothesis cannot be rejected. In all these cases, the benchmark

portfolios include Treasury bonds, corporate bonds, or both these assets. This

result is expected given the high correlation between TIPS and Treasury

bonds and TIPS and corporate bonds (see Table 1, panel B).

The conditional spanning test is not only intuitive, but it is also an

appropriate method for the present study. Investors and fund managers not

only make asset allocation decisions based on an assets risk return

characteristics, they also take into account the condition of the economy and

financial market.

The results of the conditional spanning test conducted in this study are

presented in column 3, Table 2. The results show that investors experience

statistically significant diversification benefits for any diversified portfolio.

Therefore, these results imply that during the sample period investors would

have benefited from including TIPS in their diversified portfolios.

4.2 The Case of Inflation-Linked Gilts

It may not be possible to generalize the role of TIPS in portfolio

diversification from the results in the present study for several reasons. The

empirical test of TIPS is conducted using a very short sample period. In

addition, the majority of TIPS were auctioned in late 1999 or later. During

this period, the U.S. equity market was performing poorly.9 TIPS are a new

9TheU.S. equity market had a large downturn during this period: Between August 2000 andSeptember 2002, the S&P 500 lost almost half its value.

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class of asset, and there are uncertainties related to investing in TIPS, which

may affect their return. Finally, inflation was low (i.e., 0.20% monthly) and

stable (i.e., standard deviation was only 0.21% a month) during this period,

which may also influence an investors asset allocation.

ILGs were used to test the robustness of the results presented in the

previous section. Before the introduction of TIPS in the United States and

OATi in France, ILGs constituted 80% of all inflation-linked bonds.

Currently, they constitute approximately 31%10 of all inflation-linked bonds.

Similar to TIPS, ILGs are rated AAA and Aaa by Standard and Poors and

Moodys, respectively. ILGs are linked to the UK retail price index (RPI). In

addition, data for ILGs is available from January 1981 to the present. During

the sample period, the UK inflation rate was higher and more volatile than

the U.S. inflation rate.

Benchmark assets for the robustness test include equity, government

bonds, and government bills.11 With the UK sample, return on the all-share

composite index (FTA) is used as a proxy for equity return. The return on a 3-

month UK government bill is used as a proxy for UK government bills (i.e.,

UKbill), and the Barclays Capital sterling bond index is used as a proxy for

UK Treasury bonds (i.e., UKbond). Finally, ILG represents an index of all

maturity inflation-linked gilts (available from Barclays Capital).

10See Barclay Capital.11Corporate bonds are excluded from the sample because they represent less then 1% compared to

other asset classes, and real estate is excluded from the sample because reliable REIT data would

restrict the sample period to 12 years.

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Table 3 presents the descriptive statistics for asset returns from June

1981 to August 2005. Mean real return on ILGs is lower than all other asset

classes. ILGs are also less volatile than UKbonds and FTA during the sample

period. As with TIPS, ILG returns are positively correlated to short-term

government bills and government bonds; however, unlike TIPS and the S&P

500, the results do not reveal a negative correlation between ILGs and FTA.

Results from the unconditional and conditional spanning tests are

presented in Table 4. It is assumed that investors can hold any two of three

asset classes (i.e., stocks, government bills, and government bonds) or all

three together. The results of the unconditional spanning test (presented in

column 2) show that ILGs do not improve the risk return characteristics of a

portfolio when an investor holds equity and government bills, but they do

show that ILGs enhance the risk return characteristics of an investors

portfolio for all other combinations of benchmark portfolios. As with TIPS,

the results of the conditional spanning test (presented in column 3, Table 5)

show that ILGs provide diversification benefits to investors for all

combinations of benchmark portfolios. This result supports the previous

findings that show indexed bonds (e.g., TIPS) provide additional benefits to

investors holding diversified portfolios.

4.3 Why the Puzzling TIPS Results?

The results of the present study are different from the results in

Hunter and Simons (2005) study. This difference in results may be caused by

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a difference in the sample period. Table 5 shows that the asset return

characteristics are different in their sample period (i.e., February 1997 to

August 2001) and in the rest of the period ending in August 2005.

Alternatively, the difference in results may be related to a methodological

and/or estimation problem. For example, in their unconditional test, Hunter

and Simon (2005) find that Treasury bonds cannot span the mean-variance

frontier generated by Treasury bonds and TIPS, but they obtain the opposite

result with their conditional test. Similarly, in their unconditional test, they

find that Treasury bills cannot span the mean-variance frontier generated by

Treasury bills and TIPS, but they obtain the opposite result with their

conditional test. These results contradict the theory (Hansen and Richard,

1987) that an unconditionally efficient portfolio must be conditionally

efficient, but the opposite is not true. In other words, the rejection of the

spanning hypothesis by an unconditional method implies the rejection of it by

a conditional method, if all instruments are valid.12

Although nominal bills should not be used as a substitute for indexed

bonds, Hunter and Simon (2005) conclude that Treasury bills may be a

reasonable substitute for inflation-indexed bonds, at least in periods without

any major inflation shocks (p. 366). This conclusion contradicts the findings

of Campbell and Shiller (1996) and Campbell and Viceira (2001). They point

out that it is possible to replicate the return of long-term real bonds with

12There are two reasons for a lower p-value in a conditional test: (1) An asymptotic distribution is a

poor approximation of a finite sample distribution when many instruments are used, and/or (2) the use

of conditioning information variables in a conditional test introduces noise and not information.

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short-term nominal bonds using a rollover strategy, but it is risky because

this strategy is exposed to a variation in the real interest rate.

5. Concluding Remarks

This paper investigates whether TIPS constitute a meaningful new

asset class in the sense that it increases the reward-to-risk ratio when added

to a reasonably well-diversified portfolio. In contrast to Hunter and Simon

(2005), the results of this study show that TIPS provide a diversification

benefit to investors when added to a diversified portfolio. It is assumed that

investors can hold any asset class (i.e., stocks, nominal Treasury bills,

nominal Treasury bonds, corporate bonds, and real estate) and in any

combination in a diversified portfolio. The results of the present study are

consistent with economic theory and in line with the findings of Kothari and

Shanken (2004) and Roll (2004).

Both unconditional and conditional spanning tests are used in this

study. The results of the conditional test show that the spanning hypothesis

cannot be accepted in any scenario. These results imply that TIPS constitute

a meaningful new asset class for investors.

It may not be possible to generalize the diversification benefits of TIPS

found in this study because of the short history of TIPS data and the

underperformance of the equity market during the sample period. Therefore,

ILG data from January 1981 to August 2005 is used to test the robustness of

the present studys results. The results of the robustness test show that as

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with TIPS adding ILGs to any diversified portfolio enhances its risk return

characteristics. In addition, the results of this robustness test empirically

show that the benefits of indexed bonds hold under different economic and

inflationary environments and are consistent with the prediction of economic

theories.

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Table 1

Characteristics of TIPS, Treasury Bills, Treasury Bonds, Corporate

Bonds, and S&P 500 Composite Index Real Return

This Table presents the descriptive statistics and a cross-correlation among the

monthly real returns for six major asset classes in the United States. These assetsinclude Treasury inflation protected securities (TIPS), Treasury bills (Tbills),

Treasury bonds (Tbonds), corporate bonds (Cbonds), real estate equity index (REIT),

and S&P 500 equity index (S&P 500) from March 1997 to August 2005. TIPS is an

index of all maturity TIPS available from Barclays Capital. Tbill is a 3-month

Merrill Lynch U.S. Treasury bill index. Tbond is a Merrill Lynch U.S. Treasury bond

index. Cbond is a Merrill Lynch U.S. corporate bond master index. REIT is a real

estate index from the National Association of Real Estate Investment Trust index.

S&P 500 is the S&P 500 composite index. U.S. inflation is calculated using the

urban consumer price index of U.S. city average (CPI-U).

Panel A: Descriptive Statistics

Mean Median Min Max St. DevTIPS 0.42 0.37 -5.12 4.80 1.47

Tbill 0.10 0.16 -0.50 0.60 0.26

Tbond 0.34 0.39 -4.54 3.00 1.36

Cbond 0.40 0.54 -4.52 3.50 1.38

REIT 0.75 1.47 -16.74 8.54 4.13

S&P 500 0.35 0.90 -15.72 8.70 4.78

Inflation 0.20 0.18 -0.34 0.67 0.21

Panel B: Asset Cross-correlation

TIPS Tbill Tbond Cbond REIT

Tbill 0.03Tbond 0.78 0.24

Cbond 0.73 0.21 0.87

REIT 0.17 -0.12 0.00 0.19

S&P 500 -0.16 0.13 -0.21 0.06 0.32

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Table 2

Spanning Tests of U.S. TIPS

This Table presents the results of the unconditional and conditional spanning tests.

Wald test statistics and the p-values from the spanning tests are reported. Under

the null hypothesis, the mean-variance frontier of benchmark portfolios spans the

frontier of benchmark portfolios and TIPS. The benchmark portfolio includes major

asset classes: broad equity portfolio (S&P 500), Treasury bill (Tbill), Treasury bond

(Tbond), corporate bond (Cbond), and real estate (REIT). The instruments include

slope of the yield curve between a Treasury bond and a Treasury bill, the yield

spread between a Treasury bond and TIPS, the lagged TIPS return, and the lagged

S&P return. Panels A, B, and C present the results when a benchmark portfolio

contains any two, three, or four asset classes, respectively. Panel D presents the

extreme case of a benchmark portfolio that contains all five asset classes.

Unconditional ConditionalBenchmark

FWald p(FWald) FWald p(FWald)

Panel A: Benchmark portfolio contains any two asset classesS&P 500 + Tbill 5.09 0.01 5.33 0.00

S&P 500 + Tbond 1.46 0.24 3.90 0.00

S&P 500 + Cbond 3.11 0.05 4.05 0.00

S&P 500 + REIT 195.47 0.00 91.04 0.00

Tbill + Tbond 12.44 0.00 5.90 0.00

Tbill + Cbond 5.58 0.01 8.69 0.00

Tbill + REIT 2.60 0.08 3.90 0.00

Tbond + Cbond 1.16 0.32 4.64 0.00

Tbond + REIT 0.73 0.49 4.26 0.00

Cbond + REIT 1.66 0.19 2.07 0.02

Panel B: Benchmark portfolio contains any three asset classes

S&P 500 + Tbill + Tbond 12.18 0.00 6.58 0.00

S&P 500 + Tbill + Cbond 5.87 0.00 9.52 0.00

S&P 500 + Tbill + REIT 3.69 0.03 6.54 0.00

S&P 500 + Tbond + Cbond 1.32 0.27 5.06 0.00

S&P 500 + Tbond + REIT 0.84 0.43 3.09 0.00

S&P 500 + Cbond + REIT 2.92 0.06 6.70 0.00

Tbill + Tbond + Cbond 12.09 0.00 7.28 0.00

Tbill + Tbond + REIT 7.43 0.00 6.55 0.00

Tbill + Cbond + REIT 4.49 0.01 10.10 0.00

Tbond + Cbond + REIT 0.65 0.53 3.89 0.00

Panel C: Benchmark portfolio contains any four asset classes

S&P 500 + Tbill + Tbond +Cbond 12.21 0.00 7.74 0.00S&P 500 + Tbill + Tbond +REIT 7.37 0.00 10.25 0.00

S&P 500 + Tbill + Cbond + REIT 4.41 0.01 23.88 0.00

S&P 500 + Tbond + Cbond + REIT 0.83 0.44 3.90 0.00

Tbill + Tbond + Cbond + REIT 7.41 0.00 8.17 0.00

Panel D: Benchmark portfolio includes all five asset classes

S&P 500 + Tbill + Tbond +Cbond + REIT 7.82 0.00 11.40 0.00

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Table 3

Characteristics of Major Asset Classes Real Return in the United

Kingdom

This Table presents the descriptive statistics and a cross-correlation of monthly

overlapping annual real returns for four major asset classes in the United Kingdom.These assets include ILGs, UK Treasury bills (UKbill), UK Treasury bonds

(UKbond), and the all-share composite index (FTA) from June 1981 to August 2005.

ILGs represent an index of all maturity inflation-linked gilts available from Barclays

Capital. UKbill and UKbond are the 3-month returns of a UK Treasury bill and UK

Treasury bond, respectively. UK inflation is calculated using the UK consumer price

index. Panel A presents the descriptive statistics, and panel B presents the cross-

correlation of returns among assets.

Panel A: Descriptive Statistics

Mean Median Min Max Stdev

ILG 0.29 0.30 -5.12 7.91 1.95

UKbill 0.35 0.38 -1.72 1.41 0.43UKbond 0.53 0.63 -7.23 7.08 2.00

FTA 0.74 1.38 -31.13 12.35 4.78

UK Inflation 0.33 0.32 -0.93 3.00 0.44

Panel B: Asset Cross-correlation

ILG UKbill UKbond

UKbill 0.14

UKbond 0.61 0.27

FTA 0.28 0.12 0.30

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Table 4

Spanning Tests of UK ILGs

This Table presents the result of unconditional and conditional spanning tests. Wald

test statistics and the p-values from the spanning tests are reported. Under the null

hypothesis, the mean-variance frontier of benchmark portfolios spans the frontier ofbenchmark portfolios and ILGs. The benchmark portfolio includes major asset

classes: broad equity portfolio (FTA), Treasury bill (UKbill), and Treasury bond

(UKbond). The instruments include slope of the yield curve between a Treasury bond

and a Treasury bill and the yield spread between a Treasury bond and ILG. Panel A

presents the results for a benchmark portfolio that contains any two asset classes,

and panel B presents the results for a benchmark portfolio that contains all three

asset classes.

Unconditional ConditionalBenchmark

FWald p(FWald) FWald p(FWald)

Panel A: Benchmark portfolio contains any two assets

FTSE + UKbill 2.11 0.12 6.17 0.00FTSE + UKbond 22.28 0.00 8.79 0.00

UKbill + UKbond 6.06 0.00 1.94 0.02

Panel B: Benchmark portfolio includes all three assets

FTA + UKbill + UKbond 7.07 0.00 2.35 0.00

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Table 5

US Asset Nominal Returns

This Table shows a monthly mean and standard deviation of nominal asset returns

in the United States in two periods. The first period is from February 1997 to

August 2001, and the second period is from September 2001 to August 2005.

February 1997 to August 2001 September 2001 to August 2005

Mean St.Dev Mean St. Dev

TIPS 0.53 0.77 0.71 2.00

Tbill 0.43 0.07 0.15 0.07

Tbond 0.62 1.06 0.44 1.61

Cbond 0.62 1.20 0.56 1.53

REIT 0.46 3.82 1.50 4.44

S&P 500 0.78 5.17 0.30 4.24

Inflation 0.20 0.18 0.21 0.24

Documents

Diversification Benefits of TIPS