- 0 - 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 This Draft: February 15, 2006 Comments Welcome. ## 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
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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.
##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
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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 investor’s 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 investor’s 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.
49). Therefore, Kothari and Shanken’s (2004) and Roll’s (2004) findings could
be sensitive to their sample period.
DATA
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.com 5TIPS 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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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 Roll’s (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. Treasury’s 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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METHOD
Unconditional Spanning Test
Spanning tests are used to investigate the diversification benefits of
TIPS for investors. Spanning tests investigate whether a set of K assets (i.e.,
benchmark assets) span the mean-variance efficient frontier of the N assets
(i.e., test assets) plus the K benchmark assets. Let R1t be a K-vector of the
real returns on the K benchmark assets and R2t be an N-vector of the real
returns on the N test assets. When Rt is the N+K risky asset returns at time
t, the expected returns (µ) and the covariance matrix (V) of the N+K risky
assets are
[ ]1 2 t t tR R R=
[ ] 1
2tE R
µµ
µ
= =
[ ] 11 12
21 22t
V VVar R V
V V
= =
(1)
The projection of N test assets on the space spanned by the K assets is
2 1t t tR Rα β ε= + + (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 , 0Nx N Nx NH α δ= = (3)
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where 1 1N N K K Nδ β ×= − . 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 βs) 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' 2ˆ ˆˆR (2)W R q R R qθ θ χ
− = − Ω − ∼ (4)
where × +
= 2 1
1 0 ... 01 1 ... 1
K
R ; 1 1
ˆˆˆ
K
αθ
β + ×
=
; 2 1
01
q×
=
; 1 1ˆ
K K+ × +Ω is the covariance
matrix of α and β .
In order to improve the finite sample properties, the following F-test is
used:
1,2~2 −−= KTFWF (5)
7According to Kan and Zhou’s (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.
Ferson and Schadt (1996) consider equation (6) a first-order Taylor
series approximation of a general dependence of the parameters on the
instruments. Therefore, the ith row of equation (2) can be written as
2, 1, 0 1 0 1, 1 1 1, 1 1' ( ' )t i i t i i t t i t tR a z a b R z b R e+ + + += + + + + (7)
DeRoon and Nijman (2001) show that spanning under all economic
conditions is equivalent to testing the following restriction on the parameters
estimates in equation (7):
ι=
=
=
=
0
0
1
1
0
1
0
0
i
i K
i
i
a
b
a
b
(8)
Given M instruments and K benchmark assets, the number of restrictions, J,
is equal to MK + K + 1. The F statistic is used to test the validity of the J
restrictions in equation (8):
=F( ) [ ] ( ) ( )1,~
ˆˆˆ 1
−−−−′Ω
′−
−
JKTJFJ
qRRRqR θθ (9)
where R and q are the coefficient restriction matrix in equation (8), θ is the
coefficient estimates in equation (7), and Ω is an estimated covariance matrix
of θ . With normally distributed disturbances, F is exactly distributed as
F(J,T-K-J-1).
Selected Instruments
Cochrane (2001) points out that although investors should use all the
instruments in their information set using less instruments than those
observed by investors does not bias the tests, but it does reduce the power.
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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.
PORTFOLIO DIVERSIFICATION WITH AN INFLATION-INDEXED
BOND
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.
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The results of the unconditional spanning tests are presented in
column 2, Table 2. The results show that for several benchmark portfolios the
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 asset’s 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.
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
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this period, the U.S. equity market was performing poorly.9 TIPS are a new
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 investor’s 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 Poor’s and
Moody’s, 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
9The U.S. equity market had a large downturn during this period: Between August 2000 and September 2002, the S&P 500 lost almost half its value. 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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UK Treasury bonds (i.e., UKbond). Finally, ILG represents an index of all
maturity inflation-linked gilts (available from Barclays Capital).
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 investor’s
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.
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Why the Puzzling TIPS Results?
The results of the present study are different from the results in
Hunter and Simon’s (2005) study. This difference in results may be caused by
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
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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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
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.
CONCLUSION
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
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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 study’s results. The results of the robustness test show that as
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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References
Bekaert, G., and M. Urias. 1996. Diversification, integration and emerging market closed-end funds. Journal of Finance 51: 835–869.
Campbell, J. Y., and R. J. Shiller. 1996. A scorecard for indexed government debt (NBER Working Paper 5587). National Bureau of Economic Research.
Campbell, J. Y., and L. M. Viceira. 1996. Consumption and portfolio decisions when expected returns are time-varying. Quarterly Journal of Economics 114: 433–495.
Campbell, J. Y., and L. M. Viceira. 2001. Strategic asset allocation: Portfolio choice for long-term investors. Oxford: Oxford University Press.
Chopra, V., and W. Ziemba. 1993. The effect of error in means, variances and covariances on optimal portfolio choice. Journal of Portfolio Management Winter: 6–11.
Cochrane, J. H. 1996. A cross-sectional test of an investment-based asset pricing model. Journal of Political Economy 104: 572–621.
Cochrane, J. H. 2001. Asset pricing. Princeton, NJ: Princeton University Press.
DeRoon, F., and T. Nijman. 2001. Testing for mean-variance spanning: A survey. Journal of Empirical Finance 8: 111–155.
Dybvig, P., and S. A. Ross. 1985. Performance measurement using differential information and a security market line. Journal of Finance 40: 383–399.
Ferson, W., and R. W. Schadt. 1996. Measuring fund strategy and performance in changing economic conditions. Journal of Finance 51: 425–462.
Ferson, W., and A. Siegel. 2001. The efficient use of conditioning information in portfolios. Journal of Finance 56: 967–982.
Fischer, S. 1975. The demand for indexed bonds. Journal of Political Economy 83: 509–534.
Hansen, L. P., and S. Richard. 1987. The role of conditioning information in deducing testable restrictions implied by dynamic asset pricing models. Econometrica 55: 587–613.
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Harvey, C. 1989. Time-varying conditional covariances in tests of asset pricing models. Journal of Financial Economics 24: 289–317.
Huberman, G., and S. Kandel. 1987. Mean-variance spanning. Journal of Finance 42: 873–888.
Hunter, D., and D. Simon. 2005. Are TIPS the ‘‘real’’ deal? A conditional assessment of their role in a nominal portfolio. Journal of Banking and Finance 29: 347–368.
Jaganathan, R., and Z. Wang. 2002. Empirical evaluation of asset pricing models: A comparison of SDF and beta methods. Journal of Finance 57: 2337–2367.
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Kan, R., and G. Zhou. 2001. Tests of mean-variance spanning. SSRN Working paper (http://papers.ssrn.com/sol3/papers.cfm?abstract_id=231522)
Kothari, S. P., and J. Shanken. 2004. Asset allocation with inflation-protected bonds. Financial Analysts Journal Jan/Feb: 54–70.
Newey, W. K., and K. D. West. 1987. Hypothesis testing with efficient method of moments estimation. International Economic Review 26: 777–787.
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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 assets include 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. Dev TIPS 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
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 Conditional Benchmark
FWald p(FWald) FWald p(FWald) Panel A: Benchmark portfolio contains any two asset classes
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 0.43 UKbond 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
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 of benchmark 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 Conditional Benchmark
FWald p(FWald) FWald p(FWald) Panel A: Benchmark portfolio contains any two assets
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