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Journal of Banking & Finance 24 (2000) 1433±1455
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After-tax term structures of real interest rates:
Inferences from the UK linked and non-linked
gilt markets
Andrew R. Aziz

a,*

, Eliezer Z. Prisman

b,1

a

b

Algorithmics Inc., 185 Spadina Ave., Toronto, Ont., Canada M5T 2C6
Schulich School of Business, York University, 4700 Keele St., Toronto, Ont., Canada
Received 25 September 1998; accepted 12 July 1999

Abstract
This study estimates the after-tax term structure of real interest rates using the prices
of UK linked and non-linked gilts over the period from 25 January 1986 until 25
October 1993. The impact of di€erential taxation and the existence of ``noise'' in
observed market prices is found to produce a signi®cant impact on the parameter
estimation of term structure models when compared to methods, such as Brown and
Schaefer (Brown, R., Schaefer, S., 1994. Journal of Financial Economics 35, 1±42), that
did not. Two major observations can be made regarding the estimates for spot real
interest rates. Firstly, the volatility of the short-term rate is much lower than that found
by Brown and Schaefer (1994) which provides a better ®t with the predictions of the CIR
single factor model for interest rates. Secondly, consistent with Rumsey (Rumsey, J.,
1993. An impact of the assumptions about taxes on the estimation of the properties of
interest rates. Working Paper), there appears to be some evidence to suggest that single
factor interest rate models produce a better ®t to interest rates on an after-tax basis than
on a before tax basis. Ó 2000 Published by Elsevier Science B.V. All rights reserved.
JEL classi®cation: G10
Keywords: Taxes; Real rates; CIR; Term structure; In¯ation

*

Corresponding author. Tel.: +1-416-217-4113.
E-mail addresses: aaziz@algorithmics.com (A.R. Aziz), eprisman@yorku.ca (E.Z. Prisman)
1
Tel.: +1-416-736-2100 ext: 77948.
0378-4266/00/$ - see front matter Ó 2000 Published by Elsevier Science B.V. All rights reserved.
PII: S 0 3 7 8 - 4 2 6 6 ( 9 9 ) 0 0 0 8 7 - 4

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A.R. Aziz, E.Z. Prisman / Journal of Banking & Finance 24 (2000) 1433±1455

1. Introduction
The behaviour of real interest rates is a core issue for ®nancial economics in
terms of both the dynamics of the short-term rate and the characteristics of the
overall term structure. Empirical investigations of the relationship between
nominal interest rates, real interest rates and expected in¯ation have traditionally been hindered by the fact that, of the three variables, only nominal
interest rates are able to be directly extracted from market prices. Without an
observable measure of real interest rates, it has often been convenient to
characterize this fundamental variable as both constant and exhibiting a ¯at

term structure. This assumption, however, has been disputed in several studies
including Nelson and Schwert (1977), Walsh (1987) and Rose (1988) who
conclude, by various indirect techniques, that real rates do indeed vary over
time. With the introduction of ``real return'' or ``index linked'' gilts (ILGs) in
the UK during the early 1980s, however, there now exists a means of directly
extracting real interest rates from prevailing market prices. Furthermore, the
existence of relatively liquid markets in both linked and non-linked gilts provides a means of examining the relationship between nominal interest rates,
real interest rates and in¯ation.
Although there have been a number of studies which have investigated the
properties of real interest rates by indirect methods, 2 none have attempted to
characterize a zero coupon term structure of after-tax real interest rates in a
manner such as Litzenberger and Rolfo (1984) describe for nominal interest
rates. Under this approach, the complication of di€erential taxation across
investors is addressed through the identi®cation of the marginal tax rate associated with a representative investor. The marginal tax rate of this proxy
investor is determined on the basis of the speci®c after-tax cash¯ows which
price the universe of bonds with minimum ``noise''.
Brown and Schaefer (1994) were the ®rst to estimate the before-tax real-term
structure of interest rates from a cross section of ILGs. Using the approach of
Schaefer (1981), they cast real-term structure estimation as an optimization
problem under the condition that the present value of each ILG cannot exceed

its observed price. Later in the same paper, Brown and Schaefer (1994) use
these observed ILG prices to also estimate the parameters of the CIR model for
the interest rate process.
The approach of Schaefer (1981), however, does not allow for observation
error as it treats observed market prices as the intrinsic or ``true'' prices of the
securities under consideration. Treating observed prices as ``true'' prices in a

2
Studies of this nature include Walsh (1987), Rose (1988) and Evans et al. (1993). Studies which
have directly incorporated the yields of ILGs include Pitman (1991) and Woodward (1990, 1993).

A.R. Aziz, E.Z. Prisman / Journal of Banking & Finance 24 (2000) 1433±1455

1435

market that is actually segmented, subjects the study of Brown and Schaefer
(1994) to systematic bias.
Observed prices are typically used in studies which estimate a linear valuation operator consisting of a set of stochastic discount factors (SDFs) for
equity markets. In one of a series of papers, Hansen and Jagannathan (1994)
apply the diagnostic of asset model pricing errors and recognize that there is a

set of true valuation operators which contain more than one element. Their
paper seeks to obtain some measure of the mispricing produced when a proxy
valuation operator (one not belonging to the set of true operators) is used.
They measure the mispricing by minimizing the di€erence between the proxy
and the set of true valuation operators. However, as the true set is not directly
observable, duality theory is utilized to measure the mispricing in terms of the
security prices and the portfolios (shadow prices) which are directly observable.
A related issue covered in a series of papers by Stutzer (1993, 1994) concerns the multiplicity of stochastic discount factors. In incomplete markets,
or in cases of incomplete information, many sets of SDFs will be possible. In
such cases, the question of how to choose amongst the many SDFs is of signi®cant interest, with Stutzer (1994) advocating minimization of the entropy
function.
The recent literature which has sprung from the papers of Hansen and
Jagannathan (1994, 1993), hinges on the premise that observed prices do not
allow for arbitrage opportunities. As stated by Hansen and Jagannathan
(1994), ``Except possibly when there are arbitrage opportunities present in the
data set used in the empirical investigation, the set of correctly speci®ed discount factors is non-empty and typically large''.
The question of how SDFs can be estimated in incomplete markets where
observed prices do allow for arbitrage has been largely overlooked. This is
undoubtedly due to the fact that this line of research has, thus far, primarily
focused on equity markets. For equity markets it is a very dicult task to prove

that observed prices do, indeed, allow for arbitrage. In order to do so, it would
be necessary to demonstrate that no risk neutral probability (a set of SDFs)
exists that could equate present values to observed prices.
In contrast, for ®xed income markets it is generally quite easy to
demonstrate that observed prices allow for arbitrage opportunities. For
example, consider a market for government bonds. Let A be a matrix of
payo€s if the securities are held to maturity such that aij is the payo€
from bond i in state j. In a complete market, the number of independent
securities must be equal to the number of payment periods provided the
system is consistent. For a typical bond market, however, the number of
payment periods is at least twice that of the number of bonds. Thus, a
typical bond market is incomplete and, assuming the system is consistent,
consists of many valuation operators. Every valuation operator with

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A.R. Aziz, E.Z. Prisman / Journal of Banking & Finance 24 (2000) 1433±1455

d > 0, such that d1 > d2 >


> dn and Ad ˆ P, is theoretically a possible
estimator of the term structure.

Empirically, however, it is typically impossible to ®nd even one viable valuation operator as the system is generally inconsistent due to the fact that
observed prices contain noise. This is true even if the spread between the bid
price, Pb , and the ask price, Pa , is considered. Ronn (1987) concludes that it is
still generally impossible to ®nd a valuation operator, d, even when the constraint is broadened to Pb 6 Ad 6 Pa .
These ®ndings call into question the practice of utilizing observed prices in
order to estimate the parameters of interest rate process models whose very
essence is that of no arbitrage. Brown and Schaefer (1994) estimate the parameters of the no arbitrage CIR model by using observed ILG prices which
have been shown to contain buy and hold arbitrage opportunities. This paper
suggests an improvement over this method of parameter estimation by utilizing
a set of ``true'' prices that do not allow for arbitrage opportunities. It is shown
that the di€erence between the parameters estimated in this paper compared
with that of Brown and Schaefer (1994) is signi®cant for the estimation of
interest rate derivative securities. Interest rate derivatives are of interest because the price sensitivity of these securities to errors in parameter estimation is
signi®cant.
Moreover, as another issue, Brown and Schaefer (1994) ignore tax implications even though the method of Schaefer (1981) implicitly assumes a segmented market. They assume that all ILGs are priced correctly for all tax
brackets which may not be a reasonable assumption for the gilt market. As
pointed out by Woodward (1990), the absence of capital gains taxes on gilts in
the UK will very likely lead to segmented market equilibria with distinct tax
clientele e€ects.
The purpose of this paper may be summarized as follows:

· To generate after-tax real-term structures from a cross section of ILGs using
an alternative estimation procedure to Brown and Schaefer (1994) which allows for observation error and assumes that a marginal tax bracket exists for
which all securities are priced correctly.
· To estimate the parameters of the CIR one factor model for the after-tax
interest rate process with an approach similar to Brown and Schaefer
(1994) but by using the ``true'' prices of ILGs which are implied by the above
term structure estimation. These ``true'' prices do not allow for buy and hold
arbitrage opportunities.
Despite the existence of both linked and non-linked gilts, the relationship
between nominal interest rates and real interest rates cannot be determined
in a straightforward manner. This is due to the non-contemporaneous indexation of in¯ation which, in the case of the ILG, involves a lag of eight
months. The existence of this lag implies that an ILG is e€ectively nonlinked over the eight-month period just prior to its maturity. Suitable real

A.R. Aziz, E.Z. Prisman / Journal of Banking & Finance 24 (2000) 1433±1455

1437

interest rates may still be inferred, however, by adjusting the empirically
estimated real-term structure by a proxy for the in¯ation expected over the
eight-month lag period just prior to ILG maturity. The approach used by

Brown and Schaefer (1994) to correct for the lag is to estimate expected
in¯ation with an ARIMA model. In contrast, this study uses a procedure
whereby in¯ation expectations are extracted directly from the relationship
between the nominal and real after-tax term structures. This method provides greater ¯exibility as it is able to incorporate non-seasonal price shocks
into the estimate of the relationship between nominal and real-term structures.
The outline of the paper is as follows. Section 2 describes the market and the
tax treatment for UK index linked gilts. Section 3 illustrates the construction of
after-tax real and nominal term structures while Section 4 provides estimates
for the parameters of the CIR one factor model. Section 5 provides conclusions.

2. The UK index linked gilt market
Index linked gilts (ILGs) were ®rst issued in the UK in 1981 as part of the
governments anti-in¯ation program. Both the coupon and the principal of
these securities are linked to the UK retail price index (RPI). The basis for
indexation is the RPI for the period eight months prior to gilt issue; two
months lag to enable measurement of the RPI and six months lag to enable
exact calculation of accrued interest based on a known forthcoming coupon
payment. The existence of the lag results in imperfect in¯ation protection except in the case where the in¯ation experienced eight months prior to the ILG
issue is identical to the in¯ation experienced during the last eight months of the
ILGs life. The magnitude of this imperfection diminishes as the holding period

for the security increases.
As of 1993, there had been a total of seventeen unrestricted ILG issues in the
UK (excluding an initial 1981 issue limited to pension funds), beginning in 1982
with maturity dates ranging from 1988 to 2030. ILGs were issued on a consistent basis throughout the 1980s with coupon rates varying from 2% to
4.625%. Volumes have ranged in issue size from 300 million pounds to 1 billion
pounds (see Table 1).
The total volume of ILGs has risen from 8.4% (9.5 billion pounds) of the
overall volume of outstanding government issued gilts at 31 March 1985 to
16.2% (17.5 billion pounds) at 31 March 1990. The liquidity of the market is
relatively thin, however, with a turnover of only 2.9% of the total gilt market
turnover in 1990 (de Kock, 1991). Holdings of ILGs are concentrated within
a small number of investors including, predominantly, pensions funds and

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A.R. Aziz, E.Z. Prisman / Journal of Banking & Finance 24 (2000) 1433±1455

Table 1
UK linked and non-linked marketable debta
Volume (£ Bn)

Turnover (£ Bn)

Linked debt
1985
1990

9.5
17.5

8.4%
16.2%

N/A
2.2

N/A
2.9%

Non-linked
debt
1985
1990

103.7
90.5

91.1%
81.1%

N/A
56.4

N/A
89.1%

a
The table provides volumes and turnovers for linked and non-linked gilts in terms of total sterling
marketable debt. (Source: Bank of England.)

insurance companies. Individual investors are largely con®ned to the short end
of the market.
Since 1986, both linked and unlinked gilts have been exempt from capital
gains taxes with the e€ect that lower coupon securities sold at a discount are
taxed preferentially in comparison to those with higher coupons. In general,
ILGs, whose in¯ation compensation accrues to both coupon and principal,
possess a tax advantage over non-linked gilts, whose in¯ation compensation
accrues only to the coupon which is taxed as ordinary income. Therefore, the
e€ects of di€erential taxation must be considered in any comparison of respective returns and subsequent measurement of expected in¯ation (Woodward, 1990).

3. After-tax term structures
The impact of taxes on the estimation of the term structure of interest rates
has been well documented. Pioneered by McColloch (1975), this line of research has been extended by Schaefer (1982), Jordan (1984) and Litzenberger
and Rolfo (1984) amongst others. 3 The existence of environments where ordinary income and capital gains are taxed di€erentially (even in the case where
capital gains are non-taxed) may make it impossible for investors in di€erent
tax brackets to agree on the relative pricing of bonds. In such a case, the
di€erence between the observed price of a bond and its appropriate present
value may include a tax e€ect over and above simple noise. The implication of
this for the bond market is that a non-segmented equilibrium may be unobtainable due to the existence of buy and hold tax arbitrage opportunities. A
segmented equilibrium characterised by tax clientele e€ects would be possible,
however, if frictions exist to limit arbitrage pro®t. An example of such a fric3

Katz and Prisman (1991) provide a summary of the literature in this ®eld.

A.R. Aziz, E.Z. Prisman / Journal of Banking & Finance 24 (2000) 1433±1455

1439

tion would be a restriction on the ability to sell short. In such an environment,
the resulting equilibrium is segmented in terms of investors only holding bonds
which are priced correctly for their own particular tax bracket.
Brown and Schaefer (1994) were the ®rst to estimate the before-tax real-term
structure of interest rates using cross sections of weekly ILG prices. Using the
approach of Schaefer (1981), they estimate the real-term structure as an optimization problem with the condition that the present value of each ILG cannot
exceed its observed price. In a market characterized by a ban on short sales and
a segmented equilibrium caused by di€erential taxation, this criterion would be
satis®ed by an in®nite number of possible real-term structures.
The speci®c term structure selected by the Schaefer (1981) approach is the
one that maximizes the present value of an arbitrarily chosen stream of cash¯ows ci subject to the constraint that the present value of each of the j outstanding ILGs cannot exceed their market price. This approach can be
formulated as follows:
X
di ci ;
…1†
max
di

i

subject to:
Ad 6 P;
where d and di represent the vector of discount factors and its corresponding
period i element, and where A and P represent the payo€ matrix of ILG
cash¯ows and the vector of market prices, respectively.
The use of this approach for term structure estimation may be problematic,
however. No allowance is made for observation error as it treats observed
prices as ``true prices''. This assumption seems unrealistic given the non-synchronous trading of securities in the bond market. 4 In addition, given that the
choice of the cash¯ow stream in the objective function is arbitrarily chosen
then, as pointed out by Katz and Prisman (1991), so also may be the term
structure determined by the optimization. Consequently, the estimation will be
biased in terms of the speci®c cash¯ow stream chosen and cannot be used to
identify mispriced bonds or the marginal tax rate associated with the representative investor.
In their study, Brown and Schaefer (1994) assume that no signi®cant tax
clientele e€ect exists in the ILG market and, thus, choose to ignore taxes. If this
is indeed the case, then there is no a priori reason to expect that the error
between market price and the present value contains anything beyond simple
noise. Nonetheless, their method actually estimates the term structure based on
the assumption that the prices of bonds only contain positive error terms. It is,
4

This issue is referred to in Dybvig (1989).

1440

A.R. Aziz, E.Z. Prisman / Journal of Banking & Finance 24 (2000) 1433±1455

therefore, not clear why the Schaefer (1981) approach is preferable to well
known term structure estimation approaches that are based upon the minimization of the sum of squared error terms, particularly given that Brown and
Schaefer (1994) later appeal to random price ``observation error'' in their estimation of the CIR model parameters.
As pointed out by Woodward (1990), the absence of capital gains taxes on
gilts since 1986 in the UK will very likely lead to segmented market equilibria
with distinct tax clientele e€ects. Thus, rather than assuming the absence of tax
clientele e€ects, this study utilizes the approach of McColloch (1975) to generate an after-tax term structure for ILGs based on the assumption that a given
marginal tax bracket exists for which all bonds are priced correctly for a
representative investor.
The marginal tax rate of the representative investor is determined as the
speci®c rate which produces the lowest sum of squared errors from amongst
each of the after-tax term structures estimated. This approach is based on the
Litzenberger and Rolfo (1984) study which assumes that, on a particular date,
the representative investors marginal tax rate is the same for all maturities of
gilts. In the UK gilt market, the regression is simpli®ed by the absence of
capital gains taxes which eliminates the need for the instrumental variable
adjustment utilized in McColloch (1975).
The data employed in this study are that of Woodward (1990, 1993) and
includes non-linked and ILG prices as well as the monthly RPI collected over
ninety-four periods from 25 January 1986 until 26 October 1993. Eighty-®ve
non-linked gilts and seventeen ILGs were used throughout the period with
callable and convertible gilts excluded from the sample.
The speci®c methodology used is that of Litzenberger and Rolfo (1984)
whereby the time interval to the longest gilt maturity date is partitioned into k
subintervals, each containing approximately an equal number of payment
dates, by placing k ‡ 1 knot points at t ˆ 0; 1; . . . ; T . Given that no additional
explanatory signi®cance is found beyond the third power, the spot real aftertax (marginal tax rate s) discount factor, zs0;t , for term of length t, thus, takes
the form
2
z0;t
s ˆ 1 ‡ at ‡ bt ‡

k
X
iˆ1

ci ti3 Di …t†;

where if t < tiÿ1 , then
Di …t† ˆ 0
and if t > tiÿ1 , then
Di …t† ˆ 1:
The price of an ILG can be expressed as

…2†

1441

A.R. Aziz, E.Z. Prisman / Journal of Banking & Finance 24 (2000) 1433±1455

Pn;0 ˆ

n
X
j

cs;j z0;j
s ‡ ;

…3†

where cs;j represent the period j real after-tax cash ¯ows and where,  captures
noise and tax e€ects.
The substitution of Eq. (2) into Eq. (3) produces the following regression
equation:
Pn;0 ÿ

n
X
j

cs;j ˆ a0 ‡ a

n
X
j

cs;j tj ‡ b

n
X
j

cs;j tj2 ‡

k
X

ci

iˆ1

n
X
j

3

cs;j …tj ÿ ti † ‡ :
…4†

The discount function is, thus, determined by substituting the resultant regression coecients into Eq. (2).
Regressions are run over a range of marginal tax rates with the rate which
minimizes the sum of squared error terms presumed to be that of the representative investor. For the representative investor, the  term is assumed to be
free of tax e€ects. An identical methodology is utilized to determine nominal
term structures of interest rates. The regression equation corresponding to the
determination of the nominal term structure is as follows:
Pn;0 ÿ

n
X
j

Cs;j ˆ a0 ‡ a

n
X
j

Cs;j tj ‡ b

n
X
j

Cs;j tj2 ‡

k
X
iˆ1

ci

n
X
j

3

Cs;j …tj ÿ ti † ‡ ;
…5†

where in this case, Cs;j represent the period j nominal after-tax cash ¯ows.
Unlike Woodward (1990), however, it is not assumed that the same representative investor is appropriate for both the ILG and the non-linked gilt
markets. Given that government tax laws di€ered in their treatment of linked
and non-linked gilts over this period argues for an approach that explicitly
distinguishes between the respective marginal tax rates of each class of security.
The marginal tax rates of the representative investor calculated for both
nominal gilts and ILGs are provided in Appendix A. The representative investor clearly di€ers across the linked and non-linked markets which further
underscores the importance of considering di€erential after-tax e€ects when
determining the relationship between interest rates and in¯ation. The representative investors marginal tax rate appropriate for the non-linked market
exhibits a declining trend from approximately 20% in 1986 to approximately
10% in 1993 which is consistent (although slightly lower in magnitude) with
Woodward (1990). In contrast, the tax rate appropriate for the ILG market
increases from essentially 0% in 1986 to approximately 10% in 1993.
The method of real-term structure estimation described above ignores the
eight-month lag of in¯ation indexation on ILGs. Without further re®nement,

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A.R. Aziz, E.Z. Prisman / Journal of Banking & Finance 24 (2000) 1433±1455

this approach will provide a true estimation of the real term structure only if
the expectation of in¯ation over the last eight months of the ILGs life is
equal to actual in¯ation eight months prior to ILG issue and that no corresponding in¯ation risk premium exists. If this is not the case, an in¯ation
bias will be inherent in the estimate of real spot rates which diminishes
proportionately with the length of the term. To correct for this bias, a
method of estimating in¯ation expectations for the non-indexed eight-month
period is required.
The approach used by Brown and Schaefer (1994) to correct for this bias is
to estimate expected in¯ation through the use of an ARIMA model which
acomodates the seasonality of in¯ation expectations. In contrast, this study
uses an alternative procedure to correct for the lag by extracting in¯ation expectations directly from the relationship between the estimates of the nominal
and real after-tax term structures. This method provides greater ¯exibility as it
is able to incorporate non-seasonal price shocks into the estimate of in¯ation
expectations.
The continuously compounded real, n month spot rate, k0n , extracted from
the estimated ILG term structure when n P 8 can be expressed as
 
8
…6†
…pnnÿ8 ÿ p^0ÿ8 †
k0n ˆ r0n ‡
n
and when n < 8,
k0n ˆ r0n ‡ …pn0 ÿ p^nÿ8
ÿ8 †;

…7†

r0n

where is the true n period real spot rate and where, p^ and p represent actual
in¯ation experienced prior to ILG issue and expected in¯ation over the last
months of the ILGs life, respectively. Without loss of generality it can be assumed that p also contains the in¯ation risk premium appropriate for the
period in question. For n P 8, the lag bias in the real spot rate estimate is re¯ected by the di€erence between pnnÿ8 and p^0ÿ8 . The magnitude of this bias is
clearly proportional to the term of the spot rate and at long terms to maturity,
becomes negligible. Note that for terms of eight months or less, the real rate
essentially possesses the characteristics of a nominal rate.
The continuously compounded nominal, n month spot rate, K0n , extracted
from the estimated non-linked gilt term structure is free of any lag bias and can
be simply expressed as
K0n ˆ r0n ‡ pn0 :

…8†

Under the assumption that an ``expectations hypothesis'' for the term structure
of in¯ation compensation is valid, the following relationship holds:




n
8
n
n‡8
p ‡
pn‡8 :
…9†
p0 ˆ
n‡8 0
n‡8 n

A.R. Aziz, E.Z. Prisman / Journal of Banking & Finance 24 (2000) 1433±1455

Given the following n ‡ 8 and n month nominal spot rates:




n
8
pn0 ‡
pn‡8 ;
K0n‡8 ˆ r0n‡8 ‡ p0n‡8 ˆ r0n‡8 ‡
n‡8
n‡8 n
K0n ˆ r0n ‡ pn0 ;
and the following n ‡ 8 month real spot rate:


8
…pn‡8
ÿ p^0ÿ8 †;
k0n‡8 ˆ r0n‡8 ‡
n
n‡8

1443

…10†
…11†

…12†

then the true n spot real rate derived from the estimated nominal and real-term
structures can be expressed as


 
n‡8
8 0
…13†
…K0n‡8 ÿ k0n‡8 † ‡
p^ :
r0n ˆ K0n ÿ
n
n ÿ8
This approach estimates expected future in¯ation by using the information
directly contained in the market prices of linked and non-linked gilts. The
previously noted di€erence between the representative investors marginal tax
rate appropriate for the ILG and non-linked gilt markets is critical for the
estimation and underscores the need to consider nominal and real rates on an
after-tax basis.
After-tax real and nominal term structures are estimated over monthly periods from 25 January 1986 until 25 October 1993 and are illustrated in Figs. 1
and 2.
Table 2 contains before- and after-tax mean spot rates and standard deviations over the sample period.
Several observations can be made from the estimates. First, as noted by
Brown and Schaefer (1994), real-term structures are neither ¯at nor constant
over time. The shapes of the real-term structures, while predominantly
``humped'' shaped, can be either upwardly or downwardly sloping. Most of the
volatility, however, is at the short end as spot rates for longer terms become
remarkably constant.
The impact of the eight-month lag for in¯ation indexation is apparent as the
lag adjusted real spot rates are much less variable than the unadjusted rates at
the short end of the term structure. This is not an unexpected result given that
unadjusted spot rates are highly in¯uenced by the in¯ation expected over the
eight-month period just prior to maturity.
A comparison of before-tax and after-tax rates reveals only a small reduction in the means and a negligible change in the volatilities for ILGs due to the
fact that the marginal tax bracket for these securities over much of the period
was 0%. This is not surprising given that ILGs are held predominantly by
pension funds and insurance companies, institutions which are quite likely to

1444

A.R. Aziz, E.Z. Prisman / Journal of Banking & Finance 24 (2000) 1433±1455

Fig. 1. After-tax term structure of nominal interest rates.

Fig. 2. After-tax term structure of real interest rates.

be tax-exempt. Means of after-tax spot rates were much lower for non-linked
gilts with a non-negligible decline in volatility.
Although the January 1986 until December 1989 sample period of the
Brown and Schaefer (1994) study is not identical to that of this study, some

A.R. Aziz, E.Z. Prisman / Journal of Banking & Finance 24 (2000) 1433±1455

1445

Table 2
Zero coupon spot ratesa
Term

Mean (standard deviation)
Nominal

Real (unadj.)

Real (adj.)

B/S

(2.14)
(1.87)
(1.67)
(1.39)
(1.03)
(0.71)

3.32
3.35
3.38
3.49
3.86
3.91

(1.33)
(0.93)
(0.73)
(0.51)
(0.40)
(0.33)

3.09
3.15
3.20
3.41
3.83
3.89

(1.08)
(0.81)
(0.63)
(0.49)
(0.42)
(0.37)

3.56
3.63
3.69
3.78
3.87
3.69

Panel 2. After tax
1
8.15 (2.01)
2
8.12 (1.75)
3
8.14 (1.60)
5
8.21 (1.34)
10
8.41 (1.00)
20
7.94 (0.64)

3.21
3.23
3.25
3.40
3.75
3.77

(1.33)
(0.93)
(0.73)
(0.51)
(0.39)
(0.32)

3.05
3.12
3.15
3.36
3.76
3.76

(1.01)
(0.74)
(0.61)
(0.49)
(0.40)
(0.33)

N/A
N/A
N/A
N/A
N/A
N/A

Panel 1. Before tax
1
9.80
2
9.43
3
9.35
5
9.56
10
9.66
20
8.89

(2.48)
(1.86)
(1.48)
(1.09)
(0.74)
(0.52)

a

The table provides means and standard deviations of before- and after-tax spot rates for both
nominal and real interest rates over monthly periods from 25 January 1986 until 25 October 1993.
Although the sample periods are not identical, the results of Brown and Schaefer (1994) are provided for comparison purposes.

comparisons can still be made. Their results indicate signi®cantly greater volatility at the short end of the term structure which is likely due to the di€erent
approaches used to correct for the lag in indexation. It is also noteworthy that
the Schaefer (1981) approach to term structure estimation is likely to produce
an upward bias in the estimation of spot rates. Brown and Schaefer (1994) spot
rate estimates in Table 2 show some indication of this.

4. CIR model for the real-term structure
Brown and Schaefer (1994) ®t the Cox et al. (1985) term structure model to
market prices of ILGs in order to assess the extent to which the single factor
model captures the main features highlighted by their empirical approach.
They ®nd results that are both consistent and inconsistent with the theoretical
model. On the one hand, the term structures estimated by Brown and Schaefer
(1994) consistently demonstrate the general shapes, the high correlation of
crossectional spot yields, and the low volatility of long-term yields predicted by
the CIR model. On the other hand, intertemporal stability of the CIR parameters is ®rmly rejected.
This study endeavors to estimate CIR model parameters in a manner similar
to Brown and Schaefer (1994) but to determine the theoretical term structure

1446

A.R. Aziz, E.Z. Prisman / Journal of Banking & Finance 24 (2000) 1433±1455

utilizing the ``true'' prices implied by the term structure estimation procedure
in contrast to their approach which utilizes directly observed market prices.
Given that observed prices may not be free from buy and hold arbitrage
opportunities for all investors, then their use cannot be consistent with an
equilibrium model for the term structure of interest rates. As pointed out by
Hull (1993), a model of the term structure which is comfortably within a 1%
error margin for pricing a bond could produce an error as large as 50% for
pricing a derivative security on that bond. In order to adjust for this inconsistency, Brown and Schaefer (1994) appeal to observation error and ®t
the CIR model by minimizing the sum of squared errors between the market
and ``model'' prices. This approach produces a bias, however, if a tax clientele e€ect, such as that demonstrated in the previous section, were to exist
over and above simple noise. In this study, the theoretical after-tax prices are
determined from the term structure estimation approach of this study which,
given the assumption of a representative investor, are free from clientele
e€ects.
As a ®rst step, Table 3 illustrates the correlation of cross-sectional spot rates
for both before- and after-tax real spot rates using the term structure estimation approach of this study.
Consistent with Brown and Schaefer (1994), a fairly high degree of
correlation is found for changes in real spot rates which suggests some promise
for the one factor approach to modelling real interest rates. Note that the
correlation for after-tax real rates is higher than that for before tax rates 5
which is consistent with Rumsey (1993) who ®nds that interest rate models, in
general, provide a better ®t to after-tax interest rates than to before tax interest
rates.
This study estimates the parameters of the CIR square root process in a
manner similar to Brown and Dybvig (1986) and Brown and Schaefer (1994)
by assuming the following process for the instantaneous short-term rate:
p
dr ˆ #…h ÿ r† dt ‡ r r dz;
…14†
where h is the mean, # the mean reversion coecient and r is the volatility.
Given the above process and the arbitrage pricing condition, it can be shown
that the price, P …r; t; T †, of a default free, zero coupon bond must satisfy the
following partial di€erential equation:
1
#…h ÿ r†Pr ‡ Pt ‡ r2 rPrr ÿ rP ˆ krPr ;
2

5

…15†

The after-tax results demonstrate higher correlation than both the before tax results of this
study and the before tax results of Brown and Schaefer (1994) although it must be noted that the
sample periods of the two studies are not identical.

A.R. Aziz, E.Z. Prisman / Journal of Banking & Finance 24 (2000) 1433±1455

1447

Table 3
Correlation of spot real interest ratesa
Spot rate term
2

3

5

10

20

Panel 1. After tax
1
1.00
2
0.99
3
0.95
5
0.77
10
0.67
20
0.51

1

1.00
0.98
0.85
0.64
0.57

1.00
0.93
0.72
0.66

1.00
0.86
0.73

1.00
0.86

1.00

Panel 2. Before tax
1
1.00
2
0.97
3
0.93
5
0.71
10
0.59
20
0.55

1.00
0.97
0.79
0.61
0.59

1.00
0.93
0.72
0.58

1.00
0.85
0.74

1.00
0.87

1.00

a

The table prodvides correlations across monthly before and after tax interest rate changes for spot
rates of varying terms over the period from 25 January 1986 until 25 October 1993.

where k represents the market price of risk. The terminal condition
P …r; T ; T † ˆ 1

…16†

implies that, for any s,
P …r; t; t ‡ s† ˆ A…s†eÿB…s†r ;

…17†

where


2#h=r2
2ce1=2…#‡k‡c†s
;
A…s† ˆ
…# ‡ k ‡ c†…ecs ÿ 1† ‡ 2c


2…ecs ÿ 1†
B…s† ˆ
…# ‡ k ‡ c†…ecs ÿ 1† ‡ 2c

…18†

and
cˆ

q
2
…# ‡ k† ‡ 2r2 :

…19†

The s period zero coupon spot rate de®ned by this process is
r0s ˆ

B…s†r ÿ ln…A…s††
;
s

…20†

2#h
:
#‡k‡c

…21†

where
lim rs
s!1 0

ˆ

1448

A.R. Aziz, E.Z. Prisman / Journal of Banking & Finance 24 (2000) 1433±1455

Given that prices are dependent only on the risk adjusted interest rate process
implies that the parameters to be estimated are, r, r2 , #h and # ‡ k.
The parameter values are estimated in an approach similar to Brown
and Dybvig (1986) and Brown and Schaefer (1994) by solving the following
optimization problem across j ILGs with respect to the risk adjusted CIR
parameters:
X
2
… Pbj ÿ Pj † :
…22†
min
j

This approach di€ers from the above studies in that Pj represents the theoretical price for the CIR model and Pbj represents the ``true'' price determined
from the empirical estimation procedure in the previous section rather than the
observed market price. The ``true'' price represents the observed price less the
observation error.
The CIR after-tax term structure estimates over the sample period are
illustrated in Fig. 3. with mean spot rates and standard deviations provided in
Table 4.
Similar to the conclusions reached by Brown and Schaefer (1994), the CIR
model is quite able to accommodate the general shapes of the observed term
structures. The agreement between the mean and the standard deviations of the
CIR model and the observed term structures for each term is good and is
particularly strong for longer maturities. The understatement of the real rate

Fig. 3. CIR after-tax term structure of real interest rates.

A.R. Aziz, E.Z. Prisman / Journal of Banking & Finance 24 (2000) 1433±1455

1449

Table 4
CIR zero coupon real spot ratesa
Term

CIR

1
2
3
5
10
20

3.09
3.14
3.20
3.40
3.71
3.78

Mean (standard deviation)
Est.
(0.89)
(0.78)
(0.68)
(0.50)
(0.40)
(0.32)

3.21
3.23
3.25
3.37
3.75
3.77

(1.33)
(0.93)
(0.73)
(0.51)
(0.39)
(0.32)

Est. (adj)

B/S CIR

B/S Est.

3.05
3.12
3.15
3.36
3.76
3.76

3.56
3.63
3.69
3.71
3.87
3.69

3.82
3.84
3.85
3.82
3.96
3.77

(1.01)
(0.74)
(0.61)
(0.49)
(0.40)
(0.33)

(2.48)
(1.86)
(1.48)
(1.09)
(0.74)
(0.52)

(2.89)
(2.08)
(1.58)
(1.17)
(0.81)
(0.54)

a
The table provides mean and standard deviations of after tax spot real interest rates based on the
CIR term structure model and compared to the term structure estimates utilized in this study for
the period from January 1996 to October 1993. The results are illustrated alongside those of the
similar before tax interest rate comparison found in Brown and Schaefer (1994) for the period from
January 1984 to December 1989.

standard deviation that the CIR model suggests for the shorter end is signi®cantly less than that found by Brown and Schaefer (1994) which is again likely
due to the di€erent approaches used for measuring in¯ation expectations.
The estimates of the CIR parameters are shown in Table 5. While stability of
these parameters is rejected, there appears to be less variation in the estimates
Table 5
CIR parameter estimatesa
Year

CIR parameters
#h

r2

rst

r1

r20

Panel 1. Before tax
1986
ÿ0:0059
1987
ÿ0:1274
1988
ÿ0:0930
1989
ÿ0:0477
1990
ÿ0:0745
1991
ÿ0:0171
1992
ÿ0:0212
1993
ÿ0:1153

0.00142
0.00110
0.00098
0.00092
0.00098
0.00214
0.00311
0.00310

0.0082
0.0137
0.0095
0.0068
0.0060
0.0042
0.0159
0.0342

0.0268
0.0321
0.0331
0.0271
0.0282
0.0338
0.0327
0.0340

0.0277
0.0333
0.0343
0.0284
0.0296
0.0351
0.0381
0.0362

0.0345
0.0406
0.0409
0.0387
0.0404
0.0426
0.0430
0.0355

Panel 2. After tax
1986
ÿ0:0059
1987
ÿ0:1274
1988
ÿ0:0952
1989
ÿ0:0481
1990
ÿ0:0728
1991
ÿ0:0200
1992
ÿ0:0238
1993
ÿ0:0543

0.00142
0.00110
0.00107
0.00104
0.00109
0.00175
0.00186
0.00040

0.0082
0.0137
0.0095
0.0071
0.0086
0.0042
0.0065
0.0048

0.0268
0.0321
0.0228
0.0260
0.0251
0.0219
0.0226
0.0374

0.0277
0.0333
0.0243
0.0270
0.0261
0.0284
0.0344
0.0381

0.0345
0.0406
0.0392
0.0357
0.0382
0.0376
0.0386
0.0370

#‡k

a
The table provides estimates for the CIR one factor interest rate model for both before and after
tax interest rates over the period from 25 January 1986 until 25 October 1993.

1450

A.R. Aziz, E.Z. Prisman / Journal of Banking & Finance 24 (2000) 1433±1455

for after-tax interest rates over time than for before tax rates in this study or in
that of Brown and Schaefer (1994). This is again consistent with Rumsey
(1993) and provides further support for ®tting one factor interest rate models
to after-tax interest rates.
The appropriate choice and calibration of term structure models is critical to
the accurate pricing of derivative securities. The valuation of these securities is
highly sensitive to the estimation procedures utilized. As a framework for
comparison, the parameter estimates of Brown and Schaefer (1994) are compared with the results of this paper in order to value a ®fteen year zero coupon
bond and hypothetical ®ve year call options on that bond. Three European call
options are considered with strike prices of $40, $50 and $60, respectively. The
modi®ed binomial approach of Tian (1993) is used to construct a lattice of the
CIR interest rate process. Under this approach, a path-independent binomial
tree is created from a transformation of the assumed interest rate process to
one which has constant volatility. In general, for an interest rate process described as
dr ˆ l…r; t† dt ‡ r…r; t† dz

…23†

the transformed constant volatility process can be de®ned as
dH ˆ q…r; t† dt ‡ Hr r dz;

…24†

where
1
q…r; t† ˆ Ht ‡ …l ÿ k r†Hr ‡ r2 Hrr :
2

…25†

For the CIR interest rate process an appropriate transformation is simply
Hˆ

p
r;

…26†

which implies a transformed stochastic process of
dH ˆ q dt ‡ v dz;

…27†

where
1
a1
q ˆ ‰#…h ÿ r† ÿ k rrŠHr ‡ r2 rHrr ˆ ÿ a2 H;
H
2
r
vˆ ;
2
and where

…28†

1451

A.R. Aziz, E.Z. Prisman / Journal of Banking & Finance 24 (2000) 1433±1455
Table 6
Bond and call values for CIR interest rate processa
Year

Strike

Estimated values

Brown and Schaefer

Bond

Call

Bond

Di€.
(%)

Call

Di€.
(%)

14.90
6.77
0.19

47.90

0.96

15.31
7.16
0.36

2.68
5.45
47.22

Tian modi®ed binomial lattice
1986
40
47.44
50
60
1987

40
50
60

31.22

2.05
0.01
0.00

30.93

0.94

0.93
0.00
0.00

120.43
N/A
N/A

1988

40
50
60

43.27

8.51
0.38
0.05

42.72

1.29

8.08
0.21
0.01

5.32
80.95
400.00

1989

40
50
60

52.28

17.71
9.01
0.41

52.95

1.27

18.95
10.43
1.52

6.54
13.61
73.03

a

The table provides values for a ®fteen year zero coupon bond and ®ve year European call options
of varying strike price. The estimation is based on the Tian (1993) modi®ed binomial lattice model
used to construct the CIR real interest rate process comparing Brown and Schaefer (1994) parameter estimates to those of this paper. The proportional di€erence between values estimated by
each approach is also indicated.

4#h ÿ r2
;
8

#‡k r
:
a2 ˆ
2
a1 ˆ

…29†

Tian (1993) ®nds the speed of convergence to the closed form solution to be
within approximately 0.05% for a zero coupon bond with only a ten-step lattice. 6
The values of the bond and call options estimated by both the Brown and
Schaefer (1994) approach and that of this paper are shown in Table 6.
The results demonstrate the magnitude of error inherent in a misspeci®cation of parameter estimates for an interest rate process. Although there is a fair
degree of consistency across both sets of CIR parameter estimates for the
values of a bond, the results obtained for derivatives can vary widely. The
proportional di€erences in the estimated values for the zero coupon bond range
from 0.97±1.27% compared with a range of 2.68±400.00% for values of the call
6

This is strictly true only when a1 > 1 (Tian, 1993).

1452

A.R. Aziz, E.Z. Prisman / Journal of Banking & Finance 24 (2000) 1433±1455

option. This clearly illustrates the magnitude of the pricing errors possible in
interest rate derivative securities which stem from the misspeci®cation of CIR
parameter estimates. Hence, if observed market prices do not preclude buy and
hold arbitrage opportunities, then to use these to estimate the parameters of an
interest rate process can lead to large errors when valuing interest rate derivative securities.

5. Conclusions
Utilizing an alternative approach to Brown and Schaefer (1994), this study
estimates the after-tax term structure of real interest rates from information
contained in both the UK linked and non-linked gilt markets over the period
from 25 January 1986 until 25 October 1993.
Two major observations can be made regarding the estimates for spot real
interest rates. Firstly, the volatility of the short-term rate is much lower than
that found by Brown and Schaefer (1994) which provides a better ®t with the
predictions of the CIR single factor model for interest rates. This is likely due
to the di€erent approaches used by the two studies to correct for the lag in
indexation.
Secondly, consistent with Rumsey (1993), there appears to be some evidence
to suggest that single factor interest rate models produce a better ®t to interest
rates on an after-tax basis than on a before-tax basis.
It is clear from a comparison of the results of this study and Brown and
Schaefer (1994) that the pricing of interest rate derivatives is highly sensitive to
the procedures employed to estimate the parameters embodied in interest rate
models such as CIR. It is shown that the di€erences in the techniques used to
construct term structures alone can hugely impact these estimates. This study
addressed the impact of di€erential taxation and the existence of ``noise'' in
observed market prices on term structure estimation and found signi®cant
impact on parameter estimation when compared to methods that did not. It is
clear that further research is required in order to determine the impact of a host
of other market frictions such as the e€ect of liquidity on real-term structure
estimation.

Acknowledgements
The authors would like to thank Zvi Alfasi, Alan Kimche and Uri Passy.
Special thanks are due to Eli Katz and Thomas Woodward. The usual caveat
applies. Prisman would like to thank the SSHRC of Canada and the York
University Research Authority for their ®nancial support.

A.R. Aziz, E.Z. Prisman / Journal of Banking & Finance 24 (2000) 1433±1455

Appendix A

24-Jan-86
25-Feb-86
25-Mar-86
25-Apr-86
24-May-86
24-Jun-86
24-Jul-86
26-Aug-86
25-Sep-86
25-Oct-86
25-Nov-86
27-Dec-86
24-Jan-87
25-Feb-87
25-Mar-87
25-Apr-87
26-May-87
25-Jun-87
24-Jul-87
24-Aug-87
25-Sep-87
24-Oct-87
25-Nov-87
26-Dec-87
24-Jan-88
25-Feb-88
25-Mar-88
25-Apr-88
25-May-88
25-Jun-88
24-Jul-88
25-Aug-88
24-Sep-88
25-Oct-88
25-Nov-88
24-Dec-88
24-Jan-89
25-Feb-89

ILG
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5
5
0
0
5
5
5
5
5
5
10
10
5
10
5
0
0
0
0

Gilt
20
20
15
15
20
25
20
20
20
25
25
20
15
20
15
15
20
20
20
20
20
15
20
15
20
20
15
15
15
15
15
10
15
15
10
10
10
10

24-Jan-90
25-Feb-90
25-Mar-90
25-Apr-90
25-May-90
26-Jun-90
24-Jul-90
25-Aug-90
25-Sep-90
25-Oct-90
25-Nov-90
27-Dec-90
24-Jan-91
26-Feb-91
26-Mar-91
25-Apr-91
25-May-91
25-Jun-91
24-Jul-91
25-Aug-91
25-Sep-91
25-Oct-91
26-Nov-91
27-Dec-91
24-Jan-92
25-Feb-92
25-Mar-92
25-Apr-92
25-May-92
26-Jun-92
24-Jul-92
25-Aug-92
25-Sep-92
25-Oct-92
25-Nov-92
26-Dec-92
27-Jan-93
25-Feb-93

ILG
10
5
0
0
5
5
5
10
5
5
5
5
5
10
0
0
10
10
5
5
0
0
10
15
10
10
0
5
10
15
10
10
5
5
10
5
5
5

Gilt
10
10
10
10
10
10
5
10
10
10
10
10
5
5
5
10
10
10
10
5
10
5
5
10
10
10
5
5
10
10
10
5
10
10
10
10
10
10

1453

1454

A.R. Aziz, E.Z. Prisman / Journal of Banking & Finance 24 (2000) 1433±1455

25-Mar-89
25-Apr-89
25-May-89
24-Jun-89
24-Jul-89
25-Aug-89
25-Sep-89
25-Oct-89
25-Nov-89

0
5
5
10
5
0
5
5
15

10
10
10
10
10
10
10
10
10

25-Mar-93
25-Apr-93
25-May-93
25-Jun-93
24-Jul-93
26-Aug-93
25-Sep-93
26-Oct-93

10
5
5
5
5
5
5
5

5
5
10
10
5
5
10
5

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A.R. Aziz, E.Z. Prisman / Journal of Banking & Finance 24 (2000) 1433±1455

1455

S