Abstract
Interest rates are currently at a historical
low. Since in the longer run interest rates
will return to their historical average,
this implies that bond prices are about
to fall. Popular investment advice therefore
says that investors should shorten the maturity
of their bond portfolios to minimise their
losses. This argument, however, skips over
the fact that longerdated bonds pay higher
coupons as well as the fact that a substantial
rise in interest rates may take quite some
time to occur. We examine the combined impact
of both and conclude that the current interest
rate environment in no way implies that
investors should rebalance towards shortdated
bonds. Extensive scenario analysis confirms
that in an overall portfolio context a longerdated
bond portfolio is more efficient than a
shortdated bond portfolio, especially when
longdated liabilities are present.
Introduction
A welldiversified portfolio should contain
a substantial amount of fixed income. This
leaves an important question, however. Should
investors hold short or long(er)dated bonds
and how does this depend on the level of
interest rates? Popular advice in the current
historically low interest rate environment
is that, with interest rates on their way
up and bond prices therefore on their way
down, investors should shorten the maturity
of their bond holdings to minimise losses.
However, this argument skips over the fact
that longerdated bonds pay higher coupons
as well as the fact that a substantial interest
rate rise may take quite some time to occur.
In this paper we integrate these arguments.
Bonds with longer maturities offer higher
yields. The only exception to this rule
is the case of a socalled 'inverted' yield
curve when interest rates for longer maturities
are lower than for short maturities. This
situation, however, only rarely occurs,
and if it does, it never lasts long. Longer
maturities also bear a significant market
risk. If interest rates rise, investors
with a longdated bond portfolio do not
benefit from this rate rise. On the contrary,
since the coupons on their bonds are fixed
until maturity, these bonds will become
less attractive, resulting in a drop in
value. The longer the maturity of the bonds,
the greater the loss will be.
The difficulty in specifying the optimal
maturity lies in the two contradicting effects
described above. In the current low interest
rate environment, should an investor only
hold shortdated fixed income in order to
minimise his loss in case of a rate rise,
or does the higher yield of longdated bonds
more than compensate for this risk? We demonstrate
that in the current interest rate environment,
the optimal maturity of a fixed income portfolio
will still be relatively long. Using a straightforward
analysis of the total return on a fixed
income portfolio, and taking into account
the expected rise in interest rates, we
show that longdated fixed income is expected
to outperform shortdated fixed income.
We also show what the optimal maturity would
be in other interest rate environments,
such as relatively high interest rates or
a flat yield curve. One might argue that
longdated fixed income is expected to perform
better than shortdated fixed income at
the price of higher risk. To show that this
is not the case, we perform an elaborate
scenario analysis where we look at both
the expected return and the risk of loss
of capital over a 5year horizon. We conclude
with the case where longterm liabilities
are involved. Especially in this case the
benefits of longdated bonds are quite compelling.
Current interest rates in historical
perspective
Figure 1 shows the US short (3month) and
long (10year) interest rate over the past
15 years, as well as the difference between
the two^{2}. The graph clearly illustrates
that long rates are usually higher than
short rates. The curve was inverted only
during a brief period in 1989. The fact
that longdated fixed income usually offers
a higher yield than shortdated fixed income
is often referred to as the 'yield pickup'
of longer maturities. The latter stems from
various sources, the most important being
that investors tend to require a premium
for inflation risk and lower liquidity.
Figure 1: Recent
history of US short and long rates
Click on the image for an enlarged preview
Another interesting conclusion that can
be drawn from Figure 1 is that over the
past fifteen years the long and especially
the short rate have never been as low as
is currently the case. This is true even
if one considers a longer period. Market
professionals generally assume that interest
rates tend to return to their historical
average. This effect is known as 'meanreversion'.
The short rate meanreversion level is assumed
to be approximately 4%, whereas the long
rate meanreversion level is often thought
to be around 5%^{3}. This means that in the
current interest rate environment a rise
in interest rates is far more likely than
a fall, although it should be emphasised
that a further fall in interest rates is
surely not impossible.
Click on the image for an enlarged preview
The socalled meanreversion factor
is used to specify the term on which meanreversion
is to take place. Using data over the period
19702003 we arrived at an estimate for
the meanreversion factor of 0.25. Starting
from the current yield curve this implies,
for example, that we expect the short rate
to increase from 1.3% to 2.5% over the next
two years^{4}.
Optimal maturity in the current
interest rate environment
Click on the image for an enlarged preview
Figure 2 shows the current
US yield curve and its assumed meanreversion
level. The curve is relatively steep, which
means that in the current interest rate
environment the spread between the long
and short rate, i.e. the yield pickup,
is quite high.
Figure 2: US yield
curve of June 2004 and its meanreversion
level
Figure 3: Change in portfolio valuedue
to meanreversion
The expected change in value of a fixed
income portfolio can be approximated by
multiplying its duration^{7} by the expected
change of the corresponding interest rate^{8}.
The expected change in portfolio value due
to the meanreversion effect is shown in
Figure 3, which clearly illustrates that
shortdated fixed income is relatively insensitive
to a change in interest rates. For longer
maturities the expected loss increases due
to the increase in duration. For maturities
longer than four years, however, the expected
loss decreases again. On first sight this
may seem strange, since duration increases
with maturity. From Figure 2, however, we
see that the expected change of the interest
rate is smaller the longer the time to maturity.
The 3month interest rate is approximately
3% below its meanreversion level, but the
10year interest rate is already quite close
to its meanreversion level.
Figure 4: Expected total return
(R) of a fixed income portfolio
The expected total return (R)
over a 1year horizon is shown in Figure
4. It clearly indicates that in expected
terms the yield pickup of longdated fixed
income more then outweighs the possible
loss if interest rates rise in line with
expectations.
Of course, one also has to take risk into
account. If, in a portfolio context, longdated
bonds add significantly to the overall risk
of a portfolio then the higher expected
return is simply compensation for the additional
risk incurred. Only if longdated bonds
do not increase the risk of a typical investment
portfolio we can say that such bonds are
to be preferred. We will investigate this
issue further under the section "The
Risk Factor" below. First, however,
we check on the robustness of the above
result and show how things would work out
in other interest rate environments.
Robustness of results
One question is how robust the above conclusion,
that in the current interest rate environment
a portfolio with longdated bonds leads
to a higher expected return than a portfolio
with short maturities, is with respect to
the assumed degree of meanreversion. We
therefore repeated the above analysis under
different assumptions for the meanreversion
parameter, ranging from 0.1 to 0.5. This
showed that our conclusion remains valid
even if we assume the degree of meanreversion
in interest rates to be much stronger. For
example, Figure 5 shows the expected total
return, as the sum of the current interest
rate and the expected change in market value
due to the meanreversion effect, using
a meanreversion factor equal to 0.5.
Figure 5: Expected
total return and change in value of a fixed
income portfolio in current interest rate
environment, assuming a meanreversion factor
of 0.5
Figure 6: Expected
total return and change in value of a fixed
income portfolio in current interest rate
environment, assuming a long rate meanreversion
level of 6%
We also repeated the analysis with higher
meanreversion levels for the long rate.
Again, our conclusion remains valid. Figure
6, for example, shows the expected total
return using a long rate meanreversion
level of 6% instead of 5%. The meanreversion
factor is equal to 0.25.
Optimal maturity in other interest
rate environments with a normal curve
In the analysis so far we have focused on
the current interest rate environment, in
which the yield curve is normally shaped,
interest rates are below their historical
average, and the curve is relatively steep.
But what would happen if the yield curve
was less steep, or interest rates were above
their historical average? Obviously, the
flatter the curve, the less important the
yield pickup argument will be. Likewise,
the further interest rates are from their
meanreversion level, the more important
the value argument will be.
Figure 7: Expected
total return and change in value of a fixed
income portfolio in case of a fictitious
flat yield curve with low interest rates
Consider the case of a flat yield curve
with relatively low interest rates. The
yield pickup in this case will be more
or less equal to zero. However, one would
make a loss if interest rates returned to
their historical average. Since short maturities
are less sensitive to a change in interest
rates, short maturities are in this case
preferred over longer maturities. Figure
7 depicts this conclusion graphically. Given
a relatively flat curve, significantly below
its longterm average, the expected drop
in portfolio value increases with maturity.
With hardly any yield pickup to compensate,
this translates into a similar behavior
for the expected total return. In this particular
case a shortdated fixed income portfolio
is optimal. It should be emphasised, however,
that a combination of low interest rates
and a flat curve very rarely occurs^{9}.
Figure 8: Expected
total return and change in value of a fixed
income portfolio in case of the flat US
yield curve with high interest rates of
April 1989
If the yield curve is flat but rates are
above their historical average, the optimal
strategy is again easily figured out. From
the perspective of yield pickup one would
be indifferent to either short or long maturities.
Since interest rates are expected to drop,
however, one would expect to make a profit
that increases with maturity. As a consequence,
the fixed income portfolio should clearly
have a long maturity, as is visualised in
Figure 8, which corresponds with the US
yield curve of April 1989.
Finally, if the curve is steep with interest
rates above their historical average, the
maturity of the fixed income portfolio should
of course be long. This would give the highest
yield pickup as well as the highest profit
if interest rates started to fall. It should
be emphasised though, that, as is often
the case with these kind of 'toogoodtobetrue'
situations, the combination of a steep curve
and high interest rates hardly ever occurs.
What if the yield curve was inverted?
In the previous sections we assumed the
yield curve to be normally shaped, implying
a nonnegative yield pickup. It might happen,
however, that the yield curve is inverted.
Although this usually happens in a high
interest rate environment and does not last
very long, for completeness we briefly describe
the optimal maturity in the case of a steep
inverted curve in both a high and low interest
rate environment.
Figure 9: Expected
total return and change in value of a fixed
income portfolio in case of a fictitious
steep inverted yield curve with high interest
rates
In the unlikely case of low interest rates
the maturity should clearly be short. It
gives the highest yield and no losses if
interest rates rise and return to their
historical average. A more interesting case
is a steep inverted curve with high interest
rates. The yield pickup in this case is
negative and therefore a short maturity
seems preferable. Since short maturities
are more or less insensitive to a change
in interest rates, however, longer maturities
have to be bought in order to take advantage
of the expected fall in interest rates.
Figure 9 shows that in case of a steep inverted
yield curve with relatively high interest
rates, a mediumdated fixed income portfolio
is optimal. It should be emphasised though
that this optimal maturity strongly depends
on the exact form and level of the yield
curve.
The risk factor
In the previous sections it was shown that,
in terms of 1year expected total return,
in the current interest rate environment
longdated fixed income is to be preferred
over shortdated fixed income. This conclusion,
however, ignores possible differences in
terms of risk. Only if longdated bonds
do not add to overall portfolio risk we
can truly say that they are superior to
shortdated bonds. To investigate this matter
we performed a scenario analysis in which
2000 5year scenarios were generated for
interest rates and equity returns using
a statistical model based on historical
data over the period 19702003, taking into
account correlations, crosscorrelations,
autocorrelations, and, of course, meanreversion
in interest rates^{10}. Such an extensive set
of scenarios not only provides insight in
the expected return but also in the risk
for an investor following a certain investment
strategy.
Figure 10: Riskreturn
profile for portfolios containing both equity
and fixed income with different maturities
Consider an investor who wants to invest
in both equity and fixed income. As always,
his aim is to maximise return while keeping
risk at an acceptable level. For practical
purposes, however, this is not specific
enough. The exact definition of risk and
return must depend on the specific goals
of the investor. We will assume that our
particular investor wants to maximise his
total return after five years, while at
least keeping his capital intact. This means
taking return to be the expected annual
return over a 5year period and to equate
risk to the probability that after five
years the investor's asset value is less
than at the outset. For different asset
allocation strategies, i.e. yearly rebalanced
mixtures of equity and fixed income, both
expected return and risk (as defined) are
plotted in Figure 10. The upper (blue) line
in Figure 10 reflects portfolios with shortdated
fixed income, the middle (red) line represents
portfolios with mediumdated fixed income,
and the bottom (green) line refers to portfolios
with longdated fixed income^{11}.
From Figure 10 we see, for example, that
an allocation of 30% in equity and 70% in
shortdated fixed income (the left point
of the blue line) yields an expected annual
return of 4.9% with a 2.7% probability of
capital after five years being less than
at the outset. If one invested 40% in equity
and 60% in shortdated fixed income, the
latter probability would rise to around
6% but at the same time expected return
would rise to 5.4%. Figure 10 also shows
that, holding on to the 40/60 allocation,
if we were to invest in mediumdated bonds
expected return would increase by 45 basis
points while the risk of capital after five
years being below its initial value would
drop as well. Investing in longdated fixed
income would yield even higher rewards.
Expected return would increase with almost
55 basis points, and the probability of
a drop in capital would fall from around
6% to little over 5%. Note that maturities
longer than 10 years would not significantly
improve the performance of the portfolio
any further due to the fact that the current
yield curve is relatively flat for maturities
longer than 10 years.
As an alternative to keeping the allocation
fixed at 40% equity and 60% fixed income,
we could also decide to keep the risk at
a fixed level. Keeping the probability of
capital being less than initial capital
at 6%, Figure 10 shows that the investor
should in that case invest in longdated
fixed income and at the same time enlarge
his allocation in equity from 40% to 43%.
Doing so would allow him to pick up some
more of the equity risk premium and thereby
increase his expected annual return by more
than 65 basis points, from 5.4% to little
over 6%.
The above scenario analysis clearly illustrates
that the current yield pickup is large
enough to compensate for the possible loss
if interest rates start to rise. This is
in line with the conclusion from the previous
analysis, where we only looked at expected
total return.
More dynamic strategies
The scenario analysis in the previous section
was based on a static strategy where every
year the fixed income portfolio consists
of bonds with the same maturity. It is not
difficult to incorporate dynamic strategies
that take decisions based on the state of
the economy. Every year the optimal maturity
of the fixed income portfolio is then based
on the slope of the curve and the deviation
from the meanreversion curve. Applying
these techniques would further improve the
performance of the portfolio but that is
beyond the scope of this paper.
Optimal maturity with longterm
liabilities
Our analysis so far has been assetonly,
i.e. it did not involve any liabilities.
In many cases, however, medium to longterm
liabilities play an important role on the
balance sheet of investors. This kind of
liabilities is found, for example, in private
wealth portfolios with longterm mortgages,
in charities with longterm commitments
to donate to social projects and in pension
fund and life insurance portfolios where
obligations sometimes extend far beyond
30 years. In these cases, investing in longerdated
bonds not only leads to a more efficient
asset portfolio, but also to significant
risk reduction on the total balance sheet.
After investing in longdated bonds the
assetside and the liabilityside of the
balance sheet both will have a long maturity.
As a consequence, both will react to a change
in interest rates in more or less the same
way.
Figure 11: Riskreturn profile for
portfolios containing both equity and fixed
income with different maturities. The portfolio
contains longterm liabilities in the form
of a 30year mortgage
The analysis in the previous section showed
that in an assetonly context the riskreturn
profile in many different interest rate
environments, including the current one,
is optimised by investing in longdated
bonds. The addition of longterm liabilities
makes the case for longdated investments
even stronger. Figure 11 illustrates this.
It shows the riskreturn characteristics
of our investor, who now also carries a
30year mortgage on his balance sheet, with
a notional equal to 40% of his total assets.
Investing in longdated fixed income instead
of shortdated fixed income would reduce
the risk of his capital (assets minus liabilities)
after five years being less than his initial
capital by more than 25% (from a probability
of more than 17% to a probability of less
than 13%). At the same time this would increase
his expected annual return over a 5year
horizon by almost 100 basis points.
Conclusion
An obvious strategy in the current historically
low interest rate environment is to invest
in shortdated bonds only. We showed, however,
that this is far from optimal, as it does
not take into account the spread between
long and shortterm interest rates, which
is typically positive and currently quite
large. In the current environment long maturities
should be preferred over short maturities.
Short maturity fixed income would only be
optimal if the yield curve were substantially
less steep. The following table summarises
our conclusions (assuming a normally shaped
yield curve).
In a riskreturn context, we showed that
investing in longerdated bonds not only
adds expected return, but also reduces risk
at the same time. When longterm liabilities
are present, the degree of risk reduction
is much stronger because longdated bonds
form a better hedge for such liabilities.
Still, even in these circumstances, it is
not uncommon for managers with longdated
liability portfolios to conclude from the,
statistically correct, argument that interest
rates are more likely to rise than to fall,
that they should shorten the maturity of
the asset portfolio. In this paper we have
(hopefully convincingly) shown that this
is definitively not the case.
Footnotes
 Vincent van Antwerpen
and Janwillem Engel are consultants and
Theo Kocken CEO of Cardano Risk Management
in Rotterdam, The Netherlands. Harry M.
Kat is Professor of Risk Management and
Director of the Alternative Investment
Research Centre, Cass Business School,
City University, London
 Source: Bloomberg.
 Siegel (1992) and Steehouwer
(2004), for example, present studies of
interest rates over very long time periods,
showing long term average rates around
the levels we use.
 The expectation of the
short rate increases with 0.25 *(41.3)
= 0.7 from 1.3% to 2.0% in the first year.
In the second year it increases with 0.25*(42)
= 0.5 from 2.0% to 2.5%.
 Initially, we will restrict
ourselves to expected return in determining
the optimal maturity. For a complete comparison,
however, the risk of different portfolios
should also be taken into account. This
extra dimension will be added and discussed
in section 79.
 Note that this decomposition
can also be interpreted as taking the
difference between the mean reversion
effect and the 1year forward curve from
the current yield curve. If after one
year, as a result of the mean reversion
effect, the interest rate for a certain
maturity is still below the current 1year
forward rate, then the expected total
return for that maturity is positive and
vice versa.
 Duration is defined
as the weighted average of the time to
receipt of the individual cash flows (coupon
and notional) of the bond or portfolio.
It can be used as a measure of the sensitivity
of a bond portfolio to a change of interest
rates. See for example Fabozzi and Fabozzi
(1995).
 Note that in the current
interest rate environment, where interest
rates are expected to rise, calculating
the change in value of a fixed income
portfolio using duration only, i.e. without
accounting for convexity, will somewhat
overestimate this change. Comparable arguments
apply for high interest rate environments.
 In June 2003 the US
short (3month) interest rate was just
above 1%, whereas the long (10year) interest
rate was about 3.5%. In this case both
effects (yield pickup and mean reversion)
are of similar magnitude and the expected
total return is close to zero for all
maturities.
 The scenarios are constructed
using a vector autoregressive model. For
the theoretical introduction to VARmodels
see for example Judge et al. (1985). An
application of these models in an assetliability
context can be found in part VIII of Ziemba
and Mulvey (2001).
 The shortdated fixed
income portfolio consists of bonds with
an average maturity of one year. The mediumdated
portfolio consists of bonds with an average
maturity of three years. The longdated
portfolio consists of bonds with an average
maturity of six years.
References
Fabozzi, F.J. and Fabozzi,
T.D. (1995), The Handbook of Fixed Income
Securities  Fourth edition, Irwin
Professional Publishing.
Judge, G.G., Griffiths, W.E., Hill, R.C.,
Lütkepohl, H and Lee, T.C. (1985),
The Theory and Practice of Econometrics
 Second edition, John Wiley &
Sons.
Siegel, J.J. (1992), The real rate of interest
from 18001990, Journal of Monetary
Economics, Vol. 29, pp. 227252.
Steehouwer, H. (2004), Macroeconomic
Scenarios and Reality, PhD thesis.
Ziemba, W.T. and Mulvey J.M. (2001), Worldwide
Asset and Liability Modelling, Cambridge
University Press.
