**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 longer-dated 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 short-dated bonds. Extensive scenario analysis confirms that in an overall portfolio context a longer-dated bond portfolio is more efficient than a short-dated bond portfolio, especially when long-dated liabilities are present.

**Introduction **

A well-diversified 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 longer-dated 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 so-called '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 long-dated 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 short-dated fixed income in order to minimise his loss in case of a rate rise, or does the higher yield of long-dated 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 long-dated fixed income is expected to outperform short-dated 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 long-dated fixed income is expected to perform better than short-dated 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 5-year horizon. We conclude with the case where long-term liabilities are involved. Especially in this case the benefits of long-dated bonds are quite compelling.

**Current interest rates in historical perspective **

Figure 1 shows the US short (3-month) and long (10-year) 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 long-dated fixed income usually offers a higher yield than short-dated fixed income is often referred to as the 'yield pick-up' 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 'mean-reversion'. The short rate mean-reversion level is assumed to be approximately 4%, whereas the long rate mean-reversion 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 so-called mean-reversion factor is used to specify the term on which mean-reversion is to take place. Using data over the period 1970-2003 we arrived at an estimate for the mean-reversion 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 **

Figure 2 shows the current US yield curve and its assumed mean-reversion 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 pick-up, is quite high.

**Figure 2: US yield curve of June 2004 and its mean-reversion level**

**Figure 3: Change in portfolio valuedue to mean-reversion**

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 mean-reversion effect is shown in Figure 3, which clearly illustrates that short-dated 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 3-month interest rate is approximately 3% below its mean-reversion level, but the 10-year interest rate is already quite close to its mean-reversion level.

**Figure 4: Expected total return ( R) of a fixed income portfolio
**

The expected total return (*R*) over a 1-year horizon is shown in Figure 4. It clearly indicates that in expected terms the yield pick-up of long-dated 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, long-dated 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 long-dated 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 long-dated bonds leads to a higher expected return than a portfolio with short maturities, is with respect to the assumed degree of mean-reversion. We therefore repeated the above analysis under different assumptions for the mean-reversion parameter, ranging from 0.1 to 0.5. This showed that our conclusion remains valid even if we assume the degree of mean-reversion 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 mean-reversion effect, using a mean-reversion 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 mean-reversion 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 mean-reversion level of 6%
**

We also repeated the analysis with higher mean-reversion levels for the long rate. Again, our conclusion remains valid. Figure 6, for example, shows the expected total return using a long rate mean-reversion level of 6% instead of 5%. The mean-reversion 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 pick-up argument will be. Likewise, the further interest rates are from their mean-reversion 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 pick-up 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 long-term average, the expected drop in portfolio value increases with maturity. With hardly any yield pick-up to compensate, this translates into a similar behavior for the expected total return. In this particular case a short-dated 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 pick-up 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 pick-up 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 'too-good-to-be-true' 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 non-negative yield pick-up. 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 pick-up 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 medium-dated 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 1-year expected total return, in the current interest rate environment long-dated fixed income is to be preferred over short-dated fixed income. This conclusion, however, ignores possible differences in terms of risk. Only if long-dated bonds do not add to overall portfolio risk we can truly say that they are superior to short-dated bonds. To investigate this matter we performed a scenario analysis in which 2000 5-year scenarios were generated for interest rates and equity returns using a statistical model based on historical data over the period 1970-2003, taking into account correlations, cross-correlations, auto-correlations, and, of course, mean-reversion 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: Risk-return 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 5-year 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 short-dated fixed income, the middle (red) line represents portfolios with medium-dated fixed income, and the bottom (green) line refers to portfolios with long-dated fixed income^{11}.

From Figure 10 we see, for example, that an allocation of 30% in equity and 70% in short-dated 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 short-dated 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 medium-dated 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 long-dated 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 long-dated 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 pick-up 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 mean-reversion curve. Applying these techniques would further improve the performance of the portfolio but that is beyond the scope of this paper.

**Optimal maturity with long-term liabilities **

Our analysis so far has been asset-only, i.e. it did not involve any liabilities. In many cases, however, medium to long-term liabilities play an important role on the balance sheet of investors. This kind of liabilities is found, for example, in private wealth portfolios with long-term mortgages, in charities with long-term 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 longer-dated bonds not only leads to a more efficient asset portfolio, but also to significant risk reduction on the total balance sheet. After investing in long-dated bonds the asset-side and the liability-side 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: Risk-return profile for portfolios containing both equity and fixed income with different maturities. The portfolio contains long-term liabilities in the form of a 30-year mortgage
**

The analysis in the previous section showed that in an asset-only context the risk-return profile in many different interest rate environments, including the current one, is optimised by investing in long-dated bonds. The addition of long-term liabilities makes the case for long-dated investments even stronger. Figure 11 illustrates this. It shows the risk-return characteristics of our investor, who now also carries a 30-year mortgage on his balance sheet, with a notional equal to 40% of his total assets. Investing in long-dated fixed income instead of short-dated 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 5-year horizon by almost 100 basis points.

**Conclusion **

An obvious strategy in the current historically low interest rate environment is to invest in short-dated bonds only. We showed, however, that this is far from optimal, as it does not take into account the spread between long and short-term 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 risk-return context, we showed that investing in longer-dated bonds not only adds expected return, but also reduces risk at the same time. When long-term liabilities are present, the degree of risk reduction is much stronger because long-dated bonds form a better hedge for such liabilities. Still, even in these circumstances, it is not uncommon for managers with long-dated 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.

*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 1800-1990, Journal of Monetary Economics, Vol. 29, pp. 227-252.*

*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.*

**Footnote**

^{1}
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

^{2}
Source: Bloomberg.

^{3}
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.

^{4}
The expectation of the short rate increases with 0.25 *(4-1.3) = 0.7 from 1.3% to 2.0% in the first year. In the second year it increases with 0.25*(4-2) = 0.5 from 2.0% to 2.5%.

^{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 7-9.

^{6}
Note that this decomposition can also be interpreted as taking the difference between the mean reversion effect and the 1-year 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 1-year forward rate, then the expected total return for that maturity is positive and vice versa.

^{7}
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).

^{8}
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.

^{9}
In June 2003 the US short (3-month) interest rate was just above 1%, whereas the long (10-year) interest rate was about 3.5%. In this case both effects (yield pick-up and mean reversion) are of similar magnitude and the expected total return is close to zero for all maturities.

^{10}
The scenarios are constructed using a vector autoregressive model. For the theoretical introduction to VAR-models see for example Judge et al. (1985). An application of these models in an asset-liability context can be found in part VIII of Ziemba and Mulvey (2001).

^{11}
The short-dated fixed income portfolio consists of bonds with an average maturity of one year. The medium-dated portfolio consists of bonds with an average maturity of three years. The long-dated portfolio consists of bonds with an average maturity of six years.