Testing Contingent Immunization:

Evidence from the Spanish Treasury market

Antonio Díaz

María de la O González

Eliseo Navarro

Frank S. Skinner*

Departamento de Análisis Económico y Finanzas Universidad de Castilla-La Mancha Facultad de CC Económicas y Empresariales de Albacete, Plaza de la Universidad, 1, 02071 – Albacete (Spain)

Tel: +34 967 59 92 00 Antonio.Diaz@uclm.es

MariaO.Gonzalez@uclm.es (Corresponding author) Eliseo.Navarro@uclm.es *Frank S. Skinner University of Surrey School of Management Guildford, Surrey,United Kingdom Tel: +44 (0) 148 368-6364

F.Skinner@surrey.ac.uk JEL Classification: C61, E43, G11, G12 We acknowledge the financial support provided by Junta de Comunidades de Castilla-La Mancha grant

PAI-05-074 and Ministerio de Educación y Ciencia grant SEJ2005-08931-C02-01 partially supported by

FEDER funds.

1

Testing Contingent Immunization:

Evidence from the Spanish Treasury market

Abstract

This paper tests the effectiveness of contingent immunization, a stop loss strategy

that allows portfolio managers to take advantage of their ability to forecast interest rate

movements as long as their forecasts are successful, but switches to a pure

immunisation strategy should the stop loss limit be encountered. This study uses actual

daily transactions in the Spanish Treasury market covering the period 1993-2003 and

uses performance measures that accounts for skewness and kurtosis as well as mean

variance. The main result of this paper is that contingent immunization provides

excellent performance despite its simplicity.

Keywords: Immunization; Contingent Immunization; Active portfolio management

JEL Classification: C61, E43, G11, G12

2

1. Introduction

The aim of this research is to test the effectiveness of contingent immunization

techniques. The pioneer works in this field, Leibowitz and Weinberger (1981, 1982 and

1983), developed contingent immunization as a midpoint in a risk-return framework

between pure immunization and active bond management strategies. Contingent

immunization is a stop loss strategy that allows portfolio managers to take advantage of

their ability to forecast interest rate movements as long as their forecasts are successful,

but switches to a pure immunisation strategy should the stop loss limit be encountered.

Specifically, contingent immunization consists of forming a bond portfolio with a

duration larger or smaller than the investor’s planning period depending on interest rate

expectations. If the investor thinks that interest rates are going to rise more than the

market expects she would buy a bond portfolio with a duration smaller that her planning

period and vice versa. However, if interest rates move opposite to the investor’s

expectations and the portfolio value falls to a given stop loss limit then she would

immunize and guarantee this lower limit for the eventual portfolio return. This strategy

gives contingent immunization an option like feature:1 limiting losses but preserving

upside potential if interest rates movements are favourable. Therefore contingent

immunization strategies represent an attempt to capture positive (or avoid negative)

skewness. Moreover, investors can achieve these attractive option-like distributional

characteristics synthetically by trading in government bonds without the use of costly

and illiquid options.

However, despite initial interest, there is a lack of empirical work concerning the

effectiveness of these strategies. In this sense, this paper makes a number of

3

contributions to the literature. First, this study is, to the best of our knowledge, one of

the few works where contingent immunization strategies are examined. Second, we

compare contingent immunization strategies with classical immunization, active bond

portfolio and buy and hold equity investment strategies. We borrow from the recent

hedge fund literature to measure the performance of all the strategies taking into

account four moments of the distribution of holding period returns, the mean, variance,

skewness and kurtosis. Third, we employ an extensive data set of daily actual

transactions in the Spanish Treasury market covering a ten-year period from January 4,

1993 to January 3, 2003. And finally, we measure the holding period returns of all

strategies as realistic and exact as possible by rebalancing the portfolio each time

payments are due instead of periodically and checking the portfolio value every day to

determine whether the stop loss rule should be implemented.

We find that contingent immunization strategies provide excellent results, as these

strategies are able to capture upside potential while the stop loss limit is by and large

effective in preventing large losses. We also find that bond strategies in general provide

attractive distributional properties and that by adjusting performance measures for

skewness and kurtosis the relative ranking of the “best” strategy can change. Moreover

these attractive results are achieved without the need for complex valuation models and

hedging strategies requiring the use of often illiquid interest rate derivatives.

This paper is structured as follows. First, we describe the data. Then, we

determine the structure of the portfolios and propose a model to implement alternative

contingent immunization as well as active and pure immunization strategies. Third we

introduce traditional and innovative portfolio performance measures, specifically the

Shape Ratio and adjusted Sharpe ratio that that account for mean variance, and the

4

modified Sharpe ratio that accounts for four moments of the distribution of holding

period returns. Then we present and comment on the results and finally summarize the

main conclusions.

2. Data

The data set consists of mean daily bonds, bills and repo prices of actual

transactions in the Spanish Public Debt Market over the period from January 4, 1993

until January 3, 2003. This data is provided by Banco de España and is comprised of

daily information on more than 66 different bonds during the whole sample period. 2

Bills are also included in this study as well as one-week repo market transactions.

To understand the behaviour of Spanish interest rates during the sample period we

first estimate the Spanish term structure of interest rates every day for the entire sample

period.3 The summary statistics of monthly changes of interest rates are given in Table

1. The levels of one-month, one-year and ten-year interest rates are represented in

Figure 1. Clearly there was a dramatic decrease in interest rates and a twist in the yield

curve during the sample period especially during the first half. Moreover, these dramatic

changes in interest rates are reflected in the distributional characteristics of changes in

interest rates where, especially in the first half, interest rate changes are characterised by

negative skewness and large excessive kurtosis. As expected, short rates have a greater

volatility than long-term interest rates.

Moreover we conduct a factor analysis of the yield curve.4 The first three factors

can be identified (as usual) as parallel, slope and curvature changes of the yield curve

accounting for 77.94, 15.69 and 4.68% of the total variance respectively. Compared

with other countries (see Barret, Gosnell and Heuson 2004, Driessen, Melenberg and

5

Nijman 2003, and Wolfgang et al 1999), parallel shifts seem to explain less of the

behaviour of the term structure so the risk of failure of immunization strategies due to

twists of the yield curve is greater than in other default free bond markets.5

< Insert Table 1 about here >

< Insert Figure 1 about here >

3. Portfolio design

We specify a three-year planning period and divide the sample into twenty-nine

three-year overlapping periods. Each period starts at quarterly intervals on the first

trading day of January, April, July and October from January 1993 to January 2000. The

opportunity set consists of six Treasury bonds with the highest liquidity selected from

all the Spanish government bonds outstanding.6 Two bonds must have a remaining

maturity shorter than three years, two bonds must have a maturity longer than three

years and the final two bonds must have a maturity close to three and ten years.

In contrast to prior studies (for example, Soto 2001, 2004 and Fooladi and Roberts

1992) the initial portfolio is rebalanced each time coupons are paid either by reinvesting

these payments among the existing bonds in the portfolio or in new bonds with a

duration to maintain the given strategy. 7 Díaz, Merrick and Navarro (2006) and Sarig

and Warga (1989) note that as bonds approach maturity they become less liquid.

Therefore when bonds mature, other liquid bonds replace them if we find transactions

for bonds with a suitable maturity. When we cannot we use Treasury bills. Sometimes

neither bills nor bonds are found for very short maturities. In these cases, we use one-

week repos. This means that the portfolio must be rebalanced each week until the end of

6

the holding period. In any event we use the daily mean of actual transaction prices of

bonds, bills and repos.

4. Methodology

We develop a linear program to conduct out of sample contingent and pure

immunization strategies that allows for different degrees of risk and expectations about

interest rate movements. These contingent immunization strategies are modelled

through the following linear programming set-up:

Min �=

N

jj

1

2ω (1)

subject to �=

=−N

jjj difDH

1

·ω

�=

=N

jj

1

1ω

10 ≤≤ jω

where N is the number of bonds available, �j is the weight of bond j in the portfolio, H

is the investor horizon, dif is the difference between the investor horizon and the

portfolio duration (H-D), and ( )� =⋅⋅= m

i jijijijj PtPCtD1 0 is the Fisher and Weil

duration of bond j.8 In the duration term Cij is the cash flow i generated by the bond j

(Cij > 0) due at time tij, P0(tij) is the present value of a unit zero coupon bond with

maturity at time tij and Pj is the bond price.

The objective function is set in order to choose, among those portfolios that meet

the constraints, that one with the maximum dispersion. This portfolio is the one with the

maximum degree of diversification and the lowest idiosyncratic risk. The variable dif

7

determines the degree of activeness of the strategy. We have a three-year planning

horizon and we consider six values for dif for the active strategies, -1.5, -1, -0.5, 0.5, 1,

1.5 years, that corresponds to starting durations of 4.5, 4.0, 3.5, 2.5, 2.0 and 1.5 years.

The first three would be applied if the investor expects a fall in interest rates below

forward rates and the last three if the investor expects a rise in interest rates above

forward rates. The larger the absolute value of dif the more active portfolio management

and thus, the greater the investor’s expected return but also the greater the risk. Finally

the last set of constrains indicate that short sales are not allowed in the bond market.

We proceed to run our contingent immunization strategies in the following way.

Let RH be the default-free zero coupon rate at the beginning of the holding period with

maturity of three years. This is the target return of pure immunization strategies. Let V0

be the initial and VH be the final portfolio value. When following a contingent

immunization strategy we allow the manager to undertake an active bond portfolio

management but we require that losses derived from unexpected interest rate

movements to be within a given limit. In our study, we consider three different

minimum returns: 50, 100 and 150 basis points below the target return, that is, below

RH. In the first case, for instance, the minimum final portfolio value we would demand

is VH* = V0 · (1 + RH - 0.0050)H.

To guarantee this minimum final portfolio value we check every day the current

portfolio value in the following way. Assume that t days after the initial investment,

portfolio value is Vt. We calculate, according to the interest rates outstanding at t, the

current portfolio value that we need to guarantee, at the end of the holding period, an

amount equal to VH* if we immunize our portfolio at t. This value is given by:

8

**

(1 )H

t H tH t

VV

R −−

=+

(2)

where RH-t is the time t zero coupon rate with term to maturity equal to H-t. If Vt > Vt*

we continue with the active portfolio management. On the other hand, if Vt � Vt* we

would proceed to immunize our bond portfolio making its duration equal to the

remaining time left in the three year planning period, that is, applying the linear

programming model again but replacing the first constrain to make dif equal to zero.

Note that if the portfolio value reaches this limit (Vt*) and so the portfolio is

immunized, we consolidate losses, not being able to take advantage of favourable future

interest rates moves. However recall that the aim of contingent immunization is to

eliminate the possibility of larger losses.

In order to compare contingent immunization with active management as well as

with pure immunization we also obtain the results of these two extreme strategies. In the

case of active management, we apply the portfolio selection model, described in

equation [1], but we do not require any lower limit for the final portfolio value. That is,

we do not immunize our portfolio even if we are incurring large losses. We go on with

the active management until the end of the holding period as if the investor is still

waiting for a final interest rate movement according to their initial expectations. We do

this by maintaining portfolio duration larger or smaller than the holding period. In the

case of pure immunization, we apply our portfolio selection model (equation [1]) but

make dif equal to zero (D=H). That is we proceed to immunize the portfolio over the

whole holding period. Since we have six active strategies with durations of 4.5, 4.0, 3.5,

2.5, 2.0 and 1.5, and three levels of the stop loss at 50, 100 and 150 basis points, we

9

have in total 18 different contingent immunization strategies. Counting the six active

strategies and the pure three-year immunization strategy we examine 25 strategies in all.

To achieve these objectives we have to overcome two difficulties. First, we have

to determine the exact moment when immunization should be activated (the moment

when Vt � Vt*). We check if the portfolio value has hit the stop limit every day even

though this requires a high computational effort.

Second, in the case of contingent immunization with dif larger than zero, it is

sometimes impossible to maintain the original strategy as we approach the end of the

three-year planning horizon. For instance if dif=0.5 with a holding period of three years

we must decide what to do when we approach two and a half years after the beginning

of the holding period as it is no longer possible to keep the portfolio’s duration 0.5 years

below the remaining holding period. In this case, we immunize the portfolio until the

end of the holding period provided it was not immunized before due to adverse changes

in interest rates.9

We should note that there are two factors that can cause contingent immunization

to fail in its’ objective of guaranteeing a minimum return. First immunization can fail

because our immunization strategies are based on Fisher-Weil duration and Fisher-Weil

duration requires parallel in the term structure. We know from Table 1 and Figure 1 that

the Spanish term structure experienced non-parallel shifts during the sample period.

Second, it is possible that a very sharp and sudden interest rate movement could cause

the portfolio value to fall well below the limit before we can immunize10.

5. Results

10

For each strategy we calculate the difference between the three-year zero coupon

bond yield available at the beginning of each strategy (benchmark return) and the

realized return. Table 2 provides a description of the main statistics of the benchmark

return and its three-year changes over the 29 holding periods. Moreover volatility of the

target return is much higher during the first six years, which roughly corresponds to the

convergence period of the Spanish currency to the Euro. Therefore we split the sample

period into two sub-periods to see if a difference in the behaviour of interest rates

during these two periods influences the effectiveness of the different strategies.11

< Insert Table 2 about here >

Evidently changes in the three-year interest rates are dramatic over the sample

period with a striking maximum variation of 736 basis points during the period April

1995-April 1998. Overall interest rates are declining during the sample period and this

decrease is only interrupted in two intervals, during June 1994-September 1995 and

June 1999-January 2001. This is a very interesting scenario for active management. If

an investor forecasts these movements of interest rates she can obtain an extraordinary

return by investing in long-term bonds. However, if an investor follows the wrong

strategy (in this case investing in short term bonds) the aim of contingent immunization

is to limit the losses to the stop limit.

Table 3 describes the results obtained for each strategy. In columns 2 to 4 we

show the mean of the differences between the actual return obtained when following

contingent immunization and the three-year benchmark return of the pure immunization

strategy over the 29 holding periods. As expected investing in long term bonds (dif = H-

D < 0) yield excellent results because interest rates are decreasing during the sample

period. An active bond manager forecasting these interest rates movements would enjoy

11

a very significant extra annual return, the greater the difference between duration and

investor horizon the greater the return. Thus, for dif = - 0.5 years the extra annual return

is between 80 and 83 basis points while for dif = -1.5 years this reward is between 166

and 198 basis points.

< Insert Table 3 about here >

However if the investor follows the wrong strategy and invests in short term

bonds the average losses are cut off at the stop limit. This lower limit guarantees a

minimum return of 50, 100 or 150 basis points below the target return.12 Table 3 shows

that if this minimum return is fixed at 50 basis points below the three year zero coupon

rate, the mean return when following the wrong strategy is between 24 and 34 basis

points less than the target return. With a 150 basis point stop limit the average loss

ranges between 42 and 70 basis points. We conclude that, on average, contingent

immunization is successful.

It is also interesting to see that pure immunization (dif = 0) yields an average

return of 19 basis points over the target return. This is a typical result since fixed

coupon bonds have positive convexity so that a pure immunization strategy obtains the

promised yield as a minimum or a higher amount (for details see Skinner 2005).

However we also observe a great deal of variation where the results from the pure

immunization strategy can be 64 basis points below and 83 basis points above the

benchmark return. This highlights the limitations of the pure immunization strategy and

suggests the importance of developing more accurate immunization techniques.13 These

two extreme cases correspond to the periods from October 1993 to October 1996 and

October 1999 to October 2002. Figure 1 shows that during the first period an

exceptional twist of the entire yield curve occurs where the yield curve moves from a

12

very steep downward slope to an increasing curve at the end of this period. During the

second period the yield curve flattened to become, eventually, upward sloping. This

behaviour of the yield curve can explain the relative failure of immunization techniques

based Fisher-Weil duration as Fisher-Weil duration assumes parallel shifts in the yield

curve.

When examining the range of results of contingent immunization we can see that

the stop limit is also trespassed. In fact, the maximum loss is nearly 50 basis points

below the stop limit in the three cases, specifically for a duration difference of +1.50 for

a 50 and 100 basis point stop limit and for a duration difference of +1.0 and a 50 basis

point stop limit. However, the violation of the lower limit is only serious in the case of

very active strategies when the duration difference is large. Moreover when we look at

the maximum values we can see the attractiveness of contingent immunization.

Investors following an active portfolio management can obtain nearly an annual 5 %

extra yield.

Figures 2(a) to 2(d) clearly illustrate these results. They show the actual return

from contingent immunization with a stop loss limit of 50 basis points over the 29

holding periods. The vertical axis represents the actual return above the three-year zero

coupon rate available at the beginning of each holding period and the horizontal axis

represents the unexpected change in the three year rate during the holding period. The

unexpected changes are estimated as the three-year zero coupon bond rate at the end of

each holding period minus the corresponding forward rate outstanding at the beginning

of the each holding period. To help interpret these figures the solid horizontal line

represents the performance of a perfectly immunized portfolio where the actual return is

zero basis points above the target three-year zero-coupon rate no matter what

13

unexpected change in the three-year rate actually occurs during the three year planning

horizon.

These figures illustrate the different patterns of the extra return for each strategy.

Figure 2(a) shows the most active strategy where the investor expectations are correct.

Generally this strategy achieves very good results but there are some poor results even

when interest rates unexpectedly decrease. This is because we compute unexpected

changes in interest rates at the end of the holding period. However if during the

investor’s planning period interest rates increase and the stop loss limit is reached we

immunise the portfolio thereby consolidating losses irrespective of later interest rate

movements. If we maintain the active strategy instead of immunizing it is possible that

some of these losses would have been recovered. This is in fact the “price” of

contingent immunization. Figure 2(b) is similar but with a lower degree of activeness

(dif = -0.5) the results are less striking.

< Insert Figure 2 about here >

In contrast Figures 2(c) and 2(d) provide the results of a wrong active strategy.

We can see that in the case of a very active and risky strategy (dif = +1.5) the losses are

limited. It is obvious that the stop loss limit (50 basis points) is sometimes trespassed

but this infringement is not too damaging.

Figure 3 reports the results for pure immunization and illustrates that for this

sample period, even the most passive strategy sometimes failed. Although the extra

return is close to zero in most periods sometimes the difference between the target and

actual return is over 50 basis points. Since Fisher-Weil duration is used to implement all

strategies and Fisher-Weil duration assumes parallel term structure changes, it is evident

that changes in the term structure during the sample period are much more complex and

14

this probably damages the effectiveness of both contingent and pure immunization

strategies.

< Insert Figure 3 about here >

Returning to Table 3 we compare the results of contingent immunization with

those obtained by an active strategy without a stop loss limit. The greatest loss from the

active strategy is when the investor chose a duration difference of +1.50. This loss, 103

basis points below the target return, is in contrast to the loss from the contingent

immunization strategies, which are 30, 51 and 70 basis points depending on the specific

strategy. Moreover the maximum loss from the pure active strategy is 343 basis points.

This is a much larger loss than any of the three contingent immunization strategies

considered in this study (97, 149 and 190 basis points).14 These results illustrate the

effectiveness of the stop loss limit of contingent immunization. Moreover the

possibilities of extraordinary gains are very similar in both contingent immunization and

active strategies.

Another noteworthy outcome is the minimum return obtained by active

management when following the strategy dif = -1.5. In this case, the worst result for the

active strategy is +9 basis points over the target return whereas the worst result when

following the contingent strategy is -144 basis points. This result again highlights the

price of hedging as once the stop loss limit is reached losses are consolidated and the

strategy is then unable to take advantage of later favourable interest rate movements.

6. Performance evaluation

Until now we evaluate active, contingent immunization and pure immunization

strategies by examining the potential gains and losses relative to the three-year zero-

15

coupon yield. The next step is to examine the performance of these strategies. However

the stop loss limit inherent in the contingent immunization strategy impose a “floor” on

potential losses and represents an attempt by the investor to transform the distribution of

returns to enhance positive skewness or at least avoid negative skewness. Therefore our

performance measure should account for more than simply mean variance, but also

skewness and kurtosis.

Table 4 measures the mean, standard deviation, skewness and excessive kurtosis

(compared to the normal distribution) of the 29 holding period returns for the twenty-

five bond strategies and for a buy and hold strategy for investing in the IBEX 35 index.

The IBEX 35 is a value-weighted index of the largest and most liquid 35 stocks on the

Madrid stock market.15 We expect that investors have a preference for the mean and

positive skewness and wish to avoid standard deviation and excessive (positive)

kurtosis. Hass (2007) finds that if investors have non-increasing absolute risk aversion

then excessive kurtosis is unattractive.

<<Table 4 about here>>

First, reading up the columns we see that as duration increases the mean and

standard deviation of holding period returns increase. In other words as the degree of

“activeness” of the bond strategy increases, risk, as measured by the standard deviation,

is rewarded by extra return. Moreover as we read along the row we see that as we relax

the stop loss limit the mean and standard deviation increases for strategies that hope for

decreases in interest rates (dif < 0) but the mean and variance decreases for strategies

that hope for increases in interest rates (dif > 0). As Table 1 reports that interest rates

generally decrease during the sample period it is no surprise that the strategies that hope

16

for interest rate decreases enjoyed higher mean holding period returns, but at a cost of

higher standard deviation. Overall the empirical results conform very well to the mean

variance framework.

Looking at other moments of the distribution we see that as duration increases

positive skewness decreases. In other words, another “cost” of increasing duration risk

is to reduce positive skewness as well as increase standard deviation. Reading along the

rows we are unable to determine if there us any systematic tendency of skewness or

excessive kurtosis to change as we relax the stop loss limit. Finally we note that when

compared to the buy and hold equity strategy as reported in the second column of Table

4, the duration strategies have far lower expected returns, but far more attractive

distributional characteristics with a much lower standard deviation, higher positive

skewness and lower kurtosis.16

While these results are interesting it is difficult to reach any general conclusion

regarding the performance of the strategies based on examining each of the four

moments of the distribution one by one. What is needed is a performance statistic that

accounts for all four moments of the distribution. Fortunately the hedge fund literature

suggests just such a measure.

The hedge fund literature (see Kouwenberg 2003, Gregoriou and Gueyie 2003 and

Favre and Galeano 2002) proposes adjustments to the Sharpe ratio to account for non-

normality in the return distribution through use of the value at risk VAR technology.

VAR measures the expected loss that can occur on a portfolio within a given time

interval. Portfolio managers are to set the likelihood that such a loss would occur to

17

some small value like 5 or 1%, so this technique focuses exclusively on minimising

“tail” risk. The corresponding “adjusted Sharpe ratio” ASR is as follows.

��

���

� −=

VAR

RRASR fp

Note that Rp is the average holding period return, Rf is the three-month Spanish interest

rate and VAR is a measure of risk. Therefore ASR is a reward to risk ratio like the

Sharpe ratio.

However the ASR still assumes that holding period returns are normally

distributed since VAR is measured as

σ= NVAR

where σ is the standard deviation of the holding period return and N is the number of

standard deviations associated with a given level of probability assuming that holding

period returns are normally distributed.17 For example N = 2.33 represents 2.33 standard

deviations from the mean (representing the full range of outcomes that can occur 99%

of the time assuming a normal distribution). Nevertheless the above does focus on what

we are really interested in, tail risk, and it can be extended to other moments of the

distribution. This “modified Sharpe ratio” MSR that adjusts the ASR to include the

impact of skewness and excessive kurtosis of the holding period return is shown below.

��

���

� −=

MVAR

RRMSR fp

Now MVAR is measured as follows (see Favre and Galeano, 2002).

18

σ�

��

−−−+−+= 2332 S)N5N2(361

K)N3N(241

S)1N(61

NMVAR

Notice that MVAR is simply VAR with N adjusted by the term in curved brackets.

The terms S and K refer to the skewness and kurtosis. S measures deviations in the

symmetry from the normal distribution and K measures deviations in the “peakness”

(and by implication, the “fatness” of tails) from the normal distribution. Note that if S

and K were zero, MVAR becomes VAR.

Table 5 reports the traditional Sharpe, M2, ASR and MSR ratios. The Sharpe ratio

is the traditional reward to risk ratio where the numerator is the return of the strategy at

hand above the three-month Spanish interest rate and the denominator is the standard

deviation of the strategy’s holding period return. The M2 adjusts the Sharpe ratio for the

differences in standard deviation between the IBEX 35 and the bond portfolio strategy

at hand so that we can clearly see if the equity strategy outperformed the bond strategy.

Specifically if M2 is positive, then the bond strategy is superior but if the M2 is negative,

the equity strategy is superior in the mean variance sense. If the combination of

skewness and excessive kurtosis improves performance then the MSR ratio will be

greater than the ASR ratio.

The Sharpe ratios confirm that our empirical results conform well to mean

variance theory. Specifically, as duration increases, Sharp ratios increase, and as the

stop limit increases the Sharpe ratios improve as long as duration is above the three year

benchmark, but the Sharpe ratios decrease with the stop limit for strategies where the

duration is less than the three years. We noted earlier that this later result is due to the

19

generally decreasing interest rates of the sample period. The M2 ratio clearly shows that

high duration strategies performed better than the generic buy and hold equity strategy.

The interesting question is does the attractive distributional characteristics of the

bond strategies change our perception of performance? A comparison of the ASR and

MSR ratios emphatically suggests that it does since for every strategy the MSR ratio is

higher than the corresponding ASR ratio. Moreover when comparing the MSR ratios for

bond and equity strategies we again find that high duration strategies outperform the

generic buy and hold equity strategy, but now the duration threshold where bond

strategies outperform equity is lower.18 Specifically notice that the M2 measure suggests

that the buy and hold equity strategy is superior to the bond strategies with a dif of –0.5

but now the MSR ratio suggests that these strategies are superior to the buy and hold

equity strategies. Clearly the distributional characteristics of bond holding period

returns are important as not only can performance be seen as better in absolute terms but

the relative ranking of what is the “best” strategy can change once the impact of

skewness and kurtosis is recognised.

7. Conclusions

This paper is one of the few papers that tests and compares contingent

immunization strategies with active management and pure immunization strategies. The

main conclusion is that contingent immunization does indeed provide a mid point

between pure immunization strategies and active bond portfolio management. As

claimed by the earlier studies, contingent immunization allows investors to carry on

active management but the stop loss limit is effective in limiting losses derived from

failures in predicting future interest rates.

20

Just as important however is that contingent immunisation adjusts the distribution

of holding period returns. Clearly the distributional characteristics of bond holding

period returns are important as not only can performance be seen as better but the

relative ranking of what is the “best” strategy can change once the impact of skewness

and kurtosis is recognised. We conclude that once should recognise the impact of

skewness and excessive kurtosis when measuring the performance of bond strategies.

One drawback of these strategies is inherited from the effectiveness of

immunization itself to guarantee a target return due to non-parallel shifts in the yield

curve. Additionally contingent immunization strategies do experience violations of the

stop loss limit although these violations are limited in both frequency and size. On the

other hand contingent immunization strategies do limit the potential loss from an active

bond portfolio strategy. Overall contingent immunization provides, on average,

excellent results.

It is important to note that these strategies are very simple to implement and

monitor. They provide a very flexible instrument to adjust the degree of risk assumed by

the investor and, at the same time, they give the investor much of the upside potential

available from the more risky active management strategies. Moreover contingent

immunization achieves an attractive distribution of returns without the need for complex

valuation models and hedging strategies requiring the use of often illiquid interest rate

derivatives.

21

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24

Table 1. Summary statistics of monthly changes of interest rates from term structure

estimations in the Spanish Public Debt Market

Jan 93-Jan 03 Jan 93-Dec 98 Jan 99- Jan 03

1month 1 year 10 year 1month 1 year 10 year 1month 1 year 10 year

Mean (%) -0.099 -0.089 -0.058 -0.174 -0.147 -0.102 0.012 -0.006 0.005

Stand.Dev. 0.630 0.330 0.314 0.787 0.379 0.373 0.232 0.219 0.185

Median -0.030 -0.120 -0.050 -0.173 -0.143 -0.076 0.033 0.00002 -0.012

Maximum 2.069 0.581 0.755 2.069 0.581 0.755 0.370 0.466 0.391

Minimum -4.049 -2.287 -1.142 -4.049 -2.287 -1.142 -0.591 -0.545 -0.372

Skewness -2.222 -2.483 -0.472 -1.668 -2.487 -0.229 -0.670 -0.087 0.190

Kurtosis 16.903 15.844 1.656 10.462 13.931 0.701 0.302 -0.419 -0.665

25

Table 2. Summary statistics of target returns (three-year zero coupon bond yields)

Level (%) Three year changes

Jan93-Jan03 Jan93-Dec98 Jan98-Jan03 Jan93-Jan03 Jan93-Dec98 Jan98-Jan03

Mean 6.47 8.87 4.18 -2.46 -3.60 0.07

St. Dev. 2.91 2.33 0.67 2.69 2.42 0.99

Median 5.13 8.81 4.35 -1.77 -3.49 0.16

Maximum 13.05 13.05 5.25 1.53 0.30 1.53

Mínimum 3.07 4.95 3.07 -7.36 -7.36 -1.77

Skewness 0.73 -0.13 -0.30 -0.43 -0.04 -0.59

Kurtosis -0.82 -0.87 -0.88 -1.10 -1.27 0.46

Table 3. Summary Results of Contingent Immunization Strategies for Twenty Nine Holding Three-Year

Periods (Jan 93- Jan 03)

Mean Minimum Maximum

H – D Lower bound for losses (b.p.) Lower bound for losses (b.p.) Lower bound for losses (b.p.)

(years) -50 -100 -150 Act. -50 -100 -150 Act. -50 -100 -150 Act.

-1.50 166 198 197 207 -67 -112 -144 9 483 483 483 483

-1.00 129 137 145 145 -64 -97 -4 -5 347 347 347 347

-0.50 80 83 83 83 -64 -32 -32 -32 212 212 212 212

+0.00 19 19 19 N/A -64 -64 -64 N/A 83 83 83 N/A

+0.50 -24 -39 -42 -39 -74 -118 -139 -131 43 43 43 36

+1.00 -34 -60 -76 -84 -94 -137 -179 -244 47 47 47 42

+1.50 -30 -51 -70 -103 -97 -149 -190 -343 74 74 74 53

27

Table 4: Distributional characteristics of the holding period returns of the strategies. Full sample period.

Moment Mean Variance Skewness Kurtosis

IBEX-35 0.2121 0.2052 -0.2244 -0.8765

H – D Lower bound for losses (b.p.) Lower bound for losses (b.p.) Lower bound for losses (b.p.) Lower bound for losses (b.p.)

(years) -50 -100 -150 Act. -50 -100 -150 Act. -50 -100 -150 Act. -50 -100 -150 Act.

-1.50 0.0910 0.0942 0.0941 0.0951 0.0430 0.0433 0.0434 0.0429 0.1538 0.0814 0.0848 0.0270 -1.3913 -1.5635 -1.5712 -1.5176

-1.00 0.0874 0.0882 0.0889 0.0890 0.0394 0.0389 0.0387 0.0387 0.0111 0.0709 0.0176 0.0140 -1.5476 -1.5952 -1.5621 -1.5611

-0.50 0.0824 0.0827 0.0827 0.0828 0.0347 0.0347 0.0347 0.0347 0.0115 0.0319 0.0319 0.0272 -1.5815 -1.5694 -1.5694 -1.5676

+0.00 0.0764 0.0764 0.0764 0.0764 0.0311 0.0311 0.0311 0.0311 0.0943 0.0943 0.0943 0.0943 -1.4781 -1.4781 -1.4781 -1.4781

+0.50 0.0721 0.0707 0.0703 0.0704 0.0299 0.0287 0.0282 0.0284 0.2476 0.2484 0.2784 0.2814 -1.3348 -1.2988 -1.1576 -1.1678

+1.00 0.0711 0.0685 0.0669 0.0660 0.0304 0.0285 0.0268 0.0261 0.2407 0.2265 0.2211 0.3395 -1.4976 -1.5400 -1.5395 -1.1368

+1.50 0.0716 0.0697 0.0675 0.0640 0.0300 0.0286 0.0273 0.0253 0.2316 0.2400 0.1714 0.3980 -1.5213 -1.4990 -1.6748 -1.0756

28

Table 5: Performance of the strategies. Full sample period.

Moment Sharp M2 ASR MSR

IBEX-35 0.7531 N/A 0.3232 0.3318

H – D Lower bound for losses (b.p.) Lower bound for losses (b.p.) Lower bound for losses (b.p.) Lower bound for losses (b.p.)

(years) -50 -100 -150 Act. -50 -100 -150 Act. -50 -100 -150 Act. -50 -100 -150 Act.

-1.50 0.7771 0.8474 0.8432 0.8760 0.0049 0.0194 0.0185 0.0252 0.3335 0.3637 0.3619 0.3760 0.4135 0.4463 0.4452 0.4488

-1.00 0.7574 0.7866 0.8104 0.8116 0.0009 0.0069 0.0118 0.0120 0.3251 0.3376 0.3478 0.3483 0.3871 0.4141 0.4159 0.4159

-0.50 0.7173 0.7251 0.7251 0.7269 -0.0073 -0.0057 -0.0057 -0.0054 0.3079 0.3112 0.3112 0.3120 0.3681 0.3745 0.3745 0.3747

+0.00 0.6069 0.6069 0.6069 0.6069 -0.0300 -0.0300 -0.0300 -0.0300 0.2605 0.2605 0.2605 0.2605 0.3180 0.3180 0.3180 0.3180

+0.50 0.4866 0.4570 0.4515 0.4541 -0.0547 -0.0608 -0.0619 -0.0613 0.2088 0.1961 0.1938 0.1949 0.2689 0.2515 0.2478 0.2499

+1.00 0.4454 0.3821 0.3500 0.3231 -0.0631 -0.0761 -0.0827 -0.0882 0.1912 0.1640 0.1502 0.1387 0.2506 0.2146 0.1961 0.1828

+1.50 0.4674 0.4263 0.3653 0.2571 -0.0586 -0.0635 -0.0796 -0.1018 0.2006 0.1830 0.1568 0.1103 0.2626 0.2398 0.2033 0.1492

Figure 1. Level of one-month, one-year and ten-year interest rates

0

2

4

6

8

10

12

14

16

18

2004

/01/

1993

04/0

7/19

93

04/0

1/19

94

04/0

7/19

94

04/0

1/19

95

04/0

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95

04/0

1/19

96

04/0

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96

04/0

1/19

97

04/0

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97

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1/20

00

04/0

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01

04/0

7/20

01

04/0

1/20

02

04/0

7/20

02

Date

% in

tere

st rat

es

1 month interest rate 1 year interest rate 10 years interest rate

30

Figure 2. Results of Contingent Immunization

1(a) dif = -1.5. Stop loss: 50 basis points

-100 -50

0 50

100 150 200 250 300 350 400 450 500

-900 -800 -700 -600 -500 -400 -300 -200 -100 0 100 200 Unexpected interest rate change (b.p.)

Act

ual-t

arge

t ret

urn

(b.p

.)

2(b) dif = -0.5. Stop loss: 50 basis points

-100 -50

0 50

100 150 200 250 300 350 400 450 500

-900 -800 -700 -600 -500 -400 -300 -200 -100 0 100 200 Unexpected interest rate change (b.p.)

Act

ual-T

arge

t ret

urn

(b.p

.)

31

Figure 2 (continued). Results of Contingent Immunization

2(c) dif = +0.5. Stop loss: 50 basis points

-100 -50

0 50

100 150 200 250 300 350 400 450 500

-900 -800 -700 -600 -500 -400 -300 -200 -100 0 100 200 Unexpected interest rate change (b.p.)

Act

ual-T

arge

t ret

urn

(b.p

.)

2(d) dif = +1.5. Stop loss: 50 basis points

-100 -50

0 50

100 150 200 250 300 350 400 450 500

-900 -800 -700 -600 -500 -400 -300 -200 -100 0 100 200 Unexpected interest rate change (b.p.)

Act

ual-T

arge

t ret

urn

(b.p

.)

32

Figure 3. Pure immunization strategy

1 Particularly it could be compared with barrier options. 2 See www.bde.es/banota/series.htm. Of the 66 instruments, at least 29 and at most 33 are outstanding at any point in time during the sample period. 3 Daily estimates were made using Vasicek and Fong (1982) methodology applied to mean daily prices of all bond and bills traded each day. 4 Monthly changes in one, three, six and twelve-month interest rates and two year to ten-year interest rates were used as inputs to undertake factor analysis. 5 This risk is also known as immunization risk, that is the risk of not obtaining the target return (or promised return) at the end of the investor holding period when carrying an immunization strategy due to term structure movements different from those assumed (usually parallel shifts of the term structure). 6 Liquidity is measured by its trading volume. For a more detailed analysis of the liquidity of the Spanish Treasury bond market and other related institutional issues see, for instance, Díaz, Merrick and Navarro (2006). 7 Usually, the most common assumption when testing immunization strategies is that portfolio weights are adjusted periodically. For instance, Fooladi and Roberts (1992) assume semiannual rebalancings meanwhile Soto (2001, 2004) assumes quarterly adjustments. This procedure may have an important impact especially if interest rates fluctuate sharply as it happened in the Spanish market during the period 1993-1998. 8 Our version of Fisher & Weil’s duration assumes parallel shifts in the term structure, but the term structure itself can be of any shape.

-100 -50

0 50

100 150 200 250 300 350 400 450 500

-900 -800 -700 -600 -500 -400 -300 -200 -100 0 100 200 Unexpected interest rate change (b.p.)

A

ctua

l-Tar

get r

etur

n (b

.p.)

33

9 Another possibility would be to invest in the shortest asset available, for instance in one week repo. However this strategy would incur very heavy transaction costs making this strategy extremely costly and unrealistic. 10 One could avoid this problem by checking portfolio value continuously but this would be unfeasible in practice. 11 Although the new European currency came officially into effect the first of January 1999, the convergence process towards the new monetary area was establishment before that date. The convergence of forward rates between Spanish and central European economies was accomplished at least one year before. So we find more adequate to subdivide the sample period in January 1998. 12 Recall that we have term “target return” the objective return of pure immunization strategies, in our study, the three year zero coupon bond yield at the beginning of each holding period. 13These results are in accordance with other studies where immunization has been tested using one-factor models of the term structure. For this reason, some authors suggest the use of multifactor term structure models for immunization. See for instance Elton, Gruber and Michaely (1990) and Reitano (1992) who suggest the use of a duration vector, or Navarro and Nave (1997) and Soto (2001, 2003) for its application to the Spanish Treasury market. This sort of models usually imply further constrains for the portfolio holdings adding up complexity to the portfolio selection model. 14 Note that strategies with duration lower than the investor horizon cannot be maintained for the full three year planning horizon. For example a strategy with H- D = 1.50 is limited to a period of one and a half years. Recall that, in this case, after 18 months it is not possible to keep the difference between portfolio duration and the remaining holding period equal to 1.5 years and so we proceed to immunize. Although an alternative could have been to invest in very short-term instruments (for instance in the repo market) we discarded this option for unrealistic due to the extraordinarily large number of transactions need to fulfil this strategy. 15 The holding period returns for the IBEX 35 is derived from total returns that assumes that all dividends are re-invested in the stocks just as we assume all coupons are re-invested in the bond portfolio strategies. The IBEX 35 holding period returns are calculated over precisely the same dates as the bond portfolio management strategies. 16 We split the sample into two parts to see whether this conclusion is robust to the different interest rate environments as reported earlier in Table 1 and Figure1. The conclusions we reach with respect to the bond strategies are the same. However during the first period when interest rates were sharply decreasing the equity market performed better with an extraordinarily high mean of 34% and higher positive skewness and lower excessive kurtosis that the bond strategies. Again however the standard deviation of the equity strategy was far higher than all the bond strategies. For the sake of brevity we chose not to report these results but they are available from the authors upon request. 17 Strictly speaking this is a “striped down” version of VAR as usually VAR = VpσNT0.05. However as shown in Favre and Galeano 2002 the value of the portfolio Vp appears in the numerator of the adjusted Sharpe ratio and so cancels out when the full VAR expression is included in the denominator. Therefiore we neglect the term Vp as it will cancel out anyway in the ASR. Also, the square route of time is neglected by Favre and Galeano 2002. In essence they assume that the position can be liquidated in one day so the daily earnings at risk are the same as the value at risk. 18 As discussed in note 14 we split the sample into two parts to see whether this conclusion is robust to the different interest rate environments as reported earlier in Table 1 and Figure1. The sub period results are somewhat different. During the first period when interest rates were sharply decreasing, low duration portfolios with tight stop loss limits performed better when we include the impact of skewness and excessive kurtosis than the more active strategies. Also the equity market performed better that the bond strategies. Nevertheless the conclusion is the same. Accounting for the additional distributional characteristics of skewness and kurtosis can change the relative ranking of which strategy performs best. For the sake of brevity we chose not to report these results but they are available from the authors upon request.