Using Generalized Immunization Techniques with Multiple Liabilities: Matching an Index in the UK Gilt Market. Michael Theobald * and Peter Yallup **

Transcription

1 Using Generalized Immunization Techniques with Multiple Liabilities: Matching an Index in the UK Gilt Market Michael Theobald * and Peter Yallup ** (January 2005) * Accounting and Finance Subject Group, Birmingham Business School, University of Birmingham, England ** Centre for Hedge Fund Education and Research, London Business School, England Address for correspondence: Peter Yallup Centre for Hedge Fund Education and Research London Business School Sussex Place Regent s Park London NW 4SA England PYallup@london.edu fax: (44)

2 Using Generalized Immunization Techniques with Multiple Liabilities: Matching an Index in the UK Gilt Market ABSTRACT In the standard immunization problem an immunizing portfolio of bonds is constructed which will produce enough cash to pay a given single liability at a particular time horizon in the future. We know that if the yield curve moves in a parallel fashion then this can be achieved if the immunizing portfolio has a Macaulay Duration equal to the horizon. With non-parallel movements in the yield curve a single zero coupon bond of the relevant maturity will provide superior immunization. Alternative approaches have been developed in which the standard duration measure is augmented by other statistics so that a portfolio of bonds provides immunization for non-parallel yield curve shifts in markets where zero coupon bonds of the relevant maturity are not available. In this paper we argue that, whilst the horizon/immunization problem is interesting it is of limited practical value to asset and liability management. The single dated liability problem is not a common problem. Also, in general, zero coupon bonds are usually available, or can be readily created from a medium term note programme or swap contract. In addition, these techniques can involve impractical amounts of short selling. A more relevant practical problem is one in which there are multiple liabilities at differing future times. We show how these generalized immunization techniques can be applied in the selection of immunizing portfolios to solve such problems. As an example we look at the problem of selecting an index tracking portfolio in the UK Gilt market.

3 Using Generalized Immunization Techniques with Multiple Liabilities: Matching an Index in the UK Gilt Market. Introduction The approaches to handling interest rate risk range from cash matching, through simple duration to multifactor immunization methods. While the simple duration based strategies do perform well relative to the more sophisticated techniques such as delta-gamma or factor based methods (see, for example, Jarrow and Turnbull [996], Nelson and Schaefer [983], Ho, Cadle and Theobald [200]; Elton, Gruber and Michaely [990] demonstrate how the performance of factor based bond strategies can be improved), there are a number of strong assumptions underlying their application (Bierwag, Kaufman and Khang [978], Bierwag [987] and Prisman and Shores [988] provide an extensive presentation and analysis of these matters). In particular, simple duration based immunization techniques rely upon the assumption that interest rates shift in a parallel fashion. If interest rates do not shift in this fashion, simple duration based immunization strategies will be sub-optimal. Fong and Vasicek [984] developed a measure which they call immunization risk; if this proxy is minimized, the bond portfolio will be immunized against changes in the slope of the yield curve. In other papers this approach is extended to a more general immunization concept so that more complex changes in the yield curve can be immunized (see Chambers, Carlton and McEnally [988], Crack and Nawalkha [2000], Nawalkha and Chambers [997] and de La Granville [200]). 2

4 Whilst the general immunization horizon problem is pedagogically interesting it is probably not the most useful tool for the bond portfolio manager. In this paper we suggest that the general immunization model can solve some more practical problems. 2. General Immunization Theorem The value of a bond portfolio is given by:- n B0 = c(t)exp[ G(t)] t= () Where c(t) is the cashflow on the bond portfolio at time t, and exp[-g(t)] is the discount function where G(t) is given by:- G(t) = R(t)t = t 0 f(u)du.(2) R(t) is the continuously compounded return over the interval [0,t] and f(u) is the instantaneous forward rate beginning at time u, set at time 0. D is the Macaulay Duration of a bond portfolio given by:- n D = tc(t)exp[ G(t)].(3) B 0 t= The duration is an important measure of a bond portfolio as it can be used to immunize a liability. If the value of a bond portfolio is set equal to the present value of a liability (or target amount) at some horizon date, and if the duration of a bond portfolio is equal to the investment horizon, the value of the bond portfolio will remain at least as much as the present value of the liability, when the yield curve 3

5 suffers an instantaneous shock which results in a parallel shift. In such a case the bond portfolio is described as an immunizing portfolio. However, when yields change they do not necessarily do so in a parallel fashion. Where this is the case it is possible for the present value of the target liability to exceed the value of the immunizing portfolio. As a consequence therefore there has been some interest in developing immunization measures which can immunize against more complex changes in the yield curve. Fong and Vasicek [984] developed a measure which they call immunization risk M 2 defined as:- n 2 M 2 = (t H) c(t)exp[ G(t)] B 0 t= (4) where H is the investment horizon. In addition to the basic immunization conditions above, immunization in the Fong and Vasicek [984] model requires that M 2 should be zero. Note that, provided c(t) >= 0, for all t then M 2 >= 0. That is provided all investments produce positive cashflows, then the immunization risk measure cannot be negative. If all cashflows are positive, for the M 2 statistic to be zero requires that (t-h) 2 is zero, this is the case when the asset portfolio consists of a single, zero coupon bond with its maturity matching the horizon H. The M 2 concept is empirically examined in Bierwag, Fooladi and Roberts [993]. In Fong and Vasicek [984], M 2 is described as a weighted variance of time to payments around a horizon date. The power of this result comes from the fact that, to 4

6 derive this measure Fong and Vasicek [984] did not need to assume any particular model of the (dynamics of the) term structure of interest rates. In various papers the M 2 model has been extended to further terms, Nawalkha and Chambers [997] describe the M-Vector model where M k is given by 2 :- M k n k = (t H) c(t)exp[ G(t)] B 0 t= (5) where k = {,2,3,..} For our purposes a more practical measure is the related I k model which de La Granville [200] terms Moments of Immunization (see also Prisman and Shores [988] and Crack and Nawalkha [2000] 3 ) defined as:- I k n = t B 0 t= k c(t)exp[ G(t)] (6) where k = {,2,3,..} If k= the expression reduces to that of Duration. If k=2 we get an expression similar to that of M 2 except that it does not include H. Perfect immunization would require that all moments of immunization are determined such that I k = H k, which could only be the case if the only asset in the asset portfolio were a zero coupon bond whose maturity was the same as the horizon. Very good immunization can be achieved if (as de La Granville [200] demonstrates numerically) I k =H k for k={,2,3}. To determine the amount of each bond required to immunize the horizon value against an instantaneous shock to the yield curve, requires solving four simultaneous 5

7 equations, with four unknowns. Three of the equations are, I k = H k, for k=,2,3 and the fourth is for the value of the bond portfolio being equal to the present value of the horizon liability; the four unknowns are the investments in the four bonds. There are, however, a number of practical problems in this approach. Firstly, in most developed markets zero coupon bonds are available in the secondary market or can readily be created (e.g. from a Medium Term Note programme, or an asset swap). Where a zero coupon bond is available with the same maturity as the horizon, an investment in these zeros will provide a superior (and simpler) immunization strategy. Secondly, and perhaps more importantly the approach assumes that it is possible to short sell bonds without any of the normal costs involved (e.g. bid ask spread, borrowing costs etc.) As a result in this approach immunization works well provided that we can short sell large amounts of certain bonds, in order to be very heavily invested in others. Over time the de La Granville [200] approach will also imply significant adjustments to the immunizing portfolio. Whilst the horizon/immunization problem is interesting it is of limited practical value to asset and liability management. A more practical problem is one in which there are several liabilities at differing future times. Examples would be where a portfolio is designed to provide known future payments (e.g. pension or insurance liabilities) or where a portfolio is designed to match an index (in which case the cashflows of the index define the liability). 6

8 3. Multiple Liabilities Usually it is implicitly assumed that bonds selected for the asset portfolio have only positive cashflows. But, it is possible to find bonds which incorporate negative cashflows in their structure. An example would be a partly paid bond. In a partly paid bond only a proportion of the bonds value is paid at issue, the balance is paid at various date(s) in the future. The characteristic of this type of issue is that, whilst it is partly paid, the bonds value rises (or falls) by a greater proportion in response to interest rate changes than a similar fully paid bond of the same maturity. Adding bonds with negative cashflows, or fixed rate borrowings to the portfolio of assets does not create any problems with regard to the calculation of immunization statistics. The admission of negative cashflows allows important insights into the immunization process. The immunization problem addressed by Fisher and Weil [97] and Fong and Vasicek [984] among many others is one in which a single liability cashflow is immunized. If we can admit negative cashflows into the asset portfolio we have a way of immunizing multiperiod liability cashflows. One of the liability cashflows is selected as the target cashflow of immunization (e.g. the longest maturity liability), the remaining ones (the shorter liabilities) are included as negative cashflows in the immunizing portfolio. The balance if the immunizing portfolio is then selected such that the total of the immunizing portfolio, immunizes the target cashflow. The resulting immunization problem can be written as:- 7

9 Minimize I 0 X Subject to I 0 X - J 0 Y >= V I k X - J k Y = H k X, Y >= 0 k = {,2, p} (7) Where:- I k is a ( x n) row matrix of the k th immunization moment of the n assets, X is a (n x ) column matrix of present values of the n assets. J k is a ( x [m- ]) row matrix of the k th immunization moment of the m- short liabilities, V is the present value of the long liability, Y is a ([m-] x ) column matrix of present values of the m- short liabilities and H is the horizon of the long liability. The constraint I 0 X - J 0 Y >= V ensures that the present value of the asset portfolio minus the present value of the short liabilities is equal to or exceeds the present value of the long liability (since both I 0 and J 0 are unit vectors 4 ). Furthermore the objective of the immunization (to minimize I 0 X) is to find the least cost portfolio of assets for the immunization. Note also that the non-negativity constraints ensure that short selling is not a feature of the optimization structure 5. Clearly equation system (7) could be readily solved using linear programming. With some rearrangement the problem can also be reformulated as:- 8

10 Minimize I 0 X Subject to I 0 X >= L 0 Z I k X = L k Z X, Z >= 0 k = {,2, p} (8) L k is a (m x ) row matrix of the k th immunization moment of the m liabilities (this matrix is the same as J k but with an additional last element which is H k ), Z is a (m x ) column matrix of the present values of the m liabilities (this matrix is the same as Y but with an additional last element which is V). In statistics, two distributions are equivalent if their moments are the same without reference to the type of distribution. By analogy, therefore it seems appropriate that we can say that one set of cashflows is immunized by another if the present values are equal (and opposite) and their moments of immunization agree; this is indeed captured in the above equation system, a more rigorous proof of this result is given in the Appendix. Here, immunization (of order p) requires that the moments of immunization (from 0 to p) of the asset portfolio and the liability portfolio are the same. Immunization turns out to be determined by how closely the distribution of the cashflows of assets and liabilities agree. For simple immunization only the first moments (durations) agree. For complete immunization all moments would agree, which would be the case only when the cashflows of the assets exactly match the cashflows of the liabilities (i.e. the full cash matching approach). 9

11 Hodges and Schaefer [977], suggest a linear programming model 6 as a method of selecting the least cost portfolio of bonds which would provide (match) a set of liabilities. Immunization of order p, with p selected to be sufficiently large will be expected to achieve a similar result. In the case where p is smaller a cheaper asset portfolio might be selected. In the case of p=0 (i.e. matched in terms of present value only) immunization would be expected to be poor. Immunization is thus a trade off between the asset portfolio cost and the performance of the portfolio as an immunizing portfolio. Where the liability is expected to be constant over time, this may imply selecting p to be large. Where the liability portfolio might reasonably be expected to change over time (e.g. where pension or insurance type liabilities change) a lesser value of p might be selected, as we may expect that the asset portfolio will have to be rebalanced at intervals. In the case of changing liabilities, it might be reasonable to cash match over the short term, whilst immunizing the portfolio. This would imply having a system of equations such as (7) or (8) plus some additional constraints (similar to those in Hodges and Schaefer [977]) matching the cashflows produced by the assets to the liabilities in the short term. 4. Empirical Analysis A typical requirement in the management of fixed income portfolios is to construct a portfolio which will produce a performance matching that of an index. This can be achieved by purchasing all the constituents of the index in the appropriate proportions. However, this approach can involve many small transactions, particularly when money is added to or withdrawn from the portfolio. One way of 0

12 reducing the number of transactions is to construct the portfolio such that is does not contain every constituent but rather a representative sample of bonds which would track the performance of the index in a similar fashion to strategies adopted in the equity markets. Immunization Moments provide an ideal way of selecting this representative index tracking portfolio in government bond markets via matching across moments. The construction of index tracking portfolios using Immunization Moments is empirically investigated using UK Gilt data. These Gilt data were obtained from the UK Debt Management Office (DMO) website 7. These data comprised daily end of day reference prices 8 and nominal amount issued for fixed coupon Gilts for the period from 5th December 997 to 5th December Prices for Gilt coupon strips were also obtained for the same period. Portfolios of Gilt stocks were constructed on a quarterly basis starting on the 5 th December 997 and running to 5 th September These portfolios consisted of all the bonds which had a single defined maturity date 9, had at least 3 months until maturity and which were not at that time designated as rump stocks 0. See Table and 2 for the composition of these reference portfolios at 5 th December 997 and 5 th September 2003 respectively. The amount of each Gilt in these portfolios was in direct proportion to the amount in issue on that date. These quarterly portfolios represent the Gilt market index that we will track and whose composition we assume, for simplicity, remains constant throughout the quarter. The yield on the Gilt coupon strips was used to determine an approximation to the term structure with yields being interpolated between maturities.

13 For each Gilt, the moments I 0 to I 6 were calculated using its pattern of cashflows and the term structure. The moments for the total Gilt index portfolio were also calculated from the moments of the constituents using the nominal amounts of each constituent Gilt as weights. For each quarter there were six Linear Programming problems formulated with I 0 to I, I 0 to I 2, I 0 to I 3, I 0 to I 4, I 0 to I 5 and I 0 to I 6 constrained to be equal to the moments of the Gilt index portfolio (as in equation (8)). The objective of each linear programming problem was to minimize the cost of the bonds included in the solution to this problem. Solutions to the linear programming problem (in terms of the nominal investment amounts of bonds in the solution) for 5 th December 997 and 5 th September 2003 are shown in Tables and 2 respectively. The cost of these solutions and the Immunization Moments for both the Gilt index portfolio and the optimal portfolios are shown in Tables 3 and 4. In Tables and 2 we can see that the number of bonds included in the optimal portfolios is, as anticipated, the same as the number of Immunization Moments that are constrained. In addition we can see in Table 3 and 4 that as more constrains are added the cost of the optimal portfolio increases, but even with I 0 to I 6 constrained the cost of the optimal portfolio is cheaper than the Gilt index portfolio (i.e. comprising all relevant Gilts). Table 5 shows the optimal portfolios, with I 0 to I 6 fixed, for each quarterly period from 5 th December 997 to 5 th September There is some evidence of stability from period to period in terms of the stocks selected for the optimal portfolio. In some cases the shadow price of a stock which is not optimal shows that the stock was 2

14 very close to being included in the optimal portfolio. For example, the 9.75% due 2002 Gilt was not optimal in June 998 despite being optimal in both March and September of 998. The shadow price indicates that if the stock were 0.02% cheaper, then it would have been optimal. In a practical context where we are looking at a periodic rebalancing of the optimal portfolio, a price difference of 0.02% would probably not cover the bid-ask spread for the stock concerned. It would be more appropriate to formulate the problem as one of portfolio improvement rather than portfolio selection. Here, we would incorporate the existing stock holdings into the optimization using extra variables priced at the bid-side in the objective function. In essence the optimization would be able to include any of the existing holdings at the cheaper bid-side price since the bid-side price is their opportunity cost. In Table 6 we report the standard deviation of tracking errors of the optimal portfolios over the following three month period. The portfolios are constructed to have the same value on the start date of the measurement period. The tracking error is then measured as the market value of the Gilt index minus the market value of the optimal portfolio. In Figure we can see, as an example, the tracking errors of optimal portfolios formed at 5 th December 997. In general the tracking error standard deviation reduces as the number of moments optimized is increased. The reduction is particularly apparent when compared with simple duration matching. That is, on average the tracking error standard deviation using Immunization Moments I 0 to I 2 (I 0 to I 6 ) is 58% (6%) of the tracking error standard deviation using moments I 0 to I alone. 3

15 In Table 7 we examine the stability of the tracking error. The twenty four, three month periods are each divided into two sub-periods of one and a half months. Jarque-Bera tests of normality (at the % level) of the three month data suggest that the tracking errors are not normally distributed; therefore we use non-parametric tests to compare means and standard deviations of the two sub-periods. These tests are the Kruskall-Wallis test for means and the modified Levene test (Brown and Forsythe [974]) for standard deviations. The Kruskall-Wallis statistic suggests that there is a significant difference (at the % level) in the two period means in a large number of cases (for I 0 to I fixed this is 7 out of 24 cases). In the majority of these cases the mean tracking error has fallen (i.e. the optimal portfolio has outperformed the market index). Averaging the tracking error over all cases gives a fall in the error of approximately 2 basis points between the two sub-periods. This small, but significant, result is consistent with the optimal portfolio being a cheaper portfolio than the index portfolio. It should, therefore, be expected to outperform if it is indeed also an immunizing portfolio. The modified Levene statistic suggests that there is a significant difference (at the % level) in the two period standard deviations in a large number of cases (for I 0 to I fixed this is 0 out of 24 cases). However, it seems that the standard deviation is almost equally likely to fall as to rise. Clearly the tracking error is non-stationary, and the Dickey Fuller test for a unit root suggests that we cannot reject the hypothesis of a unit root in the vast majority of cases. The size of the tracking error samples prevents a more rigorous examination of the stochastic process involved, but these statistics suggest that it would not be unreasonable to model the tracking error as a random walk with drift. 4

16 In order to gain some further understanding of how Immunization Moment matching can work we examined the impact of various synthetic changes in the term structure on the present values of the Gilt index and the optimal portfolios. If the term structure can be approximated by a polynomial function of time (see Appendix) the difference between any two polynomial functions (i.e. before and after some shock to the term structure) will also be polynomial. Here we examine changes in the term structure which are polynomial functions, t 0 (constant), t (linear), t 2 (quadratic), t 3 (cubic) and t 4 (quartic). In each case the function was scaled so that the maximum move in the term structure at any point is approximately 50 basis points (thus the coefficients of each function depend upon the maximum maturity in the Gilt index). In Table 8 we present the results of the synthetic term structure changes on the present values of the Gilt index and the optimal portfolios on 5 th December 997. Here, as we consider more elaborate changes (quadratic, cubic and quartic) it becomes apparent that more immunization moments are necessary to achieve immunization. For example, where cubic and quartic changes occur, the portfolio has zero sensitivity in all cases to interest rate changes where six moments are fixed. In Table 9 we examine similar synthetic changes on 5 th September In this case it would appear that the optimal portfolios with moments I 0 to I 2 fixed are better immunized against the more complex changes in the term structure. The reason for this may be found in Table 4. It is apparent that the optimal portfolios with moments I 0 to I 2 fixed already has moments I 0 to I 6 which are very close to those of those of the Gilt index. When the higher moments are matched this will, therefore, result in a less dramatic change in the optimal portfolio. This may also be the reason why (in Table 6) the 5

17 tracking error falls dramatically when the second moment is constrained, but the tracking error falls more slowly thereafter. When duration matching is used this will immunize parallel shifts in the term structure. Linear changes (tilts) in the term structure will require an additional second immunization moment to be matched. As we increase the order of the polynomial function giving the changes in the term structure we will require that an equal number of extra immunization moments are matched in order to achieve immunization. 5. Conclusions We have shown how Immunization Moments can be used to immunize multiple liabilities. As a practical example of this approach we have looked at the problem of constructing an index tracking portfolio in the UK Gilt market. Using Immunization Moments it was possible to construct portfolios consisting of small numbers of Gilts which would track the index with tracking errors which have much smaller standard deviations than using Macaulay Duration alone. Furthermore, the costs of the Immunization Moment matched portfolios were always less than those of the portfolio comprising all relevant Gilts (i.e. the bond market portfolio). Looking at synthetic changes in the term structure gives further insights into the way in which the Immunization Moment matching works. Higher order polynomial changes in the term structure will require a corresponding increase in the number of matched Immunization Moments for immunization to be achieved. 6

18 7

19 Appendix Assume that G(t) (where exp[-g(t)] is the discount function) can be written as a McLaurin series: G(t) = G (0) (0) + G () G (0)t + (2) (0)t 2! 2 G (m) (0)t m! m G + (m+ ) (0)( θt)t (m + )! m+ 0 < θ <.(A) Therefore G(t) can be approximated to any desired accuracy by the polynomial: G (t) A m 2 m j 0 + At + A 2t A mt = A tt j= 0 where:- ( j) G (0) A = j j! j = 0,,..., m.(a2) Let N be a time in the future such that N represents the last time either a liability in the liability portfolio or an asset (bond) in the asset portfolio will be paid. Let R(t) be the continuously compounded rate of return function and f(u) the forward rate function for instantaneous lending agreed at the time (0) that the immunization is carried out. Using (A2) the value of the asset portfolio can be written:- B = N t f (u)du N N m 0 R(t)t (A0 At... Amt ) c(t)e = c(t)e c(t) e t= t= t=.(a4) And similarly the value of the liabilities can be written:- L = N t f (u)du N N m 0 R(t)t (A0 At... Amt ) z(t)e = z(t)e z(t) e t= t= t=.(a5) where z(t) is the cashflow of the liability portfolio at time t. 8

20 For immunization we require that B 0 = L 0. In addition we require that if immediately following the immunization the term structure of interest rates undergoes a shock, both the values of asset portfolio and the liability portfolio should be affected in the same way. Since the term structure can be represented by (A2), any change in the spot structure is equivalent to changing the Aj s. Hence we require that the derivatives (with respect to the Aj s) of the value of the asset and liability portfolios be equal, that is:- B 0 A j = L 0 A j j = 0,,...,m.(A6) Differentiating (A4) with respect to Aj, we get: B A j ( ) j m 0 N (A0+ At Ajt Amt ) = c(t)e t t = j Differentiating (A5) with respect to Aj, we get: L A j ( ) j m 0 N (A0+ At Ajt Amt ) = z(t)e t t = j Hence for immunization we require that:- j = 0,,...,m j = 0,,...,m.(A7).(A8) j m N j (A0+ At Ajt Amt ) N t c(t)e = = = t t t j z(t)e (A j j m m + A t A t A t 0 j = 0,,...,m.(A9) And given that B 0 = L 0 this can also be expressed in terms of the Immunization Moments :- ) N j G(t) N t c(t)e t= t= B 0 j 0 G(t) t z(t)e = j = 0,,..., m L.(A0) 9

21 6. References Balbas, A., A. Ibanez, and S. Lopez, 2002, Dispersion Measures an Immunization Risk Measures, Journal of Banking and Finance, 26, Bierwag, G. O., 987, Duration Analysis: Managing Interest Rate Risk, (Ballinger). Bierwag, G. O., I. Fooladi, and G. S. Roberts, 993, Designing an Immunized Portfolio: Is M-Squared the Key?, Journal of Banking and Finance, 7, Bierwag, G. O., G. G. Kaufman, and C. Khang, 978, Duration and Bond Portfolio Analysis: An Overview, Journal of Financial and Quantitative Analysis, 3, Brown, M. B. and A. B. Forsythe, 974, Robust Tests for the Equality of Variances, Journal of the American Statistical Association, 69, Chambers, D. R., W. T. Carlton, and R. W. McEnally, 988, Immunizing Default- Free Bond Portfolios with a Duration Vector, Journal of Financial and Quantitative Analysis, 23, Crack, T. F. and S. K. Nawalkha, 2000, Interest Rate Sensitivities of Bond Risk Measures, Financial Analysts Journal, 56, de La Grandville, O., 200, Bond Pricing and Portfolio Analysis, (MIT Press). 20

22 Elton, E. J., M. J. Gruber and R. Michaely, 990, The Structure of Spot Rates and Immunization, Journal of Finance, 45, Fisher, L., and R. L. Weil, 97, Coping with Risk of Interest Rate Fluctuations: Returns to Bondholders from Naïve and Optimal Strategies, Journal of Business, 44, Fong, H. G., and O. A. Vasicek, 984, A Risk Minimizing Strategy for Portfolio Immunization, Journal of Finance, 39, Hodges, S. D., and S. M. Schaefer, 977, A Model for Bond Portfolio Improvement, Journal of Financial and Quantitative Analysis, 2, Ho, L., J. Cadle, and M. F. Theobald, 200, Estimation and Hedging with a One- Factor Heath-Jarrow-Morton Model, Journal of Derivatives, 8, Jarrow, R. A., and S. M. Turnbull, S., 996, Derivative Securities, (Southwestern Publishing). Nawalkha, S. K and D. R. Chambers, 997, The M-Vector Model: Derivation and Testing of Extensions to M-Square, Journal of Portfolio Management, 23, Nelson, J. and S. M. Schaefer, 983, The Dynamics of the Term Structure and Alternative Immunization Strategies, in Innovations in Bond Portfolio Management, eds. Kaufman, G., Bierwag, G. and Toevs, A., JAI Press,

23 Prisman, E. Z. and M. R. Shores, 988, Duration Measures for Specific Term Structure Estimations and Applications to Bond Portfolio Immunization, Journal of Banking and Finance, 2, Sørensen, C., 999, Dynamic Asset Allocation and Fixed Income Management, Journal of Financial and Quantitative Analysis, 34,

24 Table Liquid Fixed Coupon Gilts on 5 th December 997 and Optimal Tracking Portfolios Coupon Maturity Price Nominal I 0 to I I 0 to I 2 I 0 to I 3 I 0 to I 4 I 0 to I 5 I 0 to I 6 Issued Fixed Fixed Fixed Fixed Fixed Fixed Mar ,50 9, , , , Sep Nov , Jan , Mar , May , Aug , Nov , Mar , Jul ,7 6, , Dec , Feb , Nov , Jun , , , , Aug , Jun , Sep , Dec , Oct , Nov , Apr , Dec , Sep , Dec , Jul ,397 38, , , , , , Dec , Oct , Sep , Nov , Jul , Aug , Sep ,00 2, , , , Dec , Aug ,550 63, , , , , Jun ,500 7, , ,

25 Table 2 Liquid Fixed Coupon Gilts on 5 th September 2003 and Optimal Tracking Portfolios Coupon Maturity Price Nominal I 0 to I I 0 to I 2 I 0 to I 3 I 0 to I 4 I 0 to I 5 I 0 to I 6 Issued Fixed Fixed Fixed Fixed Fixed Fixed 5.00% 07-Jun ,504 8, , ,2.47 6, % 26-Nov , % 8-Apr , % 07-Dec , % 08-Sep , % 07-Dec , % 6-Jul ,638 58, , , , , , % 07-Dec-07.8, % 07-Mar , % 07-Mar , % 07-Dec , % 25-Nov , % 2-Jul , , % 07-Mar , % 27-Sep ,8 5.00% 07-Sep , % 07-Dec , % 25-Aug ,75 22, ,0.2 66, , , , % 07-Jun , % 07-Mar ,422 26, % 07-Dec , % 07-Jun ,829 4, , , , , % 07-Mar ,750,480.60, ,

26 Table 3 Portfolio Moments for Optimal Tracking Portfolios on 5 th December 997 Numbers in Italics were Unconstrained in the LP Gilt I 0 to I I 0 to I 2 I 0 to I 3 I 0 to I 4 I 0 to I 5 I 0 to I 6 Market Fixed Fixed Fixed Fixed Fixed Fixed Cost 244, , , , , , ,842.4 I I I I , I 4 5, , , , , , ,67.49 I 5 289, , , , , , ,66.59 I 6 5,729, ,06, ,538, ,570, ,555, ,744, ,729, Table 4 Portfolio Moments for Optimal Tracking Portfolios on 5 th September 2003 Numbers in Italics were Unconstrained in the LP Gilt I 0 to I I 0 to I 2 I 0 to I 3 I 0 to I 4 I 0 to I 5 I 0 to I 6 Market Fixed Fixed Fixed Fixed Fixed Fixed Cost 240, , , , , , ,360.8 I I I I 3 2, , , , , ,08.83 I 4 46, , , , , , , I 5,78, ,893.08,22,588.74,70,09.89,78,37.20,78,227.48,78, I 6 3,725, ,34, ,460, ,70, ,60, ,593, ,725,

27 Table 5 Optimal Tracking Portfolios - I 0 to I 6 Fixed Gilt 7Q 98 5H 98 2% 98 9H 99 2Q 99 0H 99 6% 99 0Q 99 9% 00 Dec Mar 98 Jun 98 Sep 98 Dec 98 Mar 99 Jun 99 Sep 99 (3) (2) (3) (7) () (2) (8) (4) 0 0 (2) (9) (2) 0 (8) (8) (2) 0 0 (5) (9) (7) (5) 0 (8) (9) (8) 4547 Dec 99 Mar 00 Jun 00 Sep 00 3% () (6) % (5) % 0 (6) (3) () () 0 (3) (5) % () (7) % (9) T (2) (6) (5) 0 0 (9) (5) % (2) % 03 0 () () 0 0 (9) (8) (8) (6) () (9) (4) 200 6H () 0 () H 04 0 (3) (5) 0 (6) (7) T (8) 9H 05 0 (6) (5) (2) (3) (2) (5) 8H (3) (8) (9) (8) 7T 06 0 (2) (4) () 7470 (8) (7) (9) (5) () (5) 0 0 (9) (6) 7H (3) (8) (9) 8H (5) 394 (2) (8) Q (3) 0 0 (5) (3) (2) 9% (3) (8) (5) (4) 4% 09 -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- 0 (7) 8% /- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- 5T 09 -/- -/- -/ (7) 0 (4) () (5) 6Q (3) % (3) (3) (2) % 2 -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/ (8) (7) 0 9% 2 () (5) (7) (7) () () (2) /- -/- -/- -/- -/- -/- -/- -/- -/- 8% () % 4 -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/ % (6) (6) 8T (7) (2) % (7) 24 0 (9) 5% 25 -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/ % 28 -/ (6) 4Q 32 -/- -/- -/- -/- -/- -/- -/- -/- -/- -/ Q 36 -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/- -/ Bracketed numbers are shadow prices of stocks not included where the shadow price < 0 basis points. -/- = Stock not in issue or designated rump. Dec 00 Mar 0 Jun 0 Sep 0 Dec 0 Mar 02 Jun 02 Sep 02 Dec 02 Mar 03 Jun 03 Sep 03 26

28 Table 6 The Standard Deviation of Portfolio Tracking Errors* Measured Over the Following Three Months Start I 0 to I I 0 to I 2 I 0 to I 3 I 0 to I 4 I 0 to I 5 I 0 to I 6 Date Fixed Fixed Fixed Fixed Fixed Fixed 5-Dec % % % 0.02% 0.020% % 6-Mar % % 0.028% % 0.086% 0.044% 5-Jun % 0.085% % % % % 5-Sep % 0.249% % 0.053% % 0.040% 5-Dec % 0.337% % % % 0.054% 5-Mar % % % % 0.073% % 5-Jun % 0.83% % % % 0.083% 5-Sep % 0.088% % % % % 5-Dec % % % 0.042% % 0.06% 5-Mar % % % % % % 5-Jun % % 0.039% % % 0.085% 5-Sep % % % % % 0.022% 5-Dec % % 0.034% 0.082% 0.05% 0.02% 5-Mar % % % % 0.049% % 5-Jun % % % 0.063% % 0.028% 4-Sep % % 0.030% 0.028% 0.030% 0.08% 4-Dec % % 0.097% % 0.026% 0.028% 5-Mar % % % % % 0.0% 4-Jun % 0.076% 0.036% % 0.084% 0.083% 6-Sep % % % 0.020% % 0.072% 6-Dec % % % 0.075% 0.029% 0.072% 4-Mar % % 0.054% 0.040% 0.086% 0.060% 6-Jun % 0.239% 0.072% 0.082% 0.08% % 5-Sep % 0.094% 0.06% 0.056% 0.054% % Average 0.25% % 0.045% % % 0.080% * The tracking error is the market value of the Gilt index minus the market value of the optimal portfolio 27

29 Table 7 Analysis of Portfolio Tracking Errors: Comparison of Two Sub-periods of One and a Half Months I 0 to I I 0 to I 2 I 0 to I 3 I 0 to I 4 I 0 to I 5 I 0 to I 6 Fixed Fixed Fixed Fixed Fixed Fixed Total Number of Sample Periods Number of Periods Jarque-Bera is Significant * Number of Periods Where Unit Root is not Rejected** No. of Sample Periods Where Kruskal Wallis Statistic is Significant * Of Which : U > U U < U Average Fall in Mean Tracking Error % % 0.030% % % 0.044% 0.009% Number of Sample Periods Where modified Levene Statistic is Significant * Of Which : S > S S < S Average Fall (Rise) in Tracking Error Std. Dev. % % (0.0036%) (0.007%) (0.0043%) % 0.002% U x is the Mean Tracking Error in Period x; Sx is the Standard Deviation of the Tracking Error in Period x * % Significance Level, ** 5% Significance Level 28

30 Table 8 Impact of Synthetic Term structure Changes on Portfolio Immunization on 5 th December 997 Percentage Differences Between the Present Value of the Gilt Index and Optimal Portfolios after Changes in the Term structure Term structure I 0 to I I 0 to I 2 I 0 to I 3 I 0 to I 4 I 0 to I 5 I 0 to I 6 Change (b.p.) Fixed Fixed Fixed Fixed Fixed Fixed (T=Time in years) % 0% 0% 0% 0% 0% % 0% 0% 0% 0% 0% +2 T -0.04% 0.002% -0.00% 0% 0% 0% -2 T 0.02% 0.002% -0.00% 0% 0% 0% +2(25-T) 0.06% -0.00% -0.00% 0% 0% 0% -2(25-T) % % -0.00% 0% 0% 0% +T 2 / % % % -0.00% 0% 0% -T 2 / % 0.034% % -0.00% 0% 0% (25-T) 2 / % % 0% 0% 0% 0% -(25-T) 2 / % 0.034% 0% 0.00% 0% 0% (2-T) 2 / % -0.22% % % 0% 0% -(2-T) 2 / % 0.% % % 0% 0% +T 3 / % -0.03% 0.02% -0.00% 0% 0% -T 3 / % 0.025% % -0.00% 0% 0% +(25-T) 3 / % % -0.07% 0% 0% 0% -(25-T) 3 / % 0.059% 0.08% 0.00% 0% 0% (2-T) 3 / % -0.46% -0.84% % 0.00% 0% -(2-T) 3 / % 0.27% 0.62% % 0.002% 0% +T 4 / % % 0.04% 0.00% 0.00% 0% -T 4 / % 0.008% % % 0.00% 0% +(25-T) 4 / % % % 0.004% 0% 0% -(25-T) 4 / % 0.09% 0.038% % 0.00% 0% (2-T) 4 / % 0.048% 0.07% 0.057% 0% 0% -(2-T) 4 / % % -0.23% % 0.002% 0% Numbers in bold are >= 0.0% or <= -0.0% 29

31 Table 9 Impact of Synthetic Term structure Changes on Portfolio Immunization on 5 th September 2003 Percentage Differences Between the Present Value of the Gilt Index and Optimal Portfolios after Changes in the Term structure Term structure I 0 to I I 0 to I 2 I 0 to I 3 I 0 to I 4 I 0 to I 5 I 0 to I 6 Change (b.p.) Fixed Fixed Fixed Fixed Fixed Fixed (T=Time in years) % 0% 0% 0% 0% 0% % 0% 0% 0% 0% 0% +.5 T 0.35% 0.00% 0% 0% 0% 0% -.5 T % 0.00% 0% 0% 0% 0% +.5(33-T) % -0.00% 0% 0% 0% 0% -.5(33-T) 0.345% -0.00% 0% 0% 0% 0% +T 2 / % -0.03% 0% 0% 0% 0% -T 2 / % 0.04% -0.00% 0% 0% 0% (33-T) 2 / % -0.04% 0% 0% 0% 0% -(33-T) 2 / % 0.03% 0% 0% 0% 0% (6-T) 2 / % % % % % 0% -(6-T) 2 / 5-0.5% 0.05% % % % 0% +T 3 / % -0.09% 0% -0.00% -0.00% 0% -T 3 / % 0.07% % -0.00% % 0% +(33-T) 3 / % % % 0% 0% 0% -(33-T) 3 / % 0.026% 0.00% 0% 0% 0% (6-T) 3 / % % % % % -0.00% -(6-T) 3 / % 0.07% 0.006% % % 0% +T 4 / % -0.09% 0% % % 0% -T 4 / % 0.03% % % % 0% +(33-T) 4 / % -0.03% % 0% 0% 0% -(33-T) 4 / % 0.03% 0.003% 0% 0% 0% (6-T) 4 / % % -0.00% -0.00% -0.00% -0.00% -(6-T) 4 / % 0.00% % -0.00% -0.0% -0.00% Numbers in bold are >= 0.0% or <= -0.0% 30

32 Figure Portfolio Tracking Errors* From 5 th Dec % I 0 to I Fixed 0.00% -0.05% -0.0% -0.5% 0.05% I 0 to I 2 Fixed 0.00% -0.05% -0.0% -0.5% 0.05% I 0 to I 3 Fixed 0.00% -0.05% -0.0% 0.05% I 0 to I 4 Fixed 0.00% -0.05% -0.0% 0.05% I 0 to I 5 Fixed 0.00% -0.05% -0.0% 0.05% I 0 to I 6 Fixed 0.00% -0.05% -0.0% 5 Dec 97 5 Mar 98 * The tracking error is the market value of the Gilt index minus the market value of the optimal portfolio. 3

33 Footnotes Of course a bond portfolio can consist of a single bond 2 Balbas, Ibanez and Lopez [2002] examine the situation in which only one dispersion measure is to be used. They test M 0.5, M.0, M.5, M 2.0 and M 2.5 and find that M.0 is the best single measure. 3 Chambers, Carleton and McEnally [988] consider a similar measure, the Duration Vector D k. D k is essentially I k measured one period forward. Immunization then implies D k = (H-) k for k={,2,3 } 4 5 If we substitute k=0 in (6) we find that that I 0 = and therefore I 0 and J 0 are unit vectors. Where it is considered that short selling is a practical strategy this can be incorporated explicitly in a linear programming context by adding extra variables with negative costs and negative immunization moments. 6 Sørensen [999] uses a quasi-dynamic programming approach to bond/equity asset allocation and to bond portfolio selection Gilt reference prices are the arithmetic mean of closing mid prices obtained from members of the Gilt-edged Market Makers Association after excluding outliers. For fixed coupon Gilts the outliers are any prices more than 0.5% (of par) away from the median price. 9 Thus excluding any callable or perpetual issues. 0 A rump stock is an issue for which the Gilt market makers are not obliged to quote prices due to the stock being illiquid. The DMO (or the Bank of England prior to st April 998) designates which stocks are rump stocks. 32