# Darren: Bond and Term Structure Essays

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Fin-40003 Financial Modelling Martin Diedrich

Keele University Keele, Spring 2008 m.e.diedrich@econ.keele.ac.uk

Finance Notes 7
Sequential Immunisation
This handout extends the immunisation analysis from FN4 by looking more explicitly at the sequential nature of the immunisation process. We consider a sequence of consecutive years leading up to the maturity date of the liability. At the beginning of each new year, the investor is allowed to re-balance his bondportfolio. At the end of the final year of his strategy, he cashes in all his claims and sells all remaining bonds and uses the proceeds to pay off his liability. We hope that the magnitude of the proceeds from his portfolio is very close to the size of his liability. The same material was also covered in our lectures in Week 6.

§1 The Idea of Dynamic Immunisation
Immunisation is a dynamic strategy, requiring regular updating until all liabilities have been met. 1.1 Present Value Matching over Time Consider an investor with a fixed future liability LH , due at date H. In order to be able to meet his future liability at date H, at present the investor needs to build up an asset base Q that promises to cover the liability when it becomes due. Thus, the investor wishes to ensure that at the due date (date H), his asset base Q will have a present value equal to the present value of his liability L: PV[Q](H) = PV[L](H) (at the due date H). (1.H)

The key idea of an immunisation strategy is simple: We bring about the future equality between the present value of our asset base and the present value of our liability by starting off with a similar equality at date 0, PV[Q](0) = PV[L](0) (at the present date 0), (1.0)

and then maintaining this equality over time, PV[Q](t) = PV[L](t) (at all intermediate dates t). (1.t)

1.2 Re-balancing Q Normally, maintaining the equality of the present values of Q and L requires regular updating or re-balancing of our portfolio Q. This means that

Finance Notes 7

2

we may have to alter the composition of our bond portfolio Q at various future dates t, increasing the weights of some bonds, while reducing the weights of some other bonds. The aim of such re-balancing is to ensure that any potential changes in interest rates will have the same effect on Q as on L: if my liabilities go up in value, then hopefully my asset base goes up by the same amount. Likewise, if my asset base goes down in value, then hopefully my liabilities go down by the same amount. 1.3 Duration and Immunisation We saw in FN2 and FN4 that duration gives a good measure of interest-rate risk, provided that all changes in the term structure are parallel shifts. Under parallel shifts, a bond portfolio Q that has the same present value and the same (modified) duration as the liability, PV[Q] = PV[L], and Dm [Q] = Dm [L] implies PV[Q] = PV[L] , (2)

where PV denotes present values after a parallel shift. This indicates that duration is a key measure of any sequential immunisation strategy. We require that Dm [Q](t) = Dm [L](t) at all dates t = 0, 1, 2, . . . , H, (3.t)

in addition to the present-value condition (1.t). 1.4 Self-financing Strategies If at date 0 we choose our intitial asset base Q(0) wisely, then any re-balancing at future dates t can be financed entirely out of itself; any increase in the weight of bond A, say, should be financed entirely out of corresponding sales of some other bonds in Q, say bond B. Formally, we require that at any date t, moving from an inherited portfolio Q(t − 1) to a re-balanced portfolio Q(t) should not change the overall value of Q: PV[Q(t − 1)](t) = PV[Q(t](t), at all dates t = 1, 2, . . . , H, (4)

where Q(t − 1) is our inherited bond portfolio Q at the start of trading date t (inherited from date t − 1), Q(t) is our new bond portfolio Q at the end of trading date t (after re-balancing at date t), and PVQ(t−1) and PVQ(t) are the date-t present values of the old and new portfolio under the date-t term…