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6immun - ACTSC 445 Asset-Liability Management Department of...

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ACTSC 445: Asset-Liability Management Department of Statistics and Actuarial Science, University of Waterloo Unit 6 – Immunization References (recommended readings): Chap. 3 of Financial Economics (on reserve at the library: call number HG174 .F496 1998). What is immunization? Redington (1952): Immunization implies the investment of assets in such a way that existing business is immune to a general change in the rate of interest. Fisher-Weil (1971): A portfolio of investment is immunized for a holding period if its value at the end of the holding period, regardless of the course of rates during the holding period, must be at least as it would have been had the interest rate function been constant throughout the holding period. Implication: If the realized return on an investment in bonds is sure to be at least as large as the appropriately computed yield to the horizon, then that investment is immunized. An immunization strategy is a risk management technique designed to ensure that for any small change in a specified parameter, a portfolio of debt instruments (e.g., T-bills, bonds, GICs etc) will cover a liability (or liabilities) coming due at a future date (or over a period in the future). It is a passive management technique because it takes prices as given and then tries to control the risk appropriately. (By contrast, active management techniques try to exploit changes in (1) the level of interest rates, (2) the shape of the yield curve (3) yield spreads, by using interest rate forecasts and identification of mispriced bonds) asset allocation problem (i.e., must choose assets that will produce an immunized portfolio) Single-liability case We’ll start with the case where there is only one liability in the portfolio, with corresponding cash flow of L t at some time t . The goal is to choose an asset cash flow sequence { A t , t > 0 } that will, along with L t , produce an immunized portfolio. Let’s start with an example. Example I: Suppose an insurance company faces a liability obligation of $ 1 million in 5 years. The available market instruments are: 3-year, 5-year and 7-year zero-coupon bonds, each yielding 6% annual e ff ective rate. Portfolio A: Invest $ 747,258.17 in the 5-year zero coupon bond 1
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Portfolio B: Invest the same amount (i.e. $ 747,258.17) in a 3-year zero coupon bond. The maturity value at t = 3 is $ 889,996.44. Portfolio C: Invest $ 747,258.17 in a 7-year zero coupon bond. The maturity value at t = 7 is $ 1,123,600.00. If the yields remain unchanged, then the 3 portfolios have the same value of $ 1 000 000 at time 5. To verify if these portfolios are immunized or not, we need to look at what happens if, immediately after the portfolio is acquired, the yield changes instantaneously to ˆ y and remains constant at that level. First, note that for portfolio A, this change has no impact: its value at time 5 is still $ 1 000 000. But this is not true for portfolios B and C, as Tables 1 and 2 show.
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