Based on the knowledge we gained from the previous posts on Bond portfolio immunization, it's time to look into a comprehensive example on Bond Portfolio immunization which directly brings me to an interesting problem from Investment Science by Luenberger on bond portfolio immunization - Problem 3.12

Problem 3.12

(Bond Selection) Consider the four bonds having annual payments as shown in the below table, they are traded to produce a 15% yield.

(a) Determine the price of each bond

We know that for all the bonds, Yield λ = 15%, m = 1

We know that for all the bonds, Yield λ = 15%, m = 1

For Bond A, F = 1000, C = 100, n = 3

For Bond B, F = 1000, C = 50, n = 3

For Bond C, F = 1000, C = 0, n = 3 (Zero coupon bond of 3 year duration)

For Bond D, F = 1000, C = 0, n = 1 (Zero coupon bond of 1 year duration)

For Bond B, F = 1000, C = 50, n = 3

For Bond C, F = 1000, C = 0, n = 3 (Zero coupon bond of 3 year duration)

For Bond D, F = 1000, C = 0, n = 1 (Zero coupon bond of 1 year duration)

Based on the Bond price formula discussed in the previous article, we find that,

Price of bond A, P

(b) Determine the duration of each bond (not the Modified Duration)

From on the Bond duration formula discussed in the previous article, we find that

Price of bond A, P

_{A}= $885.832, P_{B}= $771.671, P_{C}= $657.516 and P_{D}= $869.565(b) Determine the duration of each bond (not the Modified Duration)

From on the Bond duration formula discussed in the previous article, we find that

Duration of bond A, D

_{A}= 2.716, D_{B}= 2.836, D_{C}= 3 and D_{D}= 1(c) Which bond is most sensitive to a change in yield?

From the Bond Priced Sensitivity formula discussed in the previous article, we find that bond C is more sensitive to a change in yield (of course, it's a zero coupon bond with 3 year duration)

(d) Suppose you owe $2000 at the end of 2 years. Concern about interest rate risk suggests that a porfolio consisting of the bonds and the obligation should be immunized, If V

_{A}, V_{B}, V_{C}and V_{D}are the total values of bonds purchased of types A, B, C and D, respectively, what are the necessary constraints to implement the immunization? [Hint: There are two equations. (Do not Solve.)]From the bond portfolio immunization constraints discussed in the previous article, we find that

Present Value of the obligation P = 2000 / (1.15 ^ 2) = $1512.287

If V

Present Value of the obligation P = 2000 / (1.15 ^ 2) = $1512.287

If V

_{A}, V_{B}, V_{C}and V_{D}represents the value of each bond (which is equal to the price of the bond multiplied by the number of quantities of the bond purchased, see this article)V

D

2.716 * V

(e) In order to immunize the portfolio, you decide to use bond C and another bond, Which other bond should you choose? Find the amounts (in total value) of each of these to purchase

_{A}+ V_{B}+ V_{C}+ V_{D}= P = $1512.287D

_{A}* V_{A}+ D_{B}* V_{B}+ D_{C}* V_{C}+ D_{D}* V_{D}= 2 * 1512.287 or2.716 * V

_{A}+ 2.836 * V_{B}+ 3 * V_{C}+ V_{D}= 3024.574(e) In order to immunize the portfolio, you decide to use bond C and another bond, Which other bond should you choose? Find the amounts (in total value) of each of these to purchase

We know that one of the immunization constraints is that the average duration of the bond portfolio should be greater than or equal to the duration of the obligation, in this case if we select bond D with C, then the average duration (3 + 1) / 2 = 2 matches the duration of the obligation which is exactly 2 years, therefore we select bond C and bond D, then from the bond portfolio immunization constraint in section (d), we have

V

3 * V

_{C}+ V_{D}= 1512.2873 * V

_{C}+ V_{D}= 3024.574Solving, we get, V

Therefore, number of shares of bond C = V

Number of shares of bond D = V

_{C}= V_{D}= $756.14Therefore, number of shares of bond C = V

_{C}/ P_{C}= 756.14 / 657.516 = 1.15Number of shares of bond D = V

_{D}/ P_{D}= 756.14 / 869.565 = 0.869Therefore if one buys 1.15 shares of bond C and 0.869 shares of bond D, the bond portfolio will be immunized.

(f) You decided in (e) to use bond C in the immunization. Would other choices, including perhaps a combination of bonds, lead to lower cost?

Obviously not, based on the bond portfolio immunization constraints since the sum of the total value of the bonds should be equal to the present value of the obligation (see section (d)), alternate choice of bonds will not lead to lower cost, but one should make sure that the average duration of the bonds in the portfolio should be greater than or equal to the duration of the obligation.

A big post indeed, if we understand this, we can immunize any bond portfolio and ensure that we will never lose money.

Acknowledgements: Thanks to Prof. Luenberger for posting this problem in his valuable text book on Investment Science and Prof. Charles Feinstein for his lucid explanation of the bond portfolio immunization concept, this is one of the valuable lessons we learned in bond investments.

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