In this article, we will look into an important concept in bond investments, which is immunizing a bond portfolio, assume that you have a future obligation (or a financial need) to meet with your current bond investments and ensure that the future obligation is met irrespective of market conditions (changes in yield/interest rates), or in other words you want to immunize your bond portfolio against changes in yield.

Immunizing a bond portfolio is something which has practical applications in investment science in that it can be mathematically proven that when you immunize your bond portfolio, you will be protected against changes in yield (whether they go up or down), though we don't need to dig deep into the mathematics of it, in this article we will see the necessary conditions to be met for immunizing your bond portfolio.

It can be easily said that zero coupon bonds are the first choice solution for immunizing without having a portfolio, if you have a financial obligation of F after a duration of D years, you can buy a zero coupon bond with it's present value P which will pay it's face value F after D years, but assume that zero coupon bonds are not available, in which case you need to invest in a bond portfolio to meet your needs.

For the sake of simplicity, assume that there are two bonds available with

Immunizing a bond portfolio is something which has practical applications in investment science in that it can be mathematically proven that when you immunize your bond portfolio, you will be protected against changes in yield (whether they go up or down), though we don't need to dig deep into the mathematics of it, in this article we will see the necessary conditions to be met for immunizing your bond portfolio.

It can be easily said that zero coupon bonds are the first choice solution for immunizing without having a portfolio, if you have a financial obligation of F after a duration of D years, you can buy a zero coupon bond with it's present value P which will pay it's face value F after D years, but assume that zero coupon bonds are not available, in which case you need to invest in a bond portfolio to meet your needs.

For the sake of simplicity, assume that there are two bonds available with

**Bond 1:**Price = P

_{1}, Duration = D

_{1}

**Bond 2:**Price = P

_{2}, Duration = D

_{2}

Consider that you have an obligation F (with a present value of P) after D years which you wish to meet by investing in the above two bonds, then the required conditions for bond portfolio immunization are

Step 1: The duration of the obligation D should be such that

Step 1: The duration of the obligation D should be such that

(The weighted duration of the bonds should be greater than or equal to the duration of the obligation)

Step 2: Compute the present value of the obligation

Where 'm' is the number of coupon periods in an year, λ is the market yield.

**Step 3:**Buy 'x' shares of Bond 1 and 'y' shares of bond 2 such thatTherefore based on the above equations, if you have a future obligation F with a present value of P and duration D, you can meet your obligation by investing in 'x' shares of Bond 1 and 'y' shares of Bond 2 subject to the above constraints to protect against changes in yield, the above criteria for bond portfolio immunization can easily be extended to 'n' bonds.

I know it's may not be easy to understand this important concept without an example, we will look into a more practical example on how to immunize a bond portfolio in the next article.

## 1 comment:

i am not able to do following may be answer given is wrong but i need to verify..

You work for an insurance company. You have an obligation to pay $1 mln in exactly 1.5 years from today. Your goal is to provide the company with an immunized portfolio that would hedge the current obligation. The company is only interested in first-order immunization, so you do not have to deal with convexity.

There are two bonds in the market: the first bond has 1 year till maturity, pays 5% coupon rate and is traded at yield 4%. The second one has 2 years to maturity, 3% coupon and is traded at yield 4%. Assume all coupons are paid twice per year. Please provide the approximate quantities invested in every bond (in thousand dollars). Assume one can buy fractions of bonds.

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