Question

6. Immunizing a bond portfolio:

An insurance company must make a payment of $19,487 in seven years. The market interest rate is 10%. You are the portfolio manager for the insurance company and you wish to fund this obligation with a three-year zero coupon bond and perpetuities paying annual coupons. How would you immunize the obligation?

a. Calculate the duration of this liability. To get you started calculate the PV of the obligation. i.e. what is the PV of $19,487 needed in seven years discounted at 10%? The cash flow of this is identical to a zero-coupon bond. So the duration would be determined exactly the same.

b. Calculate the duration of the asset portfolio. The portfolio duration is the weighted average of duration of each component asset, with weights proportional to the funds placed in each asset. So calculate the duration of the zero-coupon bond, and also the duration of the perpetuity. (the equation for calculating the duration of a perpetuity is (1+y)/y. Where y = market interest rate. What is the duration for each?

The fraction of the portfolio invested in the zero coupon is called “w”, and the fraction invested in the perpetuity is (1-w)         w=weighted proportion.

Hint:   Asset Duration of the portfolio = [“w” X (duration of the zero-coupon bond)] + [(1-w) X the duration of the perpetuity.]

c. Find the asset mix that sets the duration of assets equal to the duration of the liability, i.e. the needed payment (7 years)   now solve for “w”. This gives you the weights to be invested in the zero-coupon bond and the weight needed to be invested in the perpetuity. What are the weights? That is “w” = ?       What are the amounts to be invested in the zero-coupon bond and the amount to be invested in the perpetuity?

d. Fully fund the obligation. You will need to fund the PV of the liability or $10,000. What amount do you need to invest it the zero-coupon bond today? And what amount do you invest in the perpetuity today?

Rebalancing the bond portfolio:

Now suppose one year has passed, and the market interest rates remain at 10%. You will need to reexamine your portfolio in #6 above. Is the position still fully funded? Is it still immunized? If not, what actions are required?

a. Examine the funding. What is the PV of the liability now? (remember, one year as gone by, so there are only 6 years left in the required $19,487)

b. What is the value now of you asset portfolio? (the zero-coupon bond and the perpetuity? That is, how much is the zero-coupon bond now worth? How much have you earned on the perpetuity? What is the amount still invested in the perpetuity? Is the obligation still fully funded?

c.

The portfolio weights must now be changed given a year has gone by. (the zero-coupon bond has only 2 years left now, while the perpetuity duration remains at 11 years. The obligation is now due in six years.

The weights must now satisfy the equation:

W X 2 + (1-w) X 11 = 6 yearswhat does “w” equal?

d.

To rebalance the portfolio and maintain the duration match, you must now invest a total of ___________ X “w” which equals ________ in the zero-coupon bond.

You have earned _____ in interest from the perpetuity, which can be invested in the zero-coupon bond. Is this enough? If not, how much more will you need?

Rebalancing the bond portfolio:

Now suppose one year has passed, and the market interest rates remain at 10%. You will need to reexamine your portfolio in #6 above. Is the position still fully funded? Is it still immunized? If not, what actions are required?

Step one: Examine the funding. What is the PV of the liability now? (remember, one year as gone by, so there are only 6 years left in the required $19,487)

What is the value now of you asset portfolio? (the zero-coupon bond and the perpetuity? That is, how much is the zero-coupon bond now worth? How much have you earned on the perpetuity? What is the amount still invested in the perpetuity? Is the obligation still fully funded?

The portfolio weights must now be changed given a year has gone by. (the zero-coupon bond has only 2 years left now, while the perpetuity duration remains at 11 years. The obligation is now due in six years.

The weights must now satisfy the equation:

W X 2 + (1-w) X 11 = 6 yearswhat does “w” equal?

To rebalance the portfolio and maintain the duration match, you must now invest a total of ___________ X “w” which equals ________ in the zero-coupon bond.

You have earned _____ in interest from the perpetuity, which can be invested in the zero-coupon bond. Is this enough? If not, how much more will you need?_

Answer 1

Immunization requires that the duration of the portfolio of assets equal the duration of the liability. We can proceed in four steps:

1. Calculate the duration of the liability: It is a single payment obligation with duration of 7 years.

2. Calculate the duration of the asset portfolio: The portfolio duration is the weighted average duration of each component asset, with weights proportional to the funds placed in each asset. The duration of the zero coupon bond is simple its maturity, 3 years. The duration of the perpetuity is (1+y)/y = 1.10/0.10 = 11 years. Asset duration = w x 3 years + (1-w) x 11 years Asset duration = Liability duration Asset duration = w x 3 years + (1-w) x 11 years =7 years i.e. w = 0.5

3. The manager should invest half of the portfolio in the zero coupon bonds and the rest half in perpetuities. This will result in an asset duration of 7 years.

4. Because the obligation has a present value of $10,000, the manager must purchase $5,000 of the zero coupon bonds and $5,000 of the perpetuities. (note that the face value of the zero will be $5000 x (1.10)3 = $6,655