Question 1 (20 Marks). make payments of $7 million in 1 year and $2 milion in 4

Question

Question 1 (20 Marks). make payments of $7 million in 1 year and $2 milion in 4 years. The yield curve is flat at 596. Required: a) If the company wants to fully fund its obligation with a zero-coupon bond, what maturity bond must it purchase? What alternative portfolio of zero coupon bonds can the company construct? Explain in detail and show al workings. (7 marks) What must be the face value and market value of the zero-coupon bond in part a)? How does this relate to your alternative in part a) for each of the zero coupon bonds? b) (8 marks)

Explanation / Answer

Solution to part a and b:

The present value of the two future payments is A = $(7 / (1 + 0.05) + 2 / (1 + 0.05)4 = $8.312071617 million.

The duration of this liability is D = [1 * 7 / (1 + 0.05) + 4 * 2 / (1 + 0.05)4] / A

                                                                = 1.593860962 years.

If the manufacturing company wants to fully fund and immunize its obligations to this customer with a single issue of a zero-coupon bond, the maturity of that bond must be the duration of the liability which is D = 1.593860962 years.

With this funding method, the manufacturing company must rebalance its portfolio regularly, as the duration of its liability may not match with the duration of the zero-coupon bond as time passes and/or when the yield curve changes.

Another funding method (cash flow matching) is to match each of its two future payments with a corresponding maturity zero coupon bond. That is: purchase 7000 units of 1 year zero coupon bonds with a total face value of $7 million to match its payment of $7 million in 1-year and purchase 2000 units of 4-year zero coupon bonds with a total face value of $2 million to match its payment of $2 million in 4 yews.

A more practical method is to fund it with a perpetuity (which is commonly available) and a portfolio of coupon bonds. The duration of a perpetuity would be (1 + 0.05) / 0.05 = 21. The next task is to find a portfolio of coupon bonds whose duration is such that the duration of the liability is between the duration of the perpetuity and the duration of the portfolio of coupon bonds. For example, let the duration of the portfolio of coupon bonds be 0.9 years. If the manufacturing company wants to fully fund and immunize its obligations to this customer with the perpetuity and a portfolio of coupon bonds with duration 0.9 years, it would have to find the appropriate weighted portfolio.

Let w be weight of the perpetuity. To find the value of w, set (w * 21 + (1 – w) * 0.9) = 1.593860962. Solving for w, we have w = (1.593860962 - 0.9) / (21 - 0.9) = 0.034520445.

That means a weighted portfolio consisting of 3.452044587% of $8.312071617 million in perpetuities and 96.54795541% of $8.312071617 million in a portfolio of coupon bonds with duration of 0.9 years.

If the manufacturing company wants to fully fund and immunize its obligations to this customer with the perpetuity and a portfolio of coupon bonds with duration of 0.9 years, it would have to buy today 0.286936418 million worth the perpetuity and 8.025135198 million worth of the portfolio of coupon bonds with duration of 0.9 years.

With this funding method, the manufacturing company must also rebalance its portfolio regularly.

The correct answer is: 1.59

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