Question

We have liabilities of $50 million per year for the next 3 years, with payments made at the end of the year. The term structure of interest rates is given by ?1 = ?2 = ?3 = 8%. We want to immunize these liabilities by using a 1-year zero coupon bond with face value of $1 million and a 3-year coupon bond with face value of $0.1 million, coupon rate 5%, and annual coupon payments. Assuming the bonds are correctly priced and risk-free.

(a) Show how to immunize the liabilities.

(b) Compare the convexity of the bond portfolio and the convexity of the liabilities. What can we conclude?

Answer 1

a) Please see the below interest payment schdedule, for principle amount of 625 $ M, $ 50 M has to pay each year.

	Y1	Y2	Y3
Liability (Amt in $M)	50	50	50
Principal Amount ($M)	625	625	625

in order to clear 625 Million Liability company has to raise 625 $ M from Bond Market.

Since ration of Zero Coupon bond and Coupon bond is not given we can consider 50 % Zero Coupon bond which wil be due in one year and 50% Coupon bond which will be dued after 3 year.

Zero Coupon Bond :

	Y1	Y2	Y3
Liability (Amt in $M)	50	50	50
Principal Amount	625	625	625
	Y1	Y2	Y3
Zero Coupon Bond (Amt in $M)	312.5
Amount Payable (Amt in $M)	328.125
Interest Payable	15.625
Coupon Bond	312.5	312.5	312.5
Coupon Interest payable	15.625	15.625	15.625
Savings on Interest	18.75	18.75	18.75

b)

Convexity relates to the interaction between a bond's price and its yield as it experiences changes in interest rates. With coupon bonds, investors rely on a metric known as duration to measure a bond's price sensitivity to changes in interest rates.

Here it is mentioned on the question that Bond will be fairly priced during the 3 year hence there is no risk.

Plan liabilities are typically positively convex: meaning that for a given level of duration, the value of liabilities will fall less when yields rise and increase more when yields fall. Essentially, being positively convex means being more right as interest rates fall and less wrong as interest rates rise