Question

A financial institution has the following liabilities which it is trying to immunize against a change in interest rates (all are priced to yield 8%):

• 5-year (annual payments) 8% coupon bonds with a total par value of $38 million.
• Three payments of $24 million, each one due at the end of the next three years.
• 12-year (annual payments) 12% coupon bonds with a total par value of $20 million.

Available for the purpose of immunization each year are a 1-year zero coupon bond (a rolling issue, new ones each year) and consol (perpetual) bonds with a 11% coupon, both yielding 10% to maturity. Assuming that you re-balance the portfolio each year immediately after any payments are made, what is the dollar amount of each hedging instrument you will hold at t=0 and t=1 to completely neutralize the institution’s exposure to interest rate changes over the coming year?

Answer 1

Duration of the liabilities calculation:

Duration of the liabilities = 3.82 years

Duration of the zero-coupon bond = 1 year (as it is a one year bond)

Duration of the perpetuity = (1+yield)/yield = (1+10%)/10% = 11 years

Let the weight of the amount put in zero-coupon bond be w. Then, the weight of the amount put in the perpetuity is (1-w).

For immunization, duration of assets = duration of liabilities

(1*w) + (11*(1-w)) = 3.82

Solving for w, we get w = 71.80%

1-w = 100%-77.44% = 28.20%

So, for immunization at T = 0, put 77.44%*PV of total liability = 71.80%*125.88 = 90.38 million in zero-coupon bonds and

28.20%*125.88 = 35.50 million in perpetuity.

Duration of liabilities at T = 1 calculation:

Duration of the liabilities at T = 1 is 3.60 years

Again, using weight w for duration of zero-coupon bond and (1-w) for duration of perpetuity, we have

1*w + 11*(1-w) = 3.60

Solving for w, we get w = 74% and 1-w = 26%

So, for immunization at T = 1, put 74%*106.51 = 78.82 million in zero-coupon bond and 26%*106.51 = 27.69 million in perpetuity.