Indexed sinking fund debentures: valuation and analysis - includes appendices - Security Design Special Issue

Financial Management (Financial Management Association), Summer, 1993 by John D. Finnerty

The put option values exhibit a different pattern. They increase monotonically in the case of the 1993A and 1993B ISFDs as the time to expiration of the out-of-the-money options increases; these increases reflect the increasing time value of the options. In the case of the other four ISFDs, the values of the put options increase monotonically to a maximum -- as the increase in time value exceeds the decrease in value due to prior redemptions -- and thereafter decrease monotonically as the redemption factor exerts greater influence than the time factor.

In Exhibit 5, I have also applied Equation (1) to calculate the price V(bond) of a conventional nonredeemable FNMA note implicit in the observed ISFD price and the calculated option values as of August 30, 1991. The 1993A and 1993B issues are scheduled to mature within three months of one another but the 1993B issue has a higher coupon and hence should have a higher implied V(bond), as is the case in Exhibit 5. The 1998A, 1999A, 1999B, and 1999C issues all mature within nine months of one another. The 1999B issue carries the highest coupon, the 1999C issue carries the lowest coupon, and the coupons on the 1998A and 1999A issues differ by just five basis points, which suggests that the 1999B issue should have the highest implied V(bond), the 1999C issue should have the lowest implied V(bond), and the 1998A and 1999A issues should have roughly equal implied V(bond) values, as again is the case in Exhibit 5.

III. Usefulness of ISFDs in Asset-Liability Management

A. Interest-Rate Sensitivity of ISFDs

Duration analysis is widely used in the fixed income markets to quantify a security's price sensitivity to interest rate changes. Duration is defined in a variety of ways, so it is important to specify which measure is being used when making duration calculations.

A security's modified duration measures its percentage price volatility. Modified duration, denoted |D.sub.m~, is defined by the formula

|D.sub.m~ = -1/P(dP/dy) = -D/(1 y), (10)

where dP/dy denotes the security's (instantaneous) rate of change in price P with respect to a change in the required market yield y and D denotes the security's time duration. The Macaulay |15~ measure of time duration is defined by the equation

|Mathematical Expression Omitted~,

where |A.sub.t~ denotes the aggregate debt service payment (principal plus interest) during period t, T denotes the maturity of the security, and y denotes the security's yield to maturity.(8)

Substituting Equation (11) into Equation (10) gives(9)

|Mathematical Expression Omitted~.

Exhibit 6 compares the modified durations of the 8.70% ISFDs and an otherwise identical five-year conventional debt issue. The modified duration of the conventional issue decreases as interest rates increase, which gives the security's price-versus-yield curve a convex shape: The market value of the bond decreases at an ever-decreasing rate -- the price-yield curve becomes flatter -- as the required market yield increases (or equivalently, increases at an ever-increasing rate -- the curve becomes steeper -- as the required market yield decreases). Within the contingency region (between the dotted lines), the modified duration of the 8.70% ISFDs is generally increasing. The 8.70% ISFDs exhibit negative convexity within this region and positive convexity outside the region. However, within each of the subintervals where the sinking fund is fixed, as well as outside the contingency region, modified duration decreases as interest rates increase, as in the case of a conventional bond.(10) For large interest rate increases (175 basis points or greater), the ISFD's price is roughly 50% more interest-rate sensitive than the price of a conventional issue. For very large interest rate decreases (375 basis points or more), the price of the conventional issue is roughly 50% more interest-rate sensitive than the ISFD's price.