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

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

C. Valuation of Imbedded Options

The options imbedded in the FNMA ISFDs are compound options because the value of each option depends upon whether any of the options corresponding to earlier sinking fund dates were exercised. The exercise value of the call option corresponding to sinking fund date t, |Mathematical Expression Omitted~, and the exercise value of the put option corresponding to sinking fund date t, |Mathematical Expression Omitted~, each as of the sinking fund date, is in each case dependent upon the principal amount of bonds B(t) outstanding immediately prior to the sinking fund date:

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

where P(r, t) denotes the market price of the ISFD, expressed in dollars per $1,000 face amount, as of sinking fund date t and |Beta~ denotes the base semiannual sinking fund percentage. The amount of bonds outstanding immediately prior to t depends upon the amounts of bonds retired on preceding sinking fund payment dates. In Equation (9a), if the call option is in-the-money, the FNMA retires a fraction |Mathematical Expression Omitted~ of the outstanding bonds, or |Mathematical Expression Omitted~ total principal amount, which exceeds the base sinking fund amount by |Mathematical Expression Omitted~. The payoff on the option is (P(r, t)/1,000) - 1 per dollar of principal amount, or |Mathematical Expression Omitted~ in the aggregate. Equation (9b) is interpreted in the same manner except that when the put option is in-the-money, the FNMA retires |Mathematical Expression Omitted~ less than the base sinking fund amount and the payoff is 1 -(P(r, t)/1,000) per dollar of principal amount.

Exhibit 5 provides the values of the strip of call options and strip of put options imbedded in each ISFD as of August 30, 1991, the last trading day in the period I studied. These options are valued by substituting |Mathematical Expression Omitted~ or |Mathematical Expression Omitted~ for P(r, t) in Equation (7) and solving the resulting partial differential equation subject to the boundary condition (9a) or (9b), respectively. On August 30, 1991, the ten-year CMT was approximately 7.80%, and it had averaged approximately 8.00% over the preceding six months so that the call options imbedded in the 1993A, 1993B, 1998A, 1999A, and 1999B FNMA ISFDs were in-the-money, the put options imbedded in those issues were out-of-the-money, and the call and put options imbedded in the 1999C FNMA ISFDs were all at-the-money (the actual sinking fund percentage would equal the base sinking fund percentage if interest rates did not change prior to the next sinking fund payment date).

The pattern of option values for each ISFD reflects two opposing forces at work: as the time to expiration increases, the value of the option would increase if the amount of bonds outstanding remained fixed; but as time passes, the operation of the sinking fund reduces the amount of bonds outstanding (except in the extreme case TABULAR DATA OMITTED TABULAR DATA OMITTED of very high interest rates), which reduces the amount of bonds B(t) available for redemption on each successive sinking fund payment date, which by Equation (9) will reduce the dollar value of the longer-term options. The tendency for longer-term call options to have a lower value than shorter-term call options will be more pronounced the deeper the call options are in-the-money. Consequently, the call option values exhibit a monotonically decreasing pattern in the case of the 1993A, 1993B, 1998A, 1999A, and 1999B ISFDs (and a generally decreasing pattern in the case of the 1999C ISFDs).