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

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

|Mathematical Expression Omitted~,

where |d.sup.FNMA~ and |d.sup.TSY~ denote the FNMA and on-the-run Treasury discount functions, respectively. Note that

|Mathematical Expression Omitted~,

where the forward rate of term T - t at time t satisfies the equation

|r.sub.f~(t; T - t) = |r.sub.s~(T)T - |r.sub.s~(t)t/T - t. A bond represents a promise to pay interest on some contractual basis and repay principal on one or more specified dates in stated amount(s). The rate at which interest is paid may be fixed or it may vary according to a particular formula. In certain situations, there are advantages to letting the interest rate float. For example, an investor whose liabilities are interest-rate sensitive and a borrower whose assets are similarly interest-rate sensitive could find it mutually advantageous to have the interest rate float according to some agreed-upon benchmark rate. A wide variety of market interest rates as well as nonfinancial indexes have been employed as the benchmark in different floating rate debt issues (see Wilson |21~).

The rate at which principal is repaid could also be either fixed or variable in accordance with some formula. But the advantages to indexing principal repayments are less obvious, at least partly because in most cases corporate issuers retain some flexibility to increase principal repayments in response to decreases in interest rates either by making optional redemptions or by exercising the option (when it is available) to "double up" on sinking fund payments. Until recently, few, if any, debt issues provided for an indexed sinking fund.

In July 1988, the Federal National Mortgage Association (FNMA) issued $500 million principal amount of 8.70% indexed sinking fund debentures (ISFDs) |5~. Five additional issues of ISFDs (one consisting of two tranches) totalling $3.175 billion principal amount followed in October and December 1988 and January, March, May, and September 1989 (|6~, |7~, |8~, |9~, and |10~). ISFDs contain an interest-rate-contingent sinking fund. Principal repayments vary inversely with market interest rates, increasing (decreasing) when interest rates decrease (increase) relative to a specified base rate. Mortgage prepayments exhibit a similar pattern. The FNMA had a $100 billion mortgage portfolio at year-end 1988 |4~. Many of these mortgages are fixed-rate securities, for which prepayments tend to increase (decrease) as interest rates decrease (increase).

This paper describes ISFDs, characterizes the interest-rate-contingent sinking fund in terms of a strip of European call options and a strip of European put options, develops a contingent claims model for valuing ISFDs and the implicit options imbedded in the contingent sinking fund, and demonstrates the usefulness of ISFDs as an asset-liability management tool. It shows that the sinking fund schedule of the ISFDs enables the FNMA to achieve a closer matching of the interest-rate sensitivities of the prices of its assets and liabilities than a conventional debt issue generally permits. ISFDs could be issued by any financial institution that invests in mortgages and wishes to match more closely these interest-rate sensitivities.

I. Description of ISFDs

Exhibit 1 provides a summary of terms for the six issues of ISFDs. They are intermediate-term debt issues that provide for semiannual sinking fund payments at the end of each interest period commencing one and one-half years from the date of issue, in the case of two of the ISFDs, and four years from the date of issue, in the case of the other four issues. All six issues are noncallable. Their sinking funds are indexed in the following manner. Consider the sinking fund for the 8.70% ISFDs. The base annual sinking fund percentage is 40%. Assuming no change in interest rates after issuance, the FNMA would redeem 20% (one-half of the 40% annual rate) of the remaining outstanding balance (as distinct from the original balance) on each semiannual sinking fund date. If interest rates change after issuance, the proportion of the outstanding balance redeemed on any sinking fund date will depend on the relationship between the average value of the ten-year United States Treasury constant maturity rate (ten-year CMT) during the semiannual interest period and 8.85%, the base rate for the 8.70% ISFDs.

Exhibit 2 compares the contingent sinking fund structures for the five-year ISFDs and the ten-year ISFDs. In each case, the sinking fund percentage depends on the average ten-year CMT. For example, consider the 8.70% ISFDs. If the average ten-year CMT is 5.10% or below (i.e., the 8.85% base rate minus 375 basis points or more) as of a particular sinking fund date, 50% of the remaining balance, the maximum possible percentage, would be repaid on that date. If the average ten-year CMT is 10.61% or above (i.e., the 8.85% base rate plus 176 basis points or more), the sinking fund percentage is zero. If the average ten-year CMT is never less than 10.61% as of any sinking fund date, the 8.70% ISFDs would have a bullet maturity.