# Indexed sinking fund debentures: valuation and analysis.

|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.

Exhibit 3 shows how the remaining outstanding balance of the 8.70% ISFDs would change over the life of the issue if the average ten-year CMT were to change, relative to the 8.85% base rate, as of the first sinking fund date by the amounts indicated and remain at the new rate for the life of the issue. For example, if the average ten-year CMT is 8.85% on each sinking fund payment date, 20% of the remaining balance would be redeemed on each sinking fund date. The remaining balance would be 80% immediately after the first sinking fund date, 64% of the original balance after the second sinking fund date, 51.2% of the original balance after the third sinking fund date, and so on.(1) The terms of the ISFDs provide that if the scheduled redemption on any sinking fund date would leave a remaining balance less than five percent of the original balance, the FNMA will immediately redeem the entire outstanding balance. Depending on the course of interest rates, the 8.70% ISFDs could be retired as early as January 1992 (3.5 years from the date of issue).

A. Ten-Year ISFDs Versus Five-Year ISFDs

The 9.80% ISFDs, 9.75% ISFDs, 9.95% ISFDs, and 9.15% ISFDs are different from the 8.70% ISFDs and 9.05% ISFDs in several important respects. They have a longer maturity (ten years); the base rate is significantly lower than the offering yield whereas it is higher for the 8.70% ISFDs and only slightly lower for the 9.05% ISFDs; the average life, calculated at the base rate, is greater (5.42 years versus 3.08 years); and the four ten-year issues have a contingent sinking fund structure that is different from that of the two five-year issues. The base annual sinking fund percentage for the five-year issues, which the FNMA sold first, is 40%. In designing the contingent sinking fund for the ten-year issues, the FNMA raised the base annual sinking fund percentage to 50%. But the FNMA set the base rate far enough below the initial offering yield that if interest rates did not change, each ten-year ISFD's annual sinking fund percentage would be 45%. In addition, the FNMA made the annual sinking fund percentage in the ten-year ISFDs relatively more sensitive to a decrease in interest rates and relatively less sensitive to an increase in interest rates than the five-year ISFD's sinking fund. For example, the five-year ISFD's annual sinking fund percentage increases by ten percent if interest rates decrease by 100 basis points but decreases by 20% if interest rates TABULAR DATA OMITTED TABULAR DATA OMITTED increase by 100 basis points. The ten-year ISFD's annual sinking fund percentage increases by 30% if interest rates decrease by 100 basis points but decreases by 15% if interest rates increase by 100 basis points. As discussed further below, the redesign resulted in the interest-rate sensitivity of the sinking fund payments matching more closely the interest-rate sensitivity of mortgage prepayments.(2)

B. Imbedded Options

The contingent sinking fund contained in the ISFDs can be explained in terms of option theory. The issuer of the ISFDs has effectively purchased a strip of European call options and a strip of European put options. The times to expiration correspond to the sinking fund dates; the strike prices are par; exercise is costless; and exercise will occur with certainty if an option is in-the-money on one of the sinking fund dates. The amounts of bonds covered by the options correspond to the amounts shown in the sinking fund percentage adjustment column in Exhibit 2. The options always work to the issuer's advantage and to the investors' disadvantage. If interest rates drop sufficiently relative to the base rate, the call options come into the money and are exercised. If interest rates rise sufficiently relative to the base rate, the put options come into the money and are exercised.

An ISFD can be characterized as a package consisting of a conventional bond and the strips of calls and puts just described. The conventional bond in this case has a sinking fund that makes semiannual payments based on the base annual sinking fund percentage and beginning when indicated in Exhibit 1. The relationship between the market price of the ISFDs, denoted P, and the values of the conventional bond and the imbedded options can be expressed as

P = V(bond) - V(call options) - V(put options), (1)

where V(bond) denotes the value of the conventional bond, V(call options) denotes the value of the strip of call options, and V(put options) denotes the value of the strip of put options.

As a consequence of the imbedded options, an 8.70% conventional issue would dominate the 8.70% ISFDs.(3) Accordingly, the conventional issue would require a lower yield. The yield differential would approximate the cost of effectively purchasing the options imbedded in the 8.70% ISFDs. At the time the 8.70% ISFDs were issued, conventional FNMA bonds with an average remaining life of 3.08 years (the average life of the 8.70% ISFDs if interest rates remain within the band from -24 basis points to +25 basis points) were trading at a yield to maturity of 8.46%. The cost to the FNMA of "purchasing" the options imbedded in the 8.70% ISFDs was thus approximately 24 basis points. The next section develops a contingent claims valuation model, which will be used to value each of the ISFDs and each of the call and put options imbedded in each ISFD.

II. ISFD Valuation Model

McConnell and Schwartz |16~, Ogden |17~, and others have developed contingent claims models to value and analyze innovative securities. I employ the Cox, Ingersoll, and Ross (CIR) |3~ one-factor arbitrage pricing model to develop a contingent claims model to value ISFDs. The CIR |3~ model is an equilibrium model of the term structure of interest rates that is consistent with an asset pricing equilibrium in which there are no arbitrage opportunities. Security values depend only upon the current short-term rate of interest and the stochastic process that will generate future values of the short-term rate. Securities are valued in a manner that ensures that riskless arbitrage is not possible among a set of securities that are fairly priced.

A. Analytical Procedure

The model rests on the following assumptions:

(A-1) Trading in securities takes place continuously in competitive, frictionless markets.

(A-2) There are no asymmetric taxes.

(A-3) The instantaneous default-free rate of interest, r(t), follows the diffusion process proposed by CIR |3~:

dr = -|Kappa~(r - |Theta~(t)) dt + |Sigma~|square root of r~dz (2)

where |Kappa~ is the mean-reversion speed, |Theta~ is the long-term or "central" rate towards which the short-term rate r is attracted, |Sigma~ is the instantaneous volatility of the short-term rate, and dz is a standard Wiener process.

(A-4) The price of any default-free (i.e., Treasury) security or any FNMA security is in each case a twice continuously differentiable function of r.

(A-5) The instantaneous market risk premium on any default-free security may be expressed as

(1/2 ||Sigma~.sup.2~|Lambda~(t)r/P)|Delta~P/|Delta~r

where |Lambda~(t) |is less than~ 0.

Equation (2) states that changes in r have a deterministic component and a random component. The deterministic component, -|Kappa~(r - |Theta~(t))dt, states that r tends to drift toward a steady-state mean value, |Theta~(t), which may itself vary over time. The rate r tends to revert to |Theta~(t) at a rate equal to |Kappa~; the magnitude of the expected adjustment equals the mean-reversion speed |Kappa~ multiplied by the magnitude of the differential, r - |Theta~(t). The random component of the change in r is, under the assumption of a standard Wiener process, distributed normally with mean zero and instantaneous variance ||Sigma~.sup.2~rdt.

Under the foregoing assumptions, the price P(r, t) of a default-free security, expressed as a function of rate and time, must satisfy the equation

1/2||Sigma~.sup.2~r||Delta~.sup.2~P/|Delta~|r.sup.2~ + |Kappa~(|Theta~(t) -r)|Delta~P/|Delta~r + |Delta~P/|Delta~t - 1/2||Sigma~.sup.2~|Lambda~(t)r|Delta~P/|Delta~r + c(r, t) - rP = 0 (3)

where c(r, t) is the cash debt service payment at time t. The solution to Equation (3) must satisfy the boundary condition

P(r, T) = 1, all r |is greater than or equal to~ 0. (4)

In order to apply Equations (3) and (4) to value a default-free security, the parameters in Equation (3) must be calibrated to the prices of default-free (i.e., Treasury) bonds observed in the marketplace. The parameters are chosen so as to produce the Treasury term structure that is consistent with the prices at which the on-the-run Treasury securities are trading on the day of estimation.(4)

Denote the discount function that corresponds to the current Treasury term structure by d(T), 0 |is less than or equal to~ T |is less than or equal to~ 30 years. Note that for the special case of a lump sum payment, Equations (3) and (4) yield an analytic solution of the form

d(r, t; T) = exp(a(t ; T) + b(t ; T)r), (5)

where d(r, t; T) denotes the value of the discount function at time t when the lump sum payment will be received at time T and the interest rate is r. The coefficients a and b solve the system of ordinary differential equations

da/dt = -|Kappa~|Theta~(t) b (6a)

db/dt = 1 + (|Kappa~ + 1/2||Sigma~.sup.2~|Lambda~(t))b -1/2||Sigma~.sup.2~|b.sup.2~ (6b)

with boundary condition a(T ; T) = b(T ; T) = 0. The Vasicek-Fong |20~ procedure is used (on a daily basis) to determine the Treasury discount function implied by the observed prices of the on-the-run Treasury securities. Then the parameters |Theta~(t) and |Lambda~(t) can be chosen so that the analytic solution (5) exactly reproduces the Treasury discount function:

d(|r.sub.0~, 0; T) = d(T), all T |is greater than or equal to~ 0,

where d(T) is today's discount function and |r.sub.0~ = -d|prime~(0) is today's short-term rate. In fact, once the mean-reversion speed |Kappa~ is specified, |Theta~(t) and |Lambda~(t) can be chosen to reproduce the discount function (and hence the term structure) for all levels of rate volatility |Sigma~, as explained in Appendix A.

The mean-reversion speed |Kappa~ and the rate volatility |Sigma~ can be estimated either from historical data (Brennan and Schwartz |2~ and Ogden |17~) or implicitly by reproducing the observed volatilities of the spot and forward rates for a range of terms (Hull and White |14~). Given the estimated riskless term structure, it is straight-forward to calculate the implied yield to maturity for a ten-year coupon-bearing riskless security, which serves as the proxy for the ten-year CMT that is needed to value the FNMA ISFDs.

The FNMA is a federal agency, but its debt obligations are not backed by the full faith and credit of the U.S. government. Nevertheless, because of the FNMA's agency status, the degree of default risk is low, which is reflected in the narrow yield differential between Treasury and FNMA debt securities of like maturity. The pricing Equation (3) can be modified to account for default risk by adding an instantaneous default risk premium (or credit spread) C, which may vary with t, to the zero-order term of the PDE:(5)

|Mathematical Expression Omitted~,

where c(r, t) represents the rate at which debt service payments are made to FNMA ISFD holders. Each debt service payment consists of interest on the remaining outstanding principal balance and the contingent sinking fund payment, which is made semiannually.

The amount of the contingent sinking fund payment depends on the relationship between the specified base rate for the ISFD issue and the average value of the ten-year CMT during the interest period. The model fits the Treasury term structure on a daily basis. Each daily term structure is then used to calculate an implied ten-year CMT. The daily ten-year CMT values for the interest period are averaged, the resulting average is compared to the base rate, and the sinking fund schedule is applied to determine the amount of the contingent sinking fund payment.

The adjustment for default risk in Equation (7) is admittedly ad hoc, although it is consistent with the manner in which Wall Street professionals adjust for it. A more rigorous approach would have to model the risk of default explicitly within a general equilibrium model, such as the one CIR |3~ developed. In view of the low degree of default risk present in FNMA debt securities, the ad hoc adjustment in Equation (7) seems a reasonable compromise that avoids having to make the assumption that FNMA debt securities are default-risk-free.

The credit spreads in Equation (7) can be inferred from the market prices of reference securities in the same default risk class, in this case, FNMA debt securities. The valuation model is first calibrated to Treasury bill and bond price data, as described in Appendix A, to estimate the parameters |Kappa~, |Sigma~, |Theta~(t) and |Lambda~(t). It is then calibrated to FNMA bond price data, as described in Appendix B, to estimate the parameters C(t). Once c(r, t) is specified appropriately, Equation (7) can be solved subject to the boundary condition (4) to value FNMA ISFDs.

To calculate the credit spreads C(t) in Equation (7) that are consistent with the prices at which noncallable FNMA debt obligations are trading in the marketplace, Equation (6a) is replaced by

da/dt = -|Kappa~|Theta~(t) b + C(t) (6a|prime~)

and the Vasicek-Fong |20~ procedure is used to choose the parameters C(t) so that the analytic solution (5) produces the FNMA term structure of interest rates most consistent with the observed market prices of noncallable FNMA debt obligations. The FNMA instantaneous default risk premium at time t, C(t), is the difference between the instantaneous forward rates implicit in the FNMA and Treasury term structures corresponding to time t. This result is demonstrated in Appendix B.

The PDE (7) must be solved numerically to obtain the value of an ISFD at some time t. One way to do this is to use finite-difference techniques (Forsythe and Wasow |12~), which solve a PDE by choosing a time step size |Delta~t and a rate grid |r.sub.1~ |is less than~ |r.sub.2~ |is less than~ ... |is less than~ |r.sub.M~ (both of which may vary over time), replacing the partial derivatives in the PDE with appropriate difference quotients, and solving the resulting sequence of systems of linear equations

|Mathematical Expression Omitted~

backward in time from maturity to today's date. In Equation (8), |Mathematical Expression Omitted~ denotes the array of approximate present values over the rate grid at time t and A and B denote PDE coefficient matrices. In general, there are two classes of finite-difference solutions: "explicit" methods for which A is the identity matrix and "implicit" methods which require the numerical solution of the linear system at each time step. While implicit methods require more computational effort per time step, they have the advantage of being "unconditionally stable," i.e., the numerical solution to the PDE converges to its asymptotic solution regardless of the choice of time step size and rate grid.

B. Valuation of FNMA ISFDs

To value the FNMA ISFDs, I use the analytical procedure described in the preceding section to calibrate the valuation model. Treasury and FNMA bond prices for each trading day between July 31, 1990 and August 31, 1991 were obtained from Street Software Technology Inc., which collects bid and ask prices as of 3:00 p.m. each trading day from Carroll McEntee and McGinley, a primary dealer in Treasury securities. The valuation model was calibrated to the average of the bid and ask prices. First, the maximum likelihood procedure described in Ogden |17~ was applied to the daily prices of the on-the-run three-month Treasury bill during the period July 31, 1990 through August 31, 1991 to obtain |Kappa~ = 0.405702 and ||Sigma~.sup.2~ = 0.053738.(6)

Second, the model was calibrated to the on-the-run Treasury securities to estimate the parameters |Theta~(t) and |Lambda~(t). In an arbitrage-free environment, the market price of each actively traded security must equal its fair value. The ten on-the-run Treasury securities are among the most actively traded debt securities. The parameters |Theta~(t) and |Lambda~(t) were chosen so that the model priced each of the on-the-run Treasury securities exactly each day. Third, the model was calibrated to the noncallable FNMA debt securities to estimate the parameters C(t). The model was recalibrated each trading day during the period July 31, 1990 through August 31, 1991. The parameters C(t) were chosen for each valuation date in such a way as to minimize the sum of the squared deviations of the prices calculated by the model from the actual market prices of all noncallable FNMA debentures.

ISFDs are distinguished from conventional bonds by the contingent nature of the sinking fund payment stream. Each ISFD debt service payment can be expressed as |Mathematical Expression Omitted~, where i(t) represents the coupon payment at time t and |Mathematical Expression Omitted~ denotes the sinking fund payment at time t, which is a function of |Mathematical Expression Omitted~, the average ten-year CMT during the semiannual interest period ending on the sinking fund payment date. Conventional sinking fund bonds typically make annual sinking fund payments, which commence after some grace period: |Mathematical Expression Omitted~ during the grace period; |Mathematical Expression Omitted~, a positive constant, on each sinking fund date, with a balloon payment |Mathematical Expression Omitted~ (with S* possibly equal to S) on the maturity date; and |Mathematical Expression Omitted~ for all other dates.

For the ISFDs, |Mathematical Expression Omitted~ for each sinking fund date is found in the following manner. First, calculate the ten-year CMT value for each trading day during the six-month interest period. Each value of the ten-year CMT is obtained by applying the estimated Treasury term structure. Second, the daily ten-year CMT values are averaged to obtain |Mathematical Expression Omitted~. Third, the amount of the sinking fund payment due at time t is calculated by finding the sinking fund payment percentage |Mathematical Expression Omitted~ that corresponds to |Mathematical Expression Omitted~ and multiplying by the amount of the ISFD issue B(t) outstanding immediately prior to the sinking fund payment date to obtain |Mathematical Expression Omitted~.

With the parameters |Kappa~, |Sigma~, C(t), |Theta~(t), and |Lambda~(t) estimated from the observed market prices of Treasury and FNMA bonds, Equation (7) is solved subject to the boundary condition (4) to calculate the estimated value of each FNMA ISFD issue. The ISFDs were valued for each trading day between July 31, 1990 and August 31, 1991, but to conserve space, only the ISFD values estimated for the last trading day of each week during this period are reported here. Exhibit 4 compares the market prices of each ISFD issue to the ISFD values estimated by applying the contingent claims valuation model.(7)

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).

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.

B. Implications for Asset-Liability Management

Why would the FNMA issue ISFDs? The composition of the FNMA's assets and the mortgage-like price performance of the ISFDs represent important clues. Exhibit 7 plots the modified durations of (i) the 8.70% ISFDs ("5-Year ISFD"), (ii) the 9.80% ISFDs ("10-Year ISFD"), (iii) five-year ("5-Year Conventional") and (iv) ten-year ("10-Year Conventional") conventional bonds, and (v) a ten percent 30-year fixed-rate mortgage ("10% 30-Year FRM"). The conventional bonds are identical to the respective ISFDs when the average ten-year CMT remains equal to the base rate throughout the life of the ISFDs.

The interest-rate sensitivity of the price of a fixed-rate mortgage (hereafter FRM) depends on the sensitivity of prepayments to interest rate changes. The mortgage prepayment speed tends to vary inversely with market interest rates (see Hayre and Mohebbi |13~). Consequently, the prices of mortgages and mortgage-backed securities generally exhibit negative convexity. In plotting the 10% 30-Year FRM curve, I assumed the following prepayment speeds: 400% of the PSA standard when interest rates drop 400 basis points, 280% PSA for a 300-basis-point drop, 200% PSA for a 200-basis-point drop, 140% PSA for a 100-basis-point drop, 100% PSA for no change in interest rates, 70% PSA when interest rates rise 100 basis points, 50% PSA for a 200-basis-point rise, 35% PSA for a 300-basis-point rise, and 25% PSA for a 400-basis-point rise.(11) These assumptions are admittedly arbitrary but they are representative of the sort of prepayment behavior FRMs exhibit in practice.

Two conclusions can be drawn from Exhibit 7. The interest-rate sensitivity of the FRM's price across a broad spectrum of interest rates more closely resembles the interest-rate sensitivity of the ISFD's price than the interest-rate sensitivity of the conventional bond's price. In particular, both the ISFD's price and the FRM's price exhibit greater interest-rate sensitivity as interest rates increase, just the opposite of how the interest-rate sensitivity of the conventional bond's price behaves. Second, the interest-rate sensitivity of the ten-year ISFD's price more closely tracks the interest-rate sensitivity of the FRM's price than does the interest-rate sensitivity of the five-year ISFD's price. As already noted, the FNMA redesigned the contingent sinking fund when it introduced the ten-year ISFD. The redesign increased (decreased) the sensitivity of the annual sinking fund percentage to a decrease (increase) in interest rates. As illustrated in Exhibit 7, the redesign resulted in the ISFD's sinking fund payments behaving more like mortgage prepayments.

A financial institution's profitability depends to a large degree on the net interest spread between the interest yield of its assets and the interest cost of its liabilities. A financial institution is said to be "perfectly duration matched" when the modified duration of its assets equals the modified duration of its liabilities. If a financial institution can remain perfectly duration matched as interest rates change, it can control how its net interest spread changes as interest rates change. But as market interest rates change, the modified durations of a financial institution's mortgage assets and its conventional fixed-rate liabilities tend to diverge. This divergence can exacerbate the change in the net interest spread unless the financial institution takes steps to reduce the duration gap.

Financing strategies previously employed to narrow a financial institution's duration gap include issuing mortgage pass-through securities and collateralized mortgage obligations (CMOs). The latter were developed in order to attract to the mortgage-backed securities market investors who are averse to mortgage prepayment risk (see Roberts et al |18~ and Spratlin et al |19~). But CMOs do not reduce overall prepayment risk; they just reallocate it from one class of investors to another.

Mortgage prepayment speeds are a function of several variables. ISFDs simplify the investors' evaluation of prepayment risk by reducing such risk to a function of a single variable, the change in the average ten-year CMT. ISFDs also limit the investors' exposure to prepayment risk to a specified contingency region. Thus, whereas mortgage pass-through securities and CMOs enable a financial institution that invests in mortgages to pass all the prepayment risk on to the investors who purchase the mortgage pass-through securities and CMOs, ISFDs effectively pass on only a portion of the prepayment risk, leaving the residual prepayment risk with the ISFD issuer. ISFDs were apparently motivated by a desire to develop a debt instrument that would pass on to investors a significant percentage -- but in all likelihood less than all -- the prepayment risk associated with investing in a portfolio of mortgages as well as an instrument whose interest-rate sensitivity would be easier for investors to quantify than the interest-rate sensitivity of mortgage pass-through securities and CMOs.

The FNMA issued five-year ISFDs first, and presumably after investors had familiarized themselves with the new security, opted for the ten-year maturity in order to achieve a closer duration matching. Depending upon one's assumptions regarding mortgage prepayments, it should be possible to adjust the maturity and sinking fund schedule of the ISFDs to achieve durations closer than the ones illustrated in Exhibit 7, that is, to achieve whatever allocation of prepayment risk is most mutually beneficial.

C. Recent Developments

The FNMA last issued ISFDs on September 7, 1989. Since that last issue, the FNMA has begun to issue two other types of debt instruments that convey asset-liability management benefits similar to those the ISFDs provided. First, the FNMA began to issue intermediate-term notes that provide just one year of call deferment. Such securities give FNMA the flexibility to redeem bonds beginning one year after issuance on whichever dates and in whatever amounts it deems appropriate -- in contrast to the fixed redemption formula contained in the ISFDs. One possible explanation for this development is that the FNMA has determined that purchasing the call option imbedded in the intermediate-term notes is cheaper than purchasing the options imbedded in the ISFDs.(12)

Second, on December 11, 1992, the FNMA issued \$150 million principal amount of 6.96% indexed redeemable medium-term notes due December 16, 1999 (IRMTNs). These notes incorporate an interest-rate-contingent sinking fund. But unlike the ISFD's sinking fund, which takes the form of a step function with large jumps in the sinking fund percentage at approximately 50-basis-point intervals as the ten-year CMT changes, the IRMTN's sinking fund makes continuous adjustments in the sinking fund percentage as the seven-year CMT changes. According to the FNMA, investors had expressed concern that the jumps in the ISFD's sinking fund percentage could make it very difficult to price an ISFD as a sinking fund date approaches if the ten-year CMT is very close to one of the break points listed in Exhibit 2. The contingent sinking fund incorporated in the IRMTNs alleviates this concern.

IV. Conclusion

This paper developed a contingent claims valuation model for ISFDs, a new type of security developed by the FNMA. ISFDs and similar financial instruments with contingent sinking funds represent an effective new tool for financial institution asset-liability management. Issuing such instruments permits a financial institution that invests in fixed-rate mortgages to match more closely the durations of its assets and liabilities.

1 The remaining balance illustrated for the no-change case would occur as long as the average ten-year CMT remained within the band from 8.61% to 9.10%.

2 Discussions with members of the FNMA's Treasurer's Department confirmed that the FNMA reengineered the contingent sinking fund to achieve this purpose.

3 Because each of the imbedded options has a nonnegative time value, one of the important implications of Equation (1) is that an issue of ISFDs would always be worth less than an otherwise identical issue of conventional bonds, even when the ten-year CMT remains equal to the base rate.

4 The on-the-run Treasury securities are ten recently issued, actively traded Treasury issues that span the Treasury securities market's 30-year maturity range. Bond market participants use the yields at which these ten issues trade as benchmarks for pricing other debt instruments.

5 It would normally be expected that C(t) would be an increasing function of t because default risk tends to be greater the more distant a payment's due date.

6 This procedure also produced an estimate of |Theta~ = 0.070396 for the 13-month period. This value was suppressed in order to permit |Theta~(t) to vary in the calibration described in Appendix A.

7 The requirement that the ISFD issue be fully repaid on any sinking fund payment date on which a scheduled redemption would reduce the remaining balance to less than five percent of the original balance introduces a time dependency that makes solving Equation (7) more complex. While this problem could be handled by introducing a second state variable, the five percent threshold is small enough that I felt justified in ignoring the five percent clean-up provision in valuing the ISFDs.

8 Macaulay duration assumes the yield curve is flat and that it shifts in parallel to its original position. Alternative measures of time duration have been proposed, but Bierwag |1~ provides empirical evidence that the Macaulay time duration measure is generally as accurate as the more sophisticated measures that are intended to take into account nonlevel term structures and nonparallel shifts in the term structure.

9 It is important to bear in mind that when the |A.sub.t~ in Equation (12) are expressed as semiannual cash flows, as in a corporate bond, y is the nominal annual rate divided by two. When the |A.sub.t~ are expressed as monthly cash flows, as in a mortgage, y is the nominal annual rate divided by 12.

10 At each interest rate where the semiannual sinking fund percentage jumps, modified duration jumps. Similarly, the price of the 8.70% ISFDs "jumps down" at each of these points.

11 The Public Securities Association (PSA) prepayment standard (referred to as 100% PSA) assumes that prepayments will occur at a 0.2% annual rate the first month the mortgage is outstanding and that the prepayment rate will increase in 0.2% increments until the thirtieth month when it will stabilize at a rate of 6.0% per annum for the remaining life of the mortgage (see Spratlin et al |19~).

12 Prior to 1990, the FNMA believed there was virtually no market for callable intermediate-term agency securities. The FNMA sought to develop such a market for its securities when it decided to improve upon the ISFDs. I would like to thank Donald Sinclair of the FNMA for useful discussions on this point.

References

1. G.O. Bierwag, Duration Analysis, Cambridge, MA, Ballinger, 1987, Chapters 11, 12.

2. M.J. Brennan and E.S. Schwartz, "An Equilibrium Model of Bond Pricing and a Test of Market Efficiency." Journal of Financial and Quantitative Analysis (September 1982), pp. 301-329.

3. J.C. Cox, J.E. Ingersoll, Jr., and S.A. Ross, "A Theory of the Term Structure of Interest Rates," Econometrica (March 1985), pp. 385-407.

4. Federal National Mortgage Association, 1988 Annual Report to Shareholders, Washington, D.C.

5. Federal National Mortgage Association, Prospectus for 8.70% Indexed Sinking Fund Debentures, Series SF-1993-A, July 7, 1988.

6. Federal National Mortgage Association, Prospectus for 9.05% Indexed Sinking Fund Debentures, Series SF-1993-B, October 4, 1988.

7. Federal National Mortgage Association, Prospectus for 9.80% Indexed Sinking Fund Debentures, Series SF-1998-A, December 5, 1988.

8. Federal National Mortgage Association, Prospectus for 9.75% Indexed Sinking Fund Debentures, Series SF-1999-A, January 30, 1989.

9. Federal National Mortgage Association, Prospectus for 9.95% Indexed Sinking Fund Debentures, Series SF-1999-B, May 2, 1989.

10. Federal National Mortgage Association, Prospectus for 9.15% Indexed Sinking Fund Debentures, Series SF-1999-C, September 7, 1989.

11. J.D. Finnerty and M. Rose, "Measuring the Relative Value of Fixed Income Securities: A New Approach," Fordham University Working Paper, September 1991.

12. G.E. Forsythe and W.R. Wasow, Finite-Difference Methods for Partial Differential Equations, New York, John Wiley & Sons, 1960.

13. L.S. Hayre and C. Mohebbi, "Mortgage Pass-Through Securities," in Advances & Innovations in the Bond and Mortgage Markets, F.J. Fabozzi (ed.), Chicago, Probus Publishing, 1989, pp. 259-304.

14. J. Hull and A. White, "Pricing Interest-Rate Derivative Securities," Review of Financial Studies (Winter 1990), pp. 573-592.

15. F.R. Macaulay, Some Theoretical Problems Suggested by the Movement of Interest Rates, Bond Yields and Stock Prices in the United States Since 1856, New York, National Bureau of Economic Research, 1938.

16. J.J. McConnell and E.S. Schwartz, "LYON Taming," Journal of Finance (July 1986), pp. 561-577.

17. J.P. Ogden, "An Analysis of Yield Curve Notes," Journal of Finance (March 1987), pp. 99-110.

18. B. Roberts, S.K. Wolf, and N. Wilt, "Advances and Innovations in the CMO Market," in Advances & Innovations in the Bond and Mortgage Markets, F.J. Fabozzi (ed.), Chicago, Probus Publishing, 1989, pp. 437-455.

19. J. Spratlin, P. Vianna, and S. Guterman, "An Investor's Guide to CMOs," in The Institutional Investor Focus on Investment Management, F.J. Fabozzi (ed.), Cambridge, MA, Ballinger, 1989, pp. 521-555.

20. O. Vasicek and H.G. Fong, "Term Structure Modelling Using Exponential Splines," Journal of Finance (May 1982), pp. 339-348.

21. R.S. Wilson, "Domestic Floating Rate and Adjustable Rate Debt Securities," in Floating Rate Instruments, F.J. Fabozzi (ed.), Chicago, Probus Publishing, 1986, pp. 5-52.

Appendix A. Stochastic Term Structure Fit(13)

The arbitrage pricing model (3) is not useful until the parameters |Kappa~, |Sigma~, |Theta~(t), and |Lambda~(t) are specified. In order to calibrate the pricing model so that the on-the-run Treasuries are repriced exactly, the theoretical discount function (5) must agree with the discount function derived from the observed current Treasury prices at all nonnegative levels of interest rate volatility. The mean-reversion speed |Kappa~ must be positive, and the term premium |Lambda~(t) must be negative in accordance with economic theory. Moreover, the parameters should be reasonably stable over time and term, i.e., they should not exhibit jump discontinuities or oscillatory behavior that is not present in the observed discount function (see Hull and White |14~).

The contingent claims model used to value the ISFDs and their imbedded options reproduces the observed discount function d(T), T |is greater than or equal to~ 0, for all nonnegative volatilities in two steps. In the first step, volatility |Sigma~ is set equal to zero. Then the central rate

|Theta~(t) = |r.sub.f~(t) + I/|Kappa~ d|r.sub.f~(t)/dt (A1)

is chosen so as to reproduce the observed discount function for any positive mean-reversion speed, where |r.sub.f~(t) denotes the instantaneous forward rate(14)

|r.sub.f~(t) = -d|prime~(t)/d(t) = -d/dtlogd(t). (A2)

Theorem 1. Let |Sigma~ = 0 and let |Theta~(t) be defined by Equation (A1). Then

d(T)/d(t) = exp(a(t ; T) + b(t ; T)|r.sub.f~(t)) (A3)

whenever 0 |is less than or equal to~ t |is less than or equal to~ T, where d(T) is the observed discount function and the coefficients a(t ; T) and b(t ; T) solve the ordinary differential equation system (6). In particular, setting t = 0 in Equation (A3) gives

d(T) = exp(a(0 ; T) + b(0 ; T)|r.sub.0~) (A4)

so that the theoretical and observed discount functions agree for all T |is greater than or equal to~ 0.(15)

Proof: The definition of |Theta~ in Equation (A1) implies

|Mathematical Expression Omitted~

for t |is less than or equal to~ s |is less than or equal to~ T. Applying Equations (A2) and (6),

|Mathematical Expression Omitted~.

Integration from t to T yields

-log d(T) + log d(t) = (a(T; T) + b(T; T)|r.sub.f~(T)) -(a(t; T) + b(t; T)|r.sub.f~(t)),

so that

log d(T)/d(t) = a(t ; T) + b(t ; T)|r.sub.f~(t).

Exponentiate to verify Equation (A3).

An interesting consequence of Equation (A1) is that the expected instantaneous future short-term rate

|Mathematical Expression Omitted~

agrees with the instantaneous forward rate |r.sub.f~(t) for all t |is greater than or equal to~ 0.(16) To verify this, set |Sigma~ = 0, take expectations on both sides of the stochastic differential Equation (2), and note that |r.sub.f~(t) satisfies the ordinary differential equation

|Mathematical Expression Omitted~,

with initial values |Mathematical Expression Omitted~.

The second step in the process of reproducing the observed discount function involves selecting the term premium |Lambda~(t) when volatility is permitted to be positive. Note that when |Sigma~ is positive, the solution to Equation (3) with c(r, t) = 0, with |Theta~(t) as above, and with zero term premium no longer agrees with the observed discount function. The size of the discrepancy between the observed and theoretical spot rate curves, i.e., between

|r.sub.s~(T) = -log d(T)/T

and

|Mathematical Expression Omitted~

increases with rate volatility |Sigma~ and term T. For an arbitrary term premium |Lambda~(t), the relation between |r.sub.s~ and |Mathematical Expression Omitted~ is given by the following theorem.

Theorem 2. When |Sigma~ |is greater than or equal to~ 0 and |Theta~(t) is defined by Equation (A1), the difference between the observed and theoretical spot rates is

|Mathematical Expression Omitted~

where |Lambda~(t) is the term premium. In particular, when |Lambda~(t) is identically zero

|Mathematical Expression Omitted~.

Proof: The theoretical discount function (5) is given by

|Mathematical Expression Omitted~.

Substitute |Theta~(t) from Equation (A1) and integrate by parts to obtain

|Mathematical Expression Omitted~,

where the boundary term equals -b(0 ; T)|r.sub.0~. Next, use the differential equation for b(t ; T) to expand the last integral

|Mathematical Expression Omitted~

where the first integral on the right is -log d(T). Combining the last three equations gives

|Mathematical Expression Omitted~,

which leads directly to Equation (A6).

Theorem 2 implies that the theoretical and observed spot rates would agree exactly if the term premium |Lambda~(t) solved the integral equation

|Mathematical Expression Omitted~

for all T |is greater than or equal to~ 0. The difficulty in extracting an analytic solution for |Lambda~(t) from Equation (A8) is that b(t ; T) depends upon |Lambda~(t) whenever rate volatility is positive. However, an approximate term premium ||Lambda~.sub.0~(t) may be obtained from Equation (A8) by setting the rate volatility equal to zero(17)

|Mathematical Expression Omitted~.

This analytic solution, which is easy to calculate, generally reduces the discrepancy between the theoretical and actual spot rates to a small fraction of a basis point for all terms T |is greater than or equal to~ 0, provided that the rate volatility is not unreasonably large. The reason setting |Lambda~(t) = ||Lambda~.sub.0~(t) reduces the error is that it increases the power of |Sigma~ on the right side of the error bound given by Equation (A6) from second-order to fourth-order. This is significant because the factor ||Sigma~.sup.2~ is very small in practice. For example, if the lognormal volatility ||Sigma~.sub.1~ and short-term rate |r.sub.0~ are both ten percent per annum, then ||Sigma~.sup.2~ = 0.001.(18)

Theorem 3. When |Theta~(t) is given by Equation (A1) and |Lambda~(t) = ||Lambda~.sub.0~(t) is given by Equation (A9)

|Mathematical Expression Omitted~

Proof: To verify Equation (A10), note that

|Mathematical Expression Omitted~.

Since

|Mathematical Expression Omitted~,

it follows that

|Mathematical Expression Omitted~.

Combine Equation (A6) and the last equation to complete the argument.

||Lambda~.sub.0~(t) can be viewed as the first term in a power series expansion of the term premium |Lambda~(t ; |Sigma~):

|Mathematical Expression Omitted~.

Analytic solutions can be obtained term-by-term for ||Lambda~.sub.n~(t), n |is greater than or equal to~ 0, by substituting the expansion (A11) into Equation (A8) and collecting the terms multiplying powers of ||Sigma~.sup.2~/2. For example, substituting ||Lambda).sub.0~(t) + ||Lambda~.sub.1~(t)||Sigma~.sup.2~/2 into Equation (A8) and solving the equation

|Mathematical Expression Omitted~

with zero volatility to obtain ||Lambda~.sub.1~(t) yields sixth-order agreement between |r.sub.s~ and |Mathematical Expression Omitted~.

The parameters |Kappa~ and |Sigma~ can be estimated either empirically, from time series of Treasury prices (see Brennan and Schwartz |2~ and Ogden |17~), or implicitly, from the current prices of derivative securities, such as debt options or interest rate cap or floor contracts.(19)

Appendix B. Estimating the FNMA Instantaneous Default Risk Premium

To calculate C(t) in Equation (7), apply the Vasicek-Fong |20~ procedure to obtain both the FNMA and on-the-run Treasury term structures, which are represented by the spot rate curves |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~, respectively. The instantaneous forward rate |r.sub.f~(t) is related to the spot rate:

|Mathematical Expression Omitted~.

The FNMA instantaneous default risk premium (or credit spread) is defined as the difference between the FNMA and on-the-run Treasury forward rates:

|Mathematical Expression Omitted~.

To verify that Equation (7) reproduces the FNMA discount function |d.sup.FNMA~(T) for this credit spread, use the analytic solution (5) with Equation (6a) replaced by Equation (6a|Prime~). Then