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Sinking fund prepurchases and the designation option.

Andrew Kalotay is an Associate Professor of Finance at Fordham University, and President of Andrew Kalotay Associates, New York, New York. Bruce Tuckman is an Assistant Professor of Finance at the Stern School of Business, New York University, New York, New York.

* Many corporate bond indentures contain sinking fund provisions. Previous work in this area has analyzed these provisions in terms of interest rate risk,(1) default risk,(2) and the "accumulation game," in which investors increase the value of a sinking fund bond issue by increasing the concentration of its ownership.(3) This paper studies sinking fund bonds when an issuer has bought some of its own bonds in anticipation of future sinking fund requirements. The analysis reveals that investor holdings decline in value as issuer prepurchases increase.

The provisions of a typical sinking fund require the issuer to retire fixed principal amounts before maturity. A simple example would be a $100 million bond issue sold in 1990 and maturing in 2000 which obligates the issuer to retire $25 million annually, beginning in 1997.(4) To facilitate these redemptions, the issuer may either purchase the required number of bonds in the market or call them at par. In the case of a call, the trustee randomly selects which of the outstanding bonds will be retired. In order to minimize the value of its outstanding bonds, i.e., to minimize its borrowing cost, the issuer will call bonds only when interest rates are relatively low. When rates are relatively high, the issuer will prefer to satisfy its requirements through market purchases.

Many issuers stay well ahead of their sinking fund requirements. Continuing with the above example, although the issuer need not retire any bonds until 1997, it might have purchased $25 million in the open market by the end of 1996. In other words, it might have prepurchased $25 million. To understand why an issuer would buy some of its own bonds and hold them in treasury, one must understand the designation option.

Indentures normally allow the issuer to designate any or all prepurchased bonds, at any time, to any future sinking fund date. In our example, the issuer could deliver the $25 million to the trustee with instructions to satisfy some future sinking fund requirement. If interest rates happen to be relatively low in 1997, this designation will have lowered the value of each bondholder's position: without the designation, $100 million of the bonds would have been outstanding and each investor would have expected one-fourth of his holding to be called through the $25 million sinking fund call. With the designation, however, only $75 million are outstanding and each investor expects one-third of his bonds to be called.

The designation option is also valuable when rates are relatively high. When bond prices are low, the issuer wants to satisfy its requirements through market purchases. But, if accumulators have bought most of the issue, the bonds will be hard to find and the issuer will be forced to buy from the accumulators at inflated prices. Returning to the example, say that accumulators have bought up $75 million face value by 1997. If the remaining $25 million were in the hands of small investors, or in the hands of loss-constrained investors,(5) the issuer might have to buy much of the $25 million requirement from the accumulators. But if the remaining $25 million were held in treasury, the issuer could designate $25 million to the requirement of 1997. In this way, the issuer postpones paying off the accumulators, thus lowering their profits and its own borrowing costs. Furthermore, the presence of the $25 million in treasury might well deter potential accumulators from hoarding the bonds in the first place.

Section I of this paper presents some summary statistics on sinking funds provisions and on the prevalence of prepurchases. Section II, taking the level of prepurchases as given, presents an algorithm for deriving the optimal designation policy and for valuing sinking fund bonds. Section III presents some numerical examples to illustrate how optimal designations of a given amount of prepurchases affect bond values. Section IV discusses market equilibrium and optimal prepurchase policy. Section V concludes.

I. Summary Statistics on Sinking Funds and Issuer Prepurchases

The management of sinking funds is important for both financial managers and investors because these provisions are quite common in the public issues of industrial firms. To support this claim, 200 bonds were randomly selected from the 1990 edition of Moody's Industrial Manual. Of the 192 bonds for which Moody's supplied sufficient information, 90, or 46.88%, had sinking fund provisions. This statistic does not tell the full story, however, because sinking funds are most common in long-term issues. Exhibit 1 reports the prevalence of sinking fund provisions as a function of original maturity. Note that about 77% of bonds issued with maturities longer than 20 years contain sinking fund provisions.
Exhibit 1. The Prevalence of Sinking Fund Provisions as a
Function of Original Maturity
Original Number of Number With % With
Maturity Observations Sinking Funds Sinking Funds
10 years or less 69 7 10.14%
11 to 20 years 39 20 51.28%
More than 20
years 78 60 76.92%
Data were collected from Moody's Industrial Manual, 1990.
The number of observations equals 186 because the original
maturity date was unavailable for some issues.

Exhibit 2 reveals that bonds of lower credit quality are more likely to contain sinking fund provisions. This fact can be explained by the Myers |10~ argument that sinking funds are designed to "reduce creditors' exposure in parallel with the expected decline in the value of assets in place." The worse the credit quality, the greater the need for this kind of protection.

To examine some of the common features of sinking fund bonds and to assess the prevalence of prepurchases, another sample, consisting of 257 sinking fund bonds, was collected from Moody's.(6)

One measure of a sinking fund provision's strength is the fraction of original principal to be retired before maturity. For example, a $100 million issue with annual sinking fund payments of $10 million retires 90% of original principal before maturity, while a $100 million issue with annual payments of $5 million retires only 45% before maturity. In the sample, the average fraction retired was 88.62% with a standard deviation of 9.96%.
Exhibit 2. The Prevalence of Sinking Fund Provisions as a
Functions of Credit Rating
 Number of Number With % With
Rating Observations Sinking Funds Sinking Funds
Aaa 14 1 7.14%
Aa 22 8 36.36%
a 66 28 42.42%
Baa 42 18 42.86%
Ba 17 10 58.82%
B 26 21 80.77%
Caa 2 1 50.00%
Data collected from Moody's Industrial Manual, 1990.
The number of observations is 189 because not all 192 bonds
were rated.

Many bond indentures include optional sinking fund payments, called "acceleration" features, which allow the issuer to call more than the required sinking fund payments at par. The "double-up" and "triple-up" options allow an issuer to call a total of twice or three times the mandatory sinking fund requirement on each date. For the sake of completeness, allowing the issuer to call 2.5 times the mandatory sinking fund requirement will be called a 2.5-up option.(7) Exhibit 3 reports the frequency of these acceleration features in this sample. The double-up feature dominates, characterizing over 81% of the issues. Note that only about seven percent of the issues contain no optional sinking fund payments.

Aside from the option to purchase bonds for the purpose of satisfying sinking fund requirements, most sinking fund bond indentures include an American call option which allows the issuer to call part or all of the issue at some schedule of prices. These exercise prices usually begin above par and decline linearly to par. All but three of the 257 bonds contain an American call feature.

Finally, this data can be used to study the prevalence of prepurchases. Consider the Crown-Zellerbach 8 7/8s of 12/15/2000. The original issue size of $125 million was to be retired through an annual sinking fund of $5 million from 1982 to 1999. If Crown-Zellerbach made no prepurchases, the principal outstanding at the end of 1989 would be $85 million, the original $125 million minus the eight $5 million payments for the years 1982 to 1989. But Moody's reports that there was only $50 million outstanding at the start of 1990. In other words, Crown-Zellerbach had prepurchased $35 million, or seven sinking fund payments.(8)
Exhibit 3. The Prevalence of Optional Sinking Fund Provisions
 Number in Sample % in Sample
No optional sinking fund 19 7.39%
Double-up option 209 81.32%
2.5-up option 11 4.28%
Triple-up 18 7.00%
Note: Data collected from Moody's Industrial Manual, 1990.

On average, the companies in the sample were 2.45 sinking fund payments ahead of schedule. Exhibit 4 shows the distribution of prepurchases across the sample. About 55% of the issues experienced prepurchases of at least one sinking fund payment.

II. The Designation Option and the Management of Prepurchases

For expository purposes, we continue with the example presented earlier. In 1990, XYZ Corporation issued $100 million of a nine percent ten-year sinking fund bond issue. The sinking fund provision requires that XYZ retire $25 million in each of the years 1997 to 2000.

Any bonds prepurchased by XYZ are put into the corporation's treasury and are governed by two rules. First, treasury bonds are considered outstanding for the purposes of a call.(9) The consequences of this rule may be understood by supposing that XYZ has purchased $25 million of its own bonds by 1997. If rates are below nine percent at that time, XYZ will probably meet its first sinking fund obligation by calling the required $25 million. Therefore, the trustee will randomly select $25 million from the $100 million outstanding and then call the selected bonds. On average, then, one-fourth of the $75 million held by investors, or $18.75 million, and one-fourth of the $25 million in treasury, or $6.25 million, will be called. But, since rates are below nine percent, the company would have preferred to call the entire $25 million from investors.
Exhibit 4. Prepurchases, in Number of Sinking Fund Payments
Number of Payments Number Percentage
Prepurchased in Sample in Sample
Less than 1 114 44.36%
Between 1 and 3 65 25.29%
Between 3 and 5 33 12.84%
Between 5 and 7 22 8.56%
More than 7 23 8.95%
Data collected from Moody's Industrial Manual.

The issuer's desire to call the entire $25 million from investors introduces the second rule about treasury bonds, namely, the designation option. In the words of a sample indenture,

... the Company may, at its option, reduce the amount of any Mandatory Sinking Fund Payment payable on any Sinking Fund Date by an amount equal to the Sinking Fund Redemption Price of Outstanding Debentures which shall be surrendered uncancelled and in transferable form by the Company to the Trustee ... together with an Officers' Certificate stating its election to use such Debentures for such purpose ...(10)

Since the "Sinking Fund Redemption Price" is almost always par, this provision allows XYZ to apply its $25 million of treasury bonds against any future sinking fund requirement. Say that it designates the $25 million to the requirement in 1999. These $25 million will be cancelled and only $75 million will remain outstanding. As a result, the subsequent call of $25 million to meet the sinking fund requirement of 1997 will result in all $25 million being called away from investors, as desired.

Designation, however, has its drawbacks. First, since the 1999 requirement has been satisfied, the issuer has forfeited its option to call $25 million at par in 1999. Second, as rates change, the right to call bonds in 1999 might be large relative to the right to call them in 1998. In other words, XYZ may wish that it had forfeited the 1998 option instead. Third, if the issue is cornered by an accumulator and rates rise, the company may wish it had kept the $25 million in treasury to meet its 1998 requirement. An optimal designation policy weighs these disadvantages against the gains from calling more of the bonds held by investors.

Another type of designation option pertains to bonds acquired through an acceleration feature or through an American call option. May the issuer choose how to designate these optionally redeemed bonds? While the designation option with respect to prepurchased bonds is included in most sinking fund bond indentures, the designation option with respect to optionally redeemed bonds may or may not be granted.(11) If not, the issuer usually has to apply any optionally redeemed bonds to the most distant, unsatisfied payment dates. For the rest of this paper, it will be assumed that the issuer has a designation option with respect to optionally redeemed bonds. The valuation technique to be presented here can be easily adjusted for those issues which do not grant this designation option.

Unlike treasury bonds, optionally redeemed bonds are not considered outstanding.(12) Thus, designating them prior to a call sacrifices flexibility without raising the number of bonds called away from investors. Consequently, optionally redeemed bonds should be designated only in order to postpone payments to accumulators.

Having described sinking funds in some detail and presented the intuition behind designation policy, the discussion turns to valuation. Two special cases are considered. The perfectly competitive case assumes that all investors own small holdings and that they do not coordinate their actions. In particular, each investor takes the quantity of issuer purchases as given when choosing that price at which he is willing to sell. The fully accumulated case, on the other hand, assumes that a single consortium of investors owns all the bonds not owned by the company. When the company offers to purchase bonds, the consortium asks for the maximum price that the company is willing to pay. While these extreme cases provide much insight, it should be noted that they only straddle more realistic ownership distributions.


The value of a sinking fund bond issue depends on its contractual features, on its ownership distribution, and on the level of interest rates. Exhibit 5 presents the notation used in the valuation procedure described below. Although not explicitly noted, variables describing ownership distribution and the rate of interest will change over time.

Let the value of $1 principal amount of the sinking fund bond held by investors at time t be |V.sub.t~ |{|a.sub.s~}, I, T~, where the dependence on the contractual provisions has been suppressed. Note that the dependence on R need not be explicitly recorded since there is an identity linking the values of {|a.sub.s~}, I, T, and R at any time. More specifically,

|Mathematical Expression Omitted~

In other words, the face value required to meet the contractual sinking fund requirements equals the face value available for that purpose, namely that held by investors, by the company in treasury, and by the trustee. The last category includes undesignated bonds acquired through optional redemptions (R), as well as designated bonds acquired through market purchases, |summation~|a.sub.s~, and optional redemptions.

The goal now is to write an expression for |V.sub.t~ as a function of |V.sub.t + |delta~t~ for some time increment |delta~t. Combining this expression with the maturity condition that |V.sub.N~ = 1 produces a recursive algorithm for computing |V.sub.t~ for any t.(13) It will be assumed throughout that future cash flows can be valued by discounting their expectation, under risk-neutral probabilities, by the risk-free rate.(14)

A. Valuation on Non-Sinking-Fund Dates

Before the first sinking fund date and between sinking fund dates, the American call feature is the only option available to the issuer. And since this option allows the issuer to call as many bonds as it pleases, there is no need to designate treasury bonds so as to raise the number of bonds which can be called away from investors. Therefore, on non-sinking-fund dates, the issuer should conserve its treasury bonds and refrain from designations. Thus, the issuer need only choose a quantity of bonds to call, C, so as to minimize the value of the outstanding bonds. As a result, the value of each $1 principal amount held by investors is

|Mathematical Expression Omitted~

where E() denotes the expectation operator under the risk-neutral probabilities and |Mathematical Expression Omitted~ denotes the number of bonds remaining in treasury after the call. Given the function |V.sub.t + |delta~t~, the optimal choice of C can be made and |V.sub.t~ can be computed.(15)

B. Valuation on Sinking Fund Dates

This discussion begins with valuation under the assumption that the company meets its redemption requirement through market purchases or designations. Then the discussion assumes that the company meets its requirement through a sinking fund call. Equipped with both results, the company chooses that action which results in the lower bond value. As one would expect, when rates are relatively high, the optimal strategy consists of market purchases or designations. When rates are relatively low, the optimal strategy is a sinking fund call.

1. Meeting the Sinking Fund Requirement Through Market Purchases or Designations

If, as assumed in this subsection, the issuer decides not to use its sinking fund call option, it will also choose not to use its acceleration option and not to use its American call option. Therefore, the issuer need only decide whether to satisfy its requirements through market purchases or through designations. As it turns out, that decision depends on the distribution of ownership of the bonds across investors.

First, assume that the bonds are competitively held and that the issuer chooses to engage in market purchases. Recalling that competitive investors take the purchase quantity, |F.sub.t~ - |a.sub.t~, as given, they will ask for that price which makes them indifferent between selling the bonds and holding them. Therefore,

|Mathematical Expression Omitted~

If, on the other hand, the issuer chooses to satisfy the sinking fund requirement by designations, the resulting value function will be the same as Equation (3a), except that the investor holding argument will be larger than I - (|F.sub.t~ - |a.sub.t~) and the treasury holding argument will not be greater than T. These changes, which reflect reduced flexibility in meeting future sinking fund requirements, result in a higher value than given by Equation (3a). Hence, when bonds are competitively held, issuers will choose market purchases over designations.

Now, assume that the bonds are fully accumulated. If T + R |is greater than or equal to~ |F.sub.t~ - |a.sub.t~, the current requirement can be met by designations alone and

|Mathematical Expression Omitted~

where |Mathematical Expression Omitted~ denotes the number of bonds remaining in treasury after these designations.

If, on the other hand, T + R |is less than~ |F.sub.t~ - |a.sub.t~, the current requirement cannot be met by designations alone. The issuer will first designate all available bonds and then purchase |F.sub.t~ - |a.sub.t~ - T - R face value from the accumulator. Since the accumulator knows that the only other way the issuer can acquire these bonds is through a sinking fund call at par, the accumulator can demand par for this quantity of bonds.(16) So, the value of the outstanding bonds in this case is

|Mathematical Expression Omitted~

Note how the relatively large number of treasury bonds available in the determination of Equation (3b) postpones the need to pay par for the |F.sub.t~ - |a.sub.t~ - T - R bonds indicated in Equation (3c).

2. Meeting the Sinking Fund Requirement Through a Call

Now assume that the issuer meets its requirement through a sinking fund call. It will choose a set of designations from treasury bonds to particular sinking fund dates, {|d.sub.s~}, a quantity of bonds to call through optional sinking fund payments, O, and a quantity of bonds to call through the American call option, C, so as to minimize the value of the outstanding bonds. Since the amount outstanding before the calls falls from |Mathematical Expression Omitted~, the fraction of investor and treasury bonds called at par is given by |Mathematical Expression Omitted~. Similarly, the fraction called at the American call price |K.sub.t~ is |Mathematical Expression Omitted~. Therefore, the value of $1 principal amount held by investors is

|Mathematical Expression Omitted~

where |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~, the number of bonds held by investors and by the corporate treasury after the designations and the calls. Note how designations raise the fraction of investor bonds to be called at the costs of giving up treasury bonds and of sacrificing future sinking fund options.

III. Numerical Examples

The examples in this section continue with the example in Section II. In 1990, the XYZ Corporation issued a nine percent, $100 million, ten-year sinking fund bond issue with mandatory payments of $25 million in each of the years 1997 to 2000. Furthermore, through an American option, the bonds may be called in 1997 at 103, in 1998 at 102, and in 1999 at 101.
Exhibit 6. Term Structure of Interest Rates for the Numerical
 Zero-Coupon Zero-Coupon
Maturity Yield Maturity Yield
1 7.000% 6 8.525%
2 7.500% 7 8.644%
3 7.875% 8 8.733%
4 8.156% 9 8.800%
5 8.367% 10 8.850%

While the valuation technique outlined in Section II can be applied to any interest rate process, implementing the procedure requires the specification of a particular process. For simplicity, this section uses a binomial model with lognormally distributed short-term rates(17) and a time step of one year. Furthermore, the model is calibrated to produce the term structure given in Exhibit 6 and a short-term rate volatility of 15%.

The sinking fund requirement begins in 1997. Exhibits 7, 8 and 9 report the value of $100 face amount of the XYZ bonds in 1997 as a function of the short-term rate and of the distribution of bond ownership. The basic message of these exhibits is that bond values decline as the number of bonds held in treasury increases.

Exhibit 7 gives bond values in 1997 under the scenario that the then prevailing short-term rate is 5.59%.(18) This rate is very low relative to the nine percent coupon, so XYZ will choose to meet its sinking fund requirement through the sinking fund call at par and to call any remaining bonds through its American option at 103. More specifically, without any bonds in treasury it will call $25 million at par and $75 million at 103, for an average value of 102.25 per investor bond. With $25 million in treasury, it will designate the $25 million to any date other than 1997 and then call $25 million at par and $50 million at 103 for an average value of 102 per bond. Similar reasoning gives the other two values in the third column. The declining values in the third column show that past prepurchases lower the value of outstanding bonds by raising the fraction which may be called under the sinking fund provision.
Exhibit 7. Value of Bonds Held by Investors in 1997 as a
Function of the Distribution of Ownership (the Short-Term Rate
is 5.59%)
Principal Principal Value When Value When
Amount Amount Investor Bonds Investor Bonds
Held Held Are Are
by in Competitively Fully
Investors Treasury Held Accumulated
$100 million $0 million $102.25 $102.25
$75 million $25 million $102.00 $102.00
$50 million $50 million $101.50 $101.50
$25 million $75 million $100.00 $100.00

The fourth column of Exhibit 7 shows that accumulated bonds are worth no more than competitively held bonds when the short rate is at this low level: the advantage of accumulating an issue accrues only when the issuer cannot purchase bonds in the market and, therefore, must negotiate with accumulators. When the issuer is calling the entire issue, however, the distribution of ownership across investors has no impact on bond values.

Exhibit 8 gives bond values when the short-term rate is 7.55%. This rate is low enough so that the issuer meets its sinking fund requirement by a call, but not low enough so that the issuer exercises its American option. When XYZ holds $75 million, it applies the full amount to satisfy the sinking fund requirements of 1998 to 2000, and calls the remaining $25 million at par. When holding $50 million, it applies the full amount to satisfy the sinking fund requirements of 1999 to 2000, and calls $25 million at par. Designating to the later dates allows the issuer to call the remaining $25 million in 1998 at par. A treasury position of $25 million allows the issuer to call one-third of investor holdings, while a treasury position of zero allows the issuer to call only one-fourth of their holdings. The values in the third column decline by less than the corresponding values in Exhibit 7 because the issuer's options are more valuable when the short rate is 5.59% than when it is 7.55%.

Comparing the third and fourth columns of Exhibit 8 reveals that, when the treasury holds $50 million or more, the value of an accumulated position equals that of a competitive position. This is because the rapid retirement of the bonds eliminates any chance that the company will eventually need to engage in market purchases. But, when the treasury holds only $25 million, the value of an accumulated position exceeds that of a competitive position because there is a chance that rates will rise before all of the bonds are retired. In other words, there is a chance that the accumulators will eventually squeeze the issuer.
Exhibit 8. Value of Bonds held by Investors in 1997 as a
Function of the Distribution of Ownership (the Short-Term Rate
is 7.55%)
Principal Principal Value When Value When
Amount Among Investor Bonds Investor Bonds
Held Held Are Are
Investors Treasury Held Accumulated
$100 million $0 million $101.51 $101.55
$75 million $25 million $101.18 $101.25
$50 million $50 million $100.67 $100.67
$25 million $75 million $100.00 $100.00

Exhibits 7 and 8 illustrate the effect of past prepurchases when rates are relatively low. Exhibit 9 illustrates the effect when rates are relatively high, namely, when the short-term rate is 10.19%. For competitive holdings, bond values decline only slightly as treasury holdings increase; since the value of the issuer's options are small, the value of calling larger and larger fractions of outstanding bonds is small. For fully accumulated holdings, however, bond values decline substantially as treasury holdings increase; using treasury bonds to meet current requirements enables the issuer to postpone making payments to accumulators.

Exhibits 7 through 9 report values as of 1997, the date of the first sinking fund payment. Exhibit 10 presents the values in 1990, the issue date, when, as can be seen from Exhibit 6, the short-term rate is seven percent. Bond values decline with the number of bonds held in treasury, for both competitive and accumulated holdings. The effect on competitive holdings is not as large as on sinking fund dates when rates are relatively low, nor is the effect on accumulated holdings as large as on sinking fund dates when rates are relatively high. This is so because the two advantages, namely, calling larger fractions of investor holdings and postponing payments to accumulators, will not be realized until 1997 and beyond. This point will be important in the equilibrium discussion of the following section.

IV. Equilibrium and Optimal Prepurchase Policy

The previous sections show how the value of sinking fund bonds decrease as treasury holdings increase and how accumulation enhances bond value. These observations lead to two questions. First, why don't issuers repurchase more of their bonds than indicated by the data in Section I? Second, why aren't all sinking fund issues accumulated by eager investors?
Exhibit 9. Value of Bonds Held by Investors in 1997 as a
Function of the Distribution of Ownership (the Short-Term Rate
is 10.19%)
Principal Principal Value When Value When
Amount Amount Investor Bonds Investor Bonds
Held Held Are Are
by in Competitively Fully
Investors Treasury Held Accumulated
$100 million $0 million $96.89 $98.47
$75 million $25 million $96.80 $97.96
$50 million $50 million $96.80 $97.31
$25 million $75 million $96.80 $96.80

In the absence of market frictions, these questions cannot be answered. There is always an incentive for both issuers and accumulators to add to their positions. The presence of market frictions, however, may dampen these incentives. An issuer incurs transaction costs when purchasing bonds in the market and when financing these purchases. These costs might exceed the advantages of holding more bonds in treasury. Similarly, from an investor's perspective, trading costs, financing costs, and the loss of diversification from buying so much of one issue might outweigh the gains from accumulation.

These costs provide the basis of a theory about the holding pattern and pricing of sinking fund bonds. Because the gains from repurchases and accumulation are relatively small at the time of issue (see Exhibit 10), neither the issuer nor the accumulator has an incentive to incur the costs and to purchase the bonds. As the bonds mature and interest rates change, however, their decisions may change. According to the exhibits in Section III, by the time the sinking fund becomes operational, treasury holdings and accumulation affect values substantially. Therefore, in a low-interest-rate environment, when the call options are particularly valuable, the issuer may very well find that the gains from repurchases exceed the accompanying transaction costs. On the other hand, in a high-interest-rate environment, both potential accumulators and the issuer have incentives to purchase sinking fund bonds. Knowing that the issuer will try to satisfy sinking fund requirements through market purchases, accumulators will want to corner the issue. In turn, the issuer will try to protect itself against accumulators by increasing its treasury holdings. In fact, given the costs of accumulation mentioned above, accumulators may not find it worthwhile to accumulate sinking fund bond issues which are protected by large treasury holdings. It follows that issuers can prevent accumulation by a carefully designed prepurchase policy.
Exhibit 10. Value of Bonds Held by Investors in 1990 When the
Short-Term Rate is 7.00% as a Function of the Distribution of
Principal Principal Value When Value When
Amount Amount Investor Bonds Investor Bonds
Held Held Are Are
by in Competitively Fully
Investors Treasury Held Accumulated
$100 million $0 million $101.44 $102.39
$75 million $25 million $101.33 $101.97
$50 million $50 million $101.18 $101.47
$25 million $75 million $100.86 $100.86

So long as market participants are aware of the effects described in this paper, market prices will reflect them. More specifically, the battle between accumulators and issuers will provide a price cushion in high-rate environments, while issuer prepurchases will depress prices in low-rate environments. Furthermore, if an issuer's borrowing rate reflects its ability to prepurchase and optimally designate bonds, then it must adopt optimal prepurchase and designation policies to break even.

It is more likely, however, that most issuers and investors have not fully understood and accounted for optimal prepurchases and designations. In that case, the ideas presented here offer opportunities to both issuers and investors striving for superior performance in a very competitive marketplace.

V. Conclusion

This paper shows how the designation of prepurchased bonds allows the issuer to minimize bond values, suggesting that issuers might be able to reduce borrowing costs by adopting an active prepurchase and designation policy. Such a policy would weigh the benefits of prepurchases and designations against the concomitant costs. This policy is likely to result in prepurchases either when interest rates have fallen well below the coupon rate or when they have risen and potential accumulators are circling the issue.(19)

Investors might profit from monitoring issuers' prepurchase and designation activities. While this paper has emphasized effects on valuation, duration is affected as well. When rates are low, prepurchases and designations reduce duration below that indicated by maturity and sinking fund structure. Conversely, when rates are high, prepurchases and optimal designations lengthen duration.

1 See Ho |4~.

2 See Ho and Singer |5~, |6~.

3 See Dunn and Spatt |3~, and Kalotay |8~.

4 In market jargon, the principal payment at maturity is not called a sinking fund payment. For ease of exposition, however, this paper will refer to all required principal payments as sinking fund payments.

5 Loss-constrained investors may not, or are extremely averse to taking book losses on the sale of securities. The most important example would be insurance companies for whom book losses reduce the size of allowed underwriting positions.

6 This sample includes every nonconvertible issue which the Moody's "Chronological List of Bonds and Notes" calls a sinking fund issue. For example, the "Martin Marietta Sinking Fund Debentures, 5 7/8s of 1992" would be included in the sample, while the "TRW Inc., Debentures, 5 1/2s of 1992" would not, even though both contain sinking fund provisions. After deleting the few bonds which were not accompanied by an adequate description, 257 bonds remained.

Convertible issues were excluded because it is difficult to distinguish between bonds no longer outstanding because of prepurchases and bonds which are no longer outstanding because of conversions.

7 These options are not cumulative, i.e., if the issuer does not choose to call more than the mandatory requirement on one date, it cannot call more than the mandatory plus optional amounts on the following date.

8 It is possible that Crown-Zellerbach has been using its double-up option, so its does not logically follow that all of the $35 million were purchased in the market. Nevertheless, market purchases are the most likely explanation. First, the double-up option will have been used when rates are relatively low, i.e., when the American call might have been used to retire all of the issue. But the fact that much of the issue is still outstanding indicates that the American call has not been used. Second, if use of the double-up option accounted for much of the prepurchases, one would expect that the higher coupons would exhibit significantly more prepurchases. In fact, the average level of prepurchases for low coupons slightly exceeds the average level for high coupons. Third, until very recently, interest rates have been high relative to the average coupon in the sample.

9 American Bar Foundation |1, p. 42~.

10 American Bar Foundation |1, p. 517~.

11 See American Bar Foundation |1, pp. 516-519~ for sample indentures of both types.

12 American Bar Foundation |1, p. 41~.

13 The usual technique of backward induction will not work here. To see this, consider the XYZ nine percent bonds in 1999, one year before maturity. The value of these bonds clearly depends on the remaining sinking fund requirement for the year 1999. If the remaining requirement is the original $25 million, XYZ has the right to call $25 million in face value. If the remaining requirement is $15 million, because XYZ had previously designated $10 million to the 1999 payment, XYZ may call only $15 million through the mandatory sinking fund. Therefore, bond values depend on actions taken before 1999. In other words, unlike most fixed-income securities, knowing the interest rate in 1999 is not enough to price a sinking fund bond issue with a designation option.

14 See Ingersoll |7, chapter 2~ for a review of this method.

15 It is assumed that investors are risk-neutral with respect to the lottery risk of a call. The justification for this assumption lies in the fact that lottery risk does not constitute market risk and is, therefore, diversifiable.

16 Dunn and Spatt |3~ provide a game-theoretic justification of this argument.

17 In the context of a discrete time model, this means that the rates on one date are multiplicative perturbations of the rates on the previous date and that volatilities are calculated from the logarithms of the rates. See, for example, Black, Derman, and Toy |2~.

18 Given the binomial model with a one-year time step, there are eight possible values for the short-term rate in 1997. Three of these, used in Exhibits 7, 8, and 9, respectively, are 5.59%, 7.55%, and 10.19%.

19 Tax and accounting considerations may very well alter the optimal prepurchase strategy. When bond prices are low, prepurchases result in tax liabilities and accounting gains. When bond prices are high, prepurchases result in tax deductions and accounting losses.


1. American Bar Foundation, Corporate Debt Financing Project, Commentaries on Model Debenture Provisions, American Bar Foundation, 1971.

2. F. Black, E. Derman, and W. Toy, "A One-Factor Model of Interest Rates and Its Application to Treasury Bond Options," Financial Analysts Journal (January-February 1990), pp. 33-39.

3. K. Dunn and C. Spatt, "A Strategic Analysis of Sinking Fund Bonds," Journal of Financial Economics (September 1984), pp. 399-423.

4. T. Ho, "The Value of a Sinking Fund Bond Provision Under Interest Rate Risk," in Recent Advances in Corporate Finance, E. Altman and M. Subrahmanyam (eds.), Richard D. Irwin Press, 1985, pp. 55-77.

5. T. Ho and R. Singer, "Bond Indenture Provisions and the Risk of Corporate Debt," Journal of Financial Economics (December 1982), pp. 375-406.

6. T. Ho and R. Singer, "The Value of Corporate Debt With a Sinking Fund Provision," Journal of Business (July 1984), pp. 315-336.

7. J. Ingersoll, Theory of Financial Decision Making, Rowman and Littlefield Publishers, 1987.

8. A. Kalotay, "On the Management of Sinking Funds," Financial Management (Summer 1981), pp. 34-40.

9. A. Kalotay and G. Williams, "The Valuation and Management of Bonds With Sinking Fund Provisions," Financial Analysts Journal (March-April 1992), pp. 59-67.

10. S. Myers, "Determinants of Corporate Borrowing," Journal of Financial Economics (November 1977), pp. 147-175.
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Title Annotation:Market Microstructure and Corporate Finance Special Issue
Author:Kalotay, Andrew; Tuckman, Bruce
Publication:Financial Management
Date:Dec 22, 1992
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