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Implied tax rates and the valuation of discount bonds
Journal of Banking and Finance 6 (1982) 145-159. North-Holland Publishing Company
IMPLIED TAX RATES AND THE VALUATION OF DISCOUNT BONDS James C. Van HORNE* Stanford University, Stanford, CA 94305, USA Received September 1980, final version received July 1981
Discount bonds afford the investor the opportunity for capital gains. If for tax reasons the market is segmented on the demand side, investors in lower and lower tax brackets must be attracted when interest rates rise and the supply of discount bonds increases. Changes in the differential tax on capital gains and interest income also should affect relative demand. Testing these hypotheses with U.S. Treasury bond data, the implied tax rate is found to vary over time in a manner consistent with market segmentation and tax law changes.
1. Introduction
Because of the capital gains attraction of discount bonds, they are valued differently in the market from bonds which trade near par. As the supply of discount bonds varies over time with changes in the overall level of interest rates, the valuation of the capital gains tax feature may vary as well. In this paper, implied individual and corporate tax rates are derived for the five-year period 1975-1979, using monthly data on Treasury bonds selling near par and a bond which sells at a discount. The results are consistent with segmentation in investor demand for discount bonds as well as with the 1976 and 1978 tax-law changes which affected capital gains. Heretofore, there have been few empirical studies of discount bonds t and none of the implied tax rates embodied in market prices over time. The paper begins with a discussion of the valuation of discount bonds and of the hypotheses to be tested, followed by the presentation of a method for determining implied tax rates. The data and their limitations are explored in the third section and the results and their implications in the fourth section. A brief summary appears at the end. *The study was supported by the Stanford Program in Finance. I am grateful to Lawrence Fok for his programming assistance on this project and to George G. Kaufman and Stephen M. Schaefer for a number of helpful comments. 1For some empirical studies, see Cook and I-Iendershott (1978), Robichek and Niebuhr (1970), and Van H o m e (1978).
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© 1982 North-Holland
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J.C. Van Home, Implied tax rates and discount bonds valuation
2. Yields and taxes
In the absence of taxes, the yield to maturity of a bond represents the promised rate of return applicable to all investors. Financial markets would be expected to equilibrate in terms of this rate. With taxes, however, a bond's yield no longer represents the relevant return to taxable investors. The reason is that the yield formula assumes that interest income and capital gains are not taxed. The favorable tax treatment of capital gains makes taxable fixed-income securities selling at a discount from their par values attractive to some taxable investors. As a result, their pre-tax yield to maturity tends to be lower than the yield of comparable bonds with higher coupon rates. The greater the discount, the greater the capital gains attraction and the lower the pre-tax yield relative to higher coupon bonds. However, the relationship between the amount of discount and the differential in yield between the discount bond and a higher current coupon bond which trades at par may not be stable over time. The market for discount bonds may be partially segmented. The basic appeal of a discount bond is to the taxable investor. The investor who pays no taxes, such as a pension fund or a University endowment fund, could well be indifferent between a dollar of interest and a dollar of capital gains where both are discounted by the market at the same rate. However, if taxable investors bid up the price of discount bonds so that they yield less before taxes than higher coupon bonds which trade at par, the non-taxable investor will have an incentive to invest in the higher coupon bond. The same reasoning applies, though to a lesser extent, to certain institutional investors which pay taxes at less than the full corporate tax rate. Discount bonds are primarily attractive to higher income individuals and to fire and casualty companies. While commercial banks typically are in high tax brackets, they are unable to avail themselves of the favorable tax rate on capital gains. Unfortunately for them, realized capital gains are treated as ordinary income when it comes to paying taxes. Therefore, commercial banks may seek bonds selling nearly at par or above. 2 Thus, the demand for discount bonds is influenced by the flow of funds to fire and casualty companies and the wherewithal of wealthy individuals to invest. On the other hand, the supply of discount bonds changes with changes in the level of interest rates. At cyclical peaks in interest rates, there are far more discount bonds than there are at cyclical troughs. By definition, at these times there are bonds issued in the past at coupon rates lower than prevailing interest rates in the market. In other words, the supply of discount bonds varies directly with the level of interest rates. 2For an analysis of the investment behavior of institutional investors with respect to taxes and likely future reinvestment rates on coupon payments, see Cramer and Hawk (1975).
J.C. Van Home, Implied tax rates and discount bonds valuation
147
If the demand for discount bonds is partially segmented, these bonds must appeal to new clienteles as the supply of them increases. As interest rates rise and the supply of discount bonds increases, investors in lower and lower tax brackets must be attracted to invest. Therefore, a finding that the implied tax rate in the market is relatively high during cyclical troughs in interest rates and low at cyclical peaks would support the idea of a segmented market for discount bonds. Some data from a Treasury survey on the ownership of discount Treasury bonds are shown in table 1. The period, 1975-1979, corresponds to the sample period used later in the paper. Unfortunately, the majority of private owners are not surveyed, as evidenced by the magnitudes shown in the last column. Of the institutions shown, state and local governments pay no taxes while mutual savings banks, life insurance companies and savings and loan associations pay only limited taxes. As we know, commercial banks are not able to avail themselves of the favorable tax rate on capital gains. Changes in the amounts of discount bonds owned by the various categories are due to two factors: actual purchases and sales of bonds; and changes in the number of existing bonds selling at a discount due to fluctuations in the level of interest rates. Over the time frame shown, there was a large increase in bonds outstanding as a result of the Treasury financing ever increasing deficits. Moreover, the proportions of discount and premium bonds changed. In late 1976, when interest rates were relatively low, a sizable number of outstanding bonds sold at a premium whereas in late 1979, when interest rates were high, virtually all outstanding bonds sold at a discount. While the survey data are far from complete, they do give a rough idea of the changing pattern of ownership over time. In addition to market segmentation, another factor affecting the valuation of discount bonds involves changes in the tax code which make capital gains less or more attractive relative to interest income. In the Tax Reform Act of 1969, the maximum capital gains ~tax was increased from 25 percent to 32.5 percent for individuals and to 30 percent for corporations. In the Tax Reform Act of 1976, the length of the holding period was increased in steps from six months to one year, and changes in the minimum and maximum taxes effectively raised the capital gains tax rate for certain individuals. Following the change, the maximum tax rate on capital gains was over 49 percent. These two tax changes lessened the attraction of discount bonds, all other things the same. Reversing this trend, the Revenue Act of 1978 lowered the capital gains tax rate. Instead of 50 percent of long-term capital gains being excluded from taxable income, 60 percent was excluded. The maximum tax rate for individuals was reduced to 28 percent. This Act made capital gains more attractive vis-a-vis interest income. An increase in the effective capital gains tax rate relative to that for interest income would be expected to be associated with an increase in the
12,432 12,614 12,791 9,627 9,968 13,798 21,288 2~670 21,664 24,195
"Source: Treasury Bulletin.
June 3~1975 December 31, 1975 June 3~1976 December 31, 1976 June 3~1977 December 31, 1977 June 3~1978 December 31, 1978 June 3~1979 December31, 1979
U.S. Gov't accounts & Federal reserve 1,985 1,859 1,779 896 1,591 1,831 2,899 3,281 3~16 3,151
378 356 299 225 262 295 330 359 306 318
460 5,394 Mutual commercial savings banks banks 2,039 1,999 2,026 1,859 2,723 2,153 2,297 2,136 1,987 1,944
287 Life ins. companies 692 688 682 603 666 884 1,176 1,316 1,345 1316
406 Casualty ins. companies 337 291 298 247 234 230 233 238 226 199
483 Savings & loans
132 137 165 278 283 386 409 641 767 602
419 Corporations
Ownership survey of treasury bonds selling at a discount 1975-1979 (in millions of dollars)."
Table 1
1,753 1,717 1,630 1,512 1,569 2,579 3,852 5,673 5,677 5,692
311 State & local governments
9,668 8,889 15,175 20,153 23,693 21,948 34,874
11,806
11,766 11,442
Others
J.C. Van Home, Implied tax rates and discount bonds valuation
149
implied tax rate embodied in bond prices, all other things the same. With the tax increases described, discount bonds would be less attractive to those in lower tax brackets vis-~i-vis interest income. On the other hand when the effective tax on capital gains vis-fi-vis interest income is reduced, discount bonds would be more attractive to those in lower tax brackets. As a result, the implied tax rate should decline, all other things the same. In what follows, we analyze implied tax rates in the Treasury bond market in relation to these two hypotheses.
3. Solving for the implied tax rate Without regard to taxes, the yield of a bond with semi-annual interest payments is found by solving the following equation for r: 2.
C/2
P ° = , =~x (1 +r/2)'
$1,000 (1 +r/2) z''
(1)
where Po = the current market price of the bond, C = coupon payment expressed on an annual basis, n = number of years to final maturity, As r would be applicable to all investors, markets would equilibrate in terms of this yield. With taxes, however, r no longer represents the relevant yield for all investors. Rational investors will take the taxes on interest income and on capital gains into account when making investment decisions. Markets then will equilibrate in terms of after-tax yields. If an investor holds a discount bond to maturity, its after-tax yield can be expressed as the discount rate r*, which equates all present value of after-tax cash returns with the current market price: 3 2,, ( c / 2 x 1 - T) ($1,000-- Po)(1 -- G) + Po (1 + r*/2)' ' P ° = t =~x (1 +r*/2)' -~
(2)
where T is the marginal tax rate on ordinary income for the investor and G is his/her marginal tax rate on capital gains. Interest payments are taxed as ordinary income while the capital gain which occurs at final maturity, denoted by ($1,000-Po), is taxed at the capital gains rate, leaving ($1,000 3A variation of this formula was first expressed by Colin and Bayer (1970) and Robichek and Niebuhr (1970).
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J.C. Van Horne, Implied tax rates and discount bonds valuation
-POX1-(3) in after-tax gain. 4 In addition, the investor 'gets back' his/her original investment of Po and this is not subject to taxation. In the decade of the seventies until late 1978, 50 percent of any long-term capital gain was exempt from taxation while the other 50 percent was subject to taxation at the ordinary income tax rate. 5 Thus, G=0.5T. For the individual then, eq. (2) can be expressed as the following up to November of 1978: ~-" (C/2)(1-- T) ($1,000- P o X 1 - 0 . 5 T ) + P o (1+,../2)2,, P ° = t =~l (l+r*/2)'
(3)
The Revenue Act of 1978 increased the exemption for capital gains from 50 percent to 60 percent. Therefore, eq. (3) needs to be modified to reflect 0.4 instead of 0.5 in the numerator of the second expression for late November, 1978 and beyond. With a variation of eq. (3), one can derive the tax rate at which the individual investor would be indifferent between the discount bond and a bond which trades at par. Before proceeding with the derivation, it is important to point out that supply and demand conditions are what determine the relative prices and yields of discount bonds and bonds traded near par. Simultaneously determined is the distribution of tax benefits between tax-exempt investors and various brackets of taxable investors. Conceptually, the tax rates implied are at the margin; they represent a lower boundary of investor tax rates for the holding of discount bonds versus bonds traded near par. Investors in higher tax brackets will benefit-from the tax advantage embodied in discount bonds. That is, they will not be indifferent between discount bonds and bonds traded near par. They will prefer discount bonds because of their higher after-tax expected return. To begin our illustration of the derivation of an implied tax rate, suppose r represents the observed yield to maturity of a bond which has the same maturity and other characteristics as the discount bond except that because of a higher coupon rate it trades at par. As a result, there is no capital gain if the bond is held to maturity. The entire yield to maturity, as represented by the interest payments, is subject to ordinary income taxes. Thus, r represents 4If the bond were bought at a premium over its face value, a capital loss is involved. The investor has two options in this case. He/she may amortize the premium over the remaining life of the instrument and deduct the prorated amount each year from ordinary income or he/she may walt until final maturity or when the bond is sold and then declare a capital loss. As the former tax treatment usually is more favourable to the investor, it is typically employed. Sin the early 1970's the maximum tax rate on capital gains was 32.5 percent. With the Tax Reform Act of 1976, the imposition of minimum and maximum taxes on unearned income effectively raised the maximum capital gains tax rate for certain individuals to over 49 percent. In 1978, the maximum tax was reduced to (1-0.6) (70 percent maximum tax bracket)=28 percent.
J.C. Van Home, Implied tax rates and discount bonds valuation
151
the before-tax yield in the market for a like bond which does not have the possibility of capital gains if held to maturity. The semi-annual after-tax return required in the market for this security is expressed as (r/2)(1- T). This rate then can be used as the after-tax return required by investors at the margin for investment in the discount bond. Accordingly, eq. (3) becomes 2.
(C/2)(1 - T)
($1,000 - PoX 1 - 0.5 T) + Po
[ 1 + (r/2)(1
P ° = ,=x ~ [1 +(r/2X1 - T)]'
- 7')] 2~
'
(4)
where, again, r is the before-tax annual rate of return in the market for a like bond which trades at par. As Po, n, C, and r are known, it is possible to solve eq. (4) for T. However, since T appears in both the numerator and the denominator, it is not possible to solve directly for it. Therefore T must be determined numerically. Computer programs were written to calculate T and these programs were used throughout. 6 As discussed, for late November of 1978 and beyond capital gains were subject to a 60 percent exemption instead of 50 percent. Consequently, the numerator of the second term of eq. (4) became 0.4T instead of 0.5Tfor these months. For the corporate investor, capital gains were taxed at a 30 percent rate prior to the Revenue Act of 1978. Therefore, eq. (4) would become _
C / 2 ( 1 - T)
Po-=, [1 +(r/2)(1- r)]
t
($1,000 - P o ) ( 1 - 0 . 3 ) + P o [1 + (r/2X1 + T)] 2"
(5)
With the Revenue Act of 1978, the corporate tax rate on capital gains became 28 percent beginning in late November of that year. As a result, eq. (5) was modified to substitute 0.28 for 0.3 in the numerator of the second expression for November, 1978 and beyond. When one solves for T in eqs. (4) and (5), based on the market price for a discount bond and the market yield for a similar maturity bond selling at par, one obtains implied tax rates in the market for individual and corporate investors at the margin. This is not to say that the T's calculated represent weighted average tax rates for individuals or for corporations. 7 Rather they represent the lower boundary tax rates to the individual and to the corporate investor at the margin which are implicit in the market prices and yields of two bonds at a moment in time. Our purpose is to derive these implied tax rates and to study their behavior over time. 6A printout of the program which was used on a D E C computer is available from the author on request. 7For an analysis of this point in a world of some market imperfections, see Schaefer (1979).
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J.C. Van Home, Implied tax rates and discount bonds valuation
4. The data used In order to hold constant default risk and to analyze bonds which are marketable and for which viable quotations are available, the Treasury bond market was used. In this market, there exist a number of discount bonds issued before 1965 with coupon rates of 4-1/4 percent or below) Many of these bonds can be used in the payment of federal estate taxes, if they are owned by the deceased at the time of death. Known as 'flower' bonds, these bonds count at their full face value in the settlement of estate taxes. Because this influence affects their valuation in addition to their capital gains attraction, these bonds were excluded from consideration. Another influence has to do with Treasury bonds which are callable. Here, the call provision is geared to final maturity where the callable Treasury bond may be called anytime from five years before final maturity up to the final maturity date. In order to avoid the influence of the call feature, we concentrate on noncallable bonds. Affecting the presence of such bonds was the 4-1/4 percent interest rate ceiling on long-term bonds imposed by the Congress. (See footnote 8.) When interest rates rose above this ceiling in the mid 1960's, the Treasury was precluded from issuing long-term debt. Relief came in 1971 when Congress authorized the Treasury to issue up to $10 billion in long-term bonds apart from the 4-1/4 percent interest rate ceiling. In 1976 and in 1977, the authorization was increased further giving the Treasury the flexibility to place debt in any maturity spectrum. The first non-callable bonds issued under this authority were the 6-1/8's of 11/15/86 in November of 1971 and the second were the 6-3/4's of 2/15/93 in January of 1973. These issues proved to have the lowest coupon rates of those issued under the Treasury's new authorization and, therefore, are likely candidates for the discount bonds to be studied. Until the mid 1970's, however, there simply were not enough bonds of like maturity which traded near par. As a result, it was not possible to find comparable higher coupon bonds on which to base r in eqs. (4) and (5). As fiscal deficits increased in the mid 1970's and long-term bonds were issued by the Treasury, maturities tended to be in the 1990's as opposed to the 1980's. For this reason, the 6-3/4's of 2/15/93 as opposed to the 6-1/8's of 11/15/86 were used as the discount bond, and the five-year period of 19751979 was studied on a monthly basis. For each month during the five-year time frame, prices were recorded from The Wall Street Journal for the last trading day of the month. The yield on SA ceiling interest rate of 4-1/4 percent was imposed by the Congress during World War I to minimize interest costs of the government in financing the war effort. After World War I, interest rates declined and the ceiling was not a constraint on long-term borrowings for over forty years. In the mid 1960's, however, interest rates increased to such levels that the constraint became binding with the result that the Treasury issued no long-term bonds.
J.C. Van Home, Implied tax rates and discount bonds valuation
153
the 6-3/4's of 2/15/93 ranged between 6.79 percent and 10.28 percent and the ask price between $74 22/32 and $99 20/32. The latter occurred in December of 1976, and, because the discount was so small, this month was excluded from consideration. The frequency distribution of prices is shown in fig. 1. For the comparison issue used for r in eqs. (4) and (5), yields were recorded on the same day as for the 6-3/4's of 2/15/93. The comparison issue selected was one which had a maturity within several years of that of 63/4's and which traded closest to par. While it would be desirable for such issues to trade exactly at par, this was not possible. Sometimes the price was slightly above and sometimes it was below. The frequency distribution of prices of the comparison issue is shown in fig. 2. It is important to point out that different comparison issues were used over time depending on the level of interest rates. Our criterion was to employ a bond of near maturity which traded closest to par.
10 t, 12
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PRICE EXCLUDING FRACTION (in dollars)
Fig. 1. Number of months the 6-3/4's of 2/15/93 traded at a particular market price. End of month 1975-1979.
Another problem has to do with the possibility of a duration effect. Two bonds of equal maturity but different coupon rates will have different durations. 9 For the higher coupon bond, a greater percentage of its present value comes from near distant coupon payments vis-/l-vis the final principal payment than is true for the lower coupon bond. Therefore, the duration of the high coupon bond is shorter than that of the low coupon one. As interest rates in general rise, the current coupon bond will have a shorter and shorter duration relative to that for the low coupon bond even though the maturity of the two issues is the same. 9For a discussion of duration as well as of its measurement, see Macaulay (1938), Hopewell and Kaufman (1973), and Reilly and Sidhu (1979).
154
J.C. Van Home, Implied tax rates and discount bonds valuation 12
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~°The formula used was:
l
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= (l+r)'/,~l(-~r)'
where C, is interest and/or principal payment at time t, (t) is length of time to the interest and/or principal payments, n is length of time to finalmaturity, and r is yield to maturity (before tax).
J.C. Van Home, Implied tax rates and discount bonds valuation I
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relative prices and yields of the two securities. Given these data problems, it must be recognized that the Ts derived are but rough estimates of the implied tax rates for individual and corporate investors in Treasury bonds.
5. The results and their implications Using eqs. (4) and (5) with the data described above, implied tax rates for individual and for corporate investors at the margin were determined. In each ease, the equation was modified in keeping with the change in tax rates caused by the Revenue Act of 1978. The implied tax rates over the 19751979 sample period are shown in the lower panels of figs. 4 and 5 for individual and for corporate investors respectively. As seen in the figures, there are fluctuations in the implied tax rates from month to month. One should note also that no implied tax rate is derived for December of 1976. At that time, the 6 - 3 / 4 ' s of 2/15/93 were selling very nearly at par. In the Upper panels of the two figures, the yield on the 6-3/4's of 2/15/93 is plotted over the 1975-1979 period. This issue is used as a proxy
156
J.C. Van Home, Implied tax rates and discount bonds valuation I
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Fig. 4. Yield to maturity of 6-3/4's of 2/15/93 and implied individual tax rate 1975-1979.
for interest-rate movements in the overall market for Treasury bonds. Although not precise, there clearly is an inverse relationship between the implied tax rates and the yield in the market for Treasury bonds. As interest rates fluctuate in the 1975-1976 period, the implied tax rates fluctuate as well. In both the individual and the corporate investor case, they fluctuate around 50 percent. As interest rates decline in late 1976 and remain low throughout 1977, the implied tax rate rises and remains quite high. For individual investors, it is in the neighborhood of 67 to 76 percent, while for corporate investors, it is generally between 65 and 74 percent. In both cases, the equations used to derive implied tax rates were unconstrained on the upside. As interest rates begin to rise steadily from late 1976 to late 1978, the implied tax rates decline steadily. With fluctuations in interest rates from the Fall of 1978 through 1979, implied tax rates fluctuate as well. For individual investors, they tend to be in the 17-26 percent range while for corporate investors, they are in the 28-36 percent range. It should be noted that the 1977 level is higher than, and the range of implied tax rates is greater than, estimates derived in studies using municipal bond yields in relation to corporate bond yields. In those studies estimates
J.C. Van Horne, Implied tax rates and discount bonds valuation
157
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Fig. 5. Yield to maturity of 6-3/4's of 2/15/93 and implied corporate tax rate 1975-1979.
range from 25 to 50 percent. 11 However, it generally is acknowledged that the municipal tax exemption feature is not fully priced in the marketplace. Put another way, municipalities do not obtain the full benefit of the feature; a portion of it goes to investors. For this reason, it is not surprising to find that an alternative procedure results at times in higher estimates of the implied tax rate than does a municipal bond-corporate bond comparison. Nonetheless, the absolute level of implied tax rate in 1977 seems too high. The patterns of implied tax rates over time for individual and corporate investors are consistent with both of the hypotheses stated earlier. If the demand for discount bonds is partially segmented, investors in lower and lower tax brackets must be attracted into the market as the supply increases with increases in interest rates. On the other hand, when interest rates decline and the supply of discount bonds shrinks, their appeal would be more concentrated in investors in high tax brackets. Thus, the finding of implied tax rates embodied in prices of like bonds with different coupon rates varying inversely with the level of interest rates 11See Kormendi and Mussa (1979), Skelton (1979), and Van Home (1978).
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J.C. Van Home, Implied tax rates and discount bonds valuation
supports the idea of a segmented market for discount bonds. By segmented, we mean that there are a limited number of investors who are able to take full advantage of the tax benefits associated with capital gains. As interest rates rise and the supply of discount bonds increases, the tax benefit is sold at the margin at a lower price in the sense that yields are higher in relation to the yield on a bond selling at par. When the supply of discount bonds declines by virtue of a drop in interest rates, the tax benefit is sold at a higher price in the market. Thus, the tax benefit associated with the discount is not invariant over time but changes with changes in the supply of discount bonds. The evidence also is consistent in direction with changes in tax code which occurred. In the Tax Reform Act of 1976, which was passed late in that year, the holding period necessary for a capital gain was increased, and changes in minimum and maximum taxes raised the effective tax rate on capital gains for wealthy individuals. As a result, the attraction of discount bonds lessened in the market overali. Their appeal would be expected to be more confined to investors in high tax brackets than before. The rather sharp increase in implied tax rates which occurred from October of 1976 to January of 1977 is consistent with this change in the tax laws. This observation needs to be tempered, of course, for the decline in interest rates which occurred beginning in October of 1976. In other words, both influences were in the same direction. The Revenue Act of 1978, effective in November of that year, made capital gains more attractive. However, here we were able to incorporate the tax changes in the equations used to compute the implied tax rates. The fact that implied tax rates fluctuate around the time of the tax change might suggest that the tax effect is incorporated in the equations. However, there is a modest drop from November to December of 1978. It is possible that some of the decline in implied tax rates which occurred throughout 1978 was in anticipation of the tax change. However, the likelihood of the favourable capital gains treatment by the Congress was not apparent until at least the Summer of 1978. The administration originally proposed a much tougher taxation of capital gains. By mid 1978, most of the decline in implied tax rates had occurred. The tax effects described above, where the appeal of a financial instrument with capital gains potential varies over time, applies to securities other than discount bonds. For example, the relative attraction of non-dividend paying common stocks would be expected to change as changes occur in the differential tax on capital gains and dividend income. To the extent the relative supply of such securities changes over time, a market segmentation tax effect may affect their valuation as well. Similar examples involving real estate, oil and gas, and other investments can be easily visualized. Discount bonds are only one of a number of vehicles for capital gains.
J.C. Van Home, Implied tax rates and discount bonds valuation
159
6. Summary In this paper, implied tax rates were derived for individual and for corporate investors at the margin from market prices and yields for a discount Treasury bond and for similar bonds which, because of a higher coupon rate, traded near par. Using the 1975-1979 period, the evidence shows that the valuation of tax benefits associated with discount bonds is not constant over time. The implied tax rate is found to vary inversely with the level of Treasury bond yields. In turn, this is consistent with a segmented market with respect to investors being able to take advantage of capital gains. As interest rates rise, the supply of discount bonds increases and investors in increasingly lower tax brackets must be attracted to them. When interest rates and supply fall, higher tax bracket investors drive down the 'price' of the capital gains tax benefit with the result that lower tax bracket investors leave the market. In addition, the evidence is consistent with the 1976 and 1978 changes in tax laws having to do with capital gains. The first change made capital gains less attractive vis-fi-vis interest income whereas the second made them more attractive. It is hoped that this paper increases our understanding of the behavior of discount bonds, a subject that has received little empirical attention in the literature. With the recent rise in interest rates, it is a subject of increasing concern. References Colin, J.W. and Richard J. Bayer, 1970, Calculation of tax effective yields for discount instruments, Journal of Financial and Quantitative Analysis 5, June, 265-273. Cook, Timothy Q. and Patric H. Hendershott, 1978, The impact of taxes, risk and the relative supplies on interest rate differentials, Journal of Finance 33, Sept., 1173-1186. Cramer, Robert H. and Stephen L. Hawk, 1975, The consideration of coupon levels, taxes, reinvestment rates, and maturity in .the investment management of financial institutions, Journal of Financial and Quantitative Analysis 10, March, 67-84. Hopewell, Michael H. and George C. Kaufman, 1973, Bond price volatility and term to maturity: A generaliT~l respectification, American Economic Review 63, Sept., 749-753. Komendi, R. and M. Mussa, 1979, The taxation of municipal bonds (The American Enterprise
Institute, Washington, DC). Livingston, Miles, 1979 Taxation and bond market equilibrium in a world of uncertain future interest rates, Journal of Financial and Quantitative Analysis 14, March, 11-28. Macaulay, Frederick R., 1938, Some theoretical problems suggested by the movement of interest rates, bond yields, and stock prices in the United States since 1856 (National Bureau of Economic Research, New York). Reilly, Frank and Rupinder S. Sidhu, 1979, The many uses of bond duration, Research paper (University of Illinois, Chicago, IL). Robichek, Alexander A. and W. David Niebuhr, 1970, Tax-induced bias in reported treasury yields, Journal of Finance 25, Dec., 1081-1090. Schaefer, Stephen M., 1979, Taxes and security market equilibrium, Research paper, Aug. (Stanford Business School, Stanford, CA). Skelton, Jeffrey L., 1979, The relative pricing of tax-exempt and taxable debt, Research paper (University of California, Berkeley, CA). Van Home, James C., 1978, Financial market rates and flows, Ch. 8 (Prentice-HaU, Englewood Cliffs, N J).
IMPLIED TAX RATES AND THE VALUATION OF DISCOUNT BONDS James C. Van HORNE* Stanford University, Stanford, CA 94305, USA Received September 1980, final version received July 1981
Discount bonds afford the investor the opportunity for capital gains. If for tax reasons the market is segmented on the demand side, investors in lower and lower tax brackets must be attracted when interest rates rise and the supply of discount bonds increases. Changes in the differential tax on capital gains and interest income also should affect relative demand. Testing these hypotheses with U.S. Treasury bond data, the implied tax rate is found to vary over time in a manner consistent with market segmentation and tax law changes.
1. Introduction
Because of the capital gains attraction of discount bonds, they are valued differently in the market from bonds which trade near par. As the supply of discount bonds varies over time with changes in the overall level of interest rates, the valuation of the capital gains tax feature may vary as well. In this paper, implied individual and corporate tax rates are derived for the five-year period 1975-1979, using monthly data on Treasury bonds selling near par and a bond which sells at a discount. The results are consistent with segmentation in investor demand for discount bonds as well as with the 1976 and 1978 tax-law changes which affected capital gains. Heretofore, there have been few empirical studies of discount bonds t and none of the implied tax rates embodied in market prices over time. The paper begins with a discussion of the valuation of discount bonds and of the hypotheses to be tested, followed by the presentation of a method for determining implied tax rates. The data and their limitations are explored in the third section and the results and their implications in the fourth section. A brief summary appears at the end. *The study was supported by the Stanford Program in Finance. I am grateful to Lawrence Fok for his programming assistance on this project and to George G. Kaufman and Stephen M. Schaefer for a number of helpful comments. 1For some empirical studies, see Cook and I-Iendershott (1978), Robichek and Niebuhr (1970), and Van H o m e (1978).
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146
J.C. Van Home, Implied tax rates and discount bonds valuation
2. Yields and taxes
In the absence of taxes, the yield to maturity of a bond represents the promised rate of return applicable to all investors. Financial markets would be expected to equilibrate in terms of this rate. With taxes, however, a bond's yield no longer represents the relevant return to taxable investors. The reason is that the yield formula assumes that interest income and capital gains are not taxed. The favorable tax treatment of capital gains makes taxable fixed-income securities selling at a discount from their par values attractive to some taxable investors. As a result, their pre-tax yield to maturity tends to be lower than the yield of comparable bonds with higher coupon rates. The greater the discount, the greater the capital gains attraction and the lower the pre-tax yield relative to higher coupon bonds. However, the relationship between the amount of discount and the differential in yield between the discount bond and a higher current coupon bond which trades at par may not be stable over time. The market for discount bonds may be partially segmented. The basic appeal of a discount bond is to the taxable investor. The investor who pays no taxes, such as a pension fund or a University endowment fund, could well be indifferent between a dollar of interest and a dollar of capital gains where both are discounted by the market at the same rate. However, if taxable investors bid up the price of discount bonds so that they yield less before taxes than higher coupon bonds which trade at par, the non-taxable investor will have an incentive to invest in the higher coupon bond. The same reasoning applies, though to a lesser extent, to certain institutional investors which pay taxes at less than the full corporate tax rate. Discount bonds are primarily attractive to higher income individuals and to fire and casualty companies. While commercial banks typically are in high tax brackets, they are unable to avail themselves of the favorable tax rate on capital gains. Unfortunately for them, realized capital gains are treated as ordinary income when it comes to paying taxes. Therefore, commercial banks may seek bonds selling nearly at par or above. 2 Thus, the demand for discount bonds is influenced by the flow of funds to fire and casualty companies and the wherewithal of wealthy individuals to invest. On the other hand, the supply of discount bonds changes with changes in the level of interest rates. At cyclical peaks in interest rates, there are far more discount bonds than there are at cyclical troughs. By definition, at these times there are bonds issued in the past at coupon rates lower than prevailing interest rates in the market. In other words, the supply of discount bonds varies directly with the level of interest rates. 2For an analysis of the investment behavior of institutional investors with respect to taxes and likely future reinvestment rates on coupon payments, see Cramer and Hawk (1975).
J.C. Van Home, Implied tax rates and discount bonds valuation
147
If the demand for discount bonds is partially segmented, these bonds must appeal to new clienteles as the supply of them increases. As interest rates rise and the supply of discount bonds increases, investors in lower and lower tax brackets must be attracted to invest. Therefore, a finding that the implied tax rate in the market is relatively high during cyclical troughs in interest rates and low at cyclical peaks would support the idea of a segmented market for discount bonds. Some data from a Treasury survey on the ownership of discount Treasury bonds are shown in table 1. The period, 1975-1979, corresponds to the sample period used later in the paper. Unfortunately, the majority of private owners are not surveyed, as evidenced by the magnitudes shown in the last column. Of the institutions shown, state and local governments pay no taxes while mutual savings banks, life insurance companies and savings and loan associations pay only limited taxes. As we know, commercial banks are not able to avail themselves of the favorable tax rate on capital gains. Changes in the amounts of discount bonds owned by the various categories are due to two factors: actual purchases and sales of bonds; and changes in the number of existing bonds selling at a discount due to fluctuations in the level of interest rates. Over the time frame shown, there was a large increase in bonds outstanding as a result of the Treasury financing ever increasing deficits. Moreover, the proportions of discount and premium bonds changed. In late 1976, when interest rates were relatively low, a sizable number of outstanding bonds sold at a premium whereas in late 1979, when interest rates were high, virtually all outstanding bonds sold at a discount. While the survey data are far from complete, they do give a rough idea of the changing pattern of ownership over time. In addition to market segmentation, another factor affecting the valuation of discount bonds involves changes in the tax code which make capital gains less or more attractive relative to interest income. In the Tax Reform Act of 1969, the maximum capital gains ~tax was increased from 25 percent to 32.5 percent for individuals and to 30 percent for corporations. In the Tax Reform Act of 1976, the length of the holding period was increased in steps from six months to one year, and changes in the minimum and maximum taxes effectively raised the capital gains tax rate for certain individuals. Following the change, the maximum tax rate on capital gains was over 49 percent. These two tax changes lessened the attraction of discount bonds, all other things the same. Reversing this trend, the Revenue Act of 1978 lowered the capital gains tax rate. Instead of 50 percent of long-term capital gains being excluded from taxable income, 60 percent was excluded. The maximum tax rate for individuals was reduced to 28 percent. This Act made capital gains more attractive vis-a-vis interest income. An increase in the effective capital gains tax rate relative to that for interest income would be expected to be associated with an increase in the
12,432 12,614 12,791 9,627 9,968 13,798 21,288 2~670 21,664 24,195
"Source: Treasury Bulletin.
June 3~1975 December 31, 1975 June 3~1976 December 31, 1976 June 3~1977 December 31, 1977 June 3~1978 December 31, 1978 June 3~1979 December31, 1979
U.S. Gov't accounts & Federal reserve 1,985 1,859 1,779 896 1,591 1,831 2,899 3,281 3~16 3,151
378 356 299 225 262 295 330 359 306 318
460 5,394 Mutual commercial savings banks banks 2,039 1,999 2,026 1,859 2,723 2,153 2,297 2,136 1,987 1,944
287 Life ins. companies 692 688 682 603 666 884 1,176 1,316 1,345 1316
406 Casualty ins. companies 337 291 298 247 234 230 233 238 226 199
483 Savings & loans
132 137 165 278 283 386 409 641 767 602
419 Corporations
Ownership survey of treasury bonds selling at a discount 1975-1979 (in millions of dollars)."
Table 1
1,753 1,717 1,630 1,512 1,569 2,579 3,852 5,673 5,677 5,692
311 State & local governments
9,668 8,889 15,175 20,153 23,693 21,948 34,874
11,806
11,766 11,442
Others
J.C. Van Home, Implied tax rates and discount bonds valuation
149
implied tax rate embodied in bond prices, all other things the same. With the tax increases described, discount bonds would be less attractive to those in lower tax brackets vis-~i-vis interest income. On the other hand when the effective tax on capital gains vis-fi-vis interest income is reduced, discount bonds would be more attractive to those in lower tax brackets. As a result, the implied tax rate should decline, all other things the same. In what follows, we analyze implied tax rates in the Treasury bond market in relation to these two hypotheses.
3. Solving for the implied tax rate Without regard to taxes, the yield of a bond with semi-annual interest payments is found by solving the following equation for r: 2.
C/2
P ° = , =~x (1 +r/2)'
$1,000 (1 +r/2) z''
(1)
where Po = the current market price of the bond, C = coupon payment expressed on an annual basis, n = number of years to final maturity, As r would be applicable to all investors, markets would equilibrate in terms of this yield. With taxes, however, r no longer represents the relevant yield for all investors. Rational investors will take the taxes on interest income and on capital gains into account when making investment decisions. Markets then will equilibrate in terms of after-tax yields. If an investor holds a discount bond to maturity, its after-tax yield can be expressed as the discount rate r*, which equates all present value of after-tax cash returns with the current market price: 3 2,, ( c / 2 x 1 - T) ($1,000-- Po)(1 -- G) + Po (1 + r*/2)' ' P ° = t =~x (1 +r*/2)' -~
(2)
where T is the marginal tax rate on ordinary income for the investor and G is his/her marginal tax rate on capital gains. Interest payments are taxed as ordinary income while the capital gain which occurs at final maturity, denoted by ($1,000-Po), is taxed at the capital gains rate, leaving ($1,000 3A variation of this formula was first expressed by Colin and Bayer (1970) and Robichek and Niebuhr (1970).
150
J.C. Van Horne, Implied tax rates and discount bonds valuation
-POX1-(3) in after-tax gain. 4 In addition, the investor 'gets back' his/her original investment of Po and this is not subject to taxation. In the decade of the seventies until late 1978, 50 percent of any long-term capital gain was exempt from taxation while the other 50 percent was subject to taxation at the ordinary income tax rate. 5 Thus, G=0.5T. For the individual then, eq. (2) can be expressed as the following up to November of 1978: ~-" (C/2)(1-- T) ($1,000- P o X 1 - 0 . 5 T ) + P o (1+,../2)2,, P ° = t =~l (l+r*/2)'
(3)
The Revenue Act of 1978 increased the exemption for capital gains from 50 percent to 60 percent. Therefore, eq. (3) needs to be modified to reflect 0.4 instead of 0.5 in the numerator of the second expression for late November, 1978 and beyond. With a variation of eq. (3), one can derive the tax rate at which the individual investor would be indifferent between the discount bond and a bond which trades at par. Before proceeding with the derivation, it is important to point out that supply and demand conditions are what determine the relative prices and yields of discount bonds and bonds traded near par. Simultaneously determined is the distribution of tax benefits between tax-exempt investors and various brackets of taxable investors. Conceptually, the tax rates implied are at the margin; they represent a lower boundary of investor tax rates for the holding of discount bonds versus bonds traded near par. Investors in higher tax brackets will benefit-from the tax advantage embodied in discount bonds. That is, they will not be indifferent between discount bonds and bonds traded near par. They will prefer discount bonds because of their higher after-tax expected return. To begin our illustration of the derivation of an implied tax rate, suppose r represents the observed yield to maturity of a bond which has the same maturity and other characteristics as the discount bond except that because of a higher coupon rate it trades at par. As a result, there is no capital gain if the bond is held to maturity. The entire yield to maturity, as represented by the interest payments, is subject to ordinary income taxes. Thus, r represents 4If the bond were bought at a premium over its face value, a capital loss is involved. The investor has two options in this case. He/she may amortize the premium over the remaining life of the instrument and deduct the prorated amount each year from ordinary income or he/she may walt until final maturity or when the bond is sold and then declare a capital loss. As the former tax treatment usually is more favourable to the investor, it is typically employed. Sin the early 1970's the maximum tax rate on capital gains was 32.5 percent. With the Tax Reform Act of 1976, the imposition of minimum and maximum taxes on unearned income effectively raised the maximum capital gains tax rate for certain individuals to over 49 percent. In 1978, the maximum tax was reduced to (1-0.6) (70 percent maximum tax bracket)=28 percent.
J.C. Van Home, Implied tax rates and discount bonds valuation
151
the before-tax yield in the market for a like bond which does not have the possibility of capital gains if held to maturity. The semi-annual after-tax return required in the market for this security is expressed as (r/2)(1- T). This rate then can be used as the after-tax return required by investors at the margin for investment in the discount bond. Accordingly, eq. (3) becomes 2.
(C/2)(1 - T)
($1,000 - PoX 1 - 0.5 T) + Po
[ 1 + (r/2)(1
P ° = ,=x ~ [1 +(r/2X1 - T)]'
- 7')] 2~
'
(4)
where, again, r is the before-tax annual rate of return in the market for a like bond which trades at par. As Po, n, C, and r are known, it is possible to solve eq. (4) for T. However, since T appears in both the numerator and the denominator, it is not possible to solve directly for it. Therefore T must be determined numerically. Computer programs were written to calculate T and these programs were used throughout. 6 As discussed, for late November of 1978 and beyond capital gains were subject to a 60 percent exemption instead of 50 percent. Consequently, the numerator of the second term of eq. (4) became 0.4T instead of 0.5Tfor these months. For the corporate investor, capital gains were taxed at a 30 percent rate prior to the Revenue Act of 1978. Therefore, eq. (4) would become _
C / 2 ( 1 - T)
Po-=, [1 +(r/2)(1- r)]
t
($1,000 - P o ) ( 1 - 0 . 3 ) + P o [1 + (r/2X1 + T)] 2"
(5)
With the Revenue Act of 1978, the corporate tax rate on capital gains became 28 percent beginning in late November of that year. As a result, eq. (5) was modified to substitute 0.28 for 0.3 in the numerator of the second expression for November, 1978 and beyond. When one solves for T in eqs. (4) and (5), based on the market price for a discount bond and the market yield for a similar maturity bond selling at par, one obtains implied tax rates in the market for individual and corporate investors at the margin. This is not to say that the T's calculated represent weighted average tax rates for individuals or for corporations. 7 Rather they represent the lower boundary tax rates to the individual and to the corporate investor at the margin which are implicit in the market prices and yields of two bonds at a moment in time. Our purpose is to derive these implied tax rates and to study their behavior over time. 6A printout of the program which was used on a D E C computer is available from the author on request. 7For an analysis of this point in a world of some market imperfections, see Schaefer (1979).
152
J.C. Van Home, Implied tax rates and discount bonds valuation
4. The data used In order to hold constant default risk and to analyze bonds which are marketable and for which viable quotations are available, the Treasury bond market was used. In this market, there exist a number of discount bonds issued before 1965 with coupon rates of 4-1/4 percent or below) Many of these bonds can be used in the payment of federal estate taxes, if they are owned by the deceased at the time of death. Known as 'flower' bonds, these bonds count at their full face value in the settlement of estate taxes. Because this influence affects their valuation in addition to their capital gains attraction, these bonds were excluded from consideration. Another influence has to do with Treasury bonds which are callable. Here, the call provision is geared to final maturity where the callable Treasury bond may be called anytime from five years before final maturity up to the final maturity date. In order to avoid the influence of the call feature, we concentrate on noncallable bonds. Affecting the presence of such bonds was the 4-1/4 percent interest rate ceiling on long-term bonds imposed by the Congress. (See footnote 8.) When interest rates rose above this ceiling in the mid 1960's, the Treasury was precluded from issuing long-term debt. Relief came in 1971 when Congress authorized the Treasury to issue up to $10 billion in long-term bonds apart from the 4-1/4 percent interest rate ceiling. In 1976 and in 1977, the authorization was increased further giving the Treasury the flexibility to place debt in any maturity spectrum. The first non-callable bonds issued under this authority were the 6-1/8's of 11/15/86 in November of 1971 and the second were the 6-3/4's of 2/15/93 in January of 1973. These issues proved to have the lowest coupon rates of those issued under the Treasury's new authorization and, therefore, are likely candidates for the discount bonds to be studied. Until the mid 1970's, however, there simply were not enough bonds of like maturity which traded near par. As a result, it was not possible to find comparable higher coupon bonds on which to base r in eqs. (4) and (5). As fiscal deficits increased in the mid 1970's and long-term bonds were issued by the Treasury, maturities tended to be in the 1990's as opposed to the 1980's. For this reason, the 6-3/4's of 2/15/93 as opposed to the 6-1/8's of 11/15/86 were used as the discount bond, and the five-year period of 19751979 was studied on a monthly basis. For each month during the five-year time frame, prices were recorded from The Wall Street Journal for the last trading day of the month. The yield on SA ceiling interest rate of 4-1/4 percent was imposed by the Congress during World War I to minimize interest costs of the government in financing the war effort. After World War I, interest rates declined and the ceiling was not a constraint on long-term borrowings for over forty years. In the mid 1960's, however, interest rates increased to such levels that the constraint became binding with the result that the Treasury issued no long-term bonds.
J.C. Van Home, Implied tax rates and discount bonds valuation
153
the 6-3/4's of 2/15/93 ranged between 6.79 percent and 10.28 percent and the ask price between $74 22/32 and $99 20/32. The latter occurred in December of 1976, and, because the discount was so small, this month was excluded from consideration. The frequency distribution of prices is shown in fig. 1. For the comparison issue used for r in eqs. (4) and (5), yields were recorded on the same day as for the 6-3/4's of 2/15/93. The comparison issue selected was one which had a maturity within several years of that of 63/4's and which traded closest to par. While it would be desirable for such issues to trade exactly at par, this was not possible. Sometimes the price was slightly above and sometimes it was below. The frequency distribution of prices of the comparison issue is shown in fig. 2. It is important to point out that different comparison issues were used over time depending on the level of interest rates. Our criterion was to employ a bond of near maturity which traded closest to par.
10 t, 12
0
-< 80
,
,
,
85
90
95
't
100
PRICE EXCLUDING FRACTION (in dollars)
Fig. 1. Number of months the 6-3/4's of 2/15/93 traded at a particular market price. End of month 1975-1979.
Another problem has to do with the possibility of a duration effect. Two bonds of equal maturity but different coupon rates will have different durations. 9 For the higher coupon bond, a greater percentage of its present value comes from near distant coupon payments vis-/l-vis the final principal payment than is true for the lower coupon bond. Therefore, the duration of the high coupon bond is shorter than that of the low coupon one. As interest rates in general rise, the current coupon bond will have a shorter and shorter duration relative to that for the low coupon bond even though the maturity of the two issues is the same. 9For a discussion of duration as well as of its measurement, see Macaulay (1938), Hopewell and Kaufman (1973), and Reilly and Sidhu (1979).
154
J.C. Van Home, Implied tax rates and discount bonds valuation 12
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<
97
98
99
100
101
102
MARKET PRICE (in dollars)
Fig. 2. Number' of months the comparison bond traded at a particular price (to the nearest 1/2 dollar). End of month 1975-1979. To determine if differences in duration affect things in any systematic way, durations were computed for both the 6"-3/4's of 2/15/93 and the comparison issue. 1° For each of the sixty observations, the duration of the comparison issue was subtracted from the duration of 6-3/4's of 2/15/93. The differences are plotted in the lower panel of fig. 3. A positive difference means the duration of the 6-3/4's was greater than the duration of the comparison issue while a negative difference depicts just the opposite. In the upper panel of the figure, the yield to maturity of the 6-3/4's of 2/15/93 is shown over the five-year period. As seen, the differences in duration are mixed over time and show no marked tendency to vary with the level of interest rates. In addition to these issues, there is the possibility that quotations for the two issues on a particular day are temporarily 'out of line' with each other. Finally, it m a y be that factors other than those discussed influence the
~°The formula used was:
l
c,
= (l+r)'/,~l(-~r)'
where C, is interest and/or principal payment at time t, (t) is length of time to the interest and/or principal payments, n is length of time to finalmaturity, and r is yield to maturity (before tax).
J.C. Van Home, Implied tax rates and discount bonds valuation I
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relative prices and yields of the two securities. Given these data problems, it must be recognized that the Ts derived are but rough estimates of the implied tax rates for individual and corporate investors in Treasury bonds.
5. The results and their implications Using eqs. (4) and (5) with the data described above, implied tax rates for individual and for corporate investors at the margin were determined. In each ease, the equation was modified in keeping with the change in tax rates caused by the Revenue Act of 1978. The implied tax rates over the 19751979 sample period are shown in the lower panels of figs. 4 and 5 for individual and for corporate investors respectively. As seen in the figures, there are fluctuations in the implied tax rates from month to month. One should note also that no implied tax rate is derived for December of 1976. At that time, the 6 - 3 / 4 ' s of 2/15/93 were selling very nearly at par. In the Upper panels of the two figures, the yield on the 6-3/4's of 2/15/93 is plotted over the 1975-1979 period. This issue is used as a proxy
156
J.C. Van Home, Implied tax rates and discount bonds valuation I
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Fig. 4. Yield to maturity of 6-3/4's of 2/15/93 and implied individual tax rate 1975-1979.
for interest-rate movements in the overall market for Treasury bonds. Although not precise, there clearly is an inverse relationship between the implied tax rates and the yield in the market for Treasury bonds. As interest rates fluctuate in the 1975-1976 period, the implied tax rates fluctuate as well. In both the individual and the corporate investor case, they fluctuate around 50 percent. As interest rates decline in late 1976 and remain low throughout 1977, the implied tax rate rises and remains quite high. For individual investors, it is in the neighborhood of 67 to 76 percent, while for corporate investors, it is generally between 65 and 74 percent. In both cases, the equations used to derive implied tax rates were unconstrained on the upside. As interest rates begin to rise steadily from late 1976 to late 1978, the implied tax rates decline steadily. With fluctuations in interest rates from the Fall of 1978 through 1979, implied tax rates fluctuate as well. For individual investors, they tend to be in the 17-26 percent range while for corporate investors, they are in the 28-36 percent range. It should be noted that the 1977 level is higher than, and the range of implied tax rates is greater than, estimates derived in studies using municipal bond yields in relation to corporate bond yields. In those studies estimates
J.C. Van Horne, Implied tax rates and discount bonds valuation
157
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Fig. 5. Yield to maturity of 6-3/4's of 2/15/93 and implied corporate tax rate 1975-1979.
range from 25 to 50 percent. 11 However, it generally is acknowledged that the municipal tax exemption feature is not fully priced in the marketplace. Put another way, municipalities do not obtain the full benefit of the feature; a portion of it goes to investors. For this reason, it is not surprising to find that an alternative procedure results at times in higher estimates of the implied tax rate than does a municipal bond-corporate bond comparison. Nonetheless, the absolute level of implied tax rate in 1977 seems too high. The patterns of implied tax rates over time for individual and corporate investors are consistent with both of the hypotheses stated earlier. If the demand for discount bonds is partially segmented, investors in lower and lower tax brackets must be attracted into the market as the supply increases with increases in interest rates. On the other hand, when interest rates decline and the supply of discount bonds shrinks, their appeal would be more concentrated in investors in high tax brackets. Thus, the finding of implied tax rates embodied in prices of like bonds with different coupon rates varying inversely with the level of interest rates 11See Kormendi and Mussa (1979), Skelton (1979), and Van Home (1978).
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J.C. Van Home, Implied tax rates and discount bonds valuation
supports the idea of a segmented market for discount bonds. By segmented, we mean that there are a limited number of investors who are able to take full advantage of the tax benefits associated with capital gains. As interest rates rise and the supply of discount bonds increases, the tax benefit is sold at the margin at a lower price in the sense that yields are higher in relation to the yield on a bond selling at par. When the supply of discount bonds declines by virtue of a drop in interest rates, the tax benefit is sold at a higher price in the market. Thus, the tax benefit associated with the discount is not invariant over time but changes with changes in the supply of discount bonds. The evidence also is consistent in direction with changes in tax code which occurred. In the Tax Reform Act of 1976, which was passed late in that year, the holding period necessary for a capital gain was increased, and changes in minimum and maximum taxes raised the effective tax rate on capital gains for wealthy individuals. As a result, the attraction of discount bonds lessened in the market overali. Their appeal would be expected to be more confined to investors in high tax brackets than before. The rather sharp increase in implied tax rates which occurred from October of 1976 to January of 1977 is consistent with this change in the tax laws. This observation needs to be tempered, of course, for the decline in interest rates which occurred beginning in October of 1976. In other words, both influences were in the same direction. The Revenue Act of 1978, effective in November of that year, made capital gains more attractive. However, here we were able to incorporate the tax changes in the equations used to compute the implied tax rates. The fact that implied tax rates fluctuate around the time of the tax change might suggest that the tax effect is incorporated in the equations. However, there is a modest drop from November to December of 1978. It is possible that some of the decline in implied tax rates which occurred throughout 1978 was in anticipation of the tax change. However, the likelihood of the favourable capital gains treatment by the Congress was not apparent until at least the Summer of 1978. The administration originally proposed a much tougher taxation of capital gains. By mid 1978, most of the decline in implied tax rates had occurred. The tax effects described above, where the appeal of a financial instrument with capital gains potential varies over time, applies to securities other than discount bonds. For example, the relative attraction of non-dividend paying common stocks would be expected to change as changes occur in the differential tax on capital gains and dividend income. To the extent the relative supply of such securities changes over time, a market segmentation tax effect may affect their valuation as well. Similar examples involving real estate, oil and gas, and other investments can be easily visualized. Discount bonds are only one of a number of vehicles for capital gains.
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6. Summary In this paper, implied tax rates were derived for individual and for corporate investors at the margin from market prices and yields for a discount Treasury bond and for similar bonds which, because of a higher coupon rate, traded near par. Using the 1975-1979 period, the evidence shows that the valuation of tax benefits associated with discount bonds is not constant over time. The implied tax rate is found to vary inversely with the level of Treasury bond yields. In turn, this is consistent with a segmented market with respect to investors being able to take advantage of capital gains. As interest rates rise, the supply of discount bonds increases and investors in increasingly lower tax brackets must be attracted to them. When interest rates and supply fall, higher tax bracket investors drive down the 'price' of the capital gains tax benefit with the result that lower tax bracket investors leave the market. In addition, the evidence is consistent with the 1976 and 1978 changes in tax laws having to do with capital gains. The first change made capital gains less attractive vis-fi-vis interest income whereas the second made them more attractive. It is hoped that this paper increases our understanding of the behavior of discount bonds, a subject that has received little empirical attention in the literature. With the recent rise in interest rates, it is a subject of increasing concern. References Colin, J.W. and Richard J. Bayer, 1970, Calculation of tax effective yields for discount instruments, Journal of Financial and Quantitative Analysis 5, June, 265-273. Cook, Timothy Q. and Patric H. Hendershott, 1978, The impact of taxes, risk and the relative supplies on interest rate differentials, Journal of Finance 33, Sept., 1173-1186. Cramer, Robert H. and Stephen L. Hawk, 1975, The consideration of coupon levels, taxes, reinvestment rates, and maturity in .the investment management of financial institutions, Journal of Financial and Quantitative Analysis 10, March, 67-84. Hopewell, Michael H. and George C. Kaufman, 1973, Bond price volatility and term to maturity: A generaliT~l respectification, American Economic Review 63, Sept., 749-753. Komendi, R. and M. Mussa, 1979, The taxation of municipal bonds (The American Enterprise
Institute, Washington, DC). Livingston, Miles, 1979 Taxation and bond market equilibrium in a world of uncertain future interest rates, Journal of Financial and Quantitative Analysis 14, March, 11-28. Macaulay, Frederick R., 1938, Some theoretical problems suggested by the movement of interest rates, bond yields, and stock prices in the United States since 1856 (National Bureau of Economic Research, New York). Reilly, Frank and Rupinder S. Sidhu, 1979, The many uses of bond duration, Research paper (University of Illinois, Chicago, IL). Robichek, Alexander A. and W. David Niebuhr, 1970, Tax-induced bias in reported treasury yields, Journal of Finance 25, Dec., 1081-1090. Schaefer, Stephen M., 1979, Taxes and security market equilibrium, Research paper, Aug. (Stanford Business School, Stanford, CA). Skelton, Jeffrey L., 1979, The relative pricing of tax-exempt and taxable debt, Research paper (University of California, Berkeley, CA). Van Home, James C., 1978, Financial market rates and flows, Ch. 8 (Prentice-HaU, Englewood Cliffs, N J).