The coupon-induced tax clientele effect in bond prices

MANAGERIAL AND DECISION ECONOMICS, VOL. 12, 367-376 ( 1 991)

The Coupon-induced Tax Clientele Effect in Bond Prices

Gerald D. Gay

and Seokchin Kim

Georgia State University, Atlanta, GA, USA

First Economic Research Institute, Seoul, South Korea

A pricing model for default-free bonds under differential taxation of coupon income and capital gains is presented which explicitly considers coupon-induced tax clienteles. Subsequent analysis provides indirect evidence in support of the existence of the coupon-induced tax clientele effect, while direct evidence is provided by analyzing differences in marginal tax rates estimated across different coupon levels for sets of US Treasury bonds with the same maturity date. The results are also generally consistent with the traditional notion that marginal tax rates are inversely related to coupon levels.

INTRODUCTION

Under differential taxation of interest income and capital gains, heterogeneous personal tax rates may lead to the formation of investor tax clienteles according to coupon levels, due to investors having differing degrees of capital gains preference. We present an after-tax pricing model for default-free debt, which considers not only differential taxation but also the existence of coupon-induced tax clienteles. Previous pricing equations such as those presented in Robichek and Neibuhr (1970) and Livingston (1979a, b) are expressed as special case$ of the model developed here.

By identifying sets of three discount bonds having the same maturity we show that discrepancies in estimates of spot rates are indirect evidence in support of the existence of the coupon-induced tax clientele effect. To provide direct evidence regarding coupon-induced tax clienteles we analyze differences in marginal tax rates estimated across coupon levels for all sets of Treasury bonds with the same maturity date trading during the 1977-86 period. The results indicate that implied marginal tax rates do appear to differ across coupon levels. To investigate the coupon-induced tax clientele effect further we regress estimated marginal tax rates against coupon levels and/or interest rate levels, and find that the estimated tax rates vary inversely with coupon rates. Van Horne’s (1982) proposition that

implied tax rates are inversely related to interest rates is re-examined. Our results provide no significant support for Van Horne’s hypothesis. Rather, the coupon level is found to be the key factor for determining the marginal tax rate.

The remainder of the paper is organized as follows. The next section develops a bond pricing model that considers not only differential taxation but also the existence of coupon-induced tax clienteles. The third section presents indirect evidence regarding the coupon-induced tax clientele effect. In the fourth section we estimate marginal tax rates implied from observed market prices for US Treas- ury bonds and provide direct evidence for the coupon-induced tax clientele effect. The final section provides a brief summary.

THE MODEL

Derivation

In order to focus on the differential taxation effect we assume, for simplicity, no transaction costs, constant marginal tax rates, and deterministic interest rates. In such a tax world both a buy-and-hold and a rollover strategy conducted over a given investment horizon should provide the same after- tax return in order for no arbitrage to occur. Consider an investor who purchases a bond with

0143-6570/9 1/050367-10$05.00 0 1991 by John Wiley & Sons, Ltd.

368 G . D. GAY AND S. KIM

coupon C and is subject to tax rates i' on ordinary income and g' on capital gains. For an initial investment of P , the sum of after-tax income from a series of short investments after n periods can be expressed as

P [ n k = 1 , , [ 1 + r k ( l -i')]-l] (1)

where i k is the kth period one-period future spot rate. The total after-tax income from an n-period discount bond with face value F and that pays coupon C at the end of each of the next n periods will be

C j = 1 , , C( 1 - 1)6, + ( F - P ) ( 1 - g') ( 2 ) where

Since the total after-tax income from the two alternative strategies should be identical, equating expressions ( 1 ) and (2) and solving for P gives

Equation (3) is our pricing formula for coupon- bearing discount bonds under differential taxation. Different marginal tax rates, i' and g', are allowed according to coupon level. Thus, the model considers not only differential taxation but also the coupon-induced tax clientele effect.

An investor who purchases a premium bond maturing in n periods has two options. He or she can either declare a capital loss at maturity or amortize the premium over the remaining life and deduct the prorated amount each year from ordinary income. In the former case the premium bond will be priced as would a discount bond. In the latter case of the amortizing tax treatment, Eqn (2) is modified as follows:

[C( 1 - i') + i'(P - F ) / n ] C 1 , dj + ( F - P ) (4)

Equating expressions ( 1 ) and (4) and solving for P gives a premium bond pricing formula:

Comparison to Other Models

To relate the above analysis to existing models, consider first the case where i'=g'=O. Then, Eqns

(3) and (5) reduce to the standard no-tax formulas. Robichek and Niebuhr (1970) derive an after-tax pricing formula that uses y = Y* ( 1 2 i) instead of r k ( l - i'), as used here, where Y* is the before-tax yield to maturity for a par bond. Thus, their formula holds only when all i k are equal to Y* and marginal tax rates are identical across coupon levels.

Consider next the pricing of zero-coupon or pure discount bonds. By setting C = 0, Eqn (3) reduces to

where io and go are the marginal tax rates on interest income and capital gains, respectively, of an investor who buys a pure discount bond. This is the pricing model for a pure discount bond if the discounted amount were subject to taxation at the capital gains tax rate.' Define R, as the n-period spot interest rate which is the yield to maturity of a pure discount bond. Mathematically, the spot rate R, is defined as the solution to

Combining the above expression with Eqn (6), the relationship between long-term spot rates and short-term future spot rates is as follows:

Consider then the case in which there are no coupon-induced clientele effects, that is, i'= io = i and g'= go = g. For that case, rearranging Eqn (7) gives

nk = 1 , n [ 1 + r k ( 1 - i)] = (1 + R,)"( 1 -9) + 9

Substituting for nk=',,[l + r k ( l - i ) ] in Eqn (3) , we have

where

This equation is the Livingston (1979a) pricing formula for discount bonds. It is a special case in which there are no coupon-induced clientele effects and the discount is taxed at the capital gains tax rate. Similarly, in the case of no coupon-induced tax

TAX CLIENTELE EFFECT 369

clientele effect, Eqn (5) can be rearranged as

(9) C(l -i)An+F[Dn-iA]

n-iA P=

where

and

D ~ { ~ k = l , n [ l + r k ( l - i ) ] } - l .

Equation (9) is Livingston’s (1979b) pricing equation for a premium bond absent coupon-induced tax clienteles, and it is another special case of the model derived here.

Table 1. Discrepancies in Estimated Two-period Spot Rates (R2) for Three Selected T- bond Triplets

Pairwise Observation Maturity coupon

Triplet date date combination RZ A 781115 791115 6.250 6.625 13.85

6.250 7.000 9.94 6.625 7.000 5.87

B 800815 810815 7.000 7.625 4.34 7.000 8.375 7.34 7.625 8.375 10.50

C 810515 820515 7.000 8.000 16.11 7.000 9.250 15.33 8.000 9.250 14.52

MODEL IMPLICATIONS

Consider two discount bonds with coupons C1 and C2 having the same maturity. From Eqn (8), their prices P1 and P2 can be expressed as

of Eqn (10) is based on Eqn (8), which assumes no coupon-induced tax clientele effect. Therefore, discrepancies in estimates of long-term spot rates according to Caks’ method may be due to the existence of coupon-induced tax clienteles caused

P1= by differential taxation and heterogeneous tax Xj=l,nC1(l -i)aj+F(l -g)

(1 +Rn)”(l-g) rates.3 and

c = 1, , c 2 (1 - i) aj + F( 1 - g) (1 + Rnl”(1 -g)

P2 =

Rearranging these equations with respect to Xj=l,naj, equating, and solving for R, gives

This is Caks’ (1977) method for estimating long- term spot rates. Different estimates of a long-term spot rate may obtain depending on which two bonds are used in the estimation. To illustrate, we identify from Treasury bond data reported in the Wall Street Journal over the 1977-86 period three observations of discount triplets (sets of three bonds having the same maturity) having exactly one year (two periods) to maturity.’

Table 1 reports estimates of R, (the two-period (one year) spot rate) determined by Eqn (10) for

ESTIMATION OF MARGINAL TAX RATES

Methodology

This section is concerned with providing direct evidence on the coupon-induced tax clientele effect by estimating and inspecting marginal tax rates estimated across different coupon levels. To begin, define:

This term may be referred to as the discount factor on the j th period after-tax income earned from holding bonds with coupon C. Furthermore, denote the price of a bond with maturity n as P,. Equations (3) and ( 5 ) for discount and premium bonds can then be expressed as

each pairwise set of bonds within each triplet. As P,=c(1-ic)cj=1,n7c; shown, the values vary greatly depending on which two bonds are selected. For example, on 15 Novem- ber 1978 the triplet maturing on 15 November 1979 provides estimates of R, of 13.85%, 9.94%, and 5.87%. Thus, one cannot tell what the true estimate

+[F-(F-Pn)g‘]7t~, if P,<F (11)

and

P, = [ C( 1 - i‘) + i‘(P, - F ) / n ] Cj = ,, $ of R, should be using this method. The derivation +Fnf, if P,>F (12)


Given that the above equations allow for the existence of coupon-induced tax clienteles, they permit the direct estimation of implied marginal tax rates across coupon levek4

Consider an investor who buys a bond at a premium, amortizes the premium over the remaining life of the instrument and deducts the prorated amount each year from ordinary income. If we know the investor's estimates of expected future spot rates, Eqn (12) can be used to solve for i'. When, however, the investor buys a bond at a discount or at a premium and plans to realize the premium as a capital loss, Eqn (11) then has two unknowns, i' and gc. The functional relationship between these two unknowns can be expressed as g' = ai', where a = 1-the percentage of the exemption for capital gains5 Substituting this relationship into Eqn (1 1) and removing g' gives

P , = C ( 1 - i') xj= 1, , 71; +[F-(F-P,)ai']n', , if P , < F (13)

Implied tax rates can be obtained by solving this equation given an appropriate value for a. Since i' appears in both the numerator and the denomin- ator in a complicated fashion, a closed-form solution cannot be derived and thus i' must be determined by a numerical search.

For the individual investor during the period of our investigation, before November 1978, 50% of long-term capital gains were exempt from taxation while the other 50% were subject to taxation at the ordinary income tax rate. Thus, a = 0.5, and therefore, gc = 0 3 ' . The exemption for capital gains was increased from 50% to 60% after October 1978 (g' = 0.4i').

Bonds with short times to maturity were used to test for the existence of clienteles. One reason for doing so is that the shorter the maturity, the less the estimation error due to changes in tax laws or interest rates, since marginal tax rates are assumed in the model to remain constant through time. More importantly, future spot rates need to be estimated in order to solve the equations for implied marginal tax rates. The longer the time to maturity, the more difficult it is to predict remote future spot rates. To minimize errors of this type and to minimize the impact of the tax-timing option effect (see Constantinides and Ingersoll, 1984), our attention is restricted to duplet and triplet sets of bonds having the same times to maturity of two periods or one year.6

Consider bonds whose time to maturity is exactly two (six-month) periods. For discount bonds or premium bonds whose premiums are to be realized as capital losses, Eqn (13) reduces to (omitting superscript c in i' for convenience):

(14) C(1-i)[2+r2(l-i)]+F(l-ai) [ 1 + r1 ( 1 - i)] [ 1 + r2(1 - i ) ] - ai

P=

For premium bonds whose premiums are to be amortized, Eqn (12) may similarly be arranged as

[C( 1 - i ) - iF /2 ] [2 + r2 (1 - i ) ] + F [ 1 + r l (1 - i)] [ 1 + r2( 1 - i ) ] - i [ 2 + r2(l - i ) ] / 2

P =

(15) In order to solve the above equations for implied marginal tax rates one must first estimate future spot rates.

Our data consist of the bid and asked prices of all duplet and triplet sets of Treasury bonds trading during the period 1977-86 and having exactly two periods remaining to maturity. Callable and flower bonds are excluded. The sample thus consists of 11 triplets and 19 duplets, i.e. 71 bonds in total (42 discount bonds and 29 premium bonds). From 1 January 1977 to 22 June 1984 the holding period for long-term capital gains treatment was one year; thereafter it became six months. In order to keep the times to maturities long term in the two-period case, data were collected for the trading day immediately prior to the coupon payment day. If the observation (coupon payment) day is a weekend or holiday, data from the first subsequent trading day are used.

To obtain spot interest rates for the first period, the Wall Street Journal bid-and-ask discount yields (D Y ) of Treasury bills whose time to maturity was six months (or nearest to six months) as of the observation day were recorded. The price per dollar of face value for the Treasury bill was computed as follows:

where D TM is the number of days to maturity of the bill and the discount yield, D Y, is the average of the bid-ask quote^.^ Thus, the first six-month period interest rate was calculated as

R , = I l =(F/P1)'182.5/DTM)- 1 (17)

While the first-period interest rate is observable, investors must estimate rates expected to prevail in subsequent periods. For the second period, two-

TAX CLIENTELE EFFECT 37 1

period spot rate estimates were obtained by observ- ing Treasury bills whose times to maturity were near one year. The two-period spot rates were computed to be

R - (,/p2)(3'565/DTM) - 1 2 -

where P2 =the price of the Treasury bill whose time to maturity was nearest one year, calculated according to Eqn (16). If markets are efficient and investors have no maturity preferences, forward rates are unbiased estimates of future spot rates in a no-tax world. Thus, the second-period future spot rate was estimated as:'

1 1 + R 2 1 + r ,

r2=-----

Results

Implied income tax rates for all duplets and triplets in the sample are reported in Table 2. Notably, 32 out of 42 discount bonds provide estimates of marginal income tax rates with interim values rang- ing from 20% to 56.2Y0, whenever the differences between the market prices and the estimated bond prices are all less than one-fourth of a basis point, i.e. 0.0025. During the sample period, the maximum tax rate on ordinary income was 70% for the pre- 1982 observations and 50% thereafter. In the case of the remaining observations where the differences are greater than 0.004, the estimated marginal tax rates are corner solutions, i.e. either zero or of maximum values.' As an example, consider the

triplet maturing on 15 August 1981. Theimplied tax rate of the bond with the 7% coupon is 12.3% while the implied tax rate of the 8.375% coupon bond is 3.6%. In both bonds the difference in price is near zero. However, the difference in price of the bond with the 7.625% coupon is twelve basis points and the estimated tax rate is 0%. The large difference in price implies that the estimates of future spot rates, which are relevant for pricing, are inappropriate. That is, they appear to have been significantly different from those of the marginal investor who sets prices. These results also imply that estimates of marginal tax rates are very sensitive to estimates of future spot rates. When we do not have reliable estimates of future spot rates we do not get reliable implied tax rates.

The divergences of estimated prices from market prices are relatively large for premium bonds-so large that the implied tax rates are extreme values. Recall that in Eqn (15) we assumed that the investor amortizes the premium over the bond's remaining life. Somewhat surprisingly, when Eqn (14) was used for both discount bonds and premium bonds, which assumes that the investor realizes a capital loss rather than amortizes the premium, better results were obtained in that the divergences became quite small." Table 3 presents these results for the cases where the divergence is less than one-fourth of a basis point. For a face value of $1000oO, this divergence corresponds to $250. With this restric- tion, the observations reduce to 55 bonds (four singles, 18 duplets and five triplets). It is now observed that the divergences are less than 0.0025

~~~ ~

Table 2. Implied Tax Rates for Two-period T-bond Duplets and Triplets in which the Investor Realizes Capital Gains and Amortizes Premiumsa

Observation date

770812

780814

781114

790214

790814

791114

8002 14

Maturity date

7808 15

7908 15

791115

800215

800815

801115

810215

r , 6.23

7.75

9.97

10.19

10.26

13.29

13.76

Coupon rl rate

6.78 7.625 8.750

8.98 6.250 6.875

10.53 6.250 6.625 7.000

10.70 4.000 6.500

9.83 6.750 9.000

12.15 . 3.500 7.125

13.34 7.000 7.375

P 101.172 102.246 98.250 98.813 96.563 97.063 97.219 95.063 96.750 97.375 99.563 93.000 95.313 94.875 95.125

DIFF

0.047 0.059 0.000 O.Oo0

-0.001 -0.001

0.000 -0.001

0.001 0.000 O.Oo0 0.002 0.000 0.000

-0.001

TAX

0.700 0.700 0.082 0.078 0.022 0.105 0.002 0.224 0.085 0.131 0.562 0.218 0.032 0.135 0.115

~~

Table 2. Observation

date

800514

800814

801 114

810213

810514

810814

811113

8205 14

820813

821112

830513

830812

831 114

840214

840514

8408 14

841 114

850214

850514

850814

851114

8602 14

860514

~~

Continued Matunfy

date

810515

810815

811115

820215

820515

8208 15

821115

830515

830815

831115

8405 15

8408 15

841115

850215

850515

8508 1 5

851115

860215

860515

860815

861115

87021 5

870515

rl

8.93

9.63

14.79

16.80

17.91

17.75

12.48

13.72

11.23

9.13

8.65

10.54

9.65

9.98

11.27

11.57

9.80

8.92

8.30

7.85

7.89

7.61

6.43

r2

9.78

10.13

13.68

14.79

16.71

17.04

12.74

14.07

13.03

9.76

9.07

11.42

10.45

10.41

12.69

12.51

10.78

9.72

9.13

8.53

8.19

7.84

6.9 1

Coupon rate

7.375 7.500 7.000 7.625 8.375 7.000 7.750 6.125 6.375 7.000 8 ,000 9.250 8.125 9.000 7.125 7.875 7.875

11.625 9.250

11.875 7.000 9.875 9.250

13.250 15.750 6.375 7.250

13.250 14.375 16.000 8.000

14.625 10.375 14.125 14.375 8.250 9.625

13.125 9.750

11.750 9.875

13.500 7.875 9.375

13.750 8.000

11.375 11.OOo 13.875 16.125 9.000

10.875 12.750 12.000 12.500 14.000

P

98.250 98.375 97.719 98.000 98.844 94.188 94.813 92.625 93.000 92.000 92.750 93.688 93.188 93.656 95.875 96.250 95.375 98.375 97.375 99.875 98.344

100.500 100.531 104.281 106.625 96.594 97.125

102.281 104.313 105.813 98.469

104.375 98.781

102.094 102.3 13 96.813 97.969

101.125 99.719

101.594 100.563 103.906 99.219

100.594 104.719 100.063 103.000 102.813 105.563 107.969 101.219 103.875 104.844 105.156 105.500 107.156

DIFF

0.099 0.091

-0.001 0.120 0.000 0.001 O.OO0 0.002

- 0.002 0.002 0.000 O.OO0 0.002 0.000 0.000

-0.001 -0.001

O.OO0 0.320 0.223 0.000 0.106 0.013 0.019 0.023 0.001

- 0.00 1 0.093

- 0.049 - 0.035

O.OO0 -0.019

0.075 0.206 0.218 0.026 0.135 0.192 0.023 0.004 0.157 0.208 0.172 0.203 0.193

-0.076 0.164 0.132 0.097

0.124 0.242 0.048 0.033 0.166

-0.185

- 0.059

TAX

0.000 0.000 0.123 0.000 0.036 0.090 0.084 0.162 0.195 0.112 0.094 0.065 0.178 0.125 0.193 0.102 0.142 0.021 0.000 0.500 0.312 0.500 0.500 0.500 0.499 0.234 0.158 0.500 0.000 0.000 0.244 0.000 0.000 0.500 0.500 0.000 0.000 0.500 0.000 0.500 0.500 0.500 0.000 0.500 0.500 0.500 0.500 0.499 0.498 0.496 0.500 0.500 0.494 0.500 0.500 0.00 1

'rl is the current one-period (six-month) spot rate estimated according to Eqn (17), r2 is the second-period future spot rate estimated according to Eqn (18), P is the market price, DIFF is the difference between the estimated and observed market price where the estimated price is determined by Eqn (15) for premium bonds or Eqn (14) for discount bonds, and TAX is the implied marginal income tax rate.

Table 3.

Observation date

770812

780814

781114

790214

790814

791 114

800214

800814

801 114

810213

810514

810814

811113

820514

821112

830513

830812

8402 14 840514

840814 841 114 850214

850514

850814 851114

860214

860514

Implied Tax Rates for Two-period T-Bond Duplets and Triplets in which the Investor Realizes Capital Gains or Losses'

Maturity date

780815

7908 15

791115

800215

8008 15

801115

810215

810815

811115

820215

8205 15

820815

821115

830515

831115

8405 15

8408 15

850215 850515

850815 851115 860215

860515

8608 15 861115

870215

870515

T I

6.23

7.75

9.97

10.19

10.26

13.29

13.76

9.63

14.79

16.80

17.91

17.75

12.48

13.72

9.13

8.65

10.54

9.98 11.27

11.57 9.80 8.92

8.30

7.85 7.89

7.61

6.43

rI 6.78

8.98

10.53

10.70

9.83

12.15

13.34

10.13

13.68

14.79

16.71

17.04

12.74

14.07

9.76

9.07

11.42

10.41 12.69

12.51 10.78 9.72

9.13

8.53 8.19

7.84

6.91

Y 6.54 6.56 8.26 8.31

10.19 10.02 10.25 9.50

10.25 9.79 9.68

11.39 12.61 13.02 13.12 9.66 9.85

13.85 13.91 14.81 14.62 16.62 16.79 17.01 16.38 16.77 11.96 12.32 13.38 13.87 8.95 9.56 8.87 8.87 8.89

10.29 10.63 11.07 9.87

12.19 12.21 12.25 10.29 9.50 9.54 8.93 8.91

8.35 8.18 8.14 7.87 7.99 7.77 6.69 6.83

Coupon

7.625 8.750 6.250 6.875 6.250 6.625 7.000 4.000 6.500 6.750 9.000 3.500 7.125 7.000 7.375 7.000 8.375 7.000 7.750 6.125 6.375 7.000 8.000 9.250 8.125 9.000 7.125 7.875 7.875

11.625 7.000 9.875 9.250

13.250 15.750 6.375 7.250

13.250 8.000

14.125 14.375 13.125 11.750 9.875

13.500 9.375

13.750 11.375 11.000 13.875 9.000

10.875 12.750 12.000 12.500

rate P DIFF

101.125 O.OO0 102.188 0.OOO 98.250 O.OO0 98.813 0.000 96.563 -0.001 97.063 -0.001 97.219 0.000 95.063 -0.001 96.750 0.001 97.375 0.000 99.563 0.000 93.000 0.002 95.313 0.000 94.875 0.000 95.125 -0.001 97.719 -0.001 98.844 0.000 94.188 0.001 94.813 0.002 92.625 0.002 93.000 -0.002 92.000 0.002 92.750 0.000 93.688 0.000 93.188 0.002 93.656 0.000 95.875 0.000 96.250 -0.001 95.375 -0.001 98.375 0.000 98.344 0.000

100.500 0.000 100.531 0.000 104.281 0.001 106.625 0.001 96.594 0.001

102.281 0,000 98.469 0.000

102.094 O.Oo0 102.313 0.000 101.125 0.000 101.594 0.000 100.563 O.Oo0

100.594 0.000

103.000 0.000 102.813 0.000

101.219 0.000 102.875 0.000 104.844 0.000 105.156 0.001 105.500 O.OO0

97.125 -0.001

103.906 -0.001

104.719 -0.001

105.563 -0.001

TAX

0.083 0.054 0.082 0.078 0.022 0.105 0.002 0.224 0.085 0.131 0.562 0.218 0.032 0.135 0.115 0.123 0.036 0.090 0.084 0.162 0.195 0.112 0.094 0.065 0.178 0.125 0.193 0.102 0.142 0.021 0.312 0.275 0.045 0.008 0.006 0.234 0.158 0.700 0.244 0.153 0.147 0.236 0.010 0.33 1 0.087 0.377 0.068 0.088 0.076 0.030 0.152 0.128 0.017 0.01 1 0.050

is the current one-period (six-month) spot rate estimated according to Eqn (17), r2 is the second-period future spot rate estimated according to Eqn (18), Y is the yield to maturity, P is the market price, DZFF is the difference between the estimated,and observed market price where the estimated price is determined by Eqn (14), and TAX is the implied marginal income tax rate.

3 74 G . D. GAY AND S. KIM

for 23 out of 29 premium bonds. Again, the estimates of the marginal tax rates fluctuate over time according to the interest rate level. The estimates for these premium bonds vary from 6% to 37.7%."

Unlike the results reported by Dymits and Mur- ray (1986), these estimates of marginal tax rates for both discount and premium bonds do not exceed the maximum limit of 70% before 1982 and 50% thereafter. Similar to Litzenberger and Rolfo's (1984) estimate of the marginal tax rate of 12.3% during November 1978 to March 1980, the average estimate using bonds sampled during that period is 14.8%. The average marginal tax rate estimate for bonds over the entire sample period is 10.9%.

In sum, estimated marginal tax rates are different across coupon levels. The hypothesis regarding the existence of the coupon-induced tax clientele effect is thus supported. We can also see that the marginal tax rate is inversely related to the coupon level, as seen in 15 out of 18 duplets and four out of five triplets.

To investigate further the coupon-induced tax clientele effect we conduct additional tests. We first run the following regression:

T= a,-+ a, C

where T is the implied tax rate for a bond having coupon C. Using the data from Table 3 we report these regression results in row (1) of Table 4. Since

the holding period for a favorable capital gains tax rate was reduced from one year to six months for securities purchased after 22 June 1984, we split our sample into two groups consisting of those observations prior to and following the date of change. In agreement with intuition, the signs of the regression coefficients are significantly negative (especially for bonds observed after 22 June 1984), which supports the argument that marginal tax rates are negatively related to coupon levels.' *

Before 1983 during the sample period, interest rates were high and virtually all the bonds in the sample sold at a discount, whereas after 1982 rates were relatively lower and a sizable number of the bonds sold at a premium. We thus re-examine Van Horne's (1982) findings regarding the relationship between tax clienteles and interest rate levels. While Van Horne tracked changes in implied tax rates with changes in yields to maturity for one specific discount bond, we regress implied tax rates from the pooled sample of discount and premium bonds on three different interest rates (yields to maturity, r; estimates of first-period spot rates, r , ; and second- period spot rates, r 2 ) as follows:

T=a,+a,r

where r refers to one of the three rates. We under- took additional regressions in order to incorporate effects that may arise from the existence of coupon-

Table 4. Regressions Results of Implied Tax Rates on Coupon Levels and/or Various Interest Rates (t-values in parentheses)

(A) Observations &lore 23 June 1984 Equation R'

( I ) T = 19.536-0.802C 0.043 (- 1.33)

( -0.18)

(0.21)

(0.08)

(2) T = 14.164-0.107 Y 0.001

(3) T=11.737+0.101 rl 0.001

(4) T= 12.342 + 0.050 r2 O.Oo0

(5) T=20.997-0.125 Y-0.805C 0.044

(6) T= 19.340+0.015r1-0.799 C 0.043

(7) T= 19.447+0.007r2-0.802C 0.043

(-0.21) (-1.31)

(0.03) (- 1.29)

(0.01) (- 1.30)

(B) Observations after 22 June 1984 Equation

T=62.502-4.303C ( -2.51)b

T= - 17.91 7 + 3.439 Y (1.61)

T = - 16.194+3.412r1

T= -15.228+3.078r2 (1.44)

(1.51) T= 32.404 + 4.182 Y-4.823 C

(2.66)b (- 3.41)'

(2.38)b (- 3.27)' T=33.466+4.261 rl -4.8i3c

T= 35.166+ 3.838rZ -4.851 C (2.5.5)b (-3.37)'

R*

0.343

0.177

0.147

0.160

0.600

0.567

0.587

T=implied tax rate, C=coupon rate, Y=yield to maturity, rl =first-period spot rate, and r2 =estimated second- period spot rate. 'Significant at the 0.01 level, two-tailed test.

Significant at the 0.05 level, two-tailed test.

TAX CLIENTELE EFFECT 375

induced tax clienteles, in the form

T= a0 +a1 r +a,C

The results of these regressions are presented in Table 4. When any of the interest rate measures are used as the only independent variable, the regression coefficients are near zero for bonds observed before 23 June 1984 and positive for observations thereafter, as indicated in rows (2)-(4). When the coupon variable is added (see rows (5)-(7)), the regression coefficients of the interest rate variables remain insignificant for observations prior to 23 June 1984 and significantly positive thereafter. In contrast, the regression coefficients of the coupon rate variable remain negative for bonds observed before 23 June 1984 and become even more significantly negative thereafter. This suggests that Van Horne's proposition of a negative relationship between interest rates and implied tax rates is not supported, and that the key factor for determining marginal tax rates is the coupon level which has a significantly negative impact.

Finally, the clientele effect will be weakened as the remaining maturity approaches the holding period for the favorable capital gains tax rate. Therefore, the more distant the bond's maturity from the holding period, the more likely a clientele effect will be detected. This is probably the reason we find relatively weak support for our hypothesis prior to 23 June 1984 and strong support thereafter. '

SUMMARY

The existence of heterogeneous tax rates may lead to the formation of tax clienteles according to coupon levels, due to investors having differing degrees of capital gains preference. As a result, marginal tax rates implicit in bond prices may differ across coupon levels. Thus, considering not only the differential taxation but also the existence of coupon-induced tax clienteles due to differing tax situ- ations of investors, we derived pricing equations for discount and premium default-free bonds. We presented evidence that implied marginal tax rates differ across coupon levels, thus providing support for the existence of coupon-induced tax clienteles. Our results also indicate a negative relationship between implied tax rates and coupon levels, but do not support Van Horne's (1982) proposition of a

negative relationship between implied tax rates and the general level of interest rates.

Acknowledgements

We have benefited from discussions with Peter Eisemann and Ernie Swift and the editorial comments of an anonymous referee.

NOTES

1. US tax law requires the purchaser of an original issue discount bond to amortize the discounted amount as regular income over the bond's life.

2. Equation (10) cannot be obtained with premium bonds.

3. Caks argues that low-coupon Treasury bonds provide higher returns than high-coupon Treasury bonds because market prices of low coupon bonds are more volatile with respect to changes in interest rates. Livingston (1979a) shows that differential price volatility and uncertain reinvestment rates for coupons will not change the form of the equilibrium bond pricing equation.

4. Previous works that estimate marginal tax rates, e.g. Robichek and Niebuhr (1970), McCulloch (1975), Van Horne (1982), Litzenberg and Rolfo (1984), and Constantinides and Ingersoll (1984), ignore the coupon-induced tax clientele effect, assuming that there exists a representative tax rate that sets all prices. Yet a priori there exists no reason why marginal tax rates should be identical across different bonds. Moreover, previous estimates of marginal tax rates varied depending on the bonds and time periods employed, which appears consistent with the existence of coupon-induced tax clienteles.

5. All previous estimations, including this study, assumed a fixed relation between the income tax rate and the capital gains tax rate. If there is a coupon- induced clientele effect, the relation might vary across coupon levels or over time periods. Moreover, we cannot know a priori what type of investor purchases, at the margin, a Treasury bond with a certain coupon level. Consider two discount bonds having the same level of coupon and adjacent maturities n and n + 1. The second bond will then be priced according to

P, + 1 = C( 1 - i') cj= 1," + 1 7r;

+ CF-(F-P,+ ,)gclx:+ 1

Estimates of ic and gc can then be obtained by solving this equation and Eqn (1 1) simultaneously. Unfortu- nately, the relevant data are not available.

6. Given accurate estimates of future spot rates, the clientele effect is more easily detected, the longer the time to maturity. Thus the detection of an effect in shorter maturity bonds would be stronger evidence in support of a coupon clientele effect.

7. The analysis was repeated using both bid and asked yields, and produced results similar to those presented here.


8. The analysis to follow was repeated using both the current two-period spot rate and the ex post second spot rate as proxies for the second period future spot rate. Both these alternatives provided poorer results.

9. As indicated in Table 2, however, the differences in the market and estimated prices have no systematic, functional relationship with coupon rates or implied tax rates.

10. This may be due to investors who purchase premium bonds with short maturities being unlikely to amortize the premium for tax purposes.

11. Most premium bond observations occurred after 1982, during which time the maximum ordinary tax rate was 50%.

12. The basis for this argument is that the lower capital gains tax rate will be more favorable to investors in high tax brackets than to investors in lower tax brackets.

13. We also estimated the implied tax rates for cases in which the times to maturity were three periods, i.e. one and a half years. Utilizing the interest rate futures market provided the most useful alternative for deriv- ing the third-period future spot rates. Compared to the two-period results, divergences between estimated and market prices were often large and thus reliable estimates of marginal tax rates were difficult to obtain. We attribute this to problems in estimating the third-period future spot rates.

REFERENCES

J. Caks (1977). The coupon effect on yield to maturity. Journal of Finance 32, March, 103-15.

G. M. Constantinides and J. E. Ingersoll, Jr (1984). Optimal bond trading with personal taxes. Journal of Financial Economics 13, September, 299-335.

L. Dymits and M. L. Murray (1986). Another look at implied tax rates. Journal of Banking and Finance 10, March, 13341.

R. H. Litzenberger and J. Rolfo (1984). An international study of tax effects on government bonds. Journal of Finance 39, March, 1-22.

M. Livingston (1979a). Taxation and bond market equilibrium in a world of uncertain future interest rates. Journal of Financial and Quantitative Analysis 14, March, 11-27.

M. Livingston (1979b). The pricing of premium bonds. Journal of Financial and Quantitative Analysis 14, September, 5 17-27.

J. H. McCulloch (1975). The tax-adjusted yield curve. Journal of Finance 30, June, 81 1-30.

A. A. Robichek and W. D. Niebuhr (1970). Tax-induced bias in reported Treasury yields. Journal of Finance 25, December, 1081-90.

J. C. Van Horne (1982). Implied tax rates and the valuation of discount bonds. Journal of Banking and Finance 6, June, 145-59.

Documents

The coupon-induced tax clientele effect in bond prices