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Another look at implied tax rates
Journal of Banking and Finance 10 (1986) 133-141. North-Holland
ANOTHER LOOK AT IMPLIED TAX RATES Lee DYMITS* University of Wisconsin, Stevens Point, WI 54481, USA
Michael L. MURRAY* The University of Iowa, Iowa City, IA 52242, USA Received September 1983, final version received April 1985 The authors include the effects of the minimum tax and the alternative tax on capital gains in calculating implied tax rates for discount Treasury bonds. They support the previous finding of Van H o m e (1982) that there is an inverse relationship between these rates and the level of Treasury bond yields. New implied tax rates, however, fall outside the actual marginal tax range. This finding is not consistent with the efficient market hypothesis.
In a 1982 article in this journal, Van Horne calculated the implied tax rates of purchasers of discount Treasury bonds. He found an inverse relationship between these rates and the level of Treasury bond yields as well as a direct relationship with changes in the tax laws. These relationships are supported in this note. However, the implied tax rates (for individual taxpayers) are recalculated incorporating the effects of the minimum tax rate and of the alternative tax on capital gains (when such existed). Although Van Horne alluded to the former, he did not include it in his formula. It should be noted that the existence of the minimum tax significantly increases the effective tax rate on capital gains for those who are subject to it, whereas the alternative tax on capital gains serves the opposite purpose. It should be also noted that the effect of either cannot be analyzed without fitting these to each investor tax situation. For this reason, filing status was added as another dimension (presuming 1 exemption for a taxpayer filing a single return, 2 for heads of households and married filing separate returns, and 4 for married filing a joint return). Within each class, various combinations of ordinary income and of capital gains were used so that the interaction of a particular tax bracket and of capital gains tax, minimum tax, etc. could be assessed and a proper
*The authors wish to thank anonymous referees for their helpful comments. 0378-4266/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)
134
L. Dymits and M.L. Murray, Implied tax rates
coefficient (T in Van Home's equation)
(c/2) (1 -- T)
( $ 1 0 0 0 - Po)(1 - 0.5 T) +Po 1 + ( r / 2 ) ( 1 - T) 2~ po=/_~x-" 1 +(r/2)(1-- T) -t
(1)
could be ascertained. This study was extended through June 1984 in order to analyze the effects of more recent changes in the tax laws and of the high volatile yields of that period. This process yielded, not surprisingly, not one but a multitude of coefficients for each filing class and each period with unique tax laws (there were altogether nine such periods between 1975 and 1984). For example, in 1975, within a single class, there were 19 different coefficients, ranging from 0.357 to 0.625. In order to keep our task manageable, this group of coefficients was reduced to only three (lowest, highest and one in between, the latter equal to that used by Van Home). These were inserted in Van Home's computer program to produce a band of implied tax rates instead of a single rate. In each figure, T1 represents the implied tax rate for an individual in the highest tax bracket, T2 is Van Home's implied tax rate for an individual who is neither subject to a minimum tax, nor can benefit from alternative tax on capital gains, and T3 is the implied tax rate for a middle income individual with the least favorable relationship of capital gains tax rate to tax rate on ordinary income. As can be seen on each figure, this band was reduced in January 1982 to T1 and T2. At that time, the number of coefficients fell to three (0.4, 0.455 and 0.5) and Van Horne's coefficient (and implied tax rate) became a boundary instead of being sandwiched in between as before. It can also be seen from the figures that implied tax rates were generally in a downtrend during the early 1980s. This may be attributable to an increase in interest rates which increased the supply of discount bonds, thus lowering their price. Interest rates, however, peaked between September 1981 and June 1982, reaching their high on the former date. It would have been natural to expect the implied tax rates to move up immediately following the reduction in supply. Instead, in January 1982 they broke through a plateau established during the previous 13 months and then formed a new, lower, plateau where they remained at least until July of 1984. This is apparently the result of the change in the tax rates on ordinary income which took effect then (most importantly the reduction in the maximum tax rate to 50 percent). This change made par value bonds more attractive to high-bracket taxpayers and thus lowered the price of discount bonds, this time via a reduced demand. Even a more remarkable picture emerges when the ranges of the implied tax rates during the periods in both studies are compared with the ranges of
L. Dymits and M.L. Murray, Implied tax rates
IMPLIED
TAX R A T E S
135
1975-79
SINGLE TAXPAYER 100 90 80 70 60
~R
v
hi lIE
50 40
2o
10
I
;
0-I JAN
...........
1975
J ...........
JAN 1 9 7 6
I ...........
JAN
1977
r
. . . . . . . . . .
JAN 1 9 7 8
I
..........
I
JAN 1 9 7 9
DATE
IMPLIED
TAX R A T E S
1980-84
SINGLE TAXPAYERS 70
60-
3
50 -i 4.0 v l.iJ l--
ry
!
-
1 0 -]
0
JAN
! [ , , , , . , , , . . , i
1980
......
JAN 1 9 8 1
, ....
JAN
i,,,.,
1982 DATE
Fig.
1
.....
,i,,,
JAN 1 9 8 3
........
i,,,.
JAN
1984
136
L. Dymits and M.L. Murray, Implied tax rates
I M P L I E D TAX R A T E S
1975-79
HEADS OF HOUSEHOLDS loo 7
.° 1 8O
A
~l
W
p,<
6O 5O 40. 3020-
I 0 -I 0
- -'r-r'r~
JAN 1975
JAN 1 9 7 6
JAN 1 9 7 7
JAN 1978
JAN 1979
DATE
I M P L I E D TAX R A T E S
1980-84
HEADS OF HOUSEHOLDS
70
80
J A
IR
4.0 -~
I--
.K
30
2o -1 I
10"-t
o -t-,-,-,JAN 1980
JAN 1981
JAN 1 9 8 2 DATE
Fig. 1 (cont.)
JAN 1 9 8 3
JAN 1 9 8 4
('luoo) I "~!~1 3/VO ll, g 6 L
NVr
~li6L
NVr
~g6L
NVr
t961
NYI"
096
NYf" 0
I "-0L
O~
0~
.,
141 A
01r d
i
-
0~
-
09
I
I
I
t MI/NIOF
/~g-0g61
- 0Z
O N r l l - I a31~l~lVlq
SR,I.V~I XVJ~ fl.~FIdI~II 3J.VO
6/.6 L NYr
8/.6L
NVr
LZ.6;
NYI"
9£6L
NVr
gL6L
NYI" 0 OL OZ OR 0"~ "4
Og 09 0/08
Ii oe O0
A'IJ.NIOT' ONr"ll=l O':IltiNYIN
6A-GL61 gfI
S~I&V~I XVJ= (I~tI~IdI~II
~a2va xv2 pa!IdtuI ',fv~nl~ "-I'l~ t my sX!ta,f(l "7
138
L. Dymits and M. L. Murray, Implied tax rates
IMPLIED TAX R A T E S
1975--79
MARRIED FILING S E P A R A T E L Y 100 90
--
80 7060 b.l IN
,it
-
T3
504O 3O 20 10--t JAN 1 9 7 5
JAN
1976
JAN
1977
JAN
1978
JAN
1979
DATE
IMPLIED TAX R A T E S
1980-84
MARRIED. FILING S E P A R A T E L Y 70
~, ,
T3 8o
50
A
4.0-
laJ I--
.( rr
30
20-
10-
o -+-,-,~ JAN
1980
JAN
19B1
JAN
1982 DATE
Fig. 1 (cont.)
JAN
19B3
JAN
1984
139
L. Dymits and M. L. Murray, Implied tax rates
Y I E L D S TO MATURITY 1 9 7 5 - 7 9 SINGLE TAXPAYER
t
9
.
.
.
.
.
.
.
7-
5 4 3 2 1 0 JAN 1 9 7 8
JAN 1 9 7 7
JAN 1 9 7 8 DATE
Y I E L D S TO MATURITY 1 9 7 5 - 7 9 HEADS OF HOUSEHOLDS
"it
9
7 8
i
3
2-R3
0 JAN 1 9 7 6 •
"
"
"
"
"
"
"
°
"
"
i
"
"
•
"
°
"
JAN 1 9 7 7
•
"
"
"
•
|
°
•
JAN 1 9 7 B
DATE Fig 1 (cont.)
"
"
"
•
•
•
•
•
L. Dymits and M. L. Murray, Implied tax rates
140
YIELDS TO MATURITY 1 9 7 5 - - 7 9 MARRIED FILING JOINTLY
7-
5
3 2 1 0
i , + , . , , + , , , i J , , , , . , , , , , I , , , I , . , , , , , I - , ' , . - , , . ,
JAN 1 9 7 5
JAN 1 9 7 8
JAN 1 9 7 7
JAH 1 9 7 8
DATE
YIELDS TO MATURITY 1 9 7 5 - - 7 9 MARRIED FILING SEPARATELY
5
R1 4
3
2
O -+----,JAN 1 9 7 5
JAN 1 9 7 6 DATE
Fig. 1 (cont.)
L. Dymits and M. L. Murray, Implied tax rates
141
actual marginal tax rates (14 percent to 70 percent until 1981, the foregoing reduced by 1.25 percent in 1981, 12 percent to 50 percent in 1982, and 11 percent to 50 percent thereafter). The greatest significance to these findings occurs whenever the implied tax rate at the higher boundary exceeds the highest actual marginal tax rate. When this is the case, not a single investor can gain anything by purchasing discount bonds. Nonetheless, higher implied rates were observed during 11 of the months in the study period. During these months, all investors could have achieved a higher return by purchasing bonds selling at or close to par. This implies that during these periods discount bonds were overpriced and par value bonds were underpriced. An intuitive explanation for this phenomenon might lie in the popularity of discount bonds as tax-saving devices. This popularity may have increased aggregate demand to a point where prices were pushed beyond equilibrium. In those months where the implied tax rates were lower than the top actual marginal tax rate, no implications can be drawn without knowing the actual marginal tax rates paid by individual investors. It might be noted, however, that the benefit of the capital gains tax rate will accrue to each and every tax-paying investor if and only if the implied tax rate at its lower boundary is less than the lowest actual marginal tax rate. This rate was observed during only seven of the 114 months studied. Since there is no reason to believe that these same results would not be obtained using any other Treasury bond, the reported results can be regarded as either an empirical anomaly or as a contradiction of the efficient market hypothesis. In either case, this finding justifies further research in view of the longevity of the event and the massive scale of the Treasury bond market.
Reference Van Home, James C., 1982, Implied tax rates and the valuation of discount bonds, Journal of Banking and Finance 6, no. 2, 145-159.
ANOTHER LOOK AT IMPLIED TAX RATES Lee DYMITS* University of Wisconsin, Stevens Point, WI 54481, USA
Michael L. MURRAY* The University of Iowa, Iowa City, IA 52242, USA Received September 1983, final version received April 1985 The authors include the effects of the minimum tax and the alternative tax on capital gains in calculating implied tax rates for discount Treasury bonds. They support the previous finding of Van H o m e (1982) that there is an inverse relationship between these rates and the level of Treasury bond yields. New implied tax rates, however, fall outside the actual marginal tax range. This finding is not consistent with the efficient market hypothesis.
In a 1982 article in this journal, Van Horne calculated the implied tax rates of purchasers of discount Treasury bonds. He found an inverse relationship between these rates and the level of Treasury bond yields as well as a direct relationship with changes in the tax laws. These relationships are supported in this note. However, the implied tax rates (for individual taxpayers) are recalculated incorporating the effects of the minimum tax rate and of the alternative tax on capital gains (when such existed). Although Van Horne alluded to the former, he did not include it in his formula. It should be noted that the existence of the minimum tax significantly increases the effective tax rate on capital gains for those who are subject to it, whereas the alternative tax on capital gains serves the opposite purpose. It should be also noted that the effect of either cannot be analyzed without fitting these to each investor tax situation. For this reason, filing status was added as another dimension (presuming 1 exemption for a taxpayer filing a single return, 2 for heads of households and married filing separate returns, and 4 for married filing a joint return). Within each class, various combinations of ordinary income and of capital gains were used so that the interaction of a particular tax bracket and of capital gains tax, minimum tax, etc. could be assessed and a proper
*The authors wish to thank anonymous referees for their helpful comments. 0378-4266/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)
134
L. Dymits and M.L. Murray, Implied tax rates
coefficient (T in Van Home's equation)
(c/2) (1 -- T)
( $ 1 0 0 0 - Po)(1 - 0.5 T) +Po 1 + ( r / 2 ) ( 1 - T) 2~ po=/_~x-" 1 +(r/2)(1-- T) -t
(1)
could be ascertained. This study was extended through June 1984 in order to analyze the effects of more recent changes in the tax laws and of the high volatile yields of that period. This process yielded, not surprisingly, not one but a multitude of coefficients for each filing class and each period with unique tax laws (there were altogether nine such periods between 1975 and 1984). For example, in 1975, within a single class, there were 19 different coefficients, ranging from 0.357 to 0.625. In order to keep our task manageable, this group of coefficients was reduced to only three (lowest, highest and one in between, the latter equal to that used by Van Home). These were inserted in Van Home's computer program to produce a band of implied tax rates instead of a single rate. In each figure, T1 represents the implied tax rate for an individual in the highest tax bracket, T2 is Van Home's implied tax rate for an individual who is neither subject to a minimum tax, nor can benefit from alternative tax on capital gains, and T3 is the implied tax rate for a middle income individual with the least favorable relationship of capital gains tax rate to tax rate on ordinary income. As can be seen on each figure, this band was reduced in January 1982 to T1 and T2. At that time, the number of coefficients fell to three (0.4, 0.455 and 0.5) and Van Horne's coefficient (and implied tax rate) became a boundary instead of being sandwiched in between as before. It can also be seen from the figures that implied tax rates were generally in a downtrend during the early 1980s. This may be attributable to an increase in interest rates which increased the supply of discount bonds, thus lowering their price. Interest rates, however, peaked between September 1981 and June 1982, reaching their high on the former date. It would have been natural to expect the implied tax rates to move up immediately following the reduction in supply. Instead, in January 1982 they broke through a plateau established during the previous 13 months and then formed a new, lower, plateau where they remained at least until July of 1984. This is apparently the result of the change in the tax rates on ordinary income which took effect then (most importantly the reduction in the maximum tax rate to 50 percent). This change made par value bonds more attractive to high-bracket taxpayers and thus lowered the price of discount bonds, this time via a reduced demand. Even a more remarkable picture emerges when the ranges of the implied tax rates during the periods in both studies are compared with the ranges of
L. Dymits and M.L. Murray, Implied tax rates
IMPLIED
TAX R A T E S
135
1975-79
SINGLE TAXPAYER 100 90 80 70 60
~R
v
hi lIE
50 40
2o
10
I
;
0-I JAN
...........
1975
J ...........
JAN 1 9 7 6
I ...........
JAN
1977
r
. . . . . . . . . .
JAN 1 9 7 8
I
..........
I
JAN 1 9 7 9
DATE
IMPLIED
TAX R A T E S
1980-84
SINGLE TAXPAYERS 70
60-
3
50 -i 4.0 v l.iJ l--
ry
!
-
1 0 -]
0
JAN
! [ , , , , . , , , . . , i
1980
......
JAN 1 9 8 1
, ....
JAN
i,,,.,
1982 DATE
Fig.
1
.....
,i,,,
JAN 1 9 8 3
........
i,,,.
JAN
1984
136
L. Dymits and M.L. Murray, Implied tax rates
I M P L I E D TAX R A T E S
1975-79
HEADS OF HOUSEHOLDS loo 7
.° 1 8O
A
~l
W
p,<
6O 5O 40. 3020-
I 0 -I 0
- -'r-r'r~
JAN 1975
JAN 1 9 7 6
JAN 1 9 7 7
JAN 1978
JAN 1979
DATE
I M P L I E D TAX R A T E S
1980-84
HEADS OF HOUSEHOLDS
70
80
J A
IR
4.0 -~
I--
.K
30
2o -1 I
10"-t
o -t-,-,-,JAN 1980
JAN 1981
JAN 1 9 8 2 DATE
Fig. 1 (cont.)
JAN 1 9 8 3
JAN 1 9 8 4
('luoo) I "~!~1 3/VO ll, g 6 L
NVr
~li6L
NVr
~g6L
NVr
t961
NYI"
096
NYf" 0
I "-0L
O~
0~
.,
141 A
01r d
i
-
0~
-
09
I
I
I
t MI/NIOF
/~g-0g61
- 0Z
O N r l l - I a31~l~lVlq
SR,I.V~I XVJ~ fl.~FIdI~II 3J.VO
6/.6 L NYr
8/.6L
NVr
LZ.6;
NYI"
9£6L
NVr
gL6L
NYI" 0 OL OZ OR 0"~ "4
Og 09 0/08
Ii oe O0
A'IJ.NIOT' ONr"ll=l O':IltiNYIN
6A-GL61 gfI
S~I&V~I XVJ= (I~tI~IdI~II
~a2va xv2 pa!IdtuI ',fv~nl~ "-I'l~ t my sX!ta,f(l "7
138
L. Dymits and M. L. Murray, Implied tax rates
IMPLIED TAX R A T E S
1975--79
MARRIED FILING S E P A R A T E L Y 100 90
--
80 7060 b.l IN
,it
-
T3
504O 3O 20 10--t JAN 1 9 7 5
JAN
1976
JAN
1977
JAN
1978
JAN
1979
DATE
IMPLIED TAX R A T E S
1980-84
MARRIED. FILING S E P A R A T E L Y 70
~, ,
T3 8o
50
A
4.0-
laJ I--
.( rr
30
20-
10-
o -+-,-,~ JAN
1980
JAN
19B1
JAN
1982 DATE
Fig. 1 (cont.)
JAN
19B3
JAN
1984
139
L. Dymits and M. L. Murray, Implied tax rates
Y I E L D S TO MATURITY 1 9 7 5 - 7 9 SINGLE TAXPAYER
t
9
.
.
.
.
.
.
.
7-
5 4 3 2 1 0 JAN 1 9 7 8
JAN 1 9 7 7
JAN 1 9 7 8 DATE
Y I E L D S TO MATURITY 1 9 7 5 - 7 9 HEADS OF HOUSEHOLDS
"it
9
7 8
i
3
2-R3
0 JAN 1 9 7 6 •
"
"
"
"
"
"
"
°
"
"
i
"
"
•
"
°
"
JAN 1 9 7 7
•
"
"
"
•
|
°
•
JAN 1 9 7 B
DATE Fig 1 (cont.)
"
"
"
•
•
•
•
•
L. Dymits and M. L. Murray, Implied tax rates
140
YIELDS TO MATURITY 1 9 7 5 - - 7 9 MARRIED FILING JOINTLY
7-
5
3 2 1 0
i , + , . , , + , , , i J , , , , . , , , , , I , , , I , . , , , , , I - , ' , . - , , . ,
JAN 1 9 7 5
JAN 1 9 7 8
JAN 1 9 7 7
JAH 1 9 7 8
DATE
YIELDS TO MATURITY 1 9 7 5 - - 7 9 MARRIED FILING SEPARATELY
5
R1 4
3
2
O -+----,JAN 1 9 7 5
JAN 1 9 7 6 DATE
Fig. 1 (cont.)
L. Dymits and M. L. Murray, Implied tax rates
141
actual marginal tax rates (14 percent to 70 percent until 1981, the foregoing reduced by 1.25 percent in 1981, 12 percent to 50 percent in 1982, and 11 percent to 50 percent thereafter). The greatest significance to these findings occurs whenever the implied tax rate at the higher boundary exceeds the highest actual marginal tax rate. When this is the case, not a single investor can gain anything by purchasing discount bonds. Nonetheless, higher implied rates were observed during 11 of the months in the study period. During these months, all investors could have achieved a higher return by purchasing bonds selling at or close to par. This implies that during these periods discount bonds were overpriced and par value bonds were underpriced. An intuitive explanation for this phenomenon might lie in the popularity of discount bonds as tax-saving devices. This popularity may have increased aggregate demand to a point where prices were pushed beyond equilibrium. In those months where the implied tax rates were lower than the top actual marginal tax rate, no implications can be drawn without knowing the actual marginal tax rates paid by individual investors. It might be noted, however, that the benefit of the capital gains tax rate will accrue to each and every tax-paying investor if and only if the implied tax rate at its lower boundary is less than the lowest actual marginal tax rate. This rate was observed during only seven of the 114 months studied. Since there is no reason to believe that these same results would not be obtained using any other Treasury bond, the reported results can be regarded as either an empirical anomaly or as a contradiction of the efficient market hypothesis. In either case, this finding justifies further research in view of the longevity of the event and the massive scale of the Treasury bond market.
Reference Van Home, James C., 1982, Implied tax rates and the valuation of discount bonds, Journal of Banking and Finance 6, no. 2, 145-159.