M
A
T
The Tax Option
in Municipal Bonds
IN
A
N
Y
FO
R
ANDREW KALOTAY AND C. DOUGLAS HOWARD
C. DOUGLAS HOWARD
LE
on market-related factors, such as the volatility
of interest rates and bid–ask spreads.
In this article, we first discuss the relevant tax considerations pertaining to municipal bonds. Next, we provide a high-level
description of the algorithm for optimally
managing the tax option. (We provide a rigorous description in the appendix.) Implementing the optimizer requires that we
specify the bond’s market price under various
interest-rate scenarios. We model these market
prices as tax-neutral values whose computation is described in the following section. In
the remainder of the article, we quantify the
value of tax options for various standard bond
structures and explore the results’ sensitivity
to the key parameters.
IT
IS
IL
LE
G
R
TI
A
L
TO
R
EP
R
O
D
U
C
E
TH
IS
is an associate professor
of Mathematics at Baruch
College at The City
University of New York
in New York, NY.
[email protected]
ur aim is to quantify the value
of the tax option embedded
in municipal bonds. We first
develop an algorithm1 that optimizes the timing of taking losses for tax purposes. Using this algorithm, we then determine
the expected tax benefit that can be extracted
over a bond’s lifetime, i.e., from issuance until
maturity or call. We show that, under realistic assumptions, the value of the tax option
embedded in a long-term bond amounts to
several points.
Constantinides and Ingersoll [1984] discuss the benefits of active bond management
over a buy-and-hold policy. In particular,
they point out that tax-driven transactions,
such as strategically selling bonds whose
value has declined, can significantly improve
performance. In this article, we quantify the
potential value of optimal tax management
over the buy-and-hold strategy for municipal
bonds.
Interest on municipal bonds is exempt
from federal income taxes. Capital gains and
losses, however, are subject to complex tax
treatment. Investors may be able to take advantage of this tax treatment by selling bonds
whose price has sufficiently declined, and
thus reduce their aggregate tax obligations. In
addition to the treatment of capital gains and
losses, the value of the tax option also depends
A
is the president of Andrew
Kalotay Associates, Inc.,
in New York, NY.
[email protected]
C
O
A NDREW K ALOTAY
94
THE TAX OPTION IN MUNICIPAL BONDS
TAX TREATMENT
OF MUNICIPAL BONDS
Ang et al. [2010] provide a thorough
discussion of the tax treatment of municipal
bonds, including original issue discounts
(OIDs) and original issue premiums (OIPs).
The initial price to public determines the
bond’s basis over its life. The investor’s tax
basis depends on the purchase price. A municipal bond transaction’s tax treatment may
depend on both the investor’s tax basis and
the bond’s basis. This interplay can be rather
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complicated for OIDs, and in the interest of brevity we
omit discussing it here.
The tax treatment of capital gains is different from
that of capital losses. Fortunately, in the case of optimal
tax option management, only capital losses matter. Capital gains occur only involuntarily, i.e., when the bond
matures or is called. In those cases, the tax treatment is
straightforward, as discussed in the following.
If a bond is issued at or above par (for our purposes), the investor’s tax basis depends only on the purchase price, P. The taxable gain on a non-OID bond
purchased at price P below par and held to maturity or
call is 100 − P. If 100 − P is modest (less than 0.25 x the
number of years remaining to maturity), the applicable
tax rate is the short- or long-term capital gains rate.
However if 100 − P exceeds the de minimis discount,
the entire gain is taxed as ordinary income. On the other
hand, if P > 100 and the bond is held to maturity, the
premium paid over par has no tax effect.
The tax treatment is more complicated if a bond is
sold prior to maturity, resulting in a capital gain or loss.
(See Ang et al. [2010].) Under optimal tax management,
selling at a gain is suboptimal,2 and therefore irrelevant
for our purposes. On the other hand, the tax treatment
of the loss resulting from a sale is critical. Long-term
losses are deductible at the long-term capital gains rate,
and short-term losses are deductible at the short-term
capital gains rate. We will assume that there are always
like capital gains for offset purposes, but we recognize
that, in practice, the tax treatment of capital losses can
be more complex.
The federal tax rates were significantly increased at
the beginning of 2013, and they may be subject to further changes. Our examples are based on the following
rates: long-term capital gain/loss at 20%; short-term
capital gain/loss at 40%, and ordinary income at 40%.
OPTIMAL MANAGEMENT
OF THE TAX OPTION
Managing taxes is a real option available to investors; we refer to it as the tax option. We offer a highlevel description of the algorithm to optimally manage
the tax option, and then examine how the value of a
bond’s embedded tax option depends on market-related
factors, such as the yield curve, interest-rate volatility,
and transaction costs. We also explore the sensitivity of
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the option value to the investor’s ability to recognize
short-term capital losses.
The analytical approach extends conventional
OAS-based valuation (Kalotay et al. [1993]) to incorporate the tax treatment of municipal bonds. The required
inputs are a market-implied (preferably issuer-specific)
optionless par yield curve and an interest-rate volatility
that together determine rate evolution. Optionless
curves are not readily available for municipal bonds.
The reported yields normally assume that the bonds
are callable at par after 10 years. Kalotay and Dorigan
[2008] discuss the approach for extracting optionless
par curves from callable curves. We assume that the
yield curve evolves according to the industry-standard
Black–Karasinski process with 0% mean reversion, i.e.,
as a lognormal process.
The value of the tax option is the difference
between a bond’s value under optimal tax management
and under the unmanaged buy-and-hold base case. If
the bond is optionless, the fair unmanaged value (i.e.,
the tax-neutral value) can be determined by finding the
price that is equal to the discounted value of the after-tax
cash f lows. For callable bonds the valuation is more
complicated, because the evolution of the yield curve
determines the call’s timing, and thus the investor’s
after-tax cash f lows.
Determining the optimum tax-management
strategy and, in turn, the value of the embedded tax
option, is considerably more challenging. Our goal is
to calculate how much tax benefit investors collectively
can extract from a bond over its lifetime. An individual
investor can determine the optimum time to sell a bond
whose value has declined, taking into account the option
value of the replacement bond. Assuming that each successive purchaser acts in a like manner, transactions take
place at progressively lower prices over the bond’s life.
The final holder, who bought the bond at the lowest
price, pays the capital gains taxes. See Exhibit 1.
Optimal option exercise requires looking into a
lattice-based future and, at each node, comparing the
trade-off between selling and waiting. The numerical
implementation is recursive valuation of a path-dependent option (Howard [1997]); details for the problem
under consideration are provided in the appendix.
Under optimal management, the investor will sell
when the resulting benefit provides full compensation
for the value of the forfeited tax option, i.e., when the
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EXHIBIT 1
Tax-Driven Sales during a Bond’s Life Cycle
efficiency of the transaction reaches 100% (Kalotay
et al. [2007]).
The price of the bond in various future market
environments is a critical consideration. In the absence of
taxes, the expected price would be the fair value. In the
case of a municipal bond, the fair value must incorporate
the investor’s tax from a capital gain. The tax-neutral
values (Kalotay [2013]), i.e., the fair tax-adjusted values
under buy and hold, provide convenient and defensible
estimates of future prices. We describe the algorithm for
estimating tax-neutral values in the following section.
Although the tax-neutral value is a theoretical
lower bound, according to empirical investigation by
Ang et al. [2010], the market prices may be even marginally lower. Our approach can be readily applied to
any reasonable pricing model, such as pretax prices (the
upper bound) or the tax-neutral prices adjusted for the
value of the tax option.
purchase price, the greater the associated taxes when
the bond is retired. Due to this dependence, in general
the tax-neutral value must be determined iteratively.
However, the calculation can be simplified if the bond
is not callable.
In the following examples, we assume an issuer
yield curve as shown in Exhibit 2. Exhibit 3 shows the
tax-neutral values and the pretax values of optionless
non-OID bonds with 10 years remaining to maturity.
They differ by the present value of the taxes payable
at maturity. Here the de minimis level is 97.50 of par
(100 − 10 × 0.25). The coupon of the bond with a
pretax value slightly above 97.5 is 2.72%. A bond purchased at this price would be subject to 20% capital
gains tax, making its after-tax value only 97.13. But in
fact, the fair value of the bond with a 2.72% coupon
would be only 96.49, because below the de minimis
threshold, the entire gain is taxed as ordinary income
at a 40% rate.
TAX-NEUTRAL VALUES
We define the tax-neutral value as the price equal
to the present value of future inf lows (i.e., interest and
principal payments), adjusted for the taxes paid on the
gain at maturity or when the bond is called. The taxneutral value depends on the purchase price: the lower the
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EXHIBIT 2
Issuer’s Optionless Par Yield Curve
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EXHIBIT 3
Tax-Neutral Values of Optionless 10-Year Bonds
EXHIBIT 4
Tax Option Embedded in a 10-Year 3% Bond Issued at Par
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EXHIBIT 5
Tax Option in a 30-Year 4.25% Non-Call 10 Bond Issued at Par
THE VALUE OF THE TAX OPTION
With the use of the algorithms we have described,
we are ready to calculate the tax option embedded in
conventional municipal bond structures. We first consider bonds issued at par and maturing in 10 and 30
years; we assume that the 30-year bond is callable at
par after 10 years. Given the current industry practice
of issuing bonds with a 5% coupon, we also take an indepth look at a callable 30-year, 5% bond. The following
exhibits display the results.
Exhibit 4 displays the value of the tax option in
a new 10-year, 3% bond issued at par. In general, the
value of the tax option increases as interest-rate volatility increases, and as transaction cost declines. Due to
its relatively short maturity, a 10-year bond exhibits a
modest value for the embedded tax option. At a 20%
interest rate volatility and 0.5% transaction cost, it is
slightly above half a point, or roughly 6 basis points in
incremental annualized return.
The value of the tax option dramatically increases
with the bond’s maturity. In Exhibit 5, we consider a
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30-year bond, callable after 10 years (30 non-call 10,
in industry parlance), issued at par. At 20% volatility
and 0.5% transaction cost, the value of the tax option
is roughly 6 points, which translates to about 35 basis
points of annualized incremental return.
The principal contributing factor is the bond’s
longer duration. As a consequence, a relatively modest
increase in the long-term rate results in a large decline
in market value, triggering the opportunity to sell the
bond at a loss and thus reduce taxes. Therefore, shortterm losses written off at a 40% rate can add significant
value. In Exhibit 6, we examine the results’ sensitivity
to the short-term capital gain/loss rate.
The final example (see Exhibit 7) displays the tax
option in a 30-year, 5% bond callable at par in year 10.
This bond is originally priced at 109.67 (yield to call
3.83%, yield to maturity 4.42%).
According to Exhibit 7, at 20% volatility and 0.5%
transaction cost, the value of the tax option is 7.28% of
par. Although the duration of the 5% bond is considerably shorter than that of a like 4.25% bond sold at par,
its tax option is much larger (7.28 versus 6.28). There
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EXHIBIT 6
Tax Option in a 30-Year, 4.25% Non-Call 10 Bond Issued at Par—the Effect of the Short-Term Capital Gain Rate
EXHIBIT 7
Tax Option in a 30-Year, 5% Non-Call 10 Bond Issued at a Premium
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are two explanations for this. First, the tax option value
is expressed as a percentage of par, rather than as a percentage of cash outlay. The second reason is that the 5%
bond is less likely to fall below the de minimis threshold,
where tax-driven trades are harder to justify due to the
depressed market price.
CONCLUSION
Investors in tax-exempt bonds may be able to
improve their after-tax performance by selling bonds
with sufficiently large value losses, i.e., by exercising
their tax option. We determined the value of the tax
option embedded in various standard bond structures. In
addition to the tax rates, the tax option’s value depends
on the bond’s effective duration, because price volatility increases the tax option’s value. A higher issue
price also increases value, because the bond is less likely
to fall below the de minimis threshold. As expected,
greater interest-rate volatility and lower transaction costs
also increase value. Under realistic assumptions, a new
long-term bond’s embedded tax option is considerable,
particularly if short-term losses can be used to offset
short-term capital gains. In that case, the value of the
tax option in a 30-year non-call 10 bond is roughly 6%.
The value is more conservative if the tax rate applicable
to short-term losses is the lower long-term rate. In this
case, the tax option in a 30NC10 declines to roughly
3%, which is still significant.
APPENDIX
THE TAX OPTION IN MUNICIPAL BONDS
In this appendix we describe the recursive algorithm
used to value the tax option. For simplicity, we assume that
the underlying bond is optionless. We further assume that
when the bond is sold under the tax option, the same bond is
immediately repurchased. In reality, a like kind bond could
be purchased to avoid the wash sales rule. The bondholder
may elect to hold the bond until maturity. If the bond was
purchased at a discount, this will generate a capital gain and
a corresponding tax payment at maturity. Let Vhold ≤ 0 denote
the present value of this tax payment, which may be easily
computed without lattice methodology. Alternatively, the
bondholder may engage in a sequence of sell/buy transactions
generating capital losses, followed by a capital gain at maturity. The resulting tax-related cash f low (TRCF) is stochastic
and must be valued using lattice methods. Let Voptimal denote
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the (expected) present value of the TRCF with optimal transacting, so Voptimal ≥ Vhold. The tax option measures the value
of the optional transactions prior to maturity, and is given
by Voption = Voptimal − Vhold. Here we discuss the computation
of Voptimal.
Lattice Notation. Out discrete valuation lattice has a
time partition 0 = t0 < t1 < t 2 < < t N , where t 0 corresponds
to the valuation date and tN corresponds to the underlying
bond’s maturity. Here time is measured in years. At each
time tn there is a finite number of possible interest rate states,
which we index by the variable i. (At t 0 there is only one such
state, which we label i = 0.) Associated with each period n
and state i, which we’ll refer to as “node (n, i),” are a singleperiod discount rate rn (i) and a corresponding single-period
discount factor, given by dn (i ) = e − rn ( )( n ++1 − n ). We let pn (i → i′)
denote the probability of transitioning from interest rate state
i at time tn to state i′ at time tn+1. The state-dependent short
rates, and thus the single-period discount factor and the transition probabilities pn (i → i′) all depend on the choice of
short-rate process (Black–Karasinski, e.g.), in conjunction
with the particular discretization in use (binomial, e.g.), as
calibrated to explain the optionless par yield curve at the
valuation date. The tax-neutral value of an optionless (or
callable) bond may be computed on this lattice using standard
recursion methodology, along with the iterative process previously described. We do this for the underlying bond and let
Vn (i) denote the tax-neutral value of the bond at node (n, i).
Here we assume that one may transact at these prices without
transaction costs.
Computing Voptimal. In addition to the interest rate state i,
we must keep track of information pertaining to the underlying bond, which we will index by k. Specifically, we will
associate two numbers, bk and τk , with the index k. The bk will
represent the most recent purchase price of the underlying
bond and the τk ’s will represent the time of purchase relative to the valuation date. A node in the augmented lattice is
now specified by (n, i, k), i.e., the state of the world at period
n is determined by the combination of the short rate and the
underlying bond’s most recent purchase price and time of
purchase.
To make the bk /τk list, we first let b 0 and τ0 denote the
purchase price and time of purchase for the underlying bond
as of the valuation date. As a convenience, we take b −1 = 100.
Then we search through the lattice, starting at node (0, 0),
and make a list of all the Vn (i) that are less than b 0. These
are candidate values for tax option transaction prices as time
elapses. Suppose b1, …, bk is the list of these values. As we
make this list, we also record the corresponding candidate
transaction times. Suppose we test node (n, i) and find that
Vn (i) < b 0 and suppose that Vn (i) is the kth item added to
the list, so bk = Vn (i). We then put τk = tn . Also, to facilitate
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computations later, record where in the list Vn (i) and tn appear:
specifically, set
k0 (n, i ) = k
(A1)
To summarize, at node (n, i, k) the time is tn , the singleperiod rate is r n (i), and the bond was last transacted (sold/
repurchased) at time τk at a price of bk .
Now let Vn n(i k ) denote the value at time tn of all TRCF
subsequent to time tn if the interest rate process is in state i
and the underlying bond is in state k. At the bond’s maturity, t N ,
we have each Vn N (i k ) = 0, as there is no subsequent cash
f low. The recursion formula to take us from period n + 1 to
period n is
Vn n(i k ) = dn (i )∑ pn (i
i′
i ′ )[taxrate ⋅ Δbasis + Vn n +1(i ′, k ′ )]
(A2)
where k′ is the branched-to bond state. Here taxrate ⋅ Δbasis is
the cash f low generated by a transaction in period n + 1 and
Vn +1(i ′, k ′ ) is the value of subsequent cash f low.
The appropriate taxrate depends on n and k. We consider
the cases n + 1 = N (the bond’s maturity) and n + 1 < N separately. If n + 1 = N, there are three possible tax rates. If bk <
100 − 0.25 ⋅ (tN − τk ), then the gain exceeds the de minimis
discount and taxrate is the ordinary rate; if not, taxrate is the
long-term capital gain rate if tN − τk ≥ 1 and the short-term
capital gain rate otherwise. If n + 1 < N, then taxrate is the
long-term capital gain rate if tn+1 − τk ≥ 1 and the short-term
capital gain rate otherwise.
The value of Δbasis depends on n, k, and k′. If k′ = k, then
there is no transaction and Δbasis = 0. Otherwise, Δbasis = b k − bk ′
where b k is the amortized basis corresponding to the original
purchase price of bk at time τk. (Amortization only affects bonds
purchased at a premium. If bk ≤ 100, then k = bk. If 100 < bk,
we will have 100 ≤ b k ≤ bk, with the precise value depending
on the elapsed time since purchase at time τk.)
The branched-to bond state, k′, is determined as follows. First,
k ′ = −1 if
n +1= N
(A3)
In this case, the Vn term in Equation (A2) is zero and the
cash f low term is zero if bk ≥ 100 (as the basis has amortized to
par by the bond’s maturity) and taxrate ⋅ (bk − 100) if bk < 100
(using here that b −1 = 100). This cash f low is never positive
and represents the settling up at the bond’s maturity (refer
back to Exhibit 1).
If Equation (A3) does not apply (so n + 1 < N), we have
k′ = k
if
Vn +1( ′ ) ≥ b k
(A4)
In this situation, the tax-neutral value of the underlying
bond in the branched-to interest rate state exceeds the current
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tax basis. There is no tax option transaction and the bond’s
state remains unchanged. This produces a cash f low of zero
in Equation (A2). If neither (A3) nor (A4) apply (so n + 1 < N
and n +1(i′′ ) < b k ), a tax option transaction is possible. We must
determine if it is, or is not optimal to transact at period n + 1.
If we choose not to transact, the bond’s state does not change
(k′ = k and Δbasis = 0), so there is no cash f low and the value
at the branched-to node (n + 1, i′, k) is
Vn no
0 Vn n +1(i ′, kk′′ ) = Vn n +1(i ′, k )
On the other hand, if we choose to transact, we have
k ′ = k0 ( + 1, i ′ )
(A5)
r e ⋅ Δbasis + Vn n +1(i ′, k ′ )
Vn yes = taxrat
(A6)
and with this k′
where Δbasis is computed with k′ as in Equation (A5). Note
that Equation (A5) holds, because the purchase price is Vn+1(i′),
which was added to the list as item number k 0 (n + 1, i′) (see
Equation (A1)). In Equation (A2), we therefore take
if Vn no > Vn yes
⎧k
k′ = ⎨
w e
⎩k0 (n + 1, i ′ ) otherwis
This, recall, obtains if neither Equation (A3) nor Equation (A4) apply.
Since the interest rate and bond states at time 0 are i = 0
and k = 0, Vn 0 (0,0) represents the value at the valuation date
of TRCF subsequent to time 0. If the tax basis of the bond
at the valuation date, b 0, exceeds the tax-neutral value of
the bond, V0 (0), it may be optimal to transact at the valuation date. We test this after the Vn recursion is complete. Let
Vn no = Vn 0 (0,0) be the value of the TRCF with no transaction
at the valuation date. If we do transact at the valuation date,
the value of the TRCF is given by
Vn yes
taxrate ⋅ ( 0 b1 ) Vn 0 (0,1)
taxrate
With V0 (0) < b 0, we will have b1 = V0 (0), as per the construction of the candidate basis list. Here taxrate is determined
as above in the second case. A transaction today therefore
produces a tax loss of b − b1 and converts the tax basis of the
underlying bond to b1, i.e., the bond state changes to k = 1.
The value of the TRCF at the valuation date is the better of
these alternatives, producing
⎧⎪Vn no
Vooptimal = ⎨
⎪⎩Vn yes
if Vn no > Vn yes,
otherwise.
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ENDNOTES
1
The results in this article were calculated using
MuniOAS™ (algorithms patent pending).
2
A rare exception is the case of a non-OID bond purchased at a deep discount, whose market value far exceeds
its accreted value.
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Howard, C. “Valuing Path-Dependent Securities: Some
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To order reprints of this article, please contact Dewey Palmieri
at [email protected] or 212-224-3675.
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