# 3. Identifying the implicit tax rate

;; 3.01
The natural first move is to read τ* off index yields, and it fails in an instructive way. Computed from the Bond Buyer 20-Bond index over the 20-year Treasury, the naive implied rate averaged {{N:naive_1950s_pct}} percent in the 1950s, {{N:naive_1960s_pct}} percent in the 1960s, {{N:naive_1970s_pct}} percent in the 1970s, {{N:naive_1980s_pct}} percent in the 1980s, and {{N:naive_1990s_pct}} percent in the 1990s. In the 2000s it fell to {{N:naive_2000s_pct}} percent, and in the 2010s it averaged {{N:naive_2010s_pct}} percent. A marginal investor in a negative tax bracket does not exist. The ratio embeds credit risk, liquidity, and the call option along with tax, and after 2008 those components swamped the tax component entirely.

;; 3.02
The full series, plotted against the top statutory rate on interest income in Section 5's historical exhibit, makes the pathology visible: long stretches after 2008 have municipal bonds out-yielding Treasuries — dealer balance sheets shrank, taxable Build America Bonds changed the supply mix, and in March 2020 the municipal market dislocated outright, with ratios far above one. Whatever a reader makes of any particular episode, the lesson is the one Section 2 stated: the naive spread is not a burden measure, and a correction built on it would produce nonsense in both directions — implausibly small burdens in the 1950s, negative burdens after 2008.

;; 3.03
Longstaff (2011) provides the cleanest identification in the literature, and the paper's headline rests on it. He uses the municipal swap market: contracts exchanging a percentage of one-week taxable rates for tax-exempt index rates, observed weekly from August 2001 to June 2008. At one-week maturity, default and call risk are negligible, and the swap term structure lets the liquidity component be stripped. The naive short-end reading — municipal swap rates averaging 85.4 percent of taxable, implying a marginal tax rate of 14.6 percent in the working-paper version — becomes, after the correction, an identified marginal tax rate averaging 38.0 percent in the published version, which the paper uses throughout.{{FN: The published abstract states the marginal tax rate "averages 38.0%" — verified against the journal's own abstract metadata (Crossref, DOI 10.1111/j.1540-6261.2011.01650.x). The NBER working-paper version (w14687), verified line by line on file, reported 41.6 percent (its Table 3: mean 0.41574, s.d. 0.03784, range 0.31369-0.49826, N = 360); the naive 14.6 percent contrast and the sample dates are from the working-paper text. Table 4 carries the working-paper figure as a transparency row; pinpoint reconciliation against the published tables awaits library access.}}{{FN: Green (1993) supplies the theory of the relation between the taxable and tax-exempt curves when bonds need not be held to maturity; we cite it at claim level from secondary discussion, as the paper itself is paywalled at the time of writing.}}

;; 3.04
Two features make 38.0 percent credible as a market-clearing tax rate rather than an artifact. It exceeds the 35 percent top federal statutory rate that prevailed for most of the sample. That is consistent with a marginal holder pricing some state tax on the taxable alternative alongside the federal rate, and inconsistent with the folk claim that the marginal municipal buyer is always under-bracketed. The instrument imposes a caution, however: the identifying market is the variable-rate demand complex, whose dominant holders are tax-exempt municipal money funds holding national paper on which state exemption attaches only in-state, so the combined-rate interpretation is one reading of an above-statute rate, not a fact about households (verify instrument description). And in the working-paper version, Longstaff notes the average sits close to the top combined federal-plus-state rates of New York and California — 42.9 and 41.7 percent — the natural habitats of the marginal tax-exempt investor (verify pinpoints).

;; 3.04b
What the market charges is not what the federal government forgoes, and the distinction runs through every number that follows. The 38.0 percent is best read as a combined rate: the marginal investor prices the exemption against federal and, in part, state taxes. A correction to the federal schedule cannot coherently exceed the federal top rate. We therefore compute every headline object twice: at the full market rate, in which case the corrected schedule is explicitly the federal schedule augmented by the entire implicit tax, accruing across levels of government; and at a federal component capped at the 35 percent federal top rate of the sample period — a bound rather than an identification, since the split depends on deductibility and on which investors price the national index, choices the data do not settle (the 3.0-point residual is the state component under the simplest additive reading). At the capped rate the top-percentile correction is {{N:fed_component_top1_delta_pp:.2f}} points and the 2021 aggregate is ${{N:fed_component_agg_2021_bn}} billion.

;; 3.05
Today's term structure is in Table 2, from three independent AAA curves published through the MSRB's public displays. At ten years the three providers land within little more than a point of one another: implied rates of {{N:tau10_bval_pct}}, {{N:tau10_ice_pct}}, and {{N:tau10_tradeweb_pct}} percent, from curves as of {{N:curve_date_long}} against the most recent Treasury constant maturity ({{N:cmt_date_long}}). The implied rate falls to {{N:tau20_bval_pct}} percent at twenty years and {{N:tau30_bval_pct}} percent at thirty.{{FN: The display curves are licensed for research use; their terms bar republication, so the published version of this paper will substitute a licensed Bloomberg BVAL pull by a coauthor, and the curve module of the replication package is written to be swapped. [WRDS robustness to be added: curves built directly from the MSRB transaction feed.]}}

@TABLE:term_structure| **Table 2. The term structure of the market-implied municipal tax rate, August 2026.** AAA municipal curves from three providers via the MSRB's public displays ({{N:curve_date_long}}); Treasury constant maturities from FRED ({{N:cmt_date_long}}). τ* = 1 − (municipal yield / Treasury yield), shown for the BVAL curve; the ICE column implies rates within roughly a point of BVAL's at each maturity, and the Tradeweb column differs by as much as three and a half points.

;; 3.05b
The Treasury benchmark is a choice, and it is the conservative one. The identification wants the default-free taxable twin of a AAA municipal curve; corporate AAA yields sit above Treasuries, so a corporate benchmark produces systematically higher implied rates — Poterba and Ramírez Verdugo (2011) report the corporate-based implicit rate at 29 percent in 2007 against 18.5 percent Treasury-based. Residual municipal credit and liquidity premia relative to Treasuries bias our measured τ* down, understating the burden — and they are not small: Schwert (2017) attributes most of the tax-adjusted municipal spread to default risk (cited at claim level; the article is paywalled at the time of writing). Every known bias in the naive readings runs the same direction; the short-end identification exists precisely to strip them. The benchmarks also differ in which taxes they identify. Treasury interest is exempt from state income tax, so a ratio against Treasuries identifies a federal-only concept; the swap identification prices against state-taxable short rates and identifies the combined rate. Part of the term structure's apparent decline is benchmark, not maturity — and read as a federal-only concept, the one-year Treasury-based rate of {{N:tau1_bval_pct}} percent sits within two points of the 40.8 percent federal top rate on interest, a coherence check on the curve's own terms.

;; 3.06
The term structure poses the question the rest of the paper must carry. The implied rate falls with maturity — from the 38.0 percent identified at Longstaff's one-week horizon to {{N:tau30_bval_pct}} percent at thirty years on today's curve. No investor's tax bracket depends on the maturity of the bond she holds. So either the long end is contaminated in ways even careful designs miss, or the long end is telling us something real about what the market prices.

;; 3.07
It is not simply contamination. Chalmers (1998) examined municipal bonds secured by escrowed Treasury obligations — default-free, non-callable — and found that they still yield too much relative to the tax story (cited from the discussions in Longstaff 2011 and Ang, Bhansali, and Xing 2010; the original is to be obtained before submission). The long-end puzzle survives clean credit controls, which rules out the easy dismissal.

;; 3.08
Part of the answer is expectations. Longstaff's own estimates separate the current short rate from the long-run forward rate, and the forward marginal tax rate is 29.4 percent — the market prices the tax wedge as declining with horizon. Poterba (1986) made the point first: the taxable-tax-exempt spread embeds expected future tax policy, not just current rates. Kueng (2014) shows municipal yield spreads move with expected future tax rates at horizons out to thirty years. What this explains is why the long-end rate is lower, not that it is illusory: under the paper's own definition, burden is the yield the holder gives up, so a wedge narrowed by expectations of lower future rates is a smaller give-up for a purchaser today, not a larger one that the market has mismeasured. Expectations tell us the composition of the long-end rate — it is a forward-looking average of tax rates rather than a reading of today's — and that is exactly why the maturity grid in Table 4 is a grid of alternative burden rates and not a set of corrections to a single true one.

;; 3.09
And part of the long end points the other way entirely. Ang, Bhansali, and Xing (2010) study municipal bonds trading at market discounts, whose price appreciation is taxed as ordinary income, and find implied tax rates of 70 to 100 percent — two to three times the statutory rate. Where the tax treatment of the marginal dollar is ordinary-income-like, the market capitalizes close to fully, at every maturity. The long end is not evidence that tax capitalization fades; it is evidence that what is being capitalized varies with the instrument.

;; 3.10
The identification choice follows. The headline uses τ* = 38.0 percent because it is the cleanest credit- and liquidity-free identification available, a choice made on identification grounds; it sits near the top of the paper's robustness grid, and the text concedes as much wherever the headline appears. Its transplantation is a second caution: the estimate is from a 2001-2008 equilibrium, and no identified short-end estimate exists for 2021 — the intervening years brought the net investment income tax, the 2017 law's rate changes and state-tax deduction cap, and the collapse in bank and insurer municipal demand, any of which could move the marginal investor's rate. The 2021-contemporaneous variant is the 2021 ten-year rate of 35.6 in Table 4; today's short end (the one-year Treasury-based 39.2) suggests the neighborhood has not moved far. The alternative anchor — matching each holder's τ* to a maturity — would require maturity composition the household data do not contain and would import the unresolved puzzle into the headline. Instead the full term structure runs through every result: Table 4 reports the corrected schedule's key objects at 38.0, at the working-paper 41.6, at the federal component, at the 2021 ten-year, at Longstaff's long-run forward rate of 29.4, and at today's ten-, twenty-, and thirty-year rates.

;; 3.11
The remaining objection is whether a one-week rate is informative about the burden on a household holding ten- and twenty-year bonds. The burden-relevant τ* is the tax component of the yield give-up at whatever maturity the household holds. At the short end that component is isolated; at the long end it is entangled with the puzzle. Using 38.0 percent everywhere amounts to assuming the tax component is flat across maturities and assigning the rest of the long-end gap to non-tax factors; using the observed curve assumes the opposite. We do not claim the two bracket the truth. The long-end ratio contains several unidentified components whose signs are not uniformly established, and the short-end estimate is transported across maturity and across two decades, so neither end is demonstrably on the correct side of the true 2021 rate. The rows in Table 4 are therefore sensitivity scenarios — each a stated assumption carried through to its consequence — and what they establish is that the correction's sign and its concentration at the top of the income distribution do not depend on which assumption a reader prefers.

;; 3.12
For the historical arc, no identified series exists before 2001. The long panel of Section 5's historical exhibit uses the Bond Buyer index because it is the only continuous municipal yield record back to the 1950s, labeled as the naive reading it is. From late 2016 the index's public series ends (the index itself continues behind the publisher's wall), and the figure splices on a AA quality curve — closer in credit quality to the index's 20-bond composite than a AAA curve — cross-checked against the AAA curve, which runs {{N:splice_aa_minus_aaa_20y_bp}} basis points richer on average over the overlap panel. The 2020 excursion in the spliced segment is the real March 2020 dislocation, not a splice artifact.
