Tax-Equivalent Yield: Comparing a Tax-Free Bond With a Taxable One

Why the formula is a division and not a mark-up

Start from the only thing that matters: what reaches your account. A taxable bond paying a yield Yt at a marginal rate t leaves you Yt × (1 − t). A tax-free bond paying Yf leaves you Yf. Set them equal and solve for the taxable yield: Yt = Yf ÷ (1 − t). That is the whole derivation, and the whole formula. At a 30 percent marginal rate, a 3.00 percent tax-free yield needs a taxable bond paying 3 ÷ 0.70 = 4.29 percent to break even, because 4.29 × 0.70 = 3.00.

The tempting shortcut is to add the tax back — 3.00 × 1.30 = 3.90 percent — and it is wrong every time. Tax is taken out of the larger number, so putting it back requires dividing by what survives, not multiplying by what is charged. The error is 0.39 points here, and it grows with the rate: at 40.8 percent the mark-up gives 4.22 percent where the correct answer is 5.07 percent, an understatement of 0.85 points. Anyone using the shortcut systematically undervalues tax-free bonds, and undervalues them most for the high-rate investors those bonds exist to attract.

Marginal, not average — and why the difference is not small

The rate in the denominator is the rate on the next unit of income, because the interest you are deciding about sits on top of everything you already earn. Where investment income is taxed on a scale this splits cleanly from the average. Take a saver with $60,000 of investment income facing a five-band savings scale of 19 percent on the first 6,000, 21 percent to 50,000 and 23 percent to 200,000: the tax is 1,140 + 9,240 + 2,300 = $12,680, an average rate of 21.13 percent, while the next dollar is taxed at 23 percent. Using the average gives a taxable-equivalent yield of 3.80 percent; the marginal gives 3.90. The saver who uses the average concludes a 3.85 percent taxable bond beats the tax-free one when in fact it loses.

Two refinements matter once you accept that. If the position is large enough for the interest to straddle a band, there is no single marginal rate at all: the correct answer converts each slice at its own rate and adds them, and using the top rate for the whole position overstates the tax-free bond. And where the tax is a single flat rate on investment income rather than a scale, the marginal and average rates coincide by construction and the distinction disappears — which is precisely why readers in flat-rate countries find the warning puzzling and readers in scale countries find it indispensable. Establish which of the two your income sits under before you type a number into the denominator.

The levies people leave out of the denominator

A marginal rate is the sum of everything charged on that next unit of taxable interest — and in most systems the headline income-tax rate is only part of it. In the United States, taxable interest above the statutory income thresholds also attracts the 3.8 percent net investment income tax, while tax-exempt municipal interest is expressly outside net investment income (IRS Topic 559). Ignoring it turns a top marginal rate of 37 into 40.8 percent: the taxable-equivalent yield of a 3.00 percent muni goes from 4.76 to 5.07 percent, and on a $100,000 position that 0.31-point gap is $306 a year of misjudged comparison.

The same pattern repeats with different names. In France the flat levy on investment income is built from an income-tax component of 12.8 percent plus social levies charged on top; a saver who divides by the 12.8 alone gets 3.44 percent where the full combined rate of 30 gives 4.29 — 0.85 points, or $850 a year on a $100,000 position, and by far the largest single error in this article. In Germany the 25 percent Abgeltungsteuer carries a solidarity surcharge of 5.5 percent of the tax, taking the combined rate to 26.375 percent, and church tax where it applies raises it further to roughly 27.8 to 28.0 percent under the formula in EStG § 32d; the taxable-equivalent yield moves from 4.07 to about 4.17 percent. None of these add-ons is large on its own. All of them are larger than the yield differences investors argue about.

What tax-advantaged fixed income actually is, market by market

Almost every English-language explanation of this formula is written about United States municipal bonds, and that framing does not travel. In the US, interest on qualifying state and local government bonds is exempt from federal income tax, an in-state issue is commonly exempt from that state's tax as well, and the exemption is what the formula exists to price. Nowhere else in the five European markets covered here is there a comparable class of retail bond whose coupon is simply untaxed, so an article that presents the muni case as universal is teaching readers the wrong instrument.

France is the market where a genuinely untaxed fixed return exists, and it is not a bond: the regulated passbook accounts — Livret A, LDDS and the LEP — pay interest free of both income tax and social levies, which is what makes their published rate directly comparable to a taxable-equivalent yield. Spain has no exempt retail bond at all; interest and coupons fall into the savings base and are taxed on a scale that runs from 19 percent upward, so the whole question becomes which band the next unit of interest lands in. Portugal taxes interest at a liberatory rate under article 71 of the IRS code, with the retirement-savings plan (PPR) the main vehicle carrying a materially lower rate on its income when it is redeemed inside the statutory conditions — which is a reduced rate, not an exemption, so the two-sided version of the formula in the next section is the one to use.

Germany and Italy diverge again, and in opposite directions. Germany has no tax-exempt bond either; what it has is an allowance, the Sparer-Pauschbetrag, which leaves a fixed amount of investment income a year untaxed and is set at 1,000 euros for a single taxpayer and 2,000 euros for a jointly assessed couple. That is a genuinely different structure, and the one-line formula handles it perfectly precisely because it takes the marginal rate: below the allowance the marginal rate is zero and no conversion is needed at all; above it the rate jumps to the full combined figure. At a 3 percent yield the allowance is used up by roughly 33,000 euros of capital, so most retail savers sit either side of that boundary rather than on one rate. Italy instead runs two substitute-tax rates — a reduced one on government securities and on issuers on the approved list, a higher one on most other financial income — so the Italian version of this question is almost never exempt-versus-taxed and almost always low-taxed-versus-high-taxed. Every one of these rates changes by legislation; check the current figure at the source before you compute anything, and if you cannot establish your own market's treatment with certainty, treat the comparison as unresolved rather than assuming someone else's rules.

The two-sided form, for when both sides are taxed

Once you accept that most markets offer reduced rates rather than exemptions, the classic formula turns out to be a special case of a more useful one. Comparing a yield Ya taxed at ta with a yield Yb taxed at tb means comparing Ya × (1 − ta) with Yb × (1 − tb). Expressed as an equivalent yield in the second instrument's tax frame: equivalent = Ya × (1 − ta) ÷ (1 − tb). Set ta = 0 and it collapses to the one-line version. Set tb = 0 and it tells you what a taxable bond is worth to a tax-free investor, which is simply its after-tax yield.

Two worked cases show why it earns its place. An Italian government bond yielding 3.00 percent under the reduced 12.5 percent rate is worth 3 × 0.875 ÷ 0.74 = 3.55 percent to someone comparing it against a corporate bond taxed at 26 percent — a corporate coupon under 3.55 percent loses to the government bond even though it looks higher on the screen. A Portuguese retirement-savings plan whose income is taxed at 8 percent on a qualifying redemption is worth 3 × 0.92 ÷ 0.72 = 3.83 percent against interest taxed at 28 percent. In both cases the conversion factor is a constant — 1.1824 and 1.2778 respectively — so once you have computed it for your own pair of rates you can apply it to any yield without redoing the algebra. And note what the factor does not include: Italy's annual stamp duty on securities accounts, or the lock-up conditions that make a PPR's 8 percent conditional in the first place. Those are separate deductions from the answer, not part of the rate.

Reading it backwards, and what it still cannot tell you

Solve the formula for t instead of Yt and you get the break-even marginal rate: t* = 1 − Yf ÷ Yt. Against a 4.00 percent taxable bond, a 3.00 percent tax-free yield breaks even at 1 − 0.75 = 25 percent — above that rate the tax-free bond wins, below it the taxable one does, and at exactly 25 percent the choice is a coin toss on tax grounds alone. The same tax-free yield breaks even at 14.29 percent against a 3.50 percent taxable bond and at 33.33 percent against a 4.50 percent one. This is the more useful direction in practice, because you usually know both yields and are trying to work out whether your own rate is high enough for the tax-free instrument to be for you.

What the formula cannot do is compare anything but tax. It says nothing about credit risk — a tax-free issuer can default, and the exemption survives nothing — nothing about duration, so a longer tax-free bond and a shorter taxable one are not the same instrument however the after-tax yields line up, and nothing about liquidity or call features. It also misses second-order tax effects that only show up in a full return: in the United States, interest on certain private-activity bonds is a preference item for the alternative minimum tax, and tax-exempt interest still counts toward the combined income that determines how much of a retiree's Social Security is taxable, so a tax-free coupon can raise a tax bill it is not itself taxed on. In every country there is some means-tested threshold or allowance the extra income touches. Treat the taxable-equivalent yield as the first screen, not the decision.

Frequently asked questions

What is the formula for taxable-equivalent yield?

Taxable-equivalent yield = tax-free yield ÷ (1 − marginal tax rate). A 3.00 percent tax-free yield at a 30 percent marginal rate is 3 ÷ 0.70 = 4.29 percent, meaning a taxable bond has to pay 4.29 percent to leave you the same money. Do not multiply by (1 + rate) instead: that gives 3.90 percent and understates the tax-free bond by 0.39 points, and the error widens as the rate rises.

Which rate do I put in — my average tax rate or my marginal one?

The marginal one, always: the interest you are deciding about is added on top of your existing income, so it is taxed at the rate that applies to the next unit, not the blended rate across everything you earn. On a five-band savings scale, a saver with 60,000 of investment income pays 12,680 in tax — an average of 21.13 percent — while the next unit is taxed at 23. The average produces a taxable-equivalent yield of 3.80 percent against the correct 3.90, which is enough to flip the decision on a 3.85 percent taxable bond. Where investment income is taxed at a single flat rate, the two coincide and the distinction is moot.

At what tax rate does a taxable bond beat a tax-free one?

Below the break-even rate t* = 1 − (tax-free yield ÷ taxable yield). A 3.00 percent tax-free yield against a 4.00 percent taxable one breaks even at 1 − 0.75 = 25 percent: under a 25 percent marginal rate the taxable bond leaves more, over it the tax-free one does. Against a 3.50 percent taxable bond the break-even falls to 14.29 percent, and against a 4.50 percent one it rises to 33.33 percent. This is why tax-free issues are priced for high-rate investors — for everyone else the taxable bond is usually the better buy on tax grounds alone.

Does every country have tax-free bonds like US municipal bonds?

No, and assuming so is the commonest way this calculation goes wrong outside the United States. The US case — state and local government bonds whose interest is exempt from federal income tax, often from the issuing state's tax too — has no direct equivalent in France, Spain, Portugal, Germany or Italy. What those markets have instead is a mix of exempt savings accounts, tax-free allowances, reduced substitute-tax rates and conditional retirement wrappers, each with its own conditions. Identify the actual vehicle in your own market, find its actual rate at the tax authority's own publication, and if you cannot establish it with certainty, do not fill in the denominator from a foreign example.

How do I compare two bonds that are both taxed, at different rates?

Use the two-sided form: equivalent yield = Ya × (1 − ta) ÷ (1 − tb), which restates the first bond's yield inside the second bond's tax frame. A 3.00 percent government bond taxed at 12.5 percent is worth 3 × 0.875 ÷ 0.74 = 3.55 percent against a corporate bond taxed at 26 percent, so any corporate coupon under 3.55 percent loses despite the bigger headline number. A retirement plan whose income is taxed at 8 percent is worth 3 × 0.92 ÷ 0.72 = 3.83 percent against interest taxed at 28 percent. The conversion factor for a given pair of rates is a constant, so compute it once and reuse it. Setting the first rate to zero recovers the classic formula.

Can a tax-free bond still raise my tax bill?

Yes, in more places than people expect, and the yield formula sees none of it. In the United States, tax-exempt interest is still counted in the combined income that determines how much of a retiree's Social Security benefit is taxable, and interest on certain private-activity bonds is a preference item for the alternative minimum tax. More generally, in every country some allowance, credit or means-tested benefit is calculated on an income measure that includes exempt income, so an exempt coupon can push you over a threshold that costs more than the exemption saved. The taxable-equivalent yield answers one narrow question about coupons and rates; whether the position changes your overall tax position is a separate calculation that needs your full return.