Pre-tax equivalent

A tax-free bond paying 8 percent and a bank deposit paying 9 percent cannot be compared as they stand. One of those numbers survives the tax return and the other does not.

There are two ways to put them on the same footing. Bring both down to a post-tax figure, or bring both up to a pre-tax figure.

The pre-tax equivalent is the second route. It asks what a taxable investment would have to pay, before tax, to leave the investor with the same money as the tax-free one.

The workbook works exactly this case. For an investor in the 20 percent bracket, a tax-free 8 percent grosses up to 10 percent. The bank deposit pays 9. So the tax-free bond wins.

The number is personal. It depends on the investor's tax bracket, so the same bond has a different pre-tax equivalent for different people.

The two routes, on the workbook's own pair. A tax-free bond paying 8 percent against a taxable bank deposit paying 9 percent, for an investor in the 20 percent bracket:

Both routes reach the same conclusion, which is the point: the answer does not depend on which platform you choose, only on consistency. In the workbook's words, the bank deposit should pay 10 percent pre-tax to be equivalent to the tax-free bond paying 8 percent.

Why tax-free bonds exist at lower coupons. The workbook explains the market logic. Tax-free bonds are issued by public sector units in the infrastructure sector. Because the interest is not taxed in the investor's hands, they are issued at a lower interest rate than taxable bonds of similar credit quality and tenor. The pre-tax equivalent is the tool that reveals whether the discount has been overdone.

The cut-off rises with the bracket. Chapter 7 tabulates the same idea as a decision rule for a 7 percent tax-free bond:

Above that cut-off, the taxable investment gives a higher post-tax return and the investor may prefer it. Below it, the tax-free bond is better. The higher the bracket, the harder a taxable alternative has to work.

The related use. Chapter 7 gives a second application of tax-free income: it can stop a retiree being tipped into the next tax bracket. If taxable income after all deductions stands at Rs 10,50,000 and that pushes the retiree from the 20 percent bracket into the 30 percent bracket, tax-free interest can supply the last Rs 50,000 of income needed while leaving them in the 20 percent bracket.

Pre-tax equivalent = tax-free rate / (1 − tax rate)

The mirror operation, the post-tax route, is taxable rate x (1 − tax rate).

Worked on the workbook's figures: 8 / (1 − 0.2) = 10 percent, and 9 x (1 − 0.2) = 7.2 percent. On Chapter 7's figures: 7 / (1 − 0.30) = 10 percent for a 30 percent bracket, 7 / (1 − 0.20) = 8.75 percent for a 20 percent bracket.

Mrs Fernandes, 61, is in the 30 percent tax bracket and has Rs 15,00,000 to place for retirement income. Two offers, similar tenor and credit quality.

• A tax-free bond of a public sector infrastructure company paying 7 percent.
• A taxable corporate deposit paying 9.5 percent.

Gross up the tax-free bond.

Pre-tax equivalent = 7 / (1 - 0.30) = 10 percent

The deposit pays 9.5 percent, which is below the 10 percent cut-off. So the bond wins.

Check it the other way and confirm.

Deposit, post-tax = 9.5 x (1 - 0.30) = 6.65 percent
Bond, post-tax    = 7 percent

In rupees, on Rs 15,00,000 for one year

The deposit's headline rate is 2.5 percentage points higher and it leaves her Rs 5,250 a year poorer.

Now change one thing: her son, in the 5 percent bracket, has the same choice. His cut-off for a 7 percent tax-free bond is only about 7.37 percent, and the deposit pays 9.5 percent.

Deposit, post-tax for him = 9.5 x (1 - 0.05) = 9.025 percent

Same two products, opposite answers, because the pre-tax equivalent belongs to the investor and not to the bond.

Chapter 5, section 5.6.1, sets up the tax-free bond against the taxable bank deposit and works both the post-tax and the Pre-Tax or Nominal return routes, ending with the sentence that the bank deposit would need 10 percent pre-tax to match a tax-free 8 percent. Chapter 7, section 7.5, turns the same arithmetic into a bracket-by-bracket cut-off table for a 7 percent tax-free bond — 7.37 percent, 8.75 percent and 10 percent.

Expect a direct computation: given a tax-free rate and a tax bracket, find the pre-tax equivalent. Expect also the comparison question where the higher headline rate is the wrong answer, and a bracket question — for which investor does the tax-free bond stop being attractive.

• Divide by (1 − tax rate) to gross up; multiply by (1 − tax rate) to net down. Using the wrong one of the two reverses the comparison.
• Compare like with like. Either both post-tax or both pre-tax — never a tax-free rate against a taxable rate as printed.
• The cut-off belongs to the investor, not to the bond. A 7 percent tax-free bond has a cut-off of 7.37 percent for a 5 percent bracket and 10 percent for a 30 percent bracket. The workbook adds that the effective return depends on the investor's tax status and the rates applicable to them.
• A tax-free bond is not a high-yield bond. It is issued at a lower coupon precisely because the interest is untaxed, so a low headline rate is expected rather than a warning sign.
• This is the inverse of a tax-adjusted return, not the same thing. The tax-adjusted return takes a taxable rate down to post-tax; the pre-tax equivalent takes a tax-free rate up to a taxable one.
• Similar risk and tenor is part of the comparison. The workbook says so explicitly. Grossing up a tax-free yield does not license a comparison with a riskier or much longer instrument.
• Tax rates and slabs in the workbook differ between the old and new regimes. Use the bracket the question gives you.