A bond pays interest at 7 percent a year. The interest is taxed at a rate of 28 percent. What interest rate would a nontaxable bond need to pay to yield the same after-tax income as the taxable bond?
Its easier to work this problem with the rates in decimal form. So at the beginning divide the given rates by 100. At the end, convert the result to a percent by multiplying by 100.
Let v = value of taxable bond.
Taxable interest = 0.07v
Tax on interest = 0.28*0.07v
After-tax income — taxable interest — tax on interest
= (0.07v) — (0.28 * 0.07v)
= (1 — 0.28) * (0.07v)
— 0.72 * 0.07v after factoring out 0.07v after doing the subtraction
Letp = interest rate on nontaxable bond (expressed as a decimal).
Interest on the non-taxable bond = pv
Let interest on non-taxable bond be the same as the after-tax income on the taxable bond. Then 0.72 * 0.07v = pv
p = 0.72 * 0.07 after dividing out v from both sides
p = 0.0504 after doing the multiplication
p = 0.0504 * 100% converting the result to a percent
p = 5.04% The tax rate of the nontaxable bond
Check for $10,000 bond
Nontaxable interest = 0.0504* 10000 = $504
Taxable interest = 0.07 *10000 = $700
Tax on $700 = 0.28 *700 = $196
After-tax interest = 700 — 196 = $504
(The condition that the after-tax interest is same as non-tax interest is satisfied.)
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