A nontaxable bond pays interest at 5 percent a year. What interest rate would a taxable bond need to pay to yield the same after-tax income as the non-taxable bond? The tax rate is 28 percent
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 = the value of taxable bond.
Letp = the interest rate of the taxable bond.
Interest on the nontaxable bond = 0.05v
Interest on the taxable bond pv
Tax on taxable interest = 0.28pv
After-tax income = taxable interest — =pv — 0.28pv
=(1 — 0.28)pv after factoring out pv after subtracting
= 0.72pv tax on taxable interest
To satisfy the condition that the interest on nontaxable bond be the same as the after tax income on the taxable bond:
0.72pv = 0.05v
0.72p = 0.05 after diving out v
p = 0.05 0.72 after dividing both sides by 0.72
p = 0.069444... after doing the division
p = 0.69444 * 100% converting the result to a percent
p = 6.94% after rounding percent to 2 places
(Notice that the solution does not depend upon the value of the bond.)
Check for $10,000 bond using the rounded value for the taxable interest rate:
Nontaxable interest = 0.05 * 10000 = $500
Taxable interest — 0.0694 * 10000 = $694
Tax on $694 = 0.28 * 694 = $194
After-tax interest = 694 — 194 = $500
(The after-tax interest is same as non-tax interest above.)
Check using a formula:
You may discover that the following formula applies:
p = r I (1 — 1)
where p is the taxable interest rate, r is the nontaxable interest rate, and t is the
tax rate. The formula works when all the are expressed as decimals.
Substituting the values for this problem into the formula gives: p = 0.05 / (1 — 0.28)
p = 0.05 / 0.72
p = 0.069444...
p = 0.06944 * 100% (expressed as percent)
p = 6.94% (rounded to 2 places)
(This checks with the values computed using the other two methods.)
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