Question 14.10: A $12,000 investment with no salvage value will return annua...
A \$12,000 investment with no salvage value will return annual benefits for six years. Assume straight- line depreciation and a 46\% income tax rate. Solve for both before- and after-tax rates of return for two situations:
1. No inflation: the annual benefits are constant at \$2,918 per year.
2. Inflation equal to 5\%: the benefits from the investment increase at this same rate, so that they continue to be the equivalent of \$2,918 in real dollars based in Year 0.
| year | annual benefits for both situations (real dollars) | No inflation, Actual Dollars Received |
5\% Inflation
Factor* |
5\% Inflation , Actual Dollars Received |
| 1 | \$2,918 | \$2,918 | (1.05)^{-1} | \$3,064 |
| 2 | 2,918 | 2,918 | (1.05)^{-2} | 3,217 |
| 3 | 2,918 | 2,918 | (1.05)^{-3} | 3,378 |
| 4 | 2,918 | 2,918 | (1.05)^{-4} | 3,547 |
| 5 | 2,918 | 2,918 | (1.05)^{-5} | 3,724 |
| 6 | 2,918 | 2,918 | (1.05)^{-6} | 3,910 |
∗May be read from the 5\% compound interest table as (F/P,5\%,n).
Before-Tax Rate of Return
Since both situations (no inflation and 5\% inflation) have an annual benefit, stated in real dollars, of \$2,918, they have the same before-tax rate of return.
PW of cost = PW of benefit
12,000 = 2,918(P/A, i, 6) ( P/A, i, 6) =\frac{12,000}{2,918}=4.11
From compound interest tables: before-tax rate of return equals 12\%.
After-Tax Rate of Return, No Inflation
| Year | Before-Tax Cash Flow |
Straight-Line Depreciation |
Taxable
Income |
46\% Income Taxes | Actual Dollars, and Real Dollars,After-Tax Cash Flow |
| 0 | -\$12,000 | -\$12,000 | |||
| 1-6 | +2,918 | \$2,000 | \$918 | -\$422 | 2,496 |
PW of cost = PW of benefit
12,000 = 2,4\%(P/A, i, 6) ( P/A, i, 6) = ( P/A, i, 6) =\frac{12,000}{2,496}=4.81
From compound interest tables: after-tax rate of return equals 6.7\%.
After-Tax Rate of Return, 5\% Inflation
| Year | Before-Tax
Cash Flow |
Straight-Line
Depreciation |
Taxable Income | 46\% Income Taxes |
Actual Dollars, After-Tax Cash Flow |
| 0 | -\$12,000 | -\$12,000 | |||
| 1 | +3,064 | \$2,000 | \$1,064 | -\$489 | +2,575 |
| 2 | +3,217 | 2,000 | 1,217 | -560 | +2,657 |
| 3 | +3,378 | 2,000 | 1,378 | -634 | +2,744 |
| 4 | +3,547 | 2,000 | 1,547 | -712 | +2,835 |
| 5 | +3,724 | 2,000 | 1,724 | -793 | +2,931 |
| 6 | +3,910 | 2,000 | 1,910 | -879 | +3,031 |
Converting to Year-0-Based Dollars and Solving for Rate of Return
| Year | Actual Dollars, After-Tax Cash Flow |
Conversion Factor |
Real Dollars, After-Tax Cash Flow |
Present Worth at 4\% |
Present Worth at 5\% |
| 0 | -\$12,000 | -\$12,000 | -\$12,000 | -\$12,000 | |
| 1 | +2,575 | ×(1.05)^{-1} = | +2,452 | +2,358 | +2,335 |
| 2 | +2,657 | ×(1.05)^{-2} = | +2,410 | +2,228 | +2,335 |
| 3 | +2,744 | ×(1.05)^{-3} = | +2,370 | +2,107 | +2,186 |
| 4 | +2,835 | ×(1.05)^{-4} = | +2,332 | +1,993 | +2,047 |
| 5 | +2,931 | ×(1.05)^{-5} = | +2,297 | +1,888 | +1,919 |
| 6 | +3,031 | ×(1.05)^{-6} = | +2,262 | \frac{+1,788}{}
+362 |
\frac{+1,688}{}
-25 |
Linear interpolation between 4\% and 5\%:
After-tax rate of return = 4\% + 1\% × [362/(362 + 25)] = 4.9\%