Question: Sami received a generous birthday gift from her parents and wants to use the money to make a smart financial move by either paying down her debt or investing it. She knows her debt is compounding interest for sure, but wonders if an investment could help her make more money than her debt is costing her, and therefore could be strategic. She wants to calculate how well her investment would have to perform to be the better choice than paying down her debt.
Her line of credit has an interest rate of 11.15% compounded monthly. The marginal tax rate on her regular income is 38.8% and it is not tax deductible. In contrast, the marginal tax rate she would pay on investment income is only 24%. Based on these factors, what would her break-even rate be (the rate her investment would have to earn to be at least as good a move as paying down the debt)?
Based on these factors, what would her break-even rate be (the point when the benefit from either choice would be equal) ? [Blank-1]
1. **State the problem:**
Sami wants to find the break-even investment rate that makes investing as beneficial as paying down her debt.
2. **Identify given data:**
- Debt interest rate (annual nominal): $11.15\%$ compounded monthly
- Marginal tax rate on regular income (debt interest): $38.8\%$ (not tax deductible)
- Marginal tax rate on investment income: $24\%$
3. **Understand the problem:**
- Paying down debt saves interest at the after-tax cost of debt.
- Investing earns interest at the after-tax return on investment.
- Break-even means after-tax cost of debt = after-tax return on investment.
4. **Calculate the effective after-tax cost of debt:**
Since the debt interest is not tax deductible, the full interest rate applies before tax, but Sami pays tax on her income separately, so the debt interest is effectively paid on after-tax income.
The effective after-tax cost of debt is:
$$ r_{debt, after-tax} = r_{debt} \times (1 - t_{income}) $$
where
$r_{debt} = 11.15\% = 0.1115$
$t_{income} = 38.8\% = 0.388$
Calculate:
$$ r_{debt, after-tax} = 0.1115 \times (1 - 0.388) = 0.1115 \times 0.612 = 0.0682 = 6.82\% $$
5. **Calculate the break-even investment rate:**
Let $r_{inv}$ be the pre-tax investment rate. The after-tax investment return is:
$$ r_{inv, after-tax} = r_{inv} \times (1 - t_{inv}) $$
where
$t_{inv} = 24\% = 0.24$
At break-even:
$$ r_{inv, after-tax} = r_{debt, after-tax} $$
Substitute:
$$ r_{inv} \times (1 - 0.24) = 0.0682 $$
$$ r_{inv} \times 0.76 = 0.0682 $$
Divide both sides by $0.76$:
$$ r_{inv} = \frac{0.0682}{0.76} $$
Show cancellation:
$$ r_{inv} = \frac{0.0682}{\cancel{0.76}} \times \frac{1}{\cancel{0.76}} = 0.0897 = 8.97\% $$
6. **Interpretation:**
The investment must earn at least $8.97\%$ pre-tax to be as good as paying down the debt with an effective after-tax cost of $6.82\%$.
**Final answer:**
The break-even investment rate is approximately **8.97\%** pre-tax.