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.
The nominal annual interest rate is $11.15\%$ compounded monthly.
The effective annual rate (EAR) of debt is:
$$
EAR_{debt} = \left(1 + \frac{0.1115}{12}\right)^{12} - 1
$$
Calculate:
$$
EAR_{debt} = \left(1 + 0.0092917\right)^{12} - 1 = (1.0092917)^{12} - 1
$$
Using approximation:
$$
(1.0092917)^{12} \approx e^{12 \times 0.00925} = e^{0.111} \approx 1.1174
$$
So,
$$
EAR_{debt} \approx 1.1174 - 1 = 0.1174 = 11.74\%
$$
Since the debt interest is not tax deductible, the after-tax cost of debt is:
$$
Cost_{debt, after-tax} = EAR_{debt} = 11.74\%
$$
5. **Calculate the after-tax return on investment:**
Let $r$ be the nominal annual investment rate (compounded annually for simplicity).
The after-tax return on investment is:
$$
Return_{investment, after-tax} = r \times (1 - 0.24) = 0.76r
$$
6. **Set break-even condition:**
$$
Return_{investment, after-tax} = Cost_{debt, after-tax}
$$
$$
0.76r = 0.1174
$$
7. **Solve for $r$:**
$$
r = \frac{0.1174}{0.76} = 0.1545 = 15.45\%
$$
**Answer:**
The break-even investment rate is approximately **15.45\%** annually.
This means Sami's investment must earn at least 15.45\% annually after taxes to be as beneficial as paying down her debt.