1. **State the problem:** We want to find the annual internal rate of return (IRR) compounded monthly for an investment that costs 12500 today and returns 25550 in 6 years. 2. **Formula used:** The future value formula for compound interest is $$A = P \left(1 + \frac{r}{n}\right)^{nt}$$ where: - $A$ is the amount after time $t$ - $P$ is the principal (initial investment) - $r$ is the annual interest rate (decimal) - $n$ is the number of compounding periods per year - $t$ is the number of years 3. **Given values:** - $P = 12500$ - $A = 25550$ - $n = 12$ (monthly compounding) - $t = 6$ 4. **Substitute values into the formula:** $$25550 = 12500 \left(1 + \frac{r}{12}\right)^{12 \times 6} = 12500 \left(1 + \frac{r}{12}\right)^{72}$$ 5. **Divide both sides by 12500:** $$\frac{25550}{12500} = \left(1 + \frac{r}{12}\right)^{72}$$ $$2.044 = \left(1 + \frac{r}{12}\right)^{72}$$ 6. **Take the 72nd root of both sides:** $$\sqrt[72]{2.044} = 1 + \frac{r}{12}$$ 7. **Calculate the 72nd root:** $$1 + \frac{r}{12} = 2.044^{\frac{1}{72}}$$ Using a calculator: $$2.044^{\frac{1}{72}} \approx 1.0102$$ 8. **Solve for $r$:** $$1 + \frac{r}{12} = 1.0102$$ $$\frac{r}{12} = 1.0102 - 1 = 0.0102$$ $$r = 0.0102 \times 12 = 0.1224$$ 9. **Convert to percentage:** $$r = 0.1224 = 12.24\%$$ **Final answer:** The annual internal rate of return compounded monthly is **12.24%**.