1. **Problem Statement:** The treasurer wants to change the compounding from semi-annually to daily but keep the same effective interest rate. The current nominal rate is 10.1% compounded semi-annually. We need to find the new nominal rate compounded daily that yields the same effective rate. 2. **Formula for Effective Annual Rate (EAR):** $$EAR = \left(1 + \frac{r}{n}\right)^n - 1$$ where $r$ is the nominal rate and $n$ is the number of compounding periods per year. 3. **Calculate the current EAR:** Given $r = 0.101$ (10.1%) and $n = 2$ (semi-annual), $$EAR = \left(1 + \frac{0.101}{2}\right)^2 - 1 = (1 + 0.0505)^2 - 1 = 1.0505^2 - 1$$ $$= 1.10302525 - 1 = 0.103025$$ So, the effective annual rate is approximately 10.3025%. 4. **Find the new nominal rate $r_{new}$ compounded daily:** We want the same EAR with $n = 365$ (daily compounding), so $$0.103025 = \left(1 + \frac{r_{new}}{365}\right)^{365} - 1$$ Add 1 to both sides: $$1.103025 = \left(1 + \frac{r_{new}}{365}\right)^{365}$$ Take the 365th root: $$\left(1.103025\right)^{\frac{1}{365}} = 1 + \frac{r_{new}}{365}$$ Calculate the left side: $$\left(1.103025\right)^{\frac{1}{365}} \approx 1.000269$$ So, $$1.000269 = 1 + \frac{r_{new}}{365}$$ Subtract 1: $$0.000269 = \frac{r_{new}}{365}$$ Multiply both sides by 365: $$r_{new} = 0.000269 \times 365 = 0.098185$$ 5. **Convert to percentage and round:** $$r_{new} = 9.8185\%$$ **Final answer:** The new nominal rate compounded daily should be approximately **9.8185%** to maintain the same effective interest rate.