1. **Problem statement:** John deposits 1000 into a fund with three different interest regimes over 10 years. We need to find the amount in the fund at the end of 10 years. 2. **Given:** - Initial principal $P=1000$ - Years 0-3: nominal annual interest rate 4% convertible quarterly - Years 3-6: constant annual force of interest 5% - Years 6-10: nominal annual discount rate 6% convertible semiannually 3. **Step 1: Calculate amount after first 3 years (quarterly compounding)** - Nominal rate $i^{(4)}=0.04$ per year compounded quarterly means quarterly rate $i_q=\frac{0.04}{4}=0.01$ - Number of quarters in 3 years: $n=3 \times 4=12$ - Amount after 3 years: $$A_3 = P(1+i_q)^n = 1000(1+0.01)^{12}$$ Calculate: $$A_3 = 1000(1.01)^{12} = 1000 \times 1.126825 = 1126.83$$ 4. **Step 2: Calculate amount after next 3 years (force of interest 5%)** - Force of interest $\delta=0.05$ per year - Amount after 3 more years (from year 3 to 6): $$A_6 = A_3 e^{\delta \times 3} = 1126.83 \times e^{0.05 \times 3} = 1126.83 \times e^{0.15}$$ Calculate: $$e^{0.15} \approx 1.161834$$ $$A_6 = 1126.83 \times 1.161834 = 1309.54$$ 5. **Step 3: Calculate amount after last 4 years (nominal discount rate 6% convertible semiannually)** - Nominal discount rate $d^{(2)}=0.06$ per year compounded semiannually - Semiannual discount rate $d_s=\frac{0.06}{2}=0.03$ - Convert discount rate to effective interest rate per half year: $$i_s = \frac{d_s}{1 - d_s} = \frac{0.03}{1 - 0.03} = \frac{0.03}{0.97} \approx 0.0309278$$ - Number of half years in 4 years: $n=4 \times 2=8$ - Amount after 4 years: $$A_{10} = A_6 (1 + i_s)^n = 1309.54 (1 + 0.0309278)^8$$ Calculate: $$(1.0309278)^8 \approx 1.27449$$ $$A_{10} = 1309.54 \times 1.27449 = 1668.17$$ **Final answer:** The amount in the fund at the end of 10 years is approximately **1668.17**. ---