1. **Problem statement:** Calculate the present value (PV) of lease payments of 700 at the beginning of every month for 9 years. 2. **Given data:** - Payment per month, $PMT = 700$ - Number of years, $n = 9$ - Total months, $N = 9 \times 12 = 108$ - Annual interest rate, $r = 5.75\% = 0.0575$ --- ### a. Present value with 5.75% compounded annually 3. Convert annual rate to effective monthly rate: $$ i = (1 + r)^{\frac{1}{12}} - 1 = (1 + 0.0575)^{\frac{1}{12}} - 1 $$ Calculate: $$ i = 1.0575^{0.0833333} - 1 \approx 0.00467 $$ 4. Since payments are at the beginning of each month, this is an annuity due. 5. Present value of annuity due formula: $$ PV = PMT \times \frac{1 - (1 + i)^{-N}}{i} \times (1 + i) $$ 6. Substitute values: $$ PV = 700 \times \frac{1 - (1 + 0.00467)^{-108}}{0.00467} \times (1 + 0.00467) $$ 7. Calculate: $$ (1 + 0.00467)^{-108} \approx 0.6007 $$ $$ \frac{1 - 0.6007}{0.00467} = \frac{0.3993}{0.00467} \approx 85.52 $$ $$ PV = 700 \times 85.52 \times 1.00467 \approx 700 \times 85.92 = 60144.00 $$ --- ### b. Present value with 5.75% compounded monthly 8. Monthly interest rate: $$ i = \frac{r}{12} = \frac{0.0575}{12} = 0.00479 $$ 9. Use the same annuity due formula: $$ PV = 700 \times \frac{1 - (1 + 0.00479)^{-108}}{0.00479} \times (1 + 0.00479) $$ 10. Calculate: $$ (1 + 0.00479)^{-108} \approx 0.5990 $$ $$ \frac{1 - 0.5990}{0.00479} = \frac{0.4010}{0.00479} \approx 83.71 $$ $$ PV = 700 \times 83.71 \times 1.00479 \approx 700 \times 84.11 = 58877.00 $$ --- **Final answers:** - a. Present value with annual compounding: $60144.00$ - b. Present value with monthly compounding: $58877.00$