# Problem Restatement - **Original Loan:** 8 annual payments of \$7,000 each, paid at end of years 1-8. - **Alternative Offer:** Skip the next 7 payments and make a single payment at end of year 8. - **Interest Rate:** 5% per year. - **Question:** What is the amount of the single payment at the end of year 8 so that the bank is indifferent? --- # Step 1: Calculate the Present Value (PV) of the 8 Payments The present value of an ordinary annuity (payments at year-end) is: \[ PV = PMT \times \frac{1 - (1 + r)^{-n}}{r} \] Where: - \( PMT = \$7,000 \) - \( r = 0.05 \) - \( n = 8 \) \[ PV = 7,000 \times \frac{1 - (1 + 0.05)^{-8}}{0.05} \] Calculate \((1 + 0.05)^{-8}\): \[ (1.05)^{-8} \approx 0.67684 \] \[ PV = 7,000 \times \frac{1 - 0.67684}{0.05} \] \[ PV = 7,000 \times \frac{0.32316}{0.05} \] \[ PV = 7,000 \times 6.4632 \approx 45,242.40 \] --- # Step 2: What Lump Sum Payment at Year 8 is Equivalent? Let \( X \) be the required lump sum in year 8. Its present value today is: \[ PV = \frac{X}{(1.05)^8} \] Set equal to the PV of the 8 payments: \[ \frac{X}{(1.05)^8} = 45,242.40 \] \[ X = 45,242.40 \times (1.05)^8 \] Calculate \((1.05)^8\): \[ (1.05)^8 \approx 1.47746 \] \[ X = 45,242.40 \times 1.47746 \approx 66,887.72 \] --- # Step 3: Final Answer **The bank would require a single payment of approximately \$66,888 at the end of 8 years.** --- ## Summary Table | Year | Original Payment | Alternative Payment | |------|-----------------|--------------------| | 1 | \$7,000 | \$0 | | 2 | \$7,000 | \$0 | | 3 | \$7,000 | \$0 | | 4 | \$7,000 | \$0 | | 5 | \$7,000 | \$0 | | 6 | \$7,000 | \$0 | | 7 | \$7,000 | \$0 | | 8 | \$7,000 | \$66,888 | ---