Amortisation and annuities Problem set B One question at a time

A loan of €15 000 charges 4.2% per year, compounded monthly, and is repaid monthly over 4 years. Find (a) the monthly payment, (b) the total repaid, (c) the total interest.

• N = 48, I(%) = 4.2, PV = 15 000, Pmt = ?, FV = 0, PpY = 12, CpY = 12
• Solve for Pmt: −340.03…
• (a) Monthly payment: €340.03.
• (b) Total repaid: 48 × 340.0299 = €16 321.44.
• (c) Interest: 16 321.44 − 15 000.00 = €1 321.44.

Dev can afford to repay £520 a month for 15 years. A bank charges 4.6% per year, compounded monthly. Find the largest loan Dev can take.

• N = 180, I(%) = 4.6, PV = ?, Pmt = −520, FV = 0, PpY = 12, CpY = 12
• Solve for PV: 67 522.48…
• The largest loan is £67 522.48.

$220 is saved at the end of every month in an account paying 3.3% per year, compounded monthly. Find the amount after 6 years.

• N = 72, I(%) = 3.3, PV = 0, Pmt = −220, FV = ?, PpY = 12, CpY = 12
• Solve for FV: 17 490.49…
• After 6 years the account holds $17 490.49.

€250 000 is paid into a fund earning 3.5% per year, compounded yearly. Equal amounts are taken out at the end of each year for 25 years, emptying the fund. Find each amount.

• N = 25, I(%) = 3.5, PV = −250 000, Pmt = ?, FV = 0, PpY = 1, CpY = 1
• Solve for Pmt: 15 168.51…
• Each year €15 168.51 can be taken out.

A student wants to have £12 000 in 5 years' time to buy a car. She pays an equal amount into an account at the end of each month. The account pays 3.6% per year, compounded monthly. (a) How much must she pay in each month? (b) How much of the £12 000 is interest?

• (a) N = 60, I(%) = 3.6, PV = 0, Pmt = ?, FV = 12000, PpY = 12, CpY = 12
• Solve for Pmt: −182.84…
• They must pay in £182.84 each month.
• (b) Total paid in = 60 × 182.8388… = £10 970.33
• Interest = 12000 − 10 970.33 = £1029.67

Pedro borrows £7500 at 7.2% per year, compounded monthly, and repays £200 a month. How many payments does he make?

• N = ?, I(%) = 7.2, PV = 7500, Pmt = −200, FV = 0, PpY = 12, CpY = 12
• Solve for N: 42.61…
• N is between 42 and 43, so 43 payments, with the last one smaller.

A fund holds £90 000 and earns 3.0% per year, compounded monthly. £1100 is taken out at the end of each month. For how many months does the money last?

• N = ?, I(%) = 3, PV = −90 000, Pmt = 1100, FV = 0, PpY = 12, CpY = 12
• Solve for N: 91.65…
• N is between 91 and 92, so there are 91 full withdrawals.

Ines borrows €6500 to buy a laptop and a desk. Plan 1 charges 8.4% per year, compounded monthly, over 2 years. Plan 2 charges 7.8% per year, compounded monthly, over 3 years. Both are repaid with equal payments at the end of each month. For each plan find (a) the monthly payment and (b) the total repaid. (c) Which plan costs less overall, and by how much?

• Plan 1: N = 24, I(%) = 8.4, PV = 6500, Pmt = ?, FV = 0, PpY = CpY = 12 → Pmt = −295.16
• Plan 2: N = 36, I(%) = 7.8, PV = 6500, Pmt = ?, FV = 0, PpY = CpY = 12 → Pmt = −203.09
• Totals: 24 × 295.1646… = €7083.95 and 36 × 203.0872… = €7311.14
• Plan 1 costs less overall, by €227.19.

A house loan of $240 000 charges 3.9% per year, compounded monthly, and is repaid monthly over 20 years. (a) Find the monthly payment. (b) Find the amount still owed after 8 years.

• N = 240, I(%) = 3.9, PV = 240 000, Pmt = ?, FV = 0, PpY = 12, CpY = 12
• Solve for Pmt: −1 441.74… (a) $1 441.74 a month.
• Keep Pmt. Set N = 96 (8 years of payments) and solve for FV.
• Solve for FV: −165 587.36…
• (b) The negative sign means money owed: $165 587.36.