When PpY and CpY are different Problem set B Worked solutions

Write down N, PpY and CpY for each of these.

1. (a) A loan repaid once a year for 5 years, interest compounded quarterly

   • N counts payments: 5 × 1 = 5
   • PpY = 1 (one payment a year), CpY = 4 (quarterly)

2. (b) Monthly savings for 4 years, interest compounded half-yearly

   • N counts payments: 4 × 12 = 48
   • PpY = 12 (monthly), CpY = 2 (half-yearly)

3. (c) Withdrawals every six months for 12 years, interest compounded quarterly

   • N counts payments: 12 × 2 = 24
   • PpY = 2 (every six months), CpY = 4 (quarterly)

A school borrows €48 000 at 5.4% per year, compounded monthly, and repays it in equal instalments at the end of each year for 7 years. Find the annual repayment.

• N = 7, PpY = 1, CpY = 12
• N = 7, I(%) = 5.4, PV = 48 000, Pmt = ?, FV = 0, PpY = 1, CpY = 12
• Solve for Pmt: −8 457.11…
• Each annual repayment is €8 457.11.

€250 is saved at the end of every month for 8 years in an account paying 4.4% per year, compounded quarterly. Find the final amount.

• N = 8 × 12 = 96, PpY = 12, CpY = 4
• N = 96, I(%) = 4.4, PV = 0, Pmt = −250, FV = ?, PpY = 12, CpY = 4
• Solve for FV: 28 684.88…
• The final amount is €28 684.88.

£120 000 earns 3.6% per year, compounded monthly. Equal withdrawals are made at the end of each quarter for 12 years, emptying the account. Find each withdrawal.

• N = 12 × 4 = 48, PpY = 4, CpY = 12
• N = 48, I(%) = 3.6, PV = −120 000, Pmt = ?, FV = 0, PpY = 4, CpY = 12
• Solve for Pmt: 3 091.71…
• Each withdrawal is £3 091.71.

A loan of $33 000 at 6% per year, compounded monthly, runs for 4 years. Compare the total repaid if it is repaid (a) monthly, (b) once a year. Explain the difference.

• (a) N = 48, I(%) = 6, PV = 33 000, Pmt = ?, FV = 0, PpY = 12, CpY = 12
• Solve for Pmt: −775.01… Total: 48 × 775.0060 = $37 200.29.
• (b) N = 4, I(%) = 6, PV = 33 000, Pmt = ?, FV = 0, PpY = 1, CpY = 12
• Solve for Pmt: −9 560.13… Total: 4 × 9 560.1343 = $38 240.54.
• Difference: $38 240.54 − $37 200.29 = $1 040.25. Paying yearly keeps the balance higher for longer, so more interest is charged.

For a loan repaid every quarter for 7 years, with interest compounded monthly, state N, PpY and CpY. Explain in a sentence why N would not change if the interest were compounded yearly instead.

• N counts payments: 7 × 4 = 28. PpY = 4 (four payments a year). CpY = 12 (interest added monthly).
• CpY is not in the calculation of N.
• If interest were compounded yearly, CpY would become 1 but N would stay 28.