When PpY and CpY are different Problem set A Worked solutions

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

1. (a) A loan repaid once a year for 8 years, interest compounded monthly

   • N = 8 × 1 = 8, PpY = 1 (once a year)
   • CpY = 12 (monthly compounding)
   • N = 8, PpY = 1, CpY = 12

2. (b) Monthly savings for 5 years, interest compounded quarterly

   • N = 5 × 12 = 60, PpY = 12 (monthly savings)
   • CpY = 4 (quarterly compounding)
   • N = 60, PpY = 12, CpY = 4

3. (c) Withdrawals every quarter for 10 years, interest compounded monthly

   • N = 10 × 4 = 40, PpY = 4 (quarterly withdrawals)
   • CpY = 12 (monthly compounding)
   • N = 40, PpY = 4, CpY = 12

A business borrows $35 000 at 4.2% per year, compounded monthly, and repays it in equal instalments at the end of each year for 6 years. Find the annual repayment.

• Payments yearly: PpY = 1. Interest monthly: CpY = 12. N = 6 × 1 = 6.
• N = 6, I(%) = 4.2, PV = 35 000, Pmt = ?, FV = 0, PpY = 1, CpY = 12
• Solve for Pmt: −6 738.04…
• The annual repayment is $6 738.04.

£300 is saved at the end of every month for 10 years in an account paying 3.6% per year, compounded quarterly. Find the final amount.

• Payments monthly: PpY = 12. Interest quarterly: CpY = 4. N = 10 × 12 = 120.
• N = 120, I(%) = 3.6, PV = 0, Pmt = −300, FV = ?, PpY = 12, CpY = 4
• Solve for FV: 43 231.36…
• The final amount is £43 231.36.

€90 000 earns 3% per year, compounded monthly. Equal withdrawals are made at the end of each quarter for 15 years, emptying the account. Find each withdrawal.

• Withdrawals quarterly: PpY = 4. Interest monthly: CpY = 12. N = 15 × 4 = 60.
• N = 60, I(%) = 3, PV = −90 000, Pmt = ?, FV = 0, PpY = 4, CpY = 12
• Solve for Pmt: 1 869.24…
• Each withdrawal is €1 869.24.

A loan of £20 000 at 5% per year, compounded monthly, runs for 5 years. Compare the total repaid if it is repaid (a) monthly, (b) once a year. Explain the difference.

• (a) Monthly: N = 5 × 12 = 60, PpY = 12, CpY = 12
• N = 60, I(%) = 5, PV = 20 000, Pmt = ?, FV = 0, PpY = 12, CpY = 12
• Solve for Pmt: −377.42…
• Total repaid = 60 × 377.42… = £22 645.48
• (b) Yearly: N = 5, PpY = 1, CpY = 12
• N = 5, I(%) = 5, PV = 20 000, Pmt = ?, FV = 0, PpY = 1, CpY = 12
• Solve for Pmt: −4 634.34…
• Total repaid = 5 × 4 634.34… = £23 171.72
• Difference = 23 171.72 − 22 645.48 = £526.24 more when repaid yearly.
• Money is repaid later, so the balance stays higher for longer and more interest is charged.

Explain in a sentence each what N, PpY and CpY count, and why changing CpY never changes N.

• N counts payments: N = years × PpY.
• PpY is the number of payments each year.
• CpY is the number of times interest is added each year.
• N counts payments, not interest periods, so changing CpY changes the interest rate per period but never N.