Finance Solver: compound interest and depreciation Problem set B Worked solutions

Write down N for each of these.

1. (a) 7 years, compounded quarterly

   • N counts payments: 7 years × 4 a year
   • N = 28

2. (b) 42 months, compounded monthly

   • Compounded monthly, so one period each month
   • N = 42

3. (c) 9 years, compounded half-yearly

   • N counts payments: 9 years × 2 a year
   • N = 18

4. (d) 4 years, compounded monthly

   • N counts payments: 4 years × 12 a year
   • N = 48

Find the final value.

1. (a) £6500 at 2.8% per year, compounded yearly, for 7 years

   • N = 7, I(%) = 2.8, PV = −6500, Pmt = 0, FV = ?, PpY = 1, CpY = 1
   • Solve for FV: 7 886.15…
   • The investment is worth £7 886.15.

2. (b) $9000 at 3.6% per year, compounded monthly, for 4 years

   • N = 48, I(%) = 3.6, PV = −9000, Pmt = 0, FV = ?, PpY = 12, CpY = 12
   • Solve for FV: 10 391.72…
   • The investment is worth $10 391.72.

How much must be invested now at 3.8% per year, compounded half-yearly, to have €12 000 in 9 years?

• N = 18, I(%) = 3.8, PV = ?, Pmt = 0, FV = 12 000, PpY = 2, CpY = 2
• Solve for PV: −8 551.57…
• The negative sign means money paid in: invest €8 551.57.

$4000 is invested at 4% per year, compounded yearly. After how many complete years is it first worth more than $6000?

• N = ?, I(%) = 4, PV = −4000, Pmt = 0, FV = 6000, PpY = 1, CpY = 1
• Solve for N: 10.34…
• N is between 10 and 11, so it passes $6000 during year 11: 11 years.

£8000 grows to £9500 in 5 years, compounded yearly. Find the annual interest rate.

• N = 5, I(%) = ?, PV = −8000, Pmt = 0, FV = 9500, PpY = 1, CpY = 1
• Solve for I(%): 3.497…
• The annual interest rate is 3.50%.

£7500 is invested at 4.2% per year, compounded monthly. How long will it take for the investment to be worth £9000? Give your answer in complete months, and then in years and months.

• N = ?, I(%) = 4.2, PV = −7500, Pmt = 0, FV = 9000, PpY = 12, CpY = 12
• Solve for N: 52.18… (months)
• Interest is added at the end of each month, so 53 months.
• 53 months = 4 years 5 months.

A used car costs $15 000 and loses 16% of its value each year. (a) Find its value after 4 years. (b) After how many complete years is it first worth less than $5000?

• N = 4, I(%) = −16, PV = −15 000, Pmt = 0, FV = ?, PpY = 1, CpY = 1
• Solve for FV: 7 468.07…
• (a) The car is worth $7 468.07.
• N = ?, I(%) = −16, PV = −15 000, Pmt = 0, FV = 5000, PpY = 1, CpY = 1
• Solve for N: 6.30… (b) N is between 6 and 7: 7 years.

£6500 is invested at 5.4% per year, compounded yearly, for 7 years. Inflation is 2.4% per year. Find the value of the investment in real terms.

• Real rate: \(5.4 - 2.4 = 3\)% per year
• N = 7, I(%) = 3, PV = −6500, Pmt = 0, FV = ?, PpY = 1, CpY = 1
• Solve for FV: 7 994.18…
• In real terms the investment is worth £7 994.18.

A club puts £9500 into a fund for 8 years. Offer 1 pays 3.6% per year, compounded annually. Offer 2 pays 3.5% per year, compounded quarterly. Which offer is worth more after 8 years, and by how much?

• Offer 1: N = 8, I(%) = 3.6, PV = −9500, Pmt = 0, FV = ?, PpY = CpY = 1
• Solve for FV: 12 606.71
• Offer 2: N = 32, I(%) = 3.5, PV = −9500, Pmt = 0, FV = ?, PpY = CpY = 4
• Solve for FV: 12 554.43
• Offer 1 gives more, by £52.27.

A tablet is bought for €800 and loses 25% of its value each year. (a) Find its value after 2 years. (b) After how many complete years is it first worth less than half of its price?

• (a) N = 2, I(%) = −25, PV = −800, Pmt = 0, FV = ?, PpY = 1, CpY = 1
• Solve for FV: 450.00 (depreciation is a negative I(%))
• (b) N = ?, I(%) = −25, PV = −800, Pmt = 0, FV = 400, PpY = 1, CpY = 1
• Solve for N: 2.41…
• After 2 years it is worth €450.00; after 3 years €337.50. So 3 complete years.

£16 000 is invested at 4.2% per year for 12 years. Find how much more it is worth if the interest is compounded quarterly rather than yearly.

• Quarterly: N = 48, I(%) = 4.2, PV = −16 000, Pmt = 0, FV = ?, PpY = 4, CpY = 4
• Solve for FV: 26 415.77…
• Yearly: N = 12, I(%) = 4.2, PV = −16 000, Pmt = 0, FV = ?, PpY = 1, CpY = 1
• Solve for FV: 26 213.96…
• Difference: £26 415.77 − £26 213.96 = £201.81.