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Strand 3 — Number

Financial Maths: Compound Interest, Annuities & Amortisation

5th Year · 6th Year (Leaving Cert)

  • ✓By the end of this lesson students will be able to calculate future and present values using compound interest and depreciation formulae.
  • ✓By the end of this lesson students will be able to determine the Annual Equivalent Rate (AER) for various compounding periods.
  • ✓By the end of this lesson students will be able to calculate the future value and present value of annuities.
  • ✓By the end of this lesson students will be able to apply financial maths principles to solve problems involving mortgages and amortisation (HL).
  • ✓By the end of this lesson students will be able to critically analyse and interpret financial information presented in real-world contexts.

Key concepts

Compound Interest

Compound interest is interest calculated on the initial principal and also on the accumulated interest from previous periods. It means that your money grows faster because the interest itself earns interest. The interest rate 'i' must be the rate per compounding period, and 't' must be the number of compounding periods.

F = P(1 + i)^t
Annual Equivalent Rate (AER)

The Annual Equivalent Rate (AER) is the actual annual rate of interest earned or paid on an investment or loan, taking into account the effect of compounding. It allows for a fair comparison of different financial products with varying compounding frequencies.

(1 + i_annual) = (1 + i_period)^n
Depreciation

Depreciation is the decrease in value of an asset over time. It is often calculated using a compound depreciation formula, where the value decreases by a fixed percentage each period. The rate 'i' is the depreciation rate per period.

F = P(1 - i)^t
Present Value (P)

Present Value (P) is the current value of a future sum of money or stream of cash flows, given a specified rate of return. It answers the question: 'How much money would I need to invest today to have a certain amount in the future?'

P = F(1 + i)^-t
Future Value (F)

Future Value (F) is the value of an asset or cash at a specified date in the future, based on its present value and a specified rate of growth. It answers the question: 'How much will my investment be worth in the future?'

F = P(1 + i)^t
Annuities

An annuity is a series of equal payments made at regular intervals over a period of time. Examples include regular savings contributions, loan repayments, or pension payments. In Leaving Cert maths, we typically deal with ordinary annuities, where payments are made at the end of each period.

Future Value of an Annuity (FVA)

The Future Value of an Annuity (FVA) is the total value of a series of regular payments at a specific point in the future, assuming each payment earns compound interest. This is useful for calculating the total amount accumulated in a savings plan.

FVA = A * [((1 + i)^t - 1) / i]
Present Value of an Annuity (PVA)

The Present Value of an Annuity (PVA) is the lump-sum amount today that is equivalent to a series of future regular payments, discounted at a specific interest rate. This is fundamental for calculating loan amounts, such as mortgages.

PVA = A * [(1 - (1 + i)^-t) / i]
Mortgages & Amortisation (HL)

A mortgage is a long-term loan used to purchase property, typically repaid by equal regular payments over many years. The loan amount is the Present Value of an Annuity (PVA), and the regular payment is 'A'. Amortisation is the process of paying off a debt over time through regular payments. Each payment consists of both interest on the outstanding balance and a portion that reduces the principal. An amortisation schedule shows how each payment is allocated and the remaining balance.

A = PVA * [i / (1 - (1 + i)^-t)]

Key facts to remember

  • 1F = P(1 + i)^t is the fundamental compound interest formula, where 'i' is the rate per period and 't' is the number of periods.
  • 2For depreciation, use F = P(1 - i)^t, where 'i' is the depreciation rate per period.
  • 3The Annual Equivalent Rate (AER) accounts for compounding and allows for fair comparison of interest rates: (1 + i_annual) = (1 + i_period)^n.
  • 4Annuities involve a series of equal payments at regular intervals. Payments are typically assumed to be at the end of each period (ordinary annuity).
  • 5Future Value of an Annuity (FVA) calculates the total accumulated amount of a series of payments.
  • 6Present Value of an Annuity (PVA) calculates the lump sum equivalent today of a series of future payments.
  • 7Mortgage repayments are calculated using the Present Value of an Annuity formula, where the loan amount is the PVA.
  • 8The remaining balance on a loan is the Present Value of the *remaining* payments.

Worked examples

Example 1

A sum of €5,000 is invested for 3 years at an annual compound interest rate of 4.5%, compounded quarterly. Calculate the future value of the investment.

IIdentify the given values: Principal (P) = €5,000, Annual interest rate = 4.5%, Time (t) = 3 years, Compounding frequency = quarterly.
IICalculate the interest rate per compounding period (i): Annual rate = 0.045. Compounded quarterly means 4 times a year. So, i = 0.045 / 4 = 0.01125.
IIICalculate the total number of compounding periods (t): 3 years * 4 periods/year = 12 periods.
IVApply the compound interest formula: F = P(1 + i)^t.
VSubstitute the values: F = 5000(1 + 0.01125)^12.
VICalculate the future value: F = 5000(1.01125)^12 = 5000 * 1.142567... = 5712.835... .

Answer

The future value of the investment is €5,712.84 (to 2 decimal places).

Always ensure the interest rate 'i' and the number of periods 't' match the compounding frequency.

Example 2

A car was bought for €28,000. Its value depreciates at an annual rate of 12%. Calculate the value of the car after 5 years. Also, determine the Annual Equivalent Rate (AER) if an investment offers a nominal annual rate of 3.8% compounded monthly.

IPart 1: Car Depreciation
IIIdentify values for depreciation: P = €28,000, i = 0.12 (annual depreciation rate), t = 5 years.
IIIApply the depreciation formula: F = P(1 - i)^t.
IVSubstitute values: F = 28000(1 - 0.12)^5 = 28000(0.88)^5.
VCalculate: F = 28000 * 0.527731968 = 14776.495... .
VIPart 2: AER Calculation
VIIIdentify values for AER: Nominal annual rate = 3.8% = 0.038. Compounded monthly, so n = 12.
VIIICalculate the monthly interest rate (i_period): i_period = 0.038 / 12.
9Apply the AER formula: (1 + i_annual) = (1 + i_period)^n.
10Substitute values: (1 + i_annual) = (1 + 0.038/12)^12.
11Calculate: (1 + i_annual) = (1.0031666...)^12 = 1.038706... .
12Solve for i_annual: i_annual = 1.038706... - 1 = 0.038706... .
13Convert to percentage: AER = 3.87% (to 2 decimal places).

Answer

The value of the car after 5 years is €14,776.50. The Annual Equivalent Rate (AER) is 3.87%.

Depreciation uses (1 - i) while appreciation uses (1 + i). Remember to convert AER to a percentage at the end.

Example 3

(HL) A couple takes out a mortgage of €300,000 over 25 years at an annual interest rate of 4.2%, compounded monthly. Calculate their monthly repayment. After 10 years, how much do they still owe on the mortgage?

IPart 1: Calculate Monthly Repayment (A)
IIIdentify values: PVA = €300,000, Annual rate = 4.2%, Time = 25 years, Compounding = monthly.
IIICalculate monthly interest rate (i): i = 0.042 / 12 = 0.0035.
IVCalculate total number of payments (t): t = 25 years * 12 months/year = 300 payments.
VUse the formula for monthly repayment (derived from PVA of an annuity): A = PVA * [i / (1 - (1 + i)^-t)].
VISubstitute values: A = 300000 * [0.0035 / (1 - (1 + 0.0035)^-300)].
VIICalculate the denominator: (1.0035)^-300 = 0.347719... . So, (1 - 0.347719...) = 0.652280... .
VIIICalculate A: A = 300000 * [0.0035 / 0.652280...] = 300000 * 0.0053658... = 1609.74... .
9Part 2: Amount Owed After 10 Years
10After 10 years, 10 * 12 = 120 payments have been made.
11The remaining balance is the Present Value of the *remaining* payments.
12Number of remaining payments = Total payments - Payments made = 300 - 120 = 180 payments.
13Use the PVA formula with the monthly repayment (A) and the remaining number of payments (t_remaining): PVA_remaining = A * [(1 - (1 + i)^-t_remaining) / i].
14Substitute values: PVA_remaining = 1609.74 * [(1 - (1 + 0.0035)^-180) / 0.0035].
15Calculate (1.0035)^-180 = 0.536906... . So, (1 - 0.536906...) = 0.463093... .
16Calculate PVA_remaining: PVA_remaining = 1609.74 * [0.463093... / 0.0035] = 1609.74 * 132.312... = 212984.72... .

Answer

The monthly repayment is €1,609.74. After 10 years, the couple still owes €212,984.72 on the mortgage.

For HL questions, understanding how to use the annuity formulae to find either the payment (A) or the present/future value is key. The remaining balance is always the present value of the *remaining* payments.

Common mistakes

  • ✗Not matching the interest rate 'i' and the number of periods 't' to the compounding frequency (e.g., using an annual rate for monthly compounding).
  • ✗Confusing Present Value and Future Value, or using the wrong formula for the context.
  • ✗Incorrectly calculating the number of periods for annuities or loans (e.g., using years instead of months for monthly payments).
  • ✗Rounding intermediate calculations too early, leading to inaccuracies in the final answer.
  • ✗Forgetting to convert interest rates from percentages to decimals (e.g., 4.5% to 0.045) before using them in formulae.

Exam tips

  • ★Read the question carefully to identify whether it's a compound interest, depreciation, annuity (PVA or FVA), or AER problem.
  • ★Always write down the formula you are using and clearly state the values for P, F, i, t, A, etc., before substituting.
  • ★Pay close attention to the compounding frequency (e.g., annually, semi-annually, quarterly, monthly) and adjust 'i' and 't' accordingly.
  • ★For HL questions involving mortgages and amortisation, remember that the loan amount is the Present Value of an Annuity, and the remaining balance is the Present Value of the *remaining* payments.
  • ★Use your calculator effectively, especially for powers and complex fractions. Avoid premature rounding.

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