Lesson Notes By Weeks and Term v5 - Grade 12

Lesson Video

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Performance objectives

• Calculate the future value (FV) of an annuity.
• Calculate the present value (PV) of an annuity.
• Apply annuity calculations to real-life scenarios like retirement and savings.
• Differentiate between ordinary annuities and annuities due.
• Analyse and compare different investment options.

Lesson summary

This week, we delve into annuities and long-term financial planning. This is a crucial area of Mathematical Literacy because it equips you with the tools to make informed decisions about your future financial security. Understanding annuities allows you to plan for retirement, save for your children's education, or invest for other long-term goals. In a South African context, where many people face financial insecurity in retirement, mastering these concepts is vital for creating a stable and prosperous future. Many South Africans rely on social grants or family support in retirement because they did not adequately plan and save during their working lives.

Lesson notes

What is an Annuity? An annuity is a series of payments made at regular intervals. These payments can be made into an investment account (for accumulation) or received from an investment account (as income).

There are two main types of annuities: Ordinary Annuity: Payments are made at the end of each period.

Annuity Due: Payments are made at the beginning of each period. The key difference lies in when the payment is made. This seemingly small difference significantly impacts the final amount, especially over long periods. Future Value of an Annuity The future value (FV) of an annuity is the total amount you will have at the end of the annuity period, considering the regular payments and the accumulated interest.

Ordinary Annuity Future Value Formula: FV = P * [((1 + i)^n - 1) / i] Where: FV = Future Value P = Periodic Payment (the regular payment amount) i = Interest rate per period (annual interest rate divided by the number of compounding periods per year) n = Number of periods (number of years multiplied by the number of compounding periods per year)

Annuity Due Future Value Formula: FV = P [((1 + i)^n - 1) / i] (1 + i) Notice the extra `(1 + i)` at the end. This accounts for the extra compounding period because the payments are made at the beginning.

Example 1: Ordinary Annuity Future Value Sipho invests R500 per month into an ordinary annuity that earns 8% interest per year, compounded monthly. He invests for 10 years. Calculate the future value of his annuity. P = R500 i = 0.08 / 12 = 0.00666667 (monthly interest rate) n = 10 12 = 120 (number of months) FV = 500 * [((1 + 0.00666667)^120 - 1) / 0.00666667] FV = 500 * [(2.219006 - 1) / 0.00666667] FV = 500 * [1.219006 / 0.00666667] FV = 500 * 182.8509 FV = R91,425.45 Therefore, Sipho will have approximately R91,425.45 after 10 years.

Example 2: Annuity Due Future Value Using the same information as in Example 1, but assuming Sipho invests at the beginning of each month (annuity due), calculate the future value. P = R500 i = 0.08 / 12 = 0.00666667 (monthly interest rate) n = 10 12 = 120 (number of months) FV = 500 [((1 + 0.00666667)^120 - 1) / 0.00666667] (1 + 0.00666667) FV = 500 182.8509 (1.00666667) FV = R91,425.45 * 1.00666667 FV = R92,035.64 Notice that the annuity due results in a higher future value (R92,035.64) than the ordinary annuity (R91,425.45). This is because each payment earns interest for an extra month. Present Value of an Annuity The present value (PV) of an annuity is the lump sum amount you need to invest today to receive a stream of payments in the future.

Ordinary Annuity Present Value Formula: PV = P * [(1 - (1 + i)^-n) / i] Where: PV = Present Value P = Periodic Payment i = Interest rate per period n = Number of periods Annuity Due Present Value Formula: PV = P [(1 - (1 + i)^-n) / i] (1 + i)

Example 3: Ordinary Annuity Present Value Thandi wants to receive R2,000 per month for 20 years after she retires. How much money does she need to invest today in an ordinary annuity that earns 9% interest per year, compounded monthly? P = R2,000 i = 0.09 / 12 = 0.0075 (monthly interest rate) n = 20 12 = 240 (number of months) PV = 2000 * [(1 - (1 + 0.0075)^-240) / 0.0075] PV = 2000 * [(1 - (0.166824)) / 0.0075] PV = 2000 * [(0.833176) / 0.0075] PV = 2000 * 111.0898 PV = R222,179.60 Thandi needs to invest approximately R222,179.60 today to receive R2,000 per month for 20 years.

Example 4: Annuity Due Present Value Using the same information as in Example 3, but assuming Thandi receives the R2,000 at the beginning of each month (annuity due), calculate the present value. P = R2,000 i = 0.09 / 12 = 0.0075 (monthly interest rate) n = 20 12 = 240 (number of months) PV = 2000 [(1 - (1 + 0.0075)^-240) / 0.0075] (1 + 0.0075) PV = 2000 111.0898 (1.0075) PV = R222,179.60 * 1.0075 PV = R223,846.95 Thandi needs to invest approximately R223,846.95 today if she receives the payments at the beginning of each month.

Long-Term Planning Considerations: Inflation: Always factor in inflation. The real value of your money decreases over time due to inflation. You may need to adjust your savings or investment amounts to account for this.

Taxes: Be aware of the tax implications of your investments and annuity payments.

Risk: Different investments carry different levels of risk. Consider your risk tolerance when making investment decisions.

Investment Fees: Research and understand any fees associated with your investments. These fees can eat into your returns. Guided Practice (With Solutions)

Question 1: John invests R1,000 at the end of each quarter (ordinary annuity) into an account that earns 12% per year, compounded quarterly. How much will he have after 5 years?

Solution: P = R1,000 i = 0.12 / 4 = 0.03 n = 5 4 = 20 FV = 1000 * [((1 + 0.03)^20 - 1) / 0.03] FV = 1000 * [(1.806111 - 1) / 0.03] FV = 1000 * [0.806111 / 0.03] FV = 1000 * 26.8704 FV = R26,870.40 John will have R26,870.40 after 5 years.