Lesson Notes By Weeks and Term v5 - Grade 12

Lesson Video

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Performance objectives

• Calculate the future value of an annuity.
• Understand the impact of interest rates and deposits.
• Compare different annuity options.
• Apply annuity calculations to real-life scenarios like retirement.

Lesson summary

Annuities and long-term financial planning are crucial for securing your financial future. In South Africa, understanding these concepts allows you to make informed decisions about investments, retirement, and large purchases. Whether you're saving for a car, a house, your children's education, or your retirement, understanding how annuities work and how to plan for the long term is essential. This week, we will explore the basics of annuities, focusing on calculating future values and understanding their implications for financial security.

Lesson notes

2. 1. What is an Annuity? An annuity is a series of equal payments made at regular intervals. It's essentially a structured savings plan where you contribute a fixed amount of money periodically, and that money grows over time due to interest accumulation. Annuities are commonly used for long-term savings goals such as retirement.

There are two main types of annuities: Ordinary Annuity: Payments are made at the end of each period (e.g., at the end of each month). This is the most common type.

Annuity Due: Payments are made at the beginning of each period (e.g., at the start of each month). We will primarily focus on ordinary annuities. 2.

2. Future Value of an Ordinary Annuity The future value (FV) of an ordinary annuity is the total value of all the payments and accumulated interest at the end of the annuity's term. The formula for calculating the future value of an ordinary annuity is: FV = P * [((1 + i)^n - 1) / i] Where: FV = Future Value P = Periodic Payment (the amount you deposit each period) 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) Important

Note: The interest rate (i) and the number of periods (n) must be expressed in the same time units (e.g., monthly, quarterly, or annually). 2.

3. Understanding the Formula Let's break down the formula: `(1 + i)^n`: This calculates the growth factor of a single initial investment over 'n' periods at an interest rate of 'i' per period. `(1 + i)^n - 1`: This subtracts the initial investment of 1, leaving only the accumulated interest from a single initial investment of 1. `((1 + i)^n - 1) / i`: This divides the total accumulated interest by the interest rate per period, essentially summing the future values of each individual payment made over the annuity's term. `P [((1 + i)^n - 1) / i]`: This multiplies the sum of future values by the periodic payment 'P', giving the total future value of the annuity. 2.

4. Worked Examples Example 1: Saving for a Car Sipho wants to save R5,000 each year for 5 years to buy a used car. The bank offers an annuity with an interest rate of 8% per year, compounded annually. What will be the future value of Sipho's annuity after 5 years? P = R5,000 i = 8% = 0.08 n = 5 years FV = 5000 * [((1 + 0.08)^5 - 1) / 0.08] FV = 5000 * [((1.08)^5 - 1) / 0.08] FV = 5000 * [(1.4693 - 1) / 0.08] FV = 5000 * [0.4693 / 0.08] FV = 5000 * 5.8666 FV = R29,333.00 Therefore, Sipho will have approximately R29,333.00 after 5 years.

Example 2: Monthly Savings for Retirement Thandi invests R800 per month into an annuity that pays an interest rate of 6% per year, compounded monthly. She plans to do this for 20 years. What will be the future value of her annuity? P = R800 i = 6% per year / 12 months = 0.06 / 12 = 0.005 per month n = 20 years 12 months = 240 months FV = 800 * [((1 + 0.005)^240 - 1) / 0.005] FV = 800 * [((1.005)^240 - 1) / 0.005] FV = 800 * [(3.3102 - 1) / 0.005] FV = 800 * [2.3102 / 0.005] FV = 800 * 462.04 FV = R369,632.00 Therefore, Thandi will have approximately R369,632.00 after 20 years.

Example 3: Comparing Interest Rates John invests R12000 yearly for 10 years. Bank A offers him 9% interest rate compounded annually while Bank B offers 8.5% interest rate compounded annually. Which bank offers a better investment in the long run?

Bank A: P = R12000 i = 9% = 0.09 n = 10 years FV = 12000 * [((1 + 0.09)^10 - 1) / 0.09] FV = 12000 * [((1.09)^10 - 1) / 0.09] FV = 12000 * [(2.3674 - 1) / 0.09] FV = 12000 * [1.3674/ 0.09] FV = 12000 * 15.1930 FV = R182,316.00 Bank B: P = R12000 i = 8.5% = 0.085 n = 10 years FV = 12000 * [((1 + 0.085)^10 - 1) / 0.085] FV = 12000 * [((1.085)^10 - 1) / 0.085] FV = 12000 * [(2.2609 - 1) / 0.085] FV = 12000 * [1.2609/ 0.085] FV = 12000 * 14.8342 FV = R178,010.40 Therefore, investing into Bank A gives a better returns in the long run. Guided Practice (With Solutions)

Question 1: Maria saves R300 per month into an annuity that earns 7% interest per year, compounded monthly. How much will she have after 3 years?

Solution: P = R300 i = 7% per year / 12 months = 0.07 / 12 = 0.005833 per month n = 3 years 12 months = 36 months FV = 300 * [((1 + 0.005833)^36 - 1) / 0.005833] FV = 300 * [((1.005833)^36 - 1) / 0.005833] FV = 300 * [(1.2333 - 1) / 0.005833] FV = 300 * [0.2333 / 0.005833] FV = 300 * 39.996 FV = R11,998.80 Maria will have approximately R11,998.80 after 3 years.

Question 2: David wants to have R100,000 in 8 years. He plans to deposit a fixed amount yearly into an annuity with a 10% annual interest rate, compounded annually. How much should he deposit each year? (Hint: This is a bit more advanced – you'll need to rearrange the formula) While we have not explicitly covered rearranging the formula in this lesson, encourage students to approximate by trial and error using the original formula.