Lesson Notes By Weeks and Term v5 - Grade 12
Finance: annuities and long-term planning – Week 8 focus
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Finance: annuities and long-term planning – Week 8 focus
Download the Lessonotes Mobile South Africa app for faster lesson access on Android and iPhone.
Subject: Mathematical Literacy
Class: Grade 12
Term: 1st Term
Week: 8
Theme: General lesson support
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This week, we delve into the critical topic of annuities and long-term financial planning. This is arguably one of the most important sections you will study in Mathematical Literacy because it provides you with the tools to make informed decisions about your financial future. South Africa presents unique economic challenges and opportunities, making sound financial planning even more vital. Understanding annuities – which are essentially a series of regular payments, either made to you (e.g., from a retirement fund) or by you (e.g., to pay off a loan) – is crucial for managing debt, saving for retirement, planning for education, and other long-term financial goals.
2.1 What is an Annuity (in our context)? In its simplest form for our purposes, an annuity is a series of equal payments made or received at regular intervals (e.g., monthly, quarterly, annually). We’re not going to be using complex annuity formulas. Instead, we'll build our understanding using concepts you already know: simple and compound interest. We will examine what happens when you save a regular amount, and how that amount grows over time, or how regular loan payments reduce the amount owed. Types of Annuities (simplified for our learning): Ordinary Annuity: Payments are made at the end of each period. This is the most common type and the one we’ll primarily focus on. For example, you pay your rent at the end of the month. Or you save a portion of your salary into a retirement fund at the end of each month.
Annuity Due: Payments are made at the beginning of each period. 2.2 Simple Interest and Savings: While compound interest is more realistic for long-term savings, let's understand the basics with simple interest first. Suppose you save a fixed amount each year. The total interest earned is calculated only on the principal and not the interest accrued.
Example 1 (Simple Interest): You save R1000 at the end of each year for 3 years. The simple interest rate is 5% per year. Let's calculate the total amount you'll have after 3 years.
Year 1: You save R
1
0
0
0. No interest earned yet.
Year 2: You save another R
1
0
0
0. Your principal is now R
2
0
0
0. Interest earned this year: 5% of R2000 = R
1
0
0. You save your R
1
0
0
0. Year 3: You save another R
1
0
0
0. Your principal is now R
3
0
0
0. Interest earned this year: 5% of R3000 = R
1
5
0. Total saved after 3 years: R3000 Total interest earned: R100 + R150 = R250 Total amount: R3000 + R250 = R3250 2.3 Compound Interest and Savings: Compound interest means that the interest earned in each period is added to the principal, and the next interest calculation is based on this new, larger principal. This "interest on interest" effect makes compound interest much more powerful for long-term savings. We will use repeated applications of the compound interest formula here rather than a single complex annuity formula.
Example 2 (Compound Interest): You save R1000 at the end of each year for 3 years. The compound interest rate is 5% per year, compounded annually.
Year 1: You save R
1
0
0
0. No interest earned yet.
Year 2: You save another R
1
0
0
0. The R1000 from year 1 earns interest: R1000 0.05 = R
5
0. You have saved R1000 + R1000 + R50 = R2050 Year 3: You save another R
1
0
0
0. Now the R2050 from Year 2 earns interest: R2050 0.05 = R102.
5
0. You have R2050 + R1000 + R102.50 = R3152.50 Total saved after 3 years: R3000 Total interest earned: R50 + R102.50 = R152.50 Total amount: R3000 + R152.50 = R3152.50 As you can see, simple interest results in higher interest earned in this particular example because of the way the calculations are done, but compound interest can result in more interest in the long run as all funds accumulate. Example 3 (Compound Interest - More realistic): Thando decides to save R500 per month into a savings account that pays 6% interest per year, compounded monthly. How much will she have after 1 year? Here, the interest rate per month is 6%/12 = 0.5% = 0.
0
0
5. Month 1: Saves R500 Month 2: Saves R
5
0
0. Month 1 amount earns interest: R500 0.005 = R2.
5
0. Total = R500 + R500 + R2.50 = R1002.50 Month 3: Saves R
5
0
0. Month 2 amount earns interest: R1002.50 0.005 = R5.
0
1. Total = R1002.50 + R500 + R5.01 = R1507.51 We will not calculate all 12 months manually, but it shows the principle. After 12 months, Thando will have approximately R6167.78 2.4 Understanding the Present Value (Simplified): The present value is the current worth of a future sum of money or stream of payments, given a specified rate of return. In the context of annuities, we might want to know how much money we need now to fund a series of future withdrawals. We can calculate this through repeated application of the present value formula.
Example 4 (Present Value): You want to withdraw R1000 at the end of each year for 2 years from a savings account. The interest rate is 5% per year, compounded annually. How much do you need to have in the account now?
Year 2 Withdrawal: To withdraw R1000 at the end of year 2, you need to have R1000/(1+0.05)^2 = R907.03 present today.
Year 1 Withdrawal: To withdraw R1000 at the end of year 1, you need to have R1000/(1+0.05)^1 = R952.38 present today. Total needed = R907.03 + R952.38 = R1859.41 Therefore, you need R1859.41 in the account now to be able to withdraw R1000 at the end of each of the next 2 years. 2.5 The Impact of Inflation: Inflation erodes the purchasing power of money over time. This is very important to consider in long-term planning. An item that costs R100 today might cost R110 in a year if the inflation rate is 10%. So, when planning for the future, you must factor in the expected inflation rate to ensure your savings keep pace with rising prices.