Lesson Notes By Weeks and Term v5 - Grade 12
Finance: annuities and long-term planning – Week 6 focus
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Finance: annuities and long-term planning – Week 6 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: 6
Theme: General lesson support
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This week, we delve into the crucial topic of annuities and long-term financial planning. Understanding annuities is vital for making informed decisions about your future financial security. Many South Africans rely on annuities to provide a regular income stream during retirement, or to fund long-term savings goals. This section helps you understand how they work, how to calculate their values, and how to compare different annuity options. Failing to plan for long-term financial needs can lead to significant challenges later in life, including financial insecurity in retirement or inability to afford essential expenses.
What is an Annuity? An annuity is a series of payments made at regular intervals over a fixed period. Think of it as a consistent flow of money, either flowing into an investment (to grow it) or flowing out of an investment (as income).
Types of Annuities: Ordinary Annuity: Payments are made at the end of each period. This is the most common type and the focus of this lesson.
Annuity Due: Payments are made at the beginning of each period. We will not be covering this type explicitly but understanding the difference is important.
Deferred Annuity: Payments start at some point in the future.
Immediate Annuity: Payments start immediately. Future Value of an Annuity (Ordinary Annuity): The future value (FV) is the total amount you'll have at the end of the annuity's term if you consistently make payments and earn interest on those payments.
The formula is: FV = P * [((1 + i)^n - 1) / i] Where: FV = Future Value P = Periodic Payment (the amount you contribute each time) i = Interest rate per period (annual interest rate divided by the number of periods per year) n = Number of periods (number of years multiplied by the number of periods per year)
Example 1: Saving for a Car Sipho wants to save up R80,000 for a used car in 3 years. He can deposit R1,800 at the end of each month into an account that pays 7.5% interest per year, compounded monthly. Will he reach his goal? P = R1,800 i = 7.5% / 12 = 0.075 / 12 = 0.00625 (monthly interest rate) n = 3 years 12 months/year = 36 months FV = 1800 * [((1 + 0.00625)^36 - 1) / 0.00625] FV = 1800 * [((1.00625)^36 - 1) / 0.00625] FV = 1800 * [(1.25024 - 1) / 0.00625] FV = 1800 * [0.25024 / 0.00625] FV = 1800 * 40.0384 FV = R72,069.12 Conclusion: Sipho will not reach his goal of R80,
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0. He will have R72,069.12 after 3 years. He needs to either save more each month, find an account with a higher interest rate, or save for a longer period. Present Value of an Annuity (Ordinary Annuity): The present value (PV) is how much a stream of future payments is worth today, considering the time value of money. It tells you how much you need to invest now to receive a specific series of payments in the future.
The formula is: PV = P * [(1 - (1 + i)^-n) / i] Where: PV = Present Value P = Periodic Payment (the amount you will receive each time) i = Interest rate per period (annual interest rate divided by the number of periods per year) n = Number of periods (number of years multiplied by the number of periods per year)
Example 2: Retirement Planning Thandi wants to receive R15,000 per month for 20 years after she retires. She expects to earn an interest rate of 6% per year, compounded monthly, on her retirement investments. How much does she need to have saved at the time of retirement to fund this annuity? P = R15,000 i = 6% / 12 = 0.06 / 12 = 0.005 (monthly interest rate) n = 20 years 12 months/year = 240 months PV = 15000 * [(1 - (1 + 0.005)^-240) / 0.005] PV = 15000 * [(1 - (1.005)^-240) / 0.005] PV = 15000 * [(1 - 0.30207) / 0.005] PV = 15000 * [0.69793 / 0.005] PV = 15000 * 139.586 PV = R2,093,790.00 Conclusion: Thandi needs to have saved R2,093,790.00 by the time she retires to receive R15,000 per month for 20 years. This highlights the importance of starting to save early!
Example 3: Loan Repayments Ayanda takes out a loan of R50,000 to start a small business. The loan has an interest rate of 12% per year, compounded monthly, and she needs to repay it over 5 years with equal monthly payments. Calculate her monthly payment. This is a present value problem. We know the present value (the loan amount) and we want to find the payment (P).
We rearrange the present value formula: P = PV / [(1 - (1 + i)^-n) / i] PV = R50,000 i = 12% / 12 = 0.01 (monthly interest rate) n = 5 years 12 months/year = 60 months P = 50000 / [(1 - (1 + 0.01)^-60) / 0.01] P = 50000 / [(1 - (1.01)^-60) / 0.01] P = 50000 / [(1 - 0.55045) / 0.01] P = 50000 / [0.44955 / 0.01] P = 50000 / 44.955 P = R1,112.22 Conclusion: Ayanda needs to pay R1,112.22 per month for 5 years to repay the R50,000 loan.
Important Considerations: Inflation: The purchasing power of money decreases over time due to inflation. When planning for long-term annuities, it's essential to factor in inflation to ensure your future income keeps pace with rising prices. The interest rate used in calculations should be a real interest rate, meaning the nominal interest rate minus the inflation rate.
Fees: Annuities often come with fees, which can reduce the overall return. Be sure to compare the fees of different annuity products before making a decision.
Investment Risk: Some annuities are linked to investments (like shares or unit trusts). The value of these annuities can fluctuate depending on the performance of the underlying investments. These are beyond the scope of this curriculum but should be noted as a factor.