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Building Towards the Future: Finding the Future Value of a General Annuity
Building Towards the Future: Finding the Future Value of a General Annuity
In Lesson 1, we explored simple annuities where payments occur at the end of each period. Now, let's broaden our horizons with general annuities! These flexible annuities allow for payments to be made at the beginning or end of each period, making them applicable to a wider range of financial scenarios.
Get ready to:
Define general annuities and understand their relevance in real-world situations.
Master the formula for calculating the future value of a general annuity.
Solve real-world problems involving the future value of general annuities, applying your knowledge to practical financial goals.
By the end of this lesson, you'll be able to confidently calculate how much your investments will grow with general annuities, empowering you to make informed financial decisions for a brighter future. Let's build that financial future!
Activity: Planning for Retirement
Activity: Planning for Retirement
Scenario: You want to retire with a lump sum of ₱1,000,000. You plan to make regular deposits at the beginning of each year into an account that earns 5% interest compounded annually.
How much should you deposit each year for the next 20 years to reach your retirement goal?
Analysis: Understanding the Formula
Analysis: Understanding the Formula
Discuss other real-world examples of using the present value of a general annuity due, such as calculating the loan amount for a mortgage or determining the periodic payments for a lease.
This lesson explores general annuities, where payments and interest compounding don't necessarily align. Let's get started!
Real-World Examples
Real-World Examples
Installment Payments: Monthly car, land, or house payments with annual interest compounding.
Debt Repayment: Semi-annual debt payments with monthly interest compounding.
Formula for Future Value of a General Ordinary Annuity
Formula for Future Value of a General Ordinary Annuity
The formula for calculating the future value (F) of a general ordinary annuity is:
Where:
F = Present Value
R = Regular Payment
i = is the equivalent interest rate per payment interval converted from the interest rate per period
n = Total number of payment periods
r = nominal rate
m1 = payment interval
m2 = the length of the compounding period
t = term of annuity
Example 1
Example 1
Problem: Cris deposits P1,000 monthly into a fund paying 6% compounded quarterly. How much will be in the fund after 15 years?
Solution:
R = P1,000
i = 6% = 0.06
m = 4 (compounded quarterly)
p = 12 (monthly payments)
n = 15 years * 12 payments/year = 180 payments
Using the formula given above, substitute all the values and compute the answer.
Therefore, Cris will have approximately P290,082.51 in the fund after 15 years.
Example 2
Example 2
Problem: A teacher saves P5,000 every six months in a bank paying 0.25% compounded monthly. How much will her savings be after 10 years?
Solution:
R = P5,000
i = 0.25% = 0.0025
m = 12 (compounded monthly)
p = 2 (semi-annual payments)
n = 10 years * 2 payments/year = 20 payments
Using the formula given above, substitute all the values and compute the answer.
Therefore, the teacher will have approximately P101,197.08 in savings after 10 years.
Example 3
Example 3
Problem: ABC Bank pays 2% interest compounded quarterly. How much will be in an account at the end of 5 years with monthly deposits of P3,000?
Solution:
R = P3,000
i = 2% = 0.02
m = 4 (compounded quarterly)
p = 12 (monthly payments)
n = 5 years * 12 payments/year = 60 payments
Using the formula given above, substitute all the values and compute the answer.
Therefore, the account will have approximately P189,126.40 at the end of 5 years.
Curious about how much your savings will grow? Here's a video lesson to understand more about finding the future value of general annuities!
Time to dive into general annuities! This assessment focuses on your understanding of general annuities and their real-world relevance. You'll be challenged to calculate the future value of general annuities in practical scenarios.
Application
Application
Instruction: Use online resources, critical thinking, and the provided information to answer the following questions. Justify your answers with explanations and calculations. Upload your documents on this google drive link: Module 2 Lesson 2 Activity Outputs
(Note: Make sure your file name will be your Section-Year-Surname-Given_Name-Module#-Lesson#-Output#, for example: [GAS11-DelaCruz-Juan-Module1-Lesson1-Output1]. Wrong file name will subject to score deduction.)
Understanding General Annuities:
Explain why the examples provided (monthly car payments with annual interest, semi-annual debt payments with monthly compounding) are considered general annuities. Have students identify the payment frequency and compounding frequency in each.
Come up with two original real-world examples of general annuities. Be sure to specify:
The payment frequency (e.g., monthly, quarterly, annually)
The interest compounding frequency (e.g., monthly, quarterly, annually)
Calculations:
Retirement Planning with a Twist: A young professional wants to start saving for retirement early. They plan to deposit P3,000 at the end of every month into an account that earns an average annual interest rate of 8%, but the interest is compounded daily.
How much will they have saved after 30 years?
Emphasize the importance of recognizing daily compounding even though the problem might initially seem like a standard monthly annuity.
Comparing Investment Strategies: You have P10,000 to invest and are considering two options:
Option A: Invest the lump sum in a high-yield savings account that offers 3% interest compounded annually.
Option B: Invest P500 at the end of every month into a mutual fund that historically yields an average of 7% compounded quarterly.
Which option will result in a higher future value after 10 years? Which option carries more risk?
Analyzing the Impact of Compounding: Using the example of Cris depositing P1,000 monthly into a fund that pays 6% compounded quarterly:
Calculate how much Cris would have if the interest were compounded:
Annually
Semi-annually
Monthly
Create a table to compare the results and discuss how the frequency of compounding affects the future value of the annuity.
Critical Thinking:
You are offered two investment plans:
Plan A: Deposit P2,000 at the beginning of each quarter for 10 years with a 5% annual interest rate compounded quarterly.
Plan B: Deposit P24,000 as a lump sum at the beginning of the 10 years with the same interest rate.
Analyze and compare the two plans. Which one would you choose and why?
Future value of annuities making sense? Time to tackle present value in Lesson 3!
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