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Unlocking the Mystery: Finding the Periodic Payment of a General Annuity
Unlocking the Mystery: Finding the Periodic Payment of a General Annuity
In previous lessons, we've explored how to calculate the future and present values of general annuities. Now, let's tackle a common financial puzzle: determining the periodic payment required to achieve a specific financial goal. Whether you're saving for retirement, a down payment on a house, or any other financial milestone, this lesson will equip you with the tools to determine the regular contributions needed.
Get ready to:
Manipulate the general annuity formulas to solve for the periodic payment.
Apply these formulas to real-world scenarios, such as calculating monthly mortgage payments or determining how much to save each month to reach a specific target.
Make informed financial decisions by understanding the relationship between periodic payments, interest rates, and time horizons.
By the end of this lesson, you'll be able to confidently calculate the periodic payments needed to achieve your financial aspirations. Let's unlock the mystery of periodic payments and pave the way to financial success!
Activity: The Savings Challenge
Activity: The Savings Challenge
Scenario: Imagine you want to save ₱500,000 in 5 years for a down payment on a house. You've found an investment opportunity that offers a 6% annual interest rate, compounded monthly.
Estimate the monthly payment you would need to make to reach your goal. Don't worry about being precise, just make a reasonable estimate based on your understanding of interest and how it grows over time.
Analysis: Understanding the Saving Dynamics
Analysis: Understanding the Saving Dynamics
How does the interest rate influence the amount you need to save each month? What if the interest rate was higher or lower?
How does the time horizon (5 years in this case) affect the required monthly payment? What if you had a longer or shorter time to save?
What other factors, besides interest rate and time, might influence your ability to reach your savings goal?
This lesson equips you with the tools to calculate periodic payments for general annuities, a valuable skill for financial planning.
Abstraction:
Abstraction:
Formulas for Periodic Payment (R) of a General Ordinary Annuity
Formulas for Periodic Payment (R) of a General Ordinary Annuity
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
Don't have a physical calculator? Use this online Scientific Calculator to solve this problems!
Example 1
Example 1
Problem: Monthly payment of the future value of P50,000 for 1 year with an interest of 10% compounded quarterly
Solution:
F = P50,000
m1 = 12 (monthly payments)
m2 = 4 (compounding payments)
t = 1 year
r = 10% = 0.10
n = 1 year * 12 payments/year = 12 payments
Convert 10% compounded quarterly to it's equivalent interest rate for monthly interval (Using the Equivalent Interest Rate (i) formula):
i = 0.00826484
*Substitute all values using the formula of the Periodic Payment when the Future Value is given above and calculate using scientific calculator.
Therefore, the required monthly payment is approximately P3,980.64.
Example 2
Example 2
Problem: To accumulate a fund of P500,000 in 3 years, how much should Aling Paring deposit in her account every 3 months if it pays an interest of 5.5% compounded annually.
Solution:
F = P500,000
m1 = 4
m2 = 1
t = 3 years
r = 0.55 or 5.5%
n = 3 years * 4 payments/year = 12 payments
Convert 5.5% compounded quarterly to it's equivalent interest rate for monthly interval (Using the Equivalent Interest Rate (i) formula):
i = 0.01347517
*Substitute all values using the formula of the Periodic Payment when the Future Value is given above and calculate using scientific calculator.
Therefore, Aling Paring should deposit approximately P38,668.13 every 3 months.
Example 3
Example 3
Problem: Nadine is the beneficiary of a P1,000,000 insurance policy. Instead of taking the money as a lump sum, she opted to receive a monthly stipend over a period of 10 years. If the insurance policy pays an interest of 5% compounded annually, what will her monthly stipend?
Solution:
P = P1,000,000
m1 = 12
m2 = 1
t = 10 years
r = 0.05
n = 10 years * 12 payments/year = 120 payments
Convert 5.5% compounded quarterly to it's equivalent interest rate for monthly interval (Using the Equivalent Interest Rate (i) formula):
i = 0.01347517
*Substitute all values using the formula of the Periodic Payment when the Present Value is given above and calculate using scientific calculator.
Therefore, Nadine's monthly stipend will be approximately P10,552.35.
Plan your finances with confidence! Here's a video lesson to understand more about calculating periodic payments for general annuities!
Ready to calculate periodic payments? This assessment challenges you to identify and utilize the correct formulas to determine the periodic or regular payments associated with general annuities in practical situations.
Application: Putting it into Practice
Application: Putting it into Practice
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 4 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 Periodic Payments of General Annuities:
Explain the difference between finding the periodic payment when given the future value versus when given the present value of a general annuity. Why are different formulas used?
Describe two real-life scenarios: one where you would need to calculate the periodic payment given the future value, and one where you would need it given the present value. Explain your reasoning.
Calculations:
Saving for a Dream Vacation: You're planning a dream vacation in 5 years and estimate it will cost P250,000. You find an investment opportunity offering 7% interest compounded semi-annually.
How much would you need to deposit at the end of every six months to reach your savings goal?
If you could only afford to deposit P15,000 every six months, how long would it take to reach your goal (adjusting for realistic partial periods)?
Car Loan Options: You want to buy a car worth P800,000. The dealership offers two financing options:
Option A: 3-year loan, 6% annual interest compounded monthly, monthly payments.
Option B: 5-year loan, 7% annual interest compounded quarterly, quarterly payments.
Calculate the periodic payment for each option. Which option best fits your budget and financial goals?
Early Retirement Planning: A person wants to retire early at age 55. They are currently 30 years old and want to have a retirement fund that will provide them with P30,000 per month for 25 years (assuming they'll live to 80). If they can earn an average of 8% annual interest compounded monthly, how much should they deposit into their retirement account at the end of each month?
Critical Thinking:
Using the example of Nadine's insurance policy: If the interest rate were higher, would her monthly stipend increase or decrease? Explain why.
You win a lottery that offers two payout options:
Option A: P5,000,000 lump sum today.
Option B: Monthly payments for 20 years, with a total payout of P10,000,000.
How would you determine the interest rate being used for Option B? At what interest rate would the two options be roughly equivalent?
Periodic payments conquered? Let's explore cash flow and market value in Lesson 5!
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