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Compound Interest: Level Up! Compounding More Than Once a Year
Compound Interest: Level Up! Compounding More Than Once a Year
Buckle up! We're about to take compound interest to the next level. In the previous lesson, we explored compounding annually, but what happens when interest is calculated more frequently – monthly, quarterly, or even daily?
Get ready to:
- Calculate compound interest, maturity value, and present value when compounding occurs multiple times per year.
- Understand key terms like "frequency of conversion" and "nominal rate."
- Solve complex financial problems involving more frequent compounding periods.
By the end of this lesson, you'll be equipped to navigate the intricacies of compound interest across various compounding frequencies, giving you a deeper understanding of its true power. Let's dive into the world of more frequent compounding!
Activity: The Impact of Compounding Frequency
Activity: The Impact of Compounding Frequency
Scenario: You are considering investing ₱10,000 for 3 years in an account that offers 6% annual interest. You have three options:
Option A: Interest compounded annually.
Option B: Interest compounded semi-annually.
Option C: Interest compounded quarterly.
Calculate the future value of your investment for each option after 3 years.
Analysis: Exploring Compounding Frequency
Analysis: Exploring Compounding Frequency
Discuss how increasing the compounding frequency (from annually to semi-annually to quarterly) affects the final amount. Explain that more frequent compounding leads to a higher return because interest is earned on interest more often.
This lesson explores the concept of compounding interest more frequently than once a year, which can significantly impact your investment growth. Let's dive into the details!
Conversion Periods and Examples
Conversion Periods and Examples
Here's a breakdown of common conversion periods:
Examples:
2% compounded annually: r = 0.02, m = 1, i = 0.02
2% compounded semi-annually: r = 0.02, m = 2, i = 0.01
2% compounded quarterly: r = 0.02, m = 4, i = 0.005
2% compounded monthly: r = 0.02, m = 12, i = 0.00166666
Maturity Value Formula
Maturity Value Formula
The formula for calculating maturity value (F) when compounding occurs 'm' times a year is:
Example 1: Quarterly Compounding
Example 1: Quarterly Compounding
Problem: Find the maturity value if P10,000 is deposited at 2% compounded quarterly for 5 years.
Solution:
P = P10,000
r = 0.02
m = 4 (quarterly compounding)
t = 5 years
F = P(1 + r/m)^(mt) = 10,000(1 + 0.02/4)^(4*5) = P11,048.96
Therefore, the maturity value after 5 years is P11,048.96.
Example 2: Monthly Compounding
Example 2: Monthly Compounding
Problem: Find the maturity value and interest if P10,000 is deposited at 2% compounded monthly for 5 years.
Solution:
P = P10,000
r = 0.02
m = 12 (monthly compounding)
t = 5 years
F = P(1 + r/m)^(mt) = 10,000(1 + 0.02/12)^(12*5) = P11,051.16
Interest (I) = F - P = P11,051.16 - P10,000 = P1,051.16
Therefore, the maturity value is P11,051.16, and the interest earned is P1,051.16.
Example 3: Loan Repayment
Example 3: Loan Repayment
Problem: Cris borrows P50,000 and promises to pay the principal and interest at 12% compounded monthly. How much must he repay after 6 years?
Solution:
P = P50,000
r = 0.12
m = 12 (monthly compounding)
t = 6 years
F = P(1 + r/m)^(mt) = 50,000(1 + 0.12/12)^(12*6) = P101,639.68
Therefore, Cris must repay P101,639.68 after 6 years.
Compounding frequency can make a big difference! Here's a video lesson to understand more about how it impacts your returns.
Think you've mastered compound interest? This assessment steps up the challenge, exploring scenarios where interest is compounded multiple times per year. Get ready to apply your knowledge to more complex calculations.
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 1 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.)
Basic Understanding
Explain the meaning of "frequency of conversion" and how it impacts the growth of an investment.
If a bank offers a nominal interest rate of 6%, what would be the interest rate per conversion period if interest is compounded: a) semi-annually, b) quarterly, c) monthly?
Why does an investment generally yield a higher return when interest is compounded more frequently, even if the nominal interest rate remains the same?
Applying the Formula
Find the maturity value of an investment of ₱25,000 at 4% compounded semi-annually for 10 years.
A savings account offers an interest rate of 1.5% compounded monthly. If you deposit ₱5,000, how much will you have in the account after 7 years?
Sarah wants to have ₱100,000 in 8 years for a down payment on a house. How much should she invest now in a certificate of deposit that pays 3% interest compounded quarterly?
Analyzing Scenarios
You are offered two investment options:
Option A: 5% compounded annually
Option B: 4.8% compounded monthly
Which option would you choose for a 5-year investment and why? Show your calculations to support your answer.
A loan of ₱100,000 is taken out with an interest rate of 8% compounded quarterly. The loan is to be repaid in 5 years.
a) Calculate the total amount to be repaid at the end of 5 years.
b) Construct an amortization schedule for the loan, showing the interest and principal payments for each quarter.
Real-World Application
Research and compare the interest rates and compounding frequencies offered by different banks in your area for savings accounts and certificates of deposit. Which bank offers the most attractive rates for your savings goals?
Choose a real-life scenario where understanding compound interest with multiple compounding periods is essential (e.g., car loan, mortgage, investment planning). Explain the importance of this concept in making informed financial decisions.
Conquered compounding frequencies? Time to find the missing pieces in Lesson 5!
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