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Traveling Back in Time: Finding the Present Value of a General Annuity
Traveling Back in Time: Finding the Present Value of a General Annuity
In the previous lesson, we journeyed into the future to determine the future value of general annuities. Now, let's reverse course and travel back in time! Lesson 3 focuses on calculating the present value of general annuities – figuring out how much a stream of future payments is worth in today's money.
Get ready to:
Master the formula for calculating the present value of a general annuity.
Apply this formula to real-world scenarios, such as determining how much to invest today to achieve a specific financial goal in the future.
Make sound financial decisions by understanding the time value of money and the impact of discounting future cash flows.
By the end of this lesson, you'll be a time-traveling financial wizard, equipped to make smart decisions about annuities and investments in the present, based on their future value. Let's turn back the clock and unlock the present value of general annuities!
Activity: The Time-Traveling Treasure
Activity: The Time-Traveling Treasure
Scenario: You've stumbled upon a time-traveling treasure map! It promises a stream of gold coins (annuity payments) delivered to you every year for a set number of years, starting in the future. However, the map only reveals the future value of this treasure. Your task is to calculate how much this future treasure is worth today (its present value).
Imagine the map promises 5 annual payments of 100 gold coins each, starting 3 years from now. Assume a "time travel interest rate" (discount rate) of 5%.
Use the present value of a general annuity formula to "travel back in time" and calculate the present value of this future stream of gold coins. Remember, the formula discounts the future value of each payment to its present value.
What is the total present value of the treasure? Is it more or less than the total future value (500 gold coins)? Why?
Analysis: Unearthing the Treasure
Analysis: Unearthing the Treasure
Why is knowing the present value of a future stream of payments important? How can this knowledge help us make better financial decisions today?
This lesson focuses on understanding and calculating the present value of general annuities, a crucial concept in finance.
Abstraction: What is the Present Value of a General Annuity?
Abstraction: What is the Present Value of a General Annuity?
The present value of a general annuity represents the current worth of a series of future payments, considering a specific interest rate and compounding period that may differ from the payment interval. Essentially, it answers: "How much money needs to be invested today to generate those future payments?"
Formula for Present Value of a General Ordinary Annuity
Formula for Present Value of a General Ordinary Annuity
The formula for calculating the present value (P) 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: Ken borrowed an amount of money from Kat. He agrees to pay the principal plus interest by paying P38,973.76 each year for 3 years. How much money did he borrow if interest is 8% compounded quarterly?
Solution:
R = P38,973.76
r = 8% = 0.08
m = 4 (compounded quarterly)
n = 3 years * 1 payment/year = 3 payments
Therefore, Ken borrowed approximately P100,000.
Example 2
Example 2
Problem: Mrs. Remoto would like to buy a television (TV) set payable for 3 months starting at the end of the month. How much is the cost of the TV set if her monthly payment is P3,000 and interest is 9% compounded semi-annually?
Solution:
R = P3,000
r = 9% = 0.09
m = 2 (compounded semi-annually)
n = 3 payments
Therefore, the cost of the TV is approximately P8,772.54.
Example 3
Example 3
Problem: A sala set is for sale at P16,000 in cash or on monthly installment P2,950 for 6 months at 12% compounded semi-annually. Which is lower: the cash price or the present value of the installment term?
Solution:
R = P2,950
r = 12% = 0.12
m = 2 (compounded semi-annually)
n = 6 payments
Therefore, the present value of the installment plan is P17,110.84, making the cash price of P16,000 cheaper.
Unlock the power of today's money! Here's a video lesson to understand more about finding the present value of general annuities!
Let's shift our focus to the present! This assessment tests your ability to apply the appropriate formulas to calculate the present value of general annuities. Get ready to solve real-world problems involving this important concept.
Application: Unlock the Present Value
Application: Unlock the Present Value
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 3 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 Present Value of General Annuities:
Explain the fundamental concept of present value in your own words. Why is it important to consider the time value of money when evaluating future cash flows?
Compare and contrast the formulas for finding the future value and present value of a general annuity. What are the key differences, and why do those differences exist?
Calculations:
Reverse Retirement Planning: Instead of figuring out how much to save, a couple knows they want P2,000,000 available when they retire in 25 years. Assuming a steady 6% annual interest rate compounded monthly, how much should they deposit at the beginning of each month to reach their goal?
Choosing the Best Loan Offer: You need a loan for P500,000 and receive two offers:
Bank A: 5 years, 4.5% annual interest compounded monthly, equal payments at the end of each month.
Bank B: 6 years, 4.8% annual interest compounded quarterly, equal payments at the end of each quarter.
Calculate the present value of each loan offer (essentially, this tells you the true cost of borrowing). Which offer is more favorable and why?
Investment Property Analysis: An investor is considering purchasing a rental property. The property is expected to generate a net income of P25,000 per quarter for the next 10 years. If the investor requires a 10% annual return compounded quarterly, what is the maximum price they should pay for the property today?
Critical Thinking:
Using the example of Mrs. Remoto buying a TV: If the store offered a 0% interest financing option for 3 months, would the present value of the TV be different? Explain your reasoning.
In the sala set example, the present value of the installment plan is higher than the cash price. Why might someone still choose the installment plan even if it means paying more overall?
Present value down? Get ready to calculate those periodic payments in Lesson 4!
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