Compounding More than Once a Year

Transcription

1 Compounding More than Once a Year by CHED on December 22, 2017 lesson duration of 5 minutes under General Mathematics generated on December 22, 2017 at 04:18 pm Tags: Simple and Compound Interest

2 Generated: Dec 23, :18 AM Compounding More than Once a Year ( 5 mins ) Written By: CHED on May 27, 2016 Subjects: General Mathematics Tags: Simple and Compound Interest Resources n/a Content Standard The learner demonstrates understanding of key concepts of simple and compound interests, and simple and general annuities. Performance Standard The learner is able to investigate, analyze and solve problems involving simple and compound interests and simple and general annuities using appropriate business and financial instruments. Learning Competencies The learner solves problems involving simple and compound interests. The learner computes interest, maturity value, future value, and present value in simple interest and compound interest environment. Introduction, Motivation 1 mins Group the students into two. Have one representative from each group `deposit' P100 into a jar or box. Group 1: This group will earn compound interest each day at a rate of 5%. Group 2: This group will earn compound interest twice a day at a rate of 2.5%. Ask the students to discuss whether one group will earn more than the other group. Let each group compute how much money they will have at the end of 5 days. Let them observe the `compounding effect' by constructing the table below. 1 / 12

3 Lesson Proper 1 mins (a) Allow the students to compare the compound amounts when compounding semi-annually and compounding annually by posing the following example: EXAMPLE 1. Given a principal of P10,000, which of the following options will yield greater interest after 5 years: OPTION A: Earn an annual interest rate of 2% at the end of the year, or OPTION B: Earn an annual interest rate of 2% in two portions 1% after 6 months, and 1% after another 6 months? Solution. OPTION A: Interest is compounded annually OPTION B: Interest is compounded semi-annually, or every 6 months. Under this option, the interest rate per conversion period 1% (2% divided by 2). 2 / 12

4 Let the students realize that interest is often compounded more than once a year (semi-annually, quarterly, and daily). If all else is equal, a more frequent compounding will result in a higher interest, which is why Option B gives a higher interest than Option A. The investment scheme in Option B introduces new concepts. Because interest is compounded twice a year, the conversion period is 6 months, and the frequency of conversion is 2. Because the investment runs for 5 years, the total number of conversion periods is 10. The nominal rate is 2% and the rate of interest for each conversion period is 1%. These terms are defined generally below. (b) Definition of Terms Let the students define the following additional terms: conversion or interest period - time between successive conversions of interest frequency of conversion (m) - number of conversion periods in one year nominal rate (i(m)) - annual rate of interest rate (j) of interest for each conversion period total number of conversion periods n n = tm = (frequency of conversion) (time in years) Note on rate notation: : r, i(m), j In earlier lessons, r was used to denote the interest rate. Now that an interest rate can refer to two rates (either nominal or rate per conversion period), the symbols i(m) and j will be used instead. (c) Provide examples of nominal rates and the corresponding frequencies of conversion and interest rate for each period. 3 / 12

5 (d) Derive the formula in finding compound amount when compounding is computed more than once a year. Let the students recall (from Lesson 25) how to compute for the compound amount when principal P is invested at an annual interest rate j compounded annually, You can modify this formula by noting that: the rate for each conversion period is in t years, interest is compounded mt times. Thus, you obtain the following formula: (e) Provide examples on compounding amounts more than once a year. 4 / 12

6 EXAMPLE 2. Find the maturity value and interest if P10,000 is deposited in a bank at 2% compounded quarterly for 5 years. Answer: The compound interest is given by Ic =F?P =11,048.96?10,000=P1, EXAMPLE 3. Find the maturity value and interest if P10,000 is deposited in a bank at 2% compounded monthly for 5 years. Solution. 5 / 12

7 Answer: The compound interest is given by Ic =F?P =11,050.79?10,000=P1, EXAMPLE 4. 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. 6 / 12

8 Answer: Thus, Cris must pay P102, after 6 years. (f) You may extend the discussion to finding present value when interest is compounded more than once a year. EXAMPLE 5. Find the present value of P50,000 due in 4 years if money is invested at 12% compounded semi- annually. Solution. 7 / 12

9 EXAMPLE 6. What is the present value of P25,000 due in 2 years and 6 months if money is worth 10% compounded quarterly? Solution. Seatwork 1 mins Seatwork 1. Complete the table by computing the interest rate per period and total number of conversion periods. 8 / 12

10 Seatwork 2. Complete the table by computing for compound amounts, compound interests and present values. Seatwork 3. Solve the following problems on compound interest. 1. Accumulate P15,000 for 2 years at 15% compounded monthly. 1. Answer: P20, How much should Kaye set aside and invest in a fund earning 2% compounded quarterly if she needs P75,000 in 15 months? 1. Answer: P73, Peter is planning to invest P100,000. Bank A is offering 5% compounded semi-annually while Bank B is offering 4.5% compounded monthly. If he plans to invest this amount for 5 years, in which bank should he invest? 1. Answer: Compound amount after 5 years: Bank A: F = P128,008.45; Bank B: F = P125,179.58; Bank A gives higher compound amount Evaluation 1 mins (a) Fill in the blanks with the correct answers. a. When money is compounded monthly, the frequency of conversion is. b. When the annual interest rate is 16% compounded quarterly the interest rate in a conversion period is. 9 / 12

11 c. If the interest rate per conversion period is 1% and money is compounded monthly, the nominal rate is. d. When the term is 3 years and 6 months and money is compounded semi-annually, the total number of conversion periods is. e. When the total number of conversion periods is 12 and the term is 6 years, then money is compounded. Answer: a. 12 b or 4% c or 12% d. 7 e. semi-annually (b) Complete the table by computing for the compound amounts, compound interests and present values. (c) Solve the following problems on compound interests. Find the compound amount due in 8 years if P200,000 is invested at 12% compounded monthly. Answer: P519, What present value, compounded quarterly at 6%, will amount to P59, in 3 years? Answer: P50, Alet borrowed P15,000 payable with interest that is compounded semi-annually at 9%. How much must she pay after 3 years? Answer: P19, How much must Angel deposit in a bank that pays 0.75% compounded quarterly so that she will have P200,000 after 15 years? Answer: P178, Suppose that you have P80,000. You decided to deposit it on a bank and will not withdraw from it for 10 years. A bank offers two types of compound interest accounts. The first account offers 6% interest compounded monthly. The second account offers 6.5% interest compounded semi-annually. Which account will you choose if you want your money to earn more? Answer: First Bank: F = P145, Second Bank = F = P151,667.03; Second bank yields more Enrichment (Optional) 1 mins 10 / 12

12 Continuous Compounding Interest can be compounded continuously like every hour, every minute or even a fraction of a second. If the number of compounding m is to increase without bound, this procedure approaches what is called continuous compounding. The formula for continuous compounding is derived as follows: EXAMPLE 7. Suppose you invested P20,000 at 3% compounded continuously. How much will you have from this investment after 6 years? 11 / 12

13 Powered by by TCPDF ( CHED.GOV.PH Hence, the amount P20,000 will become P23, if you invest it at 3% compounded continuously for 6 years. Download Teaching Guide Book 0 mins 12 / 12