CHAPTER 3 Compound Interest
Recall What can you say to the amount of interest earned in simple interest?
Do you know? An interest can also earn an interest?
Compound Interest Whenever a simple interest is added to the principal at regular intervals, and the sum become the new principal, the interest is said to be compounded.
Challenge: Give me an example of a business transaction which involves compounding or compound interest.
Do you know? Most savings accounts pay compounded interest every three months or quarterly.
Therefore, Adding interest to the principal gives more interest.
Compounded Whenever we encounter the word compounded what we mean is that the interest is added to the principal to have a new principal.
Something to think about How does the interest become compounded?
Compound Amount The final amount at the end of the term. We will denote this using majuscule letter S.
Compound Interest It refers to the difference between the compound amount S and the original Principal P.
Example 3.1: If P2,000.00 is invested at an interest rate of 8% compounded annually for 3 years, find the compound amount and interest.
Final Answer: The compound amount in 3 years is P2,519.42 and the compound interest is P519.42.
Example 3.2: Find the compound amount and compound interest if P5,000.00 is invested at 10% compounded semi-annually for 2 years.
Compounded Semi-annually It means the interest earned in 6 months is added to the principal to earn additional interest for the next 6 months.
Therefore Since we re just finding for the interest for 6 months we will divide the rate of interest by 2.
Final Answer: The compound amount semiannually for 2 years is P6,077.53 and the compound interest is P1,077.53.
Reflect: How do you find the process of computing for the compound amount and compound interest?
YAS! I feel you! It s very TEDIOUS!!!!!!
LESSON 3.2 Finding the Compound Amount and Compound Interest Using the Formula
Example 3.3 Find the accumulated value of P5,000.00 in 4 years if it is invested at 12% compounded quarterly.
Reflect: Most of the time, interest is compounded into the principal more than once a year.
Conversion/Interest Period The time between two successive conversions of interest.
Conversion: Annually 1 year Semi-annually 6 months Quarterly 3 months Monthly 1 month
Frequency of Conversion The number of conversion periods at a certain time. To denote this we will use minuscule letter m.
Frequency of Conversion: Annually 1 Semi-annually 2 Quarterly 4 Monthly 12 Daily 365/366
Nominal Rate The rate of interest in compound interest. To denote this we will use the minuscule letter j.
The rate of interest for each conversion period is denoted Formula: by i. i = nominal rate frequency of conversion Equation 3.1
The number of conversion periods in the term is denoted Formula: by n. n = time in years frequency of conversion or n = t m Equation 3.2
Example 3.2 If money is invested at 8% compounded quarterly for 3 years.
Formula for Compound Amount: Let: P = the original principal invested i = rate of interest S = compound amount of P n = number of conversion periods
Formula for Compound Amount: S = P(1 + i) n Equation 3.3
Where: S = compound amount or accumulated value of P at the end of n periods P = the original principal invested i = rate of interest = j m j = nominal rate of interest (annual rate) m = frequency of conversion n = number of conversion periods = t m t = term of investment m = frequency of conversion
Accumulation Factor In the compound amount formula, it is the factor (1 + i) n.
Example 3.3 Find the accumulated value of P5,000.00 in 4 years if it is invested at 12% compounded quarterly.
Final Answer: The accumulated value in 4 years is P8,023.53.
Do you know? The value of (1 + 0.03) 16 can be obtained using a scientific calculator or by the use of Table II.
Example 3.4 Find the compound amount and the compound interest on P10,000.00 for 9 ¼ years at 6% compounded quarterly.
Final Answer: The compound amount is P17,347.77 and the compound interest is P7,347.77.
Let s Practice: Find the interest rate (i) for each period, the total number of conversion periods (n) and the conversion period (m) at the end of the indicated time. If principal is P15,000.00, determine also the S and I. (a) 8 years at 9% compounded semi-annually (b) 12 years and 6 months at 10% compounded monthly
Assignment: Find the interest rate (i) for each period, the total number of conversion periods (n) and the conversion period (m) at the end of the indicated time. (c) 10 years and 9 months at 10.5% compounded quarterly (d) From April 1, 2015 to December 31, 2007 at 12% compounded quarterly (e) From June 1, 2002 to May 31, 2008 at 11% compounded annually
Something to think about How about when Present ValueP is missing? What should we do?
LESSON 3.3 Finding the Present Value at Compound Interest
Present Value It is an amount due in n interest periods which is invested at a given rate. We denote this using majuscule letter P.
Challenge: Derive the formula for finding Present Value P using the formula for computing for Compound Amount S.
Formula for Present Value P: P = S or (1+i) n P = S(1 + i) n Equations 3.4 and 3.5
Discount Factor This refers to the factor (1 + i) n. To discount an amount S due in n periods means to find its present value P at n periods before S is due.
Example 3.5 Find the present value of P18,500.00 due in 5 years if money is worth 8% compounded semi-annually.
Final Answer: The present value is P12,497.93.
Example 3.6 A 60 square meter house and lot is purchased on installment. The buyer makes a P110,400.00 down-payment and owes a balance of P257,600.00 payable in 5 years. Find the cash value of the house and lot if money is worth 10% compounded quarterly.
Final Answer: The cash value of the house and lot is P157,205.79.
Let s Practice: Solve the following problems: 1. Find the present value of P30,700.00 due in 6 years if money is worth 8% compounded quarterly. 2. On the birth of a son, a father wished to invest sufficient money to accumulate P2,500,000.00 by the time his son turns 21 years old. If the father invests at a rate of 10% compounded semi-annually, how much should the investment be?
LESSON 3.6 Finding the Nominal Rate
Note: The nominal rate j can be determined if S, P and n are given.
Example 3.9 At what nominal rate compounded semi-annually will P50,000.00 accumulate to P85,000.00 in 12 years?
Final Answer: The nominal rate is 4.47%
Reflect: How do you find the process for finding nominal rate?
YAS! I feel you! It s very TEDIOUS!!!!!!
Formula: Finding for j: j = ixm Equation 3.6
Formula: 1 n 1 i = S P Equation 3.7
Example 3.10 At what rate compounded quarterly, will P16,000.00 amount to P20,000.00 in 5 years?
Final Answer: The nominal rate is 4.49%
Let s Practice: Solve the following problems: 1. At what nominal rate compounded semi-annually will P18,000.00 amount to P25,000.00 in 5 years? 2. What rate compounded quarterly will double any sum of money in 9 years?
LESSON 3.7 Finding the Time
How can we find time? Using Logarithm Interpolation Method
Recall What is your idea about logarithms?
Logarithms In mathematics, the logarithm of a number is the exponent to which another fixed value, the base, must be raised to produce that number.
Logarithms The idea of logarithms is to reverse the operation of exponentiation.
Example 3.11 How long will it take for P6,500.00 to become P9,800.00 if it is invested at 8% compounded quarterly?
Final Answer: It will take 5.18 years for P6,500.00 to become P9,800.00.
Example 3.12 How long will it take for P6,000.00 to become P11,300.00 at 6% compounded semiannually?
Final Answer: It take 10.71 years for P6,000.00 to become P11,300.00.
Let s Practice: Solve the following problems: 1. How long will it take for P8,000.00 to accumulate to P9,500.00 at 8% compounded semi-annually? 2. After how many years will P21,000.00 accumulate to P42,000.00 if is its invested at 12% compounded quarterly?
JTIY Just to Inform You
Something to think about What do we usually do when we calculate for nominal rate j?
Challenge: So we can make everything else smooth and easy, can you derive/think of a direct formula for nominal rate j?
Formula for Nominal Rate: 1 j = m S P n 1 Equation 3.10
Derivation of Formula: How to find time using a formula? Let s derive the formula!
Formula for Time: t = log S P m log(1+i) Equation 3.11
Example 3.16: When is P47,600.00 due if its present value of P38,400.00 is invested at 9% compounded bimonthly?
Bimonthly This means every other month or every two months. Thus, m=6.
Final Answer The amount will be due after 2.40 years.