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PERSONAL FINANCE MBF3C Lesson #4: Present Value Using Compound Interest MBF3C Lesson #4: Present Value Using Compound Interest

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To find the principle (present value) required at the beginning of a loan or investment, you use the formula: P=A(1+i) -n Note the negative sign in the exponent Where: A is the amount, maturity value or future value of an investment or loan P is the beginning principle i is the interest rate for 1 compound period n is the total number of compound periods i = yearly interest rate number of compound period each year n = number of years number of compound period each year

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The total interest earned is calculated using I=A-P, where A is the amount and P is the principle

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Example 2:Find the principle required to have $5000 at 6.4% p.a. compounded quarterly after 7 years. Find the total interest earned Solution:Find P and I:A= 5000i= 6.4%4=6.4 ÷ 1004 =0.016 n=7 4 = 28 By Substitution: P=A1+i - n=50001+0.016 - 28=3205.872≈3205.88 The interest is: I=A - P=5000 - 3205.88=1794.12 The investment required to have $5000 in 7 years is $3205.88 and the interest will be $1794.12. Note: the discounted value means the same as the present value or principle Page 439, #1ace, 2ac, 3ac, 4, 5, 9, 10 Learning Criteria I can ____Calculate the number of compound periods (n) and periodic interest rate (i) ____Calculate the present amount or value of an investment using compound interest compounded over different periods ____ Calculate the interest earned on an investment or loan

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The time interval between the occasions at which interest is added to the account is called the compounding period. The chart below describes some of the common compounding periods:

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Rather than using a table or a graph to see how the value of an investment grows, you can use a formula.

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P is the principal, or initial value. A is the accumulated amount or future value. i is the interest rate per compounding period. n is the number of compounding periods. P is the principal, or initial value. A is the accumulated amount or future value. i is the interest rate per compounding period. n is the number of compounding periods.

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The table shows how to convert the yearly interest rate and term for various compounding periods. *continued on next slide % per year compoundedNumber of compound periods per yearValue for iValue for n Annually1 Semi-annually2 Every six months2 Quarterly4 Every three months4 Monthly12 Weekly52 Daily365

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number of times per year that interest in compounded Interest rate as a decimal AnnuallyOnce a year Semi-annually2 timesa year Quarterly4 times a year Monthly12 times a year Bi-weekly26 times a year Weekly52 times a year Daily365 times a year

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Number of times per year that interest is compounded per year Number of Years x AnnuallyOnce a year Semi-annually2 timesa year Quarterly4 times a year Monthly12 times a year Bi-weekly26 times a year Weekly52 times a year Daily365 times a year

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Example: Interest Compounded Semi-Annually Determine how much money you will have if $500 is invested for six years, at 4% per year, compounded semi-annually. Solution Interest is compounded semi-annually, meaning twice a year, for six years. There are 2 × 6, or 12, compounding periods. This can be illustrated on a timeline. Determine how much money you will have if $500 is invested for six years, at 4% per year, compounded semi-annually. Solution Interest is compounded semi-annually, meaning twice a year, for six years. There are 2 × 6, or 12, compounding periods. This can be illustrated on a timeline.

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Determine how much money you will have if $500 is invested for six years, at 4% per year, compounded semi- annually. i = 0.04 ÷ 2 = 0.02 n = 6 × 2 = 12 P = 500 A= 500(1+0.02) 12 = 634.12 After six years, you will have $634.12. i = 0.04 ÷ 2 = 0.02 n = 6 × 2 = 12 P = 500 A= 500(1+0.02) 12 = 634.12 After six years, you will have $634.12.

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Example: Interest Compounded Monthly Alice borrowed $5000 to start a small business. The interest rate on the loan was 9% per year, compounded monthly. She is expected to repay the loan in full after four years. a) How much must Alice repay? b) How much of the amount Alice repays will be interest? Alice borrowed $5000 to start a small business. The interest rate on the loan was 9% per year, compounded monthly. She is expected to repay the loan in full after four years. a) How much must Alice repay? b) How much of the amount Alice repays will be interest?

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A) How much must Alice repay? Alice borrowed $5000 to start a small business. The interest rate on the loan was 9% per year, compounded monthly. She is expected to repay the loan in full after four years. Interest is compounded monthly, meaning 12 times a year, for four years. There are 12 × 4, or 48, compounding periods. Alice borrowed $5000 to start a small business. The interest rate on the loan was 9% per year, compounded monthly. She is expected to repay the loan in full after four years. Interest is compounded monthly, meaning 12 times a year, for four years. There are 12 × 4, or 48, compounding periods.

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Alice borrowed $5000 to start a small business. The interest rate on the loan was 9% per year, compounded monthly. She is expected to repay the loan in full after four years. i = 0.09 ÷ 12 = 0.0075 n = 4 × 12 = 48 P = 5000 A= 5000(1+ 0.0075 ) 48 = 7 157.03 A lice must repay $7157.03 after four years i = 0.09 ÷ 12 = 0.0075 n = 4 × 12 = 48 P = 5000 A= 5000(1+ 0.0075 ) 48 = 7 157.03 A lice must repay $7157.03 after four years

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B) How much of the amount Alice repays will be interest? Calculate the total interest by subtracting the principal from the amount. 7157.03 - 5000 = 2157.03 Therefore, Alice will pay $2157.03 in interest. Calculate the total interest by subtracting the principal from the amount. 7157.03 - 5000 = 2157.03 Therefore, Alice will pay $2157.03 in interest.

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Key Concepts (p. 432) Compound interest can be accumulated at various intervals, such as annually, semi-annually, quarterly, and monthly. The compound interest formula A = P(1 + i) n can be used to calculate the future value, or amount – A is the future value or accumulated amount of an investment or loan. – P is the principal. – i is the interest rate, in decimal form, per interest period. – n is the number of compounding periods. Compound interest can be accumulated at various intervals, such as annually, semi-annually, quarterly, and monthly. The compound interest formula A = P(1 + i) n can be used to calculate the future value, or amount – A is the future value or accumulated amount of an investment or loan. – P is the principal. – i is the interest rate, in decimal form, per interest period. – n is the number of compounding periods.

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IN-CLASS & HOMEWORK Page 432, # 1ace, 2bc, 3bd, 4, 5, 7, 8abcd

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SUCCESS CRITERIA FOR THIS LESSON: I can ____Calculate the number of compound periods (n) and periodic interest rate (i) ____Calculate the present amount or value of an investment using compound interest compounded over different periods ____ Calculate the interest earned on an investment or loan I can ____Calculate the number of compound periods (n) and periodic interest rate (i) ____Calculate the present amount or value of an investment using compound interest compounded over different periods ____ Calculate the interest earned on an investment or loan

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