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

Note the negative sign in the exponent 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

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

Example 2:. Find the principle required to have $5000 at 6. 4% p. a 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= 5000 i=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

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:

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

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.

Number of compound periods per year The table shows how to convert the yearly interest rate and term for various compounding periods. *continued on next slide % per year compounded Number of compound periods per year Value for i Value for n Annually 1 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 1 # 𝑜𝑓 𝑦𝑒𝑎𝑟𝑠 ×1 Semi-annually 2 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 2 # 𝑜𝑓 𝑦𝑒𝑎𝑟𝑠 ×2 Every six months Quarterly 4 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 4 # 𝑜𝑓 𝑦𝑒𝑎𝑟𝑠 ×4 Every three months Monthly 12 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 12 # 𝑜𝑓 𝑦𝑒𝑎𝑟𝑠 ×12 Weekly 52 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 52 # 𝑜𝑓 𝑦𝑒𝑎𝑟𝑠 ×52 Daily 365 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 365 # 𝑜𝑓 𝑦𝑒𝑎𝑟𝑠 ×365

WHAT EXACTLY DOES THE “i” MEAN? Annually Once a year Semi-annually 2 timesa year Quarterly 4 times a year Monthly 12 times a year Bi-weekly 26 times a year Weekly 52 times a year Daily 365 times a year Interest rate as a decimal number of times per year that interest in compounded

WHAT EXACTLY DOES THE “n” MEAN? Annually Once a year Semi-annually 2 timesa year Quarterly 4 times a year Monthly 12 times a year Bi-weekly 26 times a year Weekly 52 times a year Daily 365 times a year Number of times per year that interest is compounded per year x Number of Years

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. 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.

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?

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 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 = 7157.03 Alice must repay $7157.03 after four years

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.

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.

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

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