# MATA32H3 Study Guide - Final Guide: Logarithm, Net Present Value, Integral

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10 Dec 2015

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MATA 32- Calculus for

Management I

Final Exam Study Guide

University of Toronto (UTSC) β Fall 2015

MATA32 β Lecture #1: Compound Interest

Compound Interest

Deals with money, time and interest

1 key equation:

π=π(1+π)!

P ! Principle

β’ Money invested now

r ! Periodic interest rate

β’ Interest paid per compounding period

β’ Stated in percentage or annual rate

o NOTE: Must convert percentage into a decimal

n ! Number of compounding periods

β’ Counts number of times interest is paid on investment

S ! Compound Amount

β’ Amount of money we have at the end

Interest is paid exactly at the end of compounding periods

β’ Only at the end of compounds

**For one compound period, the principle (P) plus the interest on that

1st Compound ! π1+Pr =!π!(1+π)

2nd Compound ! π!1+π+!π!1+πβπ=π!1+π1+π=π!(1+π)!

3rd Compound ! π!(1+π)!+!π!1+πβ!π=!π!1+π!β!1+π=π!(1+π)!

4th Compound ! π!(1+π)!

NOTE: IN APPLICATIONS

β’ We often have APR βaβ (Annual Percentage Rate) and frequency βkβ of

compounding; and

o How often interest is paid in a year

β’ A time βtβ period

!

!

!

!

!

!

Khl!

Compounding periods

End

Start

Fixed Period of time

n!

P $

(Principle!+!Interest!on!Principle)!

Legend:

a ! APR for period as follows

k ! Compounding frequency per year (annually)

t ! Number of years for a particular period

We have thatβ¦

!

!= r and π=πβπ‘

Thus,

π=π!(1+!

π

π)!β!

Example 1: Consider the following βinvestment schemeβ invest $10,000 at 3.05% APR

compounding monthly for a period of 5 years.

a) Find compound amount

π=10,000!1+!.!"!#

!"

!"

β11,645.17

b) Periodic rate (interest at the end of each month)

π=

π

π

=

0.0305

12

β0.002542!(0.2542%)

c) Compound Interest

Difference between beginning amount and end amount

πβπ!

β1,645.17

Return due to interest

=

1645.17

10000

β100

β16.4517%

(Over 5 years accumulated)

d) How long does it take to earn $15,000

** Express terms in units of compounding (i.e. from this example, months)**

5 years

$10,000

$11,645.17

X 60