copyright © 2008 pearson education canada11-1 chapter 11 ordinary simple annuities contemporary...

Copyright © 2008 Pearson Education Canada 11-1

Chapter 11

Ordinary Simple Annuities

Contemporary Business Mathematics With Canadian Applications

Eighth Edition S. A. Hummelbrunner/K. Suzanne Coombs

PowerPoint: D. Johnston


ObjectivesAfter completing Chapter eleven, the student

will be able to:• Distinguish different types of annuities.

• Compute the future value(or accumulated value) FV for ordinary simple annuities.

• Compute the present value (or discounted value) PV for ordinary simple annuities.

• Compute the payment, number of periods, and interest rate for ordinary simple annuities.


What Is an Annuity?

An annuity is identifiable if you have a series of regular payments, at regular intervals with a consistant interest rate.


Annuity Terminology

• payment interval - length of time between successive payments

• term of an annuity - length of time from the beginning of the first payment interval to the end of the last payment interval

• periodic rent - size of each regular payment

• annual rent - sum of periodic payments in one year


Types of Annuities

• Annuities certain -- fixed term

• Contingent annuities -- indefinite or uncertain term

• Perpetuity -- infinite number of payments


Examples of Annuities Certain

• The beginning and ending dates are known.

• Rental payments for real estate

• Lease payments on equipment

• Installment payments on loans

• Mortgage payments

• Interest payments on bonds and debentures


Examples of Contingent Annuities

• Beginning date or ending date or both are uncertain.

• Life insurance premiums

• Pension payments

• Payments from an RRSP converted into a life annuity

• Payments from a trust fund for the remaining life of a surviving spouse


Examples of Perpetuities

• Payments considered to continue forever.

• Size of periodic payment less than or equal to the periodic interest earned by a fund.

• Scholarship fund

• University endowment fund


Annuities

• Ordinary annuity -- Payments are made at the end of each payment period.

• Annuity due -- Payments are made at the beginning of each payment period.

• Deferred -- First payment is delayed for a specified time period or for a time period that may have to be calculated.


Annuities

• Simple annuity -- Interest conversion period is the same as the payment interval.

• General annuity -- Interest conversion period and payment interval are not the same.


Ordinary Simple Annuity

• Payments are made at the end of each payment interval.(i.e. Ordinary)

• Interest conversion period and payment interval are the same.(i.e. Simple)


Example of Simple Ordinary Annuity

The interest rate is 6% compounded annually

1 year 2 year 3 year 4 year 5 year

1000 1000 1000 1000 1000


Future Value of an AnnuityRate-6% compounded annually

Annual payments

FV 1000 1000 1000 1000

FV =1000(1.06)3+1000(1.06)2+1000(1.06)+1000


Future Value Formula

The formula for the FV of an ordinary simpleannuity can be derived by finding the sum of ageometric progression.

The terms in the geometric progression are thefuture values of each individual payment inthe cash flow.


Formula for Finding the Future Value of an Annuity

FVn = PMT (1+i)n - 1 i

PMT = Periodic payment i = Interest rate per conversion period n = Number of payments in annuity


Compounding or Accumulation Factor

(1+i)n – 1 i

This factor represents the accumulated valueof one dollar ($1) per payment period.


Future Value of AnnuityPayments of $1000 each are made at the endof each year for 4 years to a savings account.Find the accumulated value or future value ofthe account at the end of 4 years. The interestrate is 6% compounded annually.

FV = $1000 (1.06)4 –1 = $4374.62 .06


Present Value of Annuity

• The formula for finding the present value of an annuity is derived by finding the sum of a geometric progression.

• The geometric progression is made up of the present values of the individual payments in the cash flow.


Present Value Formula

1 1

ni

PV PMTi

PMT = Periodic payment

i = Interest rate per conversion interval

n = Number of interest conversion intervals


Discounting Factor for Present Value Equation

1 – (1+i) –n i

Discounted value of $1 per period


Example of PV 6% compounded annually

Annual year-end payments of $100

PV

100 100 100 100 PV=100(1.06) -1+100(1.06) -2+100(1.06) -3+100(1.06) –4 (continued)


Finding PV Using the Formula

PV = 100 1 – (1.06) –4 = $346.51 .06

PMT = 100 , i = .06, n=4


Present Value Application

The purchase of a home requires a down payment plus periodic payments on a mortgage loan.

Cash Value = Down Payment + Present Value of Periodic Payments


Finding Cash Value

A vacation property is bought for $3000 down and payments of $1000 at the end of each six months for 12 years. The interest rate is 7% compounded semi-annually. Cash Value = 3000+1000 1- (1.035) –24 .035 = 3000 + 16,058.37 = $19,058.37 (continued)


Electronic Calculator Solution for PV of Mortgage Payments


Home purchased for $100,000Assume a rate of 8% compounded semi-annually with monthly payments over 25

years.

Down Payment

Mortgage Principal

Total Interest Paid

$5,000 $95,000 $122,516

$10,000 $90,000 $116,068

$25,000 $75,000 $96,723


Summary

The simple ordinary annuity satisfies the following two conditions: 1. The payments are made at the end of the interest

conversion interval with the first payment at the end of the first interval.(i.e. Ordinary)

2. The payment period interval and the interest

conversion interval are equal.(i.e. Simple)

copyright © 2008 pearson education canada11-1 chapter 11 ordinary simple annuities contemporary...

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