copyright © 2008 pearson education canada 12-1 chapter 12 ordinary general annuities contemporary...

Copyright © 2008 Pearson Education Canada 12-1

Chapter 12

Ordinary General Annuities

Contemporary Business Mathematics With Canadian Applications

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

PowerPoint: D. Johnston


ObjectivesAfter completing chapter twelve, the student

will be able to:

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

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

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


General Annuity

• Length of interest conversion period is different from the length of the payment interval.

• In Canada, home mortgages are usually compounded semi-annually and payments are often made on a monthly, semi-monthly, or weekly basis.


Ordinary General Annuity

“Ordinary” means that the payments are made or received at the end of each payment interval. “General” means that the payment interval and interest conversion intervals are different.


Relationship between Payment Interval and Interest Conversion

Period

number of interest conversions per year

number of payments per yearc


Two Cases-Ordinary General Annuities

Case 1-- c is a fraction less than 1. The interest conversion period is longer than the payment period. Each payment interval contains only a fraction of one conversion period.

Case 2—c has a value greater than 1. The interest conversion period is shorter than the payment period. Each payment period contains more than one conversion period.


Determining the value of c

Conversion period

Payment interval

c

Monthly Semi-annual 126

2c

Monthly Quarterly 12

34

c Semi-annual Monthly 2 1

12 6c


Effective Rate of Interest per Payment Period

Depending on the length of the paymentinterval, the effective interest rate per paymentperiod may be a monthly, quarterly, semi-annual, or annual rate.

p = (1+i)c - 1


Calculating Effective Rate p

Joan deposits $1000 at the end of each half year into a savings account paying interest at 6% compounded monthly. Find the effective rate p. There are 12 interest conversion intervals per year and 2 payment intervals per year.

12

62

c

p = (1+.06/12) 6 – 1 = 3.038%

(continued)


Formula for the Future Value of an Ordinary General

Annuity

1 ) 1npFV PMT

p

where p = (1+i)c - 1


Finding FV or Accumulated ValueFind the accumulated value of payments of $1250 made at the end of each quarter for 8 years if interest is 5.5% compounded annually. Step 1 – Determine effective rate p. c = ¼ = 0.25 p= (1.055) 0.25 – 1 = .0134752 (continued)


Find the Future Value Using the Future Value Formula

FV = 1250 (1.0134752)32 – 1) = $49,599.22 .0134752

where p = .0134752 (continued)


Electronic Calculator Solution


Present Value of Ordinary General Annuity

PVnc = PMT 1 - (1 + p) –n p

where p =(1+i)c - 1


Finding PV of an AnnuityA 25-year mortgage on a home requires payments of $619.94 at the end of each month. The interest rate is 9.5% compounded semi-annually. Find the mortgage principal which is the present value (PV) of the cash flow. Step 1 – Calculate the value of p. p = (1.0475) (1/6) – 1 = .007764383 (continued)


Calculating PV

Step 2 – Calculate the PV of the annuity using P = .007764383.

PV = 619.94 1 - (1.007764383) –300 .007764383

= $72,000.01 (continued)


Electronic Calculator Solution


Using a Preprogrammed Calculator to Find PV or FV of an Ordinary General Annuity

Step 1 - Set P/Y as number of payments per year.

Step 2 - Set C/Y as number of compounding periods per year.

Step 3 -Determine PV or FV as required.


Finding PMT When the Future Value Is Known

FVnc = PMT (1 + p)n - 1 p

Divide both sides by the coefficient of PMT.

PMT = FVnc p (1+p)n -1


Find PMT Using the Electronic Calculator

When using a preprogrammed financecalculator, you can find PMT by entering 5known values (FV, N, I/Y, P/Y, C/Y) andpressing CPT PMT.

Recall that the PMT value will be negative aspayments are considered to be cash outflows.


Finding PMT When Present Value Is Known

PVnc = PMT 1 – (1 + p) –n p

Divide both sides by the coefficient of PMT.

PMT = PVncp 1 – (1+p) -n


Finding PMT on Electronic Calculator When PV Is Known

When using a preprogrammed financialcalculator, you can enter five known values(PV, N, I/Y, P/Y, C/Y) and press CPT PMT.


Finding i Irina deposited $150 in a savings account at the end of each month for 60 months. If the accumulated value of the deposits was $10000 and interest was compounded semi-annually, what was the nominal annual rate of interest? FV= 10,000 P/Y = 12 PMT = -150 C/Y = 2 N = 60 PV=0 Once this is entered you then press CPT followed by I/Y. In this case the answer will be 4.255410% compounded semi-annually. (continued)


Calculator solution (continued)


Summary

• In an ordinary general annuity, the payment interval and interest conversion interval do not coincide.(i.e. General)

• It is necessary to calculate an equivalent rate of interest so that the payment interval and interest conversion interval do coincide.

copyright © 2008 pearson education canada 12-1 chapter 12 ordinary general annuities contemporary...

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