Copyright © 2008 Pearson Education Canada13-1 Chapter 13 Annuities Due, Deferred Annuities, and Perpetuities Contemporary Business Mathematics With Canadian Applications Eighth Edition S. A. Hummelbrunner/K. Suzanne Coombs PowerPoint: D. Johnston
Copyright © 2008 Pearson Education Canada13-2 Objectives After completing chapter thirteen, the student will be able to: Compute the future value, present value, periodic payment, term, and interest rate for simple annuities due. Compute the future value, present value, periodic payment, term, and interest rate for general annuities due. (continued)
Copyright © 2008 Pearson Education Canada13-3 Objectives (continued) Compute the future value, present value, periodic payment, term, and interest rate for ordinary deferred annuities and deferred annuities due. Compute the present value, periodic payment, and interest rate for ordinary perpetuities, perpetuities, and deferred perpetuities.
Copyright © 2008 Pearson Education Canada13-4 Annuities Due
Copyright © 2008 Pearson Education Canada13-5 Simple Annuities Due “Due” means that the payments are made at the beginning of the payment interval. “Simple” means that the payment interval and interest conversion interval are the same.
Copyright © 2008 Pearson Education Canada13-6 Simple Annuity Due 6% compounded annually Annual $100 payments made at beginning of year
Copyright © 2008 Pearson Education Canada13-7 Formula for FV of Simple Annuity Due
Copyright © 2008 Pearson Education Canada13-8 Calculating the FV of a Simple Annuity Due
Copyright © 2008 Pearson Education Canada13-9 Calculating Interest Earned (continued)
Copyright © 2008 Pearson Education Canada13-10 PV of Simple Annuity Due 6% compounded annually Annual payments made at beginning of year
Copyright © 2008 Pearson Education Canada13-11 Formula for Present Value of Simple Annuity Due
Copyright © 2008 Pearson Education Canada13-12 Calculating PV of an Annuity Due
Copyright © 2008 Pearson Education Canada13-13 Calculator Solution
Copyright © 2008 Pearson Education Canada13-14 Finding PMT for a Simple Annuity Due PMT is found the same way as we did for Simple Annuities. Just remember to include the (1+i) in your calculations. Be able to solve for PMT both algebraically and with your calculator. The calculator is used to verify that your manual calculation is correct.
Copyright © 2008 Pearson Education Canada13-15 Finding n for a Simple Annuity Due The is also done the same way as we did for Simple Annuities. Include the (1+i) additional factor in your calculations. Your must be able to solve this manually using natural logarithms but you should also be able to solve it on your calculator to verify if your manual calculation is correct.
Copyright © 2008 Pearson Education Canada13-16 Finding i for a Simple Annuity Due Appendix II of the CD-ROM explains how to solve for I manually using an approach known as “Trial and Error” You should usually solve for I using the calculator just as we did for Simple Annuities.
Copyright © 2008 Pearson Education Canada13-17 General Annuities Due General Annuities Due are identical to Simple Annuities Due, the only exception being that the interest conversion periods are not the same as the frequency of payments. Because of this you use the same formula you used for General Annuities, adding the factor (1+p) (continued)
Copyright © 2008 Pearson Education Canada13-18 General Annuities Due (continued) The solving for general annuiteis due is the same as it was for General Annuities. Just remember the additional (1+p) in the formula. This applies to solving for Future Value, Present Value, PMT, n and i.
Copyright © 2008 Pearson Education Canada13-19 Deferred Annuity A deferred annuity is one in which the first payment is made at a time later than the end of the first payment interval. Time period from “now” to starting point of term of annuity is called the period of deferment. The symbol d is used to represent the number of compounding periods in the period of deferment.
Copyright © 2008 Pearson Education Canada13-20 Deferred Annuity Types As discussed earlier, we can have ordinary deferred annuities (both the simple and the general cases), and deferred annuities due (again with both the simple and general cases being possible)
Copyright © 2008 Pearson Education Canada13-21 Future Value of Deferred Annuity
Copyright © 2008 Pearson Education Canada13-22 Present Value of a Deferred Annuity
Copyright © 2008 Pearson Education Canada13-23 Finding the PV of a Deferred Annuity
Copyright © 2008 Pearson Education Canada13-24 Perpetuity Periodic payments begin on a fixed date and continue indefinitely. Since there is no end to the term, it is not possible to determine the FV of a perpetuity.
Copyright © 2008 Pearson Education Canada13-25 Examples of Perpetuities Dividends on preferred shares Interest payments on permanently invested funds. Consolidated annuities
Copyright © 2008 Pearson Education Canada13-26 Graph of Ordinary Perpetuity
Copyright © 2008 Pearson Education Canada13-27 PV of Ordinary Simple Perpetuity
Copyright © 2008 Pearson Education Canada13-28 Present Value of Ordinary General Perpetuity
Copyright © 2008 Pearson Education Canada13-29 Finding the PV of a Perpetuity
Copyright © 2008 Pearson Education Canada13-30 Finding the PV of a General Perpetuity
Copyright © 2008 Pearson Education Canada13-31 PV of Simple Perpetuity Due
Copyright © 2008 Pearson Education Canada13-32 PV of General Perpetuity Due
Copyright © 2008 Pearson Education Canada13-33 Summary