Time Value of Money, Part 5 Present Value aueof An Annuity Intermediate Accounting I Dr. Chula King 1 Learning Outcomes The concept of present value Present value of an annuity Ordinary annuity versus annuity due How to set up the problem How to use the tables to solve the problem. 2 Present Value Refers to the amount of money that I need to invest today (in the present) to produce a known future value or values. 3 1
Annuity Equal periodic amount 1/1/X1 1/1/X2 1/1/X3 1/1/X4 10,000 10,000 10,000 1/1/X1 1/1/X2 1/1/X3 1/1/X4 12,000 15,000 10,000 1/1/X1 7/1/X1 7/1/X2 10/1/X2 10,000 10,000 10,000 4 Present Value of an Annuity Present value of an annuity determines the annuity value at the present time. Types of annuity Ordinary Odi Annuity An annuity whose payments occur at the end of each period; determines the present value one period before the first payment. Annuity Due An annuity whose payments occur at the beginning of each period; determines the present value at the time of the first payment, like a down payment. 5 Present Value of an Ordinary Annuity An annuity whose payments occur at the end of each period. One period before the first payment 6 2
Example What is the present value of a three year ordinary annuity of 100 at 10% compounded annually? 7 A Pictorial Representation 0 1 2 3 + 90.91 1.10 190.91 173.555 1.10 273.555 248.686 1.10 8 Table Solution Time value of money tables 1: Future value of a single sum 2: Present value of a single sum 3: Future value of an ordinary annuity 4: Present value of an ordinary annuity 5: Future value of an annuity due 6: Present value of an annuity due 9 3
Reading the Table Select the correct table Select the column corresponding to the interest rate per period Read down that column to the row that corresponds to the number of payments Take the resultant factor, and multiply it by the annuity amount The result is the present value of an annuity! 10 Example What is the present value of a three year ordinary annuity of 100 at 10% compounded annually? 11 Present Value of An Ordinary Annuity (Table 4, n=3, i=10%) Determines the present value one period before the first payment. 12 4
Present Value of an Ordinary Annuity of $1 (n) i Periods 2% 4% 6% 8% 10% 1 0.98039 0.96154 0.94340 0.92583 0.90909 2 1.94156 1.89969 1.83339 1.78326 1.73554 3 2.88388 2.77509 2.67301 2.57710 2.48685 4 3.80773 3.62990 3.46511 3.31213 3.16987 5 4.71346 4.45182 4.21236 3.99271 3.79079 6 5.60143 5.24214 4.91732 4.62288 4.35526 13 Present Value of An Ordinary Annuity Table 4, (n=3, i=10%) 248.685 = 100 x 2.48685 Determines the present value one period before the first payment. 14 Example Sally Rogers wants to accumulate a sum of money to pay for graduate school. She wants to invest a single amount today in a savings account earning 8% interest compounded annually, such that she can withdraw $20,000 a year for four years, with the first withdrawal occurring one year from today. How much must she invest? 15 5
Pictorial Representation of Example. 0 1 2 3 4 20,000 20,000 20,000 20,000 Table 4, (n=4, i=8%) Determines the present value one period before the first payment. 16 Present Value of an Ordinary Annuity of $1 (n) i Periods 2% 4% 6% 8% 10% 1 0.98039 0.96154 0.94340 0.92583 0.90909 2 1.94156 1.89969 1.83339 1.78326 1.73554 3 2.88388 2.77509 2.67301 2.57710 2.48685 4 3.80773 3.62990 3.46511 3.31213 3.16987 5 4.71346 4.45182 4.21236 3.99271 3.79079 6 5.60143 5.24214 4.91732 4.62288 4.35526 17 Pictorial Representation of Example. 0 1 2 3 4 20,000 20,000 20,000 20,000 Table 4, (n=4, i=8%) 66,243 = 20,000 x 3.31213 Determines the present value one period before the first payment. 18 6
Present Value of an Annuity Due An annuity whose payments occur at the beginning of each period. The present value is determined at the time of the first payment. X 19 What is the present value of a three year annuity due of 100 at 10% compounded annually? Example 20 Present Value of an Annuity Due Table 6, n = 3, i = 10% X Determines the present value at the time of the first payment. 21 7
Present Value of an Annuity Due of $1 (n) i Periods 2% 4% 6% 8% 10% 1 1.00000 1.00000 1.00000 1.00000 1.00000 2 1.98039 1.86154 1.94340 1.92593 1.90909 3 2.94156 2.88609 2.83339 2.78326 2.73554 4 3.88388 3.77509 3.67301 3.57710 3.48685 5 4.80773 4.62990 4.46511 4.31213 4.16987 6 5.71346 5.45182 5.21236 4.99271 4.79079 22 Present Value of an Annuity Due Table 6, n = 3, i = 10% 273.55 = 2.73554 x 100 X Determines the present value at the time of the first payment. 23 PV of an Ordinary Annuity versus Annuity Due Ordinary Annuity 0 1 2 3 Annuity Due 0 PMT PMT PMT 1 2 3 PMT PMT PMT X 24 8
The Next Step Exercises 6 2, 6 3, 6 4, 6 7, 6 8, 6 11, 6 12, 6 17 Problems 6 1, 6 2, 6 3, 6 4, 6 5, 6 13 25 9