Time Value of
Money, Part 5
Present Value of
1ese t a ue o
An Annuity
Intermediate Accounting II Dr. Chula KingLearning Outcomes
• The concept of present value • Present value of an annuity • Ordinary annuity versus annuity due 2 © Dr. Chula King All Rights Reserved y y y • How to set up the problem • How to use the tables to solve the problem.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 © Dr. Chula King All Rights ReservedAnnuity
• Equal periodic amount 1/1/X1 1/1/X2 1/1/X3 1/1/X4 | | | | 10,000 10,000 10,000 4 © Dr. Chula King All Rights Reserved , , , 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 10,000Present Value of an Annuity
• Present value of an annuity determines the annuity value at the present time. • Types of annuity O di A i A i h 5 © Dr. Chula King All Rights Reserved –Ordinary 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.Present Value of an Ordinary Annuity
• An annuity whose payments occur at the end of each period. ….. 0 1 2 3 |‐‐‐‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐‐‐‐‐| 100 100 100 One period before theExample
• What is the present value of a three year ordinary annuity of 100 at 10% compounded annually? 7 © Dr. Chula King All Rights ReservedA Pictorial Representation
0 1 2 3 |‐‐‐‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐‐‐‐‐| 100 100 100 8 © Dr. Chula King All Rights Reserved + 90.91 ÷1.10 190.91 173.555 ÷ 1.10 273.555 248.686 ÷ 1.10Table Solution
• Time value of money tables –1: Future value of a single sum –2: Present value of a single sum3: Future value of an ordinary annuity
9 © Dr. Chula King All Rights Reserved –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
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
10 © Dr. Chula King All Rights Reserved 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!
Example
• What is the present value of a three year ordinary annuity of 100 at 10% compounded annually? 11 © Dr. Chula King All Rights ReservedPresent Value of An Ordinary Annuity
….. 0 1 2 3 |‐‐‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐‐‐‐| 100 100 100 (Table 4, n=3, i=10%)Determines the present value one period before the first payment.
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 13 © Dr. Chula King All Rights Reserved 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 4.35526Present Value of An Ordinary Annuity
….. 0 1 2 3 |‐‐‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐‐‐‐| 100 100 100 14 © Dr. Chula King All Rights Reserved Table 4, (n=3, i=10%) 248.685 = 100 x 2.48685Determines the present value one period before the first payment.
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
15 © Dr. Chula King All Rights Reserved 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?
Pictorial Representation of Example
… . 0 1 2 3 4 |‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐| 20,000 20,000 20,000 20,000 16 © Dr. Chula King All Rights Reserved Table 4, (n=4, i=8%)Determines the present value one period before the first payment.
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 17 © Dr. Chula King All Rights Reserved 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 4.35526Pictorial 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 3.31213Present 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 19 © Dr. Chula King All Rights Reserved the first payment. ….. 0 1 2 3 |‐‐‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐‐‐‐‐| 100 100 100X X
Example
• What is the present value of a three year annuity due of 100 at 10% compounded annually? 20 © Dr. Chula King All Rights ReservedPresent Value of an Annuity Due
….. 0 1 2 3 |‐‐‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐‐‐‐‐| 100 100 100X X
21 © Dr. Chula King All Rights Reserved Table 6, n = 3, i = 10%Determines the present value at the time of the first payment.
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 22 © Dr. Chula King All Rights Reserved 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 4.79079Present Value of an Annuity Due
….. 0 1 2 3 |‐‐‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐‐‐‐‐| 100 100 100X X
23 © Dr. Chula King All Rights Reserved Table 6, n = 3, i = 10% 273.55 = 2.73554 x 100Determines the present value at the time of the first payment.
PV of an Ordinary Annuity versus Annuity Due Ordinary Annuity 0 1 2 3 PMT PMT PMT 0 1 2 3 Annuity Due
