Time Value of Money, Part 4 Future Value aueof An Annuity. Learning Outcomes. Future Value

Time Value of 

Money, Part 4

Future Value of 

1

utu e a ue o

An Annuity

Intermediate Accounting I Dr. Chula King

Learning Outcomes

• The concept of future value • Future 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.

Future Value

• Refers to the amount of money to which an  investment will grow over a finite period of  time at a given interest rate 

• The cash value of an investment at a particular

3

© Dr. Chula King All Rights Reserved

• The cash value of an investment at a particular  time in the future. 

Annuity

• Equal periodic amount 1/1/X1 1/1/X2 1/1/X3 1/1/X4 | | | | 10,000 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 | | | | 10,000 12,000 15,000 10,000 1/1/X1 7/1/X1 7/1/X2 10/1/X2 | | | | 10,000 10,000 10,000 10,000

Future Value of an Annuity

• Future value of an annuity determines the  annuity value at a future time • Types of annuity –Ordinary Annuity – An annuity whose deposits occur  5 © Dr. Chula King All Rights Reserved at the end of each period; determines the future  value immediately after the last deposit. –Annuity Due – An annuity whose deposits occur at the  beginning of each period; determines the future  value one period after the last deposit.

Future Value of an Ordinary Annuity

• An annuity whose deposits occur at the end of  each period. ….. 0

X

X

1 2 3 |‐‐‐‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐‐‐‐‐| 100 100 100

Example

• What is the future value of a three year  ordinary annuity of 100 at 10% compounded  annually? 7 © Dr. Chula King All Rights Reserved

A Pictorial Representation

0 1 2 3 |‐‐‐‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐‐‐‐‐| 100

X

X

100 100 8 © Dr. Chula King All Rights Reserved x 1.10  110 210 x 1.10  231 331 Immediately after the last deposit + +

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

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 (OA: 1.0000; AD: 1+  Interest) • Select the column corresponding to the interest  rate per period 10 © Dr. Chula King All Rights Reserved p p • Read down that column to the row that  corresponds to the number of deposits • Take the resultant factor, and multiply it by the  annuity amount • The result is the future value of an ordinary  annuity or annuity due!

Example

• What is the future value of a three year  ordinary annuity of 100 at 10% compounded  annually? 11 © Dr. Chula King All Rights Reserved

Future Value of An Ordinary Annuity

….. 0 1 2 3 |‐‐‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐‐‐‐| 100 100      100

X

X

n = # Deposits (n=3, i=10%)

Future Value of an Ordinary Annuity of $1

(n) i Periods 2% 4% 6% 8% 10% 1 1 00000 1 00000 1 00000 1 00000 1 00000 13 © Dr. Chula King All Rights Reserved 1 1.00000 1.00000 1.00000 1.00000 1.00000 2 2.02000 2.04000 2.06000 2.08000 2.10000 3 3.06040 3.12160 3.18360 3.24640 3.31000 4 4.12161 4.24646 4.37462 4.50611 4.64100 5 5.20404 5.41632 5.63709 5.86660 6.10510 6 6.30812 6.63298 6.97532 7.33592 7.71561

Future Value of An Ordinary Annuity

….. 0 1 2 3 |‐‐‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐‐‐‐| 100 100      100

X

X

n = # Deposits 14 © Dr. Chula King All Rights Reserved (n=3, i=10%) 3.3100 x 100 = 331.00

Immediately after the last deposit

Example

• Amos Adams is 60 years old today.  When he  retires in five years at the age of 65, he would  like  to have $100,000 available to supplement  his social security Beginning on his 61st

15 © Dr. Chula King All Rights Reserved his social security.  Beginning on his 61 birthday, Amos plans to make five equal  annual deposits in a fund that earns 6%  compounded annually.  How much must each  deposit be?

Pictorial Representation of Example

….60      61 62 63      64      65  |‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐| 100,000

X

X

16 © Dr. Chula King All Rights Reserved X        X X X X (n=5, i=6%)

Future Value of an Ordinary Annuity of $1

(n) i Periods 2% 4% 6% 8% 10% 1 1 00000 1 00000 1 00000 1 00000 1 00000 17 © Dr. Chula King All Rights Reserved 1 1.00000 1.00000 1.00000 1.00000 1.00000 2 2.02000 2.04000 2.06000 2.08000 2.10000 3 3.06040 3.12160 3.18360 3.24640 3.31000 4 4.12161 4.24646 4.37462 4.50611 4.64100 5 5.20404 5.41632 5.63709 5.86660 6.10510 6 6.30812 6.63298 6.97532 7.33592 7.71561

The Solution

….60      61 62 63      64      65  |‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐| 100,000

X

X

X       X X X X (n=5, i=6%) X x 5.63709 = 100,000 X = 100,000 ÷ 5.63709 = $17,740

Future Value of an Annuity Due

• An annuity whose deposits occur at the  beginning of each period. • The future value is determined one period after  19 © Dr. Chula King All Rights Reserved the last deposit. ….. 0 1 2 3 |‐‐‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐‐‐‐‐| 100 100 100

Example

• What is the value of a three year annuity due  of 100 at 10% compounded annually? 20 © Dr. Chula King All Rights Reserved

The Solution

….. 0 1 2 3 |‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐| 100 100        100 21 © Dr. Chula King All Rights Reserved (Table 5, n=3, i=10%)

Future Value of an Annuity Due of $1

(n) i Periods 2% 4% 6% 8% 10% 1 1 0200 1 0400 1 0600 1 0800 1 1000 22 © Dr. Chula King All Rights Reserved 1 1.0200 1.0400 1.0600 1.0800 1.1000 2 2.0604 2.1216 2.1836 2.2464 2.3100 3 3.1216 3.2465 3.3746 3.5061 3.6410 4 4.2040 4.4163 4.6371 4.8666 5.1051 5 5.3081 5.6330 5.9753 6.3359 6.7156 6 6.4343 6.8983 7.3938 7.9228 8.4872

The Solution

….. 0 1 2 3 |‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐| 100 100       100 23 © Dr. Chula King All Rights Reserved (Table 5, n=3, i=10%) 3.6410 x 100 = 364.1 FV of Ordinary Annuity versus Annuity Due Ordinary Annuity 0

X X

1 2 3

DEP DEP DEP

0 1 2 3

Concluding Comments

• Part 5:  Present Value of an Annuity –Ordinary Annuity –Annuity Due 25 © Dr. Chula King All Rights Reserved