# Future Value of an Ordinary Annuity by pengxuebo

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Appendix D

Compound
Interest
2

Objectives
1. Understand simple interest and compound
interest.
2. Compute and use the future value of a
single sum.
3. Compute and use the present value of a
single sum.
4. Compute and use the future value of an
ordinary annuity.
5. Compute and use the future value of an
annuity due.    Continued
3

Objectives
6. Compute and use the present value of an
ordinary annuity.
7. Compute and use the present value of an
annuity due.
8. Compute and use the present value of a
deferred ordinary annuity.
9. Explain the conceptual issues regarding the
use of present value in financial reporting.
4

Simple Interest
Simple interest is interest
on the original principal
regardless of the number
of time periods that have
passed.

Interest = Principal x Rate x Time
5

Compound Interest
Compound interest is the interest
that accrues on both the principal
and the past unpaid accrued
interest.
6

Compound Interest
Value at                               Value at
Beginning                   Compound    End of
Period   of Quarter x Rate   x Time = Interest   Quarter
1st qtr. \$10,000.00 x 0.12 x 1/4 \$ 300.00 \$10,300.00
2nd qtr. 10,300.00 x 0.12 x 1/4    309.00 10,609.00
3rd qtr. 10,609.00 x 0.12 x 1/4    318.27 10,927.27
4th qtr. 10,927.27 x 0.12 x 1/4    327.82 11,255.09
5th qtr. 11,255.09 x 0.12 x 1/4    337.65 11,592.74
Compound interest on \$10,000 at
12% compounded quarterly for
5 quarters………………………... \$1,592.74
7

Future Value of a Single Sum at
Compound Interest
One thousand dollars is invested in a savings account
on December 31, 2004. What will be the amount in
the savings account on December 31, 2008 if interest
at 14% is compounded annually each year?
How much will be in the
\$1,000 is invested
savings account (the future
on this date
value) on this date?

Dec. 31,    Dec. 31,     Dec. 31,     Dec. 31,     Dec. 31,
2004        2005         2006         2007         2008
8

Future Value of a Single Sum at
Compound Interest
(1)       (2)              (3)               (4)
Annual          Future Value
Value at        Compound             at End
Beginning of       Interest          of Year
Year      Year        (Col. 2 x 0.14)   (Col. 2 + Col. 3)
2005   \$1,000.00         \$140.00        \$1,140.00
2006    1,140.00          159.60         1,299.60
2007    1,299.60          181.94         1,481.54
2008    1,481.54          207.42         1,688.96
9

Future Value of a Single Sum at
Compound Interest
Formula Approach
n
ƒ = p(1 + i)
where ƒ = future value of a single sum at compound
interest i and n periods
p = principal sum (present value)
i = interest rate for each of the stated time
periods
n = number of time periods
10

Future Value of a Single Sum at
Compound Interest
Formula Approach

f = p(1 + i) n
4
fn=4, i=14 = (1.14)
f = \$1,000(1.688960) = \$1,688.96
11

Future Value of a Single Sum at
Compound Interest
Table Approach

This time we will use a table to
determine how much \$1,000 will
accumulate to in four years at 14%
compounded annually.
12

Future Value of a Single Sum at
Compound Interest
Table Approach

Using Table 1 (the future value
of 1) at the end of Appendix D,
determine the table value for an
annual interest rate of 14
percent and four periods.
13

Future Value of a Single Sum at
Compound Interest
Table Approach
n      8.0%     9.0%      10.0%      12.0%      14.0%      16.0%
1   1.080000 1.090000   1.100000   1.120000   1.140000   1.160000
2   1.166400 1.188100   1.210000   1.254400   1.299600   1.345600
3   1.259712 1.295029   1.331000   1.404928   1.481544   1.560896
4   1.360489 1.411582   1.464100   1.573519   1.688960
1.688960 1.810639
5   1.469328 1.538624   1.610510   1.762342   1.925415   2.100342
6   1.586874 1.677100   1.771561   1.973823   2.194973   2.436396
14

Future Value of a Single Sum at
Compound Interest
Table Approach

One thousand dollars times
1.688960 equals the future
value, or \$1,688.96.
15

Present Value of a Single Sum
If \$1,000 is worth \$1,688.96 when it earns 14%
compounded annually for 4 years, then it follows
that \$1,688.96 to be received in 4 years from now is
worth \$1,000 now at time period zero.

\$1,000 (the
present value)                   For \$1,688.96 to be
must be invested                 received on this date
on this date

Dec. 31,    Dec. 31,     Dec. 31,      Dec. 31,      Dec. 31,
2004        2005         2006          2007          2008
16

Present Value of a Single Sum
Formula Approach
1
p = f (1 + i) n
Where p = present value of any given future value due
in the future
ƒ = future value
i = interest rate for each of the stated time
periods
n = number of time periods
17

Present Value of a Single Sum
Formula Approach

1
p n=4, i=14 =   (1 .14) 4
= 0.592080

p = \$1,688.96(0.592080) = \$1,000.00
18

Present Value of a Single Sum
Table Approach

Use 14% 3, the present
Find Tableand four
value of obtain end
periods to1, at thethe of
Appendix
table value. D.
19

Present Value of a Single Sum
Table Approach

n       8.0%     9.0%       10.0%      12.0%      14.0%      16.0%
1    0.925926 0.917431    0.909091   0.892857   0.877193   0.862069
2    0.857339 0.841680    0.826446   0.797194   0.769468   0.743163
3    0.793832 0.772183    0.751315   0.711780   0.674972   0.640658
4    0.735030 0.708425    0.683013   0.635518   0.592080
0.592080 0.552291
5    0.680583 0.649931    0.620921   0.567427   0.519369   0.476113
6    0.630170 0.596267    0.564474   0.506631   0.455587   0.410442
20

Present Value of a Single Sum
Table Approach

\$1,688.96 times
0.592080 equals
\$1,000.
21

Future Value of an
Ordinary Annuity
The future value
Debbi Whitten wants to calculate the future value of
of an ordinary
four cash flows of \$1,000, each with interest
annuity
compounded annually at 14%, where the first cash is
flow is made on December 31, 2004.  determined
immediately
after the last cash
flow
\$1,000      \$1,000        \$1,000       \$1,000

Dec. 31,    Dec. 31,      Dec. 31,     Dec. 31,
2004        2005          2006         2007
22

Future Value of an
Ordinary Annuity
Formula Approach
n
(1 + i) - 1
Fo = C
i
Where Fo= future value of an ordinary annuity of a
series of cash flows of any amount
C = amount of each cash flow
n = number of cash flows
i = interest rate for each of the stated time
periods
23

Future Value of an
Ordinary Annuity
Formula Approach

(1 .14)4 – 1
Fo = n=4, i=14 =                  = 4.921144
0.14

Fo = \$1,000(4.921144) = \$4,921.14
24

Future Value of an
Ordinary Annuity
Table Approach
Using the same data—four
Go to Table 2, the future value of
equal annual cash flows of
an ordinary annuity of 1. Read
\$1,000 beginning on December
31, 2004 and an interest rate and
the table value for n equals 4 of i
equals 14%.
14 percent.
25

Future Value of an
Ordinary Annuity
Table Approach

n      8.0%     9.0%       10.0%      12.0%      14.0%      16.0%
1   1.000000 1.000000    1.000000   1.000000   1.000000   1.000000
2   2.080000 2.090000    2.100000   2.120000   2.140000   2.160000
3   3.246400 3.278100    3.310000   3.374400   3.439600   3.505600
4   4.506112 4.573129    4.641000   4.779328   4.921144
4.921144 5.066496
5   5.866601 5.984711    6.105100   6.352847   6.610104   6.877135
6   7.335929 7.523335    7.715610   8.115189   8.535519   8.977477
26

Future Value of an
Ordinary Annuity
So, cash flow of \$1,000 each at
14% at the end of 2004, 2005,
2006, and 2007 will accumulate to
a future value of \$4,921.14.

\$1,000 x 4.921144 = \$4,921.14
27

Future Value of an
Annuity Due
Solutions Approach

How much will be in the fund on
this date, which is 1 period after
the last cash flow in the series?
\$1,000     \$1,000        \$1,000        \$1,000

Dec. 31,   Dec. 31,      Dec. 31,     Dec. 31,
2004       2005          2006         2007
28

Future Value of an
Annuity Due
Solutions Approach
Step 1:
In the ordinary annuity table (Table 2),
look up the value of n + 1 cash flows at
14% or the value of 5 cash flows at 14%.
29

Future Value of an
Annuity Due
Solutions Approach
n      8.0%     9.0%      10.0%      12.0%      14.0%      16.0%
1   1.000000 1.000000   1.000000   1.000000   1.000000   1.000000
2   2.080000 2.090000   2.100000   2.120000   2.140000   2.160000
3   3.246400 3.278100   3.310000   3.374400   3.439600   3.505600
4   4.506112 4.573129   4.641000   4.779328   4.921144   5.066496
5   5.866601 5.984711   6.105100   6.352847   6.610104
6.610104 6.877135
6   7.335929 7.523335   7.715610   8.115189   8.535519   8.977477
30

Future Value of an
Annuity Due
Solutions Approach
Step 1:
In the ordinary annuity table (Table 2),
look up the value of n + 1 cash flows at
14% or the value of 5 cash flows at 14%.   6.610104
Step 2:
Subtract 1 without interest.               (1.000000)
Table value                                5.610104
31

Future Value of an
Annuity Due
Solutions Approach
Step 3:
Multiply the amount of each cash flow
(\$1,000) by the table value from Step 2.

Fd = \$1,000(5.610104) = \$5,610.10
32

Future Value of an
Annuity Due

So, if \$1,000 is deposited
…a cumulative total of
annually for four years
\$5,610 can be withdrawn
beginning on December
on December 31, 2008.
31, 2004…
33

Present Value of an
Ordinary Annuity
Table Approach
Kyle Vasby wants to calculate the present value on
January 1, 2004 (one period before the first cash flow)
of four future withdrawals (cash flows) of \$1,000 each,
with the first withdrawal being made on December 31,
2004. Assume again an interest rate of 14%.

\$1,000       \$1,000       \$1,000       \$1,000

Dec. 31,     Dec. 31,     Dec. 31,     Dec. 31,
2004         2005         2006         2007
34

Present Value of an
Ordinary Annuity

Go to Table 4, the present value of an
ordinary annuity of 1. Read the table
value for n equals 4 and i equals 14%.
35

Present Value of an
Ordinary Annuity
Table Approach

n      8.0%     9.0%       10.0%      12.0%      14.0%      16.0%
1   0.925926 0.917431    0.909091   0.892857   0.877193   0.862069
2   1.783265 1.759111    1.735537   1.690051   1.646661   1.605232
3   2.577097 2.531295    2.486852   2.401831   2.321632   2.245890
4   3.312127 3.239720    3.169865   3.037349   2.913712
2.913712 2.798181
5   3.992710 3.889651    3.790787   3.604776   3.433081   3.274294
6   4.622880 4.485919    4.355261   4.111407   3.888668   3.684736
36

Present Value of an
Ordinary Annuity
Table Approach

One thousand dollars times
2.913713 equals \$2,913.71. So,
the present value of this ordinary
annuity is \$2,913.71.
37

Present Value of an Annuity Due

Table Approach
Barbara Livingston wants to calculate the present
value of an annuity on December 31, 2004, which
will permit four annual future receipts of \$1,004
each, the first to be received on December 31, 2004.

\$1,000       \$1,000       \$1,000       \$1,000

Dec. 31,    Dec. 31,     Dec. 31,     Dec. 31,
2004        2005         2006         2007
38

Present Value of an Annuity Due

Table Approach

Step 1:
In the ordinary annuity table (Table 4),
look up the value of n – 1 cash flows at
14% or the value of 3 cash flows at 14%.
39

Present Value of an Annuity Due

Table Approach

n      8.0%     9.0%      10.0%      12.0%      14.0%      16.0%
1   0.925926 0.917431   0.909091   0.892857   0.877193   0.862069
2   1.783265 1.759111   1,735537   1.690051   1.546661   1.605232
3   2.577097 2.531295   2.485852   2.402831   2.321632
2.321632 2.245890
4   3.312127 3.329720   3.159865   3.037349   2.913712   2.798181
5   3.992710 3.889651   3.790787   3.604776   3.443081   3.274294
6   4.622880 4.485919   4.355261   4.111407   3.888668   3.684736
40

Present Value of an Annuity Due

Table Approach

Step 1:
In the ordinary annuity table (Table 4),
look up the value of n – 1 cash flows at
14% or the value of 3 cash flows at 14%.   2.321632
Step 2:
3.321632
41

Present Value of an Annuity Due

Table Approach

One thousand dollars times
3.321632 equals \$3,321.63.
So, this is the present value
of an ordinary annuity due.
42

Present Value of a Deferred
Ordinary Annuity
Table Approach

Helen Swain buys an annuity on January 1,
2004 that yields her four annual payments of
\$1,000 each, with the first payment on January
1, 2008. The interest rate is 14% compounded
annually. What is the cost of the annuity?
43

Present Value of a Deferred
Ordinary Annuity
The present              Table Approach
value of the
deferred                     \$1,000    \$1,000    \$1,000     \$1,000
annuity is
determined                     Jan. 1,   Jan. 1,               Jan. 1,
Jan. 1,
on this date                    2008      2009                  2011
2010

Jan.1, Jan. 1, Jan. 1, Jan. 1,
2004 2005 2006 2007
\$1,000 x 2.913712(n=4,
i=14) = \$2,913.71
44

Present Value of a Deferred
Ordinary Annuity
The present              Table Approach
value of the
deferred
annuity is                      \$2,913.71
determined
on this date

Jan.1, Jan. 1, Jan. 1, Jan. 1,
2004 2005 2006 2007
\$2,913.71 x 0.674972 =
\$1,966.67
45

Present Value of a Deferred
Ordinary Annuity
If Helen buys an annuity for
\$1,966.67 on January 1, 2004, she
can make four equal annual
\$1,000 withdrawals (cash flows)
beginning on January 1, 2008.
46

Appendix D

The End
47

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