The Time Value of Money Future Value and Present Value .docx

The Time Value of
Money: Future Value
and Present Value
Computations
"If I deposit $10,000 today, how much will I have for a down
payment on a house in five years?"
"Will $2,000 saved a year give me enough money when I
retire?"
"How much must I save today to have enough for my children's
post-secondary education?"
As introduced in Chapter 1 and used to measure financial
opportunity costs in other chapters,
the time value of money, more commonly referred to as interest,
is the cost of money that is bor-
rowed or lent. Interest can be compared to rent, the cost of
using an apartment or other item.
The time value of money is based on the fact that a dollar
received today is worth more than a
dollar that will be received one year from today because the
dollar received today can be saved
or invested and will be worth more than a dollar a year from
today Similarly, a dollar that will
be received one year from today is currently worth less than a
dollar today.
The time value of money has two major components: future
value and present value. Future
value computations, which are also referred to as compounding,
yield the amount to which a

current sum will increase based on a certain interest rate and
period of time, Present value,
which is calculated through a process called discounting, is the
current value of a future sum
based on a certain interest rate and period of time.
In future value problems, you are given an amount to save or
invest and you calculate the
amount that will be available at some future date. With present
value problems, you are given the
amount that will be available at some future date and you
calculate the current value of that amount
Both future value and present value computations are based on
basic interest rate calculations.
FINANCIAL CALCULATORS
Currently, financial calculators, with time value of money
functions built in, are widely used to
calculate future value, present values, and annuities, For the
following examples, we will use the
Texas Instruments BA II Plus financial calculator, which is
recommended by the Canadian
Institute of Financial Planning.
When using the BA II Plus calculator to solve time value of
money problems, you will be
working with the TVM keys that include:
CPT — Compute key used to initiate financial calculations once
all values are inputted
— Number of periods
— Interest rate per period
— Present value

PMT — Amount of payment, used only for annuities
— Future value
Enter values for PV, PMT, and FV as negative if they represent
cash outflows (e.g., investing
a sum of money) or as positive if they represent cash inflows
(e.g., receiving the proceeds of an
investment). To convert a positive number to a negative
number, enter the number and then
press the +/— key.
37
Part 1 PLANNING YOUR PERSONAL FINANCES
The examples that are shown in this chapter assume that interest
is compounded
annually and that there is only one cash flow per period. To
reflect this, we must set the number
of payments and compounding per period to 1 (the default
setting is 12). To do this, press in turn
the I 2!'°) button (yellow), the 1 I/Y button (for the PLY
function shown above it), the
number 1, the ENTER button, the 1 2ND , button again, and
finally the I CPT button
(for the quit function above it). Before using any financial
calculator, we strongly recommend that
you consult the instruction manual that accompanies it and
attempt the examples shown there.

Now let's try a problem. What is the future value of $100 after
three years at a 10 percent
annual interest rate? Remember that an investment of money is
considered to be an outflow of
cash; therefore, the present value of $100 should be entered as a
negative number.
First, you must enter the data. Remember that an investment of
money is considered to be
an outflow of cash; therefore, the $100 should be entered as a
negative number.
3
10
100 + / —
I IN
PV
0 PMT (optional if registers are cleared)
To find the solution, the future value, press , and the future
value of 133.1 is
displayed.
FUTURE VALUE OF A SINGLE AMOUNT
The future value of an amount consists of the original amount
plus compound interest. This
calculation involves the following elements:
FV = Future value

PV = Present value
i = Interest rate
n = Number of time periods
The formula and financial calculator computations are as
follows:
(Note: These financial calculator notations may require slightly
different keystrokes when
using various brands and models.)
Future Value of a Single Amount
Formula Table Financial Calculator
FV = PV(1 +
Using Exhibit 1B-1:
FV = PV (Table Factor)
PV L IN N PMT , 1 CPT FV1
Example A: The future value of $1 at 10 percent after three
years is $1.33. This amount is calculated as follows:
$1.33 = $(1.001 + 0.10)3 Using Exhibit 1B-1: 1 1 PV ,
10E 3 N ,OLPMT ,I CPT 1.33
$1.33 = $1.00(1.33)
Future value tables are available to help you determine
compounded interest amounts (see Exhibit 1B-1).
Looking at Exhibit 1B-1 for 10 percent and three years, you can
see that $1 is worth $1.33 at that time. For
other amounts, multiply the table factor by the original amount.
This process may be viewed as follows:

Example B: If your savings of $400 earns 12 percent,
compounded monthly, over a year and a half, use the
table factor for 1 percent for 18 time periods; the future value
is:
$478.46 = $400(1 + 0.01)18 -$478.40-= $400(1.196) 400
PV 12/12 = 1.5 x 12 = 18 I N
01 PMT1 I CPT', -:FV 478.46
FV = Annuity -
FV = Annuity (Table Factor)
I PMT I N IIY [ PV CPT L FV (1 + - 1 Using
Exhibit 1B-2:
Appendix 1B The Time Value of Money: Future Value and
Present Value Computations
Sample Problem 1 What is the future value of $800 at 8 percent
after six years?
Sample Problem 2 How much would you have in savings if you
kept $200 on deposit for
eight years at 8 percent, compounded semi-annually?
FUTURE VALUE OF A SERIES OF EQUAL
AMOUNTS (AN ANNUITY)
Future value may also be calculated for a situation in which
regular additions are made to sav-

ings. The formula and financial calculator computations are as
follows:
Future Value of a Series of Payments
This calculation assumes that: (1) each deposit is for the same
amount; (2) the interest rate is the same for
each time period; and (3) the deposits are made at the end of
each time period.
Example C: The future value of three $1 deposits made at the
end of the next three years, earning 10 percent
interest, is $3.31. This is calculated as follows:
(1 + 0.10)3 - 1 Using Exhibit 1B-2:
$3.31 -
0.10 $3.31 = $1 x 3.31
- 1 PMT, 31 N ,i0( IN ,
FV 3.31
This may be viewed as follows:
Example D: If you plan to deposit $40 a year for 10 years,
earning 8 percent compounded annually, the future
value of this amount is:
$40(1 + 0.08)" - 1
$579.46 =

0.08
Using Exhibit 1B-2:
$579.48 = $40(14.487)
- 40 PMT 10
[ CPT_[J L 579.46
01 IN ,o( PV ,
Sample Problem 3 What is the future value of an annual deposit
of $230 earning
6 percent for 15 years?
Sample Problem 4 What amount would you have in a retirement
account if you made
annual deposits of $375 for 25 years earning 12 percent,
compounded annually?
PRESENT VALUE OF A SINGLE AMOUNT
If you want to know how much you need to deposit now to
receive a certain amount in the
future, the formula and financial calculator computations are as
follows:
Present Value of a Single Amount
FV
PV -

(1 + On
Using Exhibit 1B-3:
PV = FV (Table Factor)
I N IN , I PMT t CPT_ PV V
Example E: The present value of $1 to be received three years
from now based on a 10 percent interest rate is
calculated as follows:
$1 Using Exhibit 1B-3:
$0.75 -
(1 + 0.10)3 $0.75 = $1(0.751)
1 FV , 10[ IN ,0 I PMT ,
I CPT I PV —75131
(continued)
1
1
(1 + 0.14)10
0.14 _
Using Exhibit 1B-4: 100 PMf, 101 N , 141 IN . 0 1 FV ,
$521.60 = $100(5.216)
1 CPT I PV ' 521.61156
$521.61 = $100

Part 1 PLANNING YOUR PERSONAL FINANCES
Present value tables are available to assist you in this process
(see Exhibit 1B-3). Notice that $1 at 10 percent
for three years has a present value of $0.75. For amounts other
than $1, multiply the table factor by the
amount involved.
Example F: If you want to have $300 seven years from now and
your savings earn 10 percent, compounded
semi-annually (5 percent for 14 time periods), finding how
much you would have to deposit today is
calculated as follows:
$300 Using Exhibit 1B-3:
$151.52 =
(1 + 0.05)14 $151.50 = $300(0.505)
300 FV , 7 x 2 = 141 N , 10/2 = 1 I/Y ,
0 t PMT ,1 CPT; PV -151.52
Sample Problem 5 What is the present value of $2,200 earning
15 percent for eight years?
Sample Problem 6 To have $6,000 for a child's education in 10
years, what amount should
a parent deposit in a savings account that earns 12 percent,
compounded quarterly?
PRESENT VALUE OF A SERIES OF EQUAL
AMOUNTS (AN ANNUITY)
The final time value of money situation allows you to receive an

amount at the end of each time
period for a certain number of periods. The formula and
financial calculator computations are
as follows:
Present Value of a Series of Payments
1
1 Using Exhibit 1B-4:
(1 + On PV = Annuity (Table Factor)
PV = Annuity )<
Example G: The present value of a $1 withdrawal at the end of
the next three years would be $2.49, for
money earning 10 percent. This would be calculated as follows:
1
(1 + 0.10)3
0.10 _
Using Exhibit 1B-4:
$2.49 = $1(2.487)
1IPMT,31 N , 101 ,O1 FV ,
CPT PV -2.48685
$2.49 = $1

This same amount appears in Exhibit 1B-4 for 10 percent and
three time periods. To use the table for other
situations, multiply the table factor by the amount to be
withdrawn each year.
Example H: If you wish to withdraw $100 at the end of each
year for 10 years from an account that earns
14 percent, compounded annually, what amount must you
deposit now?
Sample Problem 7 What is the present value of a withdrawal of
$200 at the end of each
year for 14 years with an interest rate of 7 percent?
P T' L IN 11-FV 1 CPT j TV
ample Problem 8
$650 at the end of e
ow much would you have to deposit now to be able to withdraw
year for 20 years from an account that earns 11 percent?
Appendix 1B The Time Value of Money: Future Value and
Present Value Computations-
Present value tables can also be used to determine instalment
payments for a loan as
follows:

USING PRESENT VALUE TO DETERMINE
LOAN PAYMENTS
Present Value to Determine Loan Payments
Table
Financial Calculator
Example I: If you borrow $1,000 with a 6 percent interest rate
to be repaid in three
equal payments at the end of the next three years, the payments
will be $374.11. This
is calculated as follows:
Sample Problem 9 What would be the annual payment amount
for a $20,000, 10-year
loan at 7 percent?
$1,000
— $374.11
2,673
1000 6 3
374.10981
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