FIN 200:
Personal Finance
Topic 3-The Time Value of Money
Larry Schrenk, Instructor
Learning Objectives
3.
Explain the reasons for a time value of
money. ▪
Explain compounding and discounting.
Define a lump sum payment.
4.
Calculate the present and future value . ▪
1.
2.
The Time Value of Money
Time Value of Money


Why a Time Value of Money?
Components▪



Opportunity Cost
Inflation
Risk
NOTE: I will use ‘cash flow’ (CF) as a general term to
designate any flow of money positive or negative.▪
Compounding
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Compounding (Saving)



Compounding is calculating the amount you
will have in the future: if, for example, you put
money in a savings account.
If you make a deposit, called the present
value (PV), how much will you have after N
years if you get I/Y interest rate per year?▫
The answer is the future value (FV).
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Future Value (FV)



Compounding
This is the equivalent to a one-time deposit in
a savings account.
If I put in $100.00 today, how much will I have
in...



One year?
Ten years?
One hundred years?
Note on Percentages

Percentages can be expressed in




Integer form 8%, or
Decimal form 0.08.
These are mathematically the same.
But calculators normally have a ‘percentage
convention’ when you do financial
calculations:

If the interest rate is 12%, you should type 12 (the
calculator assumes the ‘%’).▫
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Future Value (FV)

Calculating the Future Value


How much do I have after one year?
If the interest rate (I/Y) is 10%, then
$100.00 × (1 + 10%) = $100.00 × 1.1 = $110.00▫


I multiply the original sum by 1, because I still have
my original deposit ($100.00) and I also multiply by
0.10 to calculate the additional interest ($10.00).
NOTE: In calculations the red font will indicate the
solution or emphasize a certain part of a calculation
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Future Value (FV)

Calculating the Future Value



How much do I have after two years?▪
At the beginning of the second year I have
$110.00.
If the interest rate is 10%, then in two years:
$110.00 × (1 + 10%) = $110.00 × 1.1 = $121.00▪
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Compound Interest


Compound Interest: Interest is ‘compound’ because
we get interest on previous interest.
How do we get $121.00 after two years?
Original amount:
$100.00▪
Interest in first year:
$10.00
Interest in second year:
$10.00
Interest in the second year
on the interest from the first
year $10.00 × (1 + 10%):
$1.00
TOTAL
$121.00▪
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Simple Interest

Simple interest is when you do not receive
interest on the previous interest.

Deposit $100 at a simple interest rate of 10%.





Deposit
Year 1
Year 2
Year 3
$100
$110
$120
$130...
Simple interest is rarely used in modern
financial calculations because it
underestimates the true value of an
investment over several periods.
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Future Value (FV)

Calculating the Future Value


How much do I have after more years?
We can generalize the technique, so that each year
the value increases by 1 + I/Y (here 1 + 10%).
Year
Formula
0
Value
$100.00
1
$100.00(1.10) =
$110.00
2
$100.00(1.10)(1.10) =
$121.00
3
$100.00(1.10)(1.10)(1.10) =
$133.10
4
$100.00(1.10)(1.10)(1.10)(1.10) =
$146.41
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Timelines

If the timing of cash flows is ever confusing,
use a timeline:
I/Y
I/Y
1
0
I/Y
I/Y
2
3
4
FV
PV
10%
0
$100.00
10%
1
10%
10%
2
3
4
???
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Future Value Formula

We could construct a formula:
FV  PV 1  I / Y 




N
FV is the value of our money in year N.
PV is how much we invest now.
I/Y is the interest rate each year.
N is the number of years we let it grow.
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Calculations

Possible Methods of Calculation



Formulae–Complicated
Tables (Textbook)–Confusing
Calculator!
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Calculator Help



In examples I use one particular calculator,
but fortunately most financial calculators work
is almost the same way.
If you get into trouble, first try reading the
manual, though these can be very confusing.
If you can’t figure it out, didn't get frustrated,
instead...
Come to office hours. Bring your calculator
and the manual.
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Calculator Buttons

For Now…






FV = Future Value
PV = Present Value
N = Number of Payments
I/Y, I = Interest Rate
CPT = Compute (only on the TI)
Later…


PMT = Payment
P/Y = Payments per Year
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Future Value with a Calculator
How much do we have after 4 years if we
begin with $200 and the interest rate is
12%?▪

Input 4, Press N
2.
Input 12, Press I/Y
3.
Input 200, press +/-, press PV (you get -200)
(Why negative? In a minute)
4.
Press CPT, FV to get 314.70, i.e., $314.70
NOTES: 1) Calculators assume the % when you
press the I/Y key (do not input 12% as 0.12), 2)
some calculators do not require the CPT key,
and 3) the order of the inputs does not matter.▪
1.
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Future Value with a Calculator
0
I/Y
1
I/Y
2
I/Y
I/Y
3
FV
PV
Number of
Periods
N
Annual
Interest
Present
Value
Future
Value
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Future Value with a Calculator
12%
0
12%
1
2
12%
12%
3
4
$314.70
$200.00
4
12
-200
314.70
Remember to press CPT, before FV (TI Only).
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Essential Note: Clearing/Resetting

When you start a new problem, remove any values
the calculator may hold from the last calculation.



You can ‘clear’ selected values. This is the process of
returning them to the default (usually 0 for numeric values).
The more thorough solution is to ‘reset’ your calculator which
clears all values, e.g., you will lose any numbers held in
memory.
Do not assume that turning your calculator off and on
clears all the values.
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TI and HP Calculators

Reset/Clear the TI





[2nd ]
[RESET]
[ENTER]
“RST
0.00”
Reset/Clear the HP


[Orange]
[C ALL]
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Why the Negative?

We calculate:
FV  PV 1  I / Y 

N
The calculator calculates:
FV  PV 1  I / Y   0
N

For the latter calculation, one and only one of the cash
flows we input must be negative, but it does not matter
which one.
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Compounding Practice Problems



How much is $350.00 worth in 5 years if the
interest rate is 9%?▪
$538.52
How much is $400.00 worth in 15 years if the
interest rate is 11%?
$1,913.84
How much is $1.00 worth in 100 years if the
interest rate is 15%?
$1,174,313.45▪
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Discounting
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Discounting





Discounting is calculating the current value
(PV) of money coming in the future.
What is the current value (PV) of money (FV)
I expect to receive in N years given I/Y
interest?
The answer is the present value (PV).
If someone promises me $100.00 next year,
how much is that worth today?
Or how much would I need to save today to
have $100.00 next year?
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Discounting

Discounting is the exact opposite of
compounding.


More technically, discounting is the inverse of
compounding.
If I start with $100.00, compound it and then
discount it (using the same values, e.g., N), I
get the original $100.00.
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Discounting

What is the value today of $100.00 I receive it
in...



One year?
Ten years?
One hundred years?
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Discounting

Calculating the Present Value


How much is money worth if I receive it in one
year?
If the interest rate (I/Y) is 10%, then
$100.00/(1 + 10%) = $100.00/1.1 = $90.91


All I did was change the ‘×’ to ‘/’ in the formula.
I divide the original future value by 1 + 10%,
because 10% is the growth of money over time.
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Discounting

We can generalize this for money coming at
different times:
Year
Formula
0
Value
$100.00
1
$100.00/(1.10) =
$90.91
2
$100.00/(1.10)(1.10) =
$82.64
3
$100.00/(1.10)(1.10)(1.10) =
$75.13
4
$100.00/(1.10)(1.10)(1.10)(1.10) =
$68.30
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Timelines

Again, if the timing of cash flows is ever
confusing, use a time line:
I/Y
I/Y
1
0
I/Y
I/Y
2
3
4
FV
PV
10%
0
???
10%
1
10%
10%
2
3
4
$100.00
32 (of 40`)
Present Value Formula

We could construct a formula:
PV 





FV
1  I / Y 
N
FV is the value of our money in year N.
PV is how much we invest now.
I/Y is the interest rate each year.
N is the number of years we let it grow.
But again we will just use a calculator.
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Present Value with a Calculator
How much is $200 received in 4 years worth
now, if we the interest rate is 12%?▪

Input 4, press N
2.
Input 12, press I/Y
3.
Input 200, press +/-, press FV (you get -200)
4.
Press CPT, PV to get 127.10, i.e., $127.10
NOTES: 1) Calculators assume the % when you
press the I/Y key (do not input 12% as 0.12), 2)
some calculators do not require the CPT key, and
3) the order of the inputs does not matter. ▪
1.
34 (of 40`)
Present Value with a Calculator
12%
0
12%
1
2
12%
12%
3
4
$200.00
$127.10
4
12
127.10
-200
Remember to press CPT, before FV (if necessary).
35 (of 40`)
Discounting Practice Problems



How much is $350.00 received in 5 years
worth if the interest rate is 9%?▪
$227.48
How much is $400.00 received in 15 years
worth if the interest rate is 11%?
$83.60
How much is $1,000,000 received in 100
years worth if the interest rate is 15%?
85 cents!▪
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Ethical Dilemma (Chap. 3, 76.16)

Cindy and Jack have budgeted $300 per month for car
payments. A salesman, Herb, insists that they look at a more
expensive car with payments of $500 per month. They can only
afford the expensive car by discontinuing a $200 monthly
retirement contribution. Since they plan to retire in 30 years, Herb
explains that they would only need to stop the $200 monthly
payments for the five years of the car loan and calculates that the
$12,000 in lost contributions could be made up over the
remaining 25 years by increasing their monthly contribution by
only $40 per month.
 a. Comment on the ethics of a salesperson who attempts to talk
customers into spending more than they had originally planned
and budgeted.
 b. Is Herb correct in his calculation?
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