Topic 2: Time Value of Money

Topic Overview
Lesson 2.1
2.1.1 Time Value of Money
2.1.2 Future Value and Present Value
2.1.3 Compounding
2.1.4 Discounting
2.1.5 Moving Cash Flows Across
Time
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 2
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2.1.1 Time value of money
 We saw in Topic 1 that the corporate
objective is to maximise the value of the firm
COST-BENEFIT ANALYSIS
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 3
 However, financial decision-making involves
more than just comparing numbers of dollars
 To evaluate the costs and benefits of a
decision, we must value the options in the
same terms – cash today
Dollar
value of
benefits
Dollar
value of
costs
The decision will
increase the value
of the firm > 
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2.1.1 Time value of money
 It is difficult to make decisions involving cash
flows at different points in time
 This is because money has time value
 This means that a dollar today is worth more
than a dollar tomorrow (because today’s
dollar can be invested and will grow to more
than one dollar tomorrow)
 Hence, the interest rate is a measure of the
time value of money at any given time
CASH FLOWS AT DIFFERENT TIMES
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 4
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2.1.1 Time value of money
CASH FLOWS AT DIFFERENT TIMES
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 5
Today One year
‐$100,000 +$105,000
Would you undertake this investment?
A Yes B No C Don’t know
Today One year
‐$100,000 +$105,000
 Would you undertake this investment?
 It is impossible to answer this question
without knowing the interest rate – the rate
of return you could get from the next best
alternative investment
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2.1.1 Time value of money
CASH FLOWS AT DIFFERENT TIMES
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 6
Today One year
‐$100,000 Investment +$105,000
Would you undertake this investment?
A Yes B No C Don’t know
Today One year
‐$100,000 Investment +$105,000
 Would you undertake this investment if the
interest rate is 10% p.a.?
‐$100,000 Bank +$110,000
 No. $100,000 invested at 10% would be
worth $110,000 in one year
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2.1.1 Time value of money
 The previous shows that if the interest rate
is 10% p.a., $100,000 today is equivalent
to $110,000 in one year
 More generally, we use r to denote the
interest rate for a given period, and
therefore we can exchange $1 today for
(1 + r) dollars in the future (since they are
equivalent in value)
 (1 + r) is the interest rate factor which can
be used to convert cash flows across time
CASH FLOWS AT DIFFERENT TIMES
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 7
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2.1.2 Future Value and Present Value
 We can use the interest rate factor to
convert from a cash flow today to an
equivalent cash flow one period in the
future (the future value):
FUTURE VALUE OF A CASH FLOW TODAY
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 8
FV  C1 r 
 The future value of $100,000 invested at
10% p.a. for one year is equal to:
FV  100,0001 0.10  $110,000
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2.1.2 Future Value and Present Value
 We can also use the interest rate factor to
convert a future cash flow to a present value
PRESENT VALUE OF A FUTURE CASH FLOW
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 9
1
PV C
r


 The present value of $110,000 received in
one year, at 10% p.a. interest, is equal to:
110,000 $100,000
1 0.10
PV  

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2.1.2 Future Value and Present Value
 The present value of the investment offering
a payment of $105,000 in 1 year is equal to:
PRESENT VALUE OF A FUTURE CASH FLOW
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 10
 We have therefore converted the benefit of
the investment ($105,000 in one year) to an
equivalent amount of cash today
 Since the cost ($100,000) is also in terms of
cash today, we can directly compare them
105,000 $95,454.55
1 0.10
PV  

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2.1.3 Compounding
Rule 1
 It is only possible to compare or combine
values at the same point in time
Rule 2
 To calculate the future value of a cash flow,
you must compound it
Rule 3
 To calculate the value of a cash flow at an
earlier point in time, you must discount it
RULES OF FINANCIAL DECISION-MAKING
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 11
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2.1.3 Compounding
 Compounding means calculating a future
value over many periods by repeatedly
applying the calculation we did earlier
 Consider an investment of $1000 at 10%
p.a. for 2 years:
COMPOUNDING
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 12
0 2
$1000 FV2
1
FV1
 Interest is earned in the 1st year, and then the
total amount earns interest in the 2nd year
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2.1.3 Compounding
 In each period, interest is earned on the
present value and all interest earned in prior
periods – this can result in substantial growth
over long periods of time
 E.g. $1000 at 10% p.a. for 40 years
Without compounding, this will grow to
$1000 + (40 x $100) = $5000
With annual compounding, this will grow to
$45,259
COMPOUNDING
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 13
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2.1.4 Discounting
 Discounting means calculating a present
value, many periods earlier, by repeatedly
applying the calculation we did earlier
 Consider $1000 to be received in 2 years,
where the interest rate is 10% p.a.
DISCOUNTING
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 14
0 2
PV0 $1000
1
PV1
 The $1000 is discounted once to get PV1,
and this value is discounted again to get PV0
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2.1.4 Discounting
 As we shall see in later topics, present value
is calculated more often in finance than
future value
 Most financial decisions involve comparing
cash flows that occur at different points in
time
 The usual way we do is to convert them all
to today’s dollars – i.e. discount them to a
present value – so that a comparison can
be made
DISCOUNTING
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 15
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2.1.5 Moving Cash Flows Across Time
 We will use a number of formulas from the
text book throughout the remainder of this
topic to solve time value of money problems
 However, we can see from what we have
done so far that we can move any cash flow
at any period of time to an equivalent cash
flow at any other period of time
 This is done by either compounding or
discounting, depending on which direction
we need to move in time
MOVING CASH FLOWS ACROSS TIME
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 16
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2.1.5 Moving Cash Flows Across Time
 Given a cash flow at a particular point in
time, if you need to find an equivalent cash
flow at a later point in time (i.e. move to the
right along the time line):
MOVING CASH LATER IN TIME
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 17
you multiply the cash flow by the interest rate
factor (1 + r) a total of n times, where n is the
number of periods you want to move in time.
$
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2.1.5 Moving Cash Flows Across Time
 Given a cash flow at a particular point in
time, if you need to find an equivalent cash
flow at an earlier point in time (i.e. move to
the left along the time line):
MOVING CASH EARLIER IN TIME
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 18
you divide the cash flow by the interest rate
factor (1 + r) a total of n times, where n is the
number of periods you want to move in time.
$
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Topic Overview
Lesson 2.2
2.2.1 Future Value of a Single
Cash Flow
2.2.2 Present Value of a Single
Cash Flow
2.2.3 Frequency of Compounding
2.2.4 Continuous Compounding
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 19
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2.2.1 Future Value of a Single Cash Flow
 The general formula for the future value of a
single cash flow is:
FUTURE VALUE OF A SINGLE CASH FLOW
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 20
where C = the cash flow
r = the interest rate per period
n = the number of periods
FVn = the future value of cash flow
C in period n
1 n
n FV  C   r (3.1)
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2.2.2 Present Value of a Single Cash Flow
 The general formula for the present value of
a single cash flow is:
PRESENT VALUE OF A SINGLE CASH FLOW
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 21
where C = the future cash flow
r = the interest rate per period
n = the number of periods
PV = the present value of cash
flow C paid in n periods
1 n
PV C
r

   (3.2)
1 
1
n
n
PV C C r
r
    

These two
versions are
mathematically
equivalent, but
you may find
one or the
other more
convenient in
some
situations
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2.2.3 Frequency of Compounding
 It is important to note that the r and n relate
to compounding periods – not years
 A compounding period is the length of time
over which interest accrues, and is often less
than a year
 Since interest rates are usually quoted as
per annum (p.a.) – i.e. per year – and
periods of time are also usually quoted in
years, we need to convert the interest rate
and the time period in order to find r and n
FREQUENCY OF COMPOUNDING
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 22
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2.2.3 Frequency of Compounding
FREQUENCY OF COMPOUNDING
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 23
365 times a year
For example, interest could compound:
52 times a year
26 times a year
12 times a year
4 times a year
Daily
Weekly
Fortnightly
Monthly
Quarterly
Semi‐annually Twice a year
To calculate r in the PV
and FV formulas:
To calculate n in the PV
and FV formulas:
Divide the annual interest
rate by the number of
periods per year
Multiply the number of
years by the number of
periods per year
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2.2.4 Continuous Compounding
 In theory, compounding periods could
become infinitely small, and the number of
compounding periods infinitely large
 This is called continuous compounding
 Although interest can’t literally be calculated
in this way, many financial valuations
involving growth in value other than from
interest (e.g. share price changes) are based
on the concept of continuous compounding
CONTINUOUS COMPOUNDING
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 24
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2.2.4 Continuous Compounding
 The formula for future value of a cash flow,
based on continuous compounding is:
CONTINUOUS COMPOUNDING
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 25
where FV = the future value
C = the present cash flow
e = the base for natural
logarithms (≈ 2.7182818)
APR = The Annual Percentage Rate
Y = the number of years
FV  C  eAPRY
Note that this
could be a
fraction
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2.2.4 Continuous Compounding
 The formula for present value of a cash flow,
based on continuous compounding is:
CONTINUOUS COMPOUNDING
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 26
where PV = the present value
C = the future cash flow
e ≈ 2.7182818
APR = the Annual Percentage Rate
Y = the number of years
APR Y
PV C
e  
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Topic Overview
Lesson 2.3
2.3.1 Multiple Cash Flows
– Future Value
2.3.2 Multiple Cash Flows
– Present Value
2.3.3 Multiple Cash Flows –
Equivalence of Present and
Future Value
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 27
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2.3.1 Multiple cash flows
 Generally, if we need to find the future
value of a series of cash flows, we can do
so by repeatedly using the formula for the
future value of a single cash flow (3.1)
 This converts all cash flows to a common
point in time (Rule 2)
 Only then can the cash flows be summed
to find the future value of the entire series
of cash flows (Rule 1)
FUTURE VALUE OF MULTIPLE CASH FLOWS
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 28
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2.3.2 Multiple cash flows
 Similarly, if we need to find the present
value of a series of cash flows, we can do
so by repeatedly using the formula for the
present value of a single cash flow (3.2)
 This converts all cash flows to a common
point in time – time 0, or today (Rule 3)
 Then the cash flows can be summed to
find the present value of the entire series of
cash flows (Rule 1)
PRESENT VALUE OF MULTIPLE CASH FLOWS
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 29
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2.3.3 Multiple cash flows
 If we calculate the present value of a serious
of cash flows, and calculate the future value of
that present value, it will be equal to the future
value of the series of cash flows
 It can be shown that we would be indifferent
(would consider to be of equal value):
 A series of cash flows
 The present value of the cash flows
 The future value of the cash flows
(with the last two received as a single sum)
EQUIVALENCE OF PRESENT & FUTURE VALUE
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 30
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Topic Overview
Lesson 2.4
2.4.1 Present Value of a Perpetuity
2.4.2 Present Value of a Growing
Perpetuity
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 31
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2.4.1 Present Value of a Perpetuity
 Many applications of financial mathematics
involve periodic cash flows where the size
of the cash flow remains constant, or grows
at a constant rate
 Valuing these cash flows would be tedious,
using the techniques in the previous
section, but fortunately there are some
short-cuts which enable these valuations to
be done with just one or two calculations
MULTIPLE CASH FLOWS
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 32
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2.4.1 Present Value of a Perpetuity
 One such application involves a
perpetuity, which is constant stream of
equal cash flows continuing forever
PERPETUITIES
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 33
0 2
C
1
C C
3
 Note that the first cash flow occurs at the
end of the first period of the perpetuity – the
formulas we will use for perpetuities and
annuities will be based on this assumption
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2.4.1 Present Value of a Perpetuity
 It makes no sense to talk about the future
value of a perpetuity – this would be an
infinite number that would occur an infinite
period of time in the future – but we can
calculate the present value of a perpetuity
 This is because cash flows in the far distant
future are discounted so heavily so that
their contribution to the present value of the
perpetuity approaches zero
PRESENT VALUE OF A PERPETUITY
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 34
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2.4.1 Present Value of a Perpetuity
 The formula for the present value of a
perpetuity is:
PRESENT VALUE OF A PERPETUITY
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 35
where PV = the present value
C = the constant cash flow (with
the first cash flow occurring
one period in the future)
r = the interest rate per period
PV C
r
 (4.4)
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2.4.2 Present value of a perpetuity
 Sometimes we need to find the present
value of an infinite stream of cash flows that
will grow at a constant rate in perpetuity
 The most common application is the
valuation of shares based on constantly
growing dividends (covered in Topic 3)
 We can find the present value as long as the
future cash flows are being discounted at a
greater rate than their growth rate
PRESENT VALUE OF A
GROWING PERPETUITY
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 36
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2.4.2 Present value of a perpetuity
 The formula for the present
value of a growing perpetuity is:
PRESENT VALUE OF A
GROWING PERPETUITY
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 37
where C1 = first cash flow (one period
in the future)
r = the interest rate per period
g = the growth rate (where r > g)
1 PV C
r g


(4.7)
C0 the most recent cash flow
C1 = the next cash flow (in one period)
  1 0 C C 1 g PV
r g 
 
 We can use
this second
version if we
know the
current cash
flow, but not
the next one.
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Although not required for this subject, many
students find a financial calculator easier to use
in many situations instead of formulas, or use it
to double check the results from using formulas
The following slides demonstrate how to solve
selected Lecture Examples from Topic 2 using a
financial calculator
Buttons shown are for the Texas Instruments
BAII Plus, although others are similar
FINANCIAL CALCULATOR GUIDE
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 38
Appendix 2.1
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There are five TVM (time value of money)
buttons on a financial calculator:
FINANCIAL CALCULATOR GUIDE
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 39
For TVM calculations, you input four of the
variables and ask the calculator to compute the
value of the fifth variable
In many situations, you only need three input
variables, so the fourth can be set to zero or
you can clear all TVM values before you begin
Appendix 2.1
N I/Y PV PMT FV
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Lecture Example 2.1
What is the future value of $1000 invested for 4
years at an interest rate of 8% p.a.?
FINANCIAL CALCULATOR GUIDE
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 40
The last step computes the future
value, which appears as a negative
number (if the present value is positive, the future
value is negative, and vice versa)
Appendix 2.1
2ND FV
4 N
8 I/Y
1 0
CPT FV
0 0 PV
This first step clears all TVM values from previous entries
The next three steps input the number of periods, the
interest rate per period and the present value, respectively
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Lecture Example 2.2
What is the present value of $4000 to be
received in 7 years if the interest rate is 5% p.a.?
FINANCIAL CALCULATOR GUIDE
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 41
Appendix 2.1
2ND FV
7 N
5 I/Y
4 0
CPT PV
0 0 FV
2ND FV
0 PMT
For the remaining examples shown in this appendix,
it is assumed that you will clear all TVM values using
this step, or else explicitly enter 0 for the variable
not being used. For example, instead of …
you could enter…
In this example and get the same result.
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Appendix 2.1
Lecture Example 2.3
What is the future value of $1000 invested for 4
years at an interest rate of 8% p.a., if interest is
compounded semi-annually?
FINANCIAL CALCULATOR GUIDE
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 42
8 N
4 I/Y
1 0
CPT FV
0 0 PV
If interest compounds more than once per year, we
need to work out r (in this case, 4%) and n (8 years)
and enter the problem as if it was 4% p.a. for 8 years
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Rounding errors
It is important to avoid rounding errors
whenever possible. Minor rounding errors will
not be penalised in this subject if it is clear that
the correct technique has been used, but major
rounding errors (especially errors so great that
it hard to be sure that the correct technique has
been used) will be penalised.
GENERAL CALCULATOR GUIDE
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 43
Appendix 2.2
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Rounding errors
The best way to avoid rounding errors is to
never write down an intermediate answer and
then later enter it in your calculator. You should
keep intermediate answers in the calculator.
The following slides show three different ways
to solve Lecture Example 2.23 (using the BAII
Plus, but others are similar), avoiding the
necessity to write down intermediate answers
and therefore avoiding rounding errors.
GENERAL CALCULATOR GUIDE
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 44
Appendix 2.2
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Algebraic Operating System
Before looking at these examples, it is important
to understand two different ways in which
calculators perform operations:
Chain – Operations are carried out in the order
in which they are entered
AOS – Operations are carried out in the
mathematically correct order (e.g.
multiplication is done before addition)
GENERAL CALCULATOR GUIDE
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 45
Appendix 2.2
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Algebraic Operating System
Scientific calculators use AOS. Simple
calculators, and many financial calculators,
generally use the Chain system. The BAII Plus,
and most advanced financial calculators, can
be set to either.
All of the following examples assume your
calculator is set to use AOS. If your calculator
uses the Chain system you probably need to
use a lot more brackets to get the right answer.
GENERAL CALCULATOR GUIDE
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 46
Appendix 2.2
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Algebraic Operating System
The following is a simple test to determine
which system your calculator uses (or has been
set to use). Enter the following calculation:
If your answer is 9, your calculator uses the
Chain system. If your answer is 7 (which is the
correct answer) your calculator uses AOS.
GENERAL CALCULATOR GUIDE
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 47
Appendix 2.2
1 + 2 × 3 =
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Algebraic Operating System
To set the BAII Plus to AOS, use the following
steps:
GENERAL CALCULATOR GUIDE
FIN1FOF Fundamentals of Finance – Topic 2 – Time Value of Money 48
Appendix 2.2
↑
2ND
2ND CPT
2ND .
If you want to change the number of
decimal places, at this point enter
where n in this case is the number of
decimal places you want (e.g. 4).
ENTER
n ENTER