Time Value Of Money

1
Time Value Of Money
2
Objectives
You will learn 1. That money has a time value
which is defined by the interest rate. 2. Money
is no longer thought of in terms of one dimension
(how much) but rather is two dimensional (how
much and when). 3. A cash flow diagram (see
figure 2.2 page 34) is the best way to represent
an investment opportunity. 4. How a present sum
of money (P) can produce a future sum of money
(F) by earning interest (moving a single sum of
money forward in time).
3
Objectives
5. How to find the present value (P) of a known
future value (F) by removing or discounting the
interest (moving a single sum of money backward
in time). 6. How to obtain the present worth (P)
or future worth (F) of a uniform series of cash
flows (A).
4
2.1 Introduction

• When making a decision, you will usually consider
  criteria such as quality, safety, customer
  service, and return on investment.
• Usually one of the most important criteria is
  economic performance.
• Among the economic performance characteristics
  normally considered are initial investment
  requirements, return on investment, and the cash
  flow (CF) profile.
• Since the cash flow profiles are usually quite
  different among the several design alternatives,
  in order to compare the economic performances of
  the alternatives, one must compensate for the
  differences in the timing of cash flows.

5
2.1 Introduction

• A fundamental concept underlies much of the
  material covered in the text money has a time
  value.
• The value of a given sum of money depends not
  only on the amount of money but when the money is
  received.
• Money has a time value because if you have a
  dollar now it can be put into an investment (bank
  account, bond, stock, or other) and produce a
  value greater than one dollar at some time in the
  future.
• The amount of the future value depends on the
  interest rate or rate of return earned on your
  investment.

6
Example - Agreeing to pay, but never saying when

• If someone agrees to pay you 1000 but does not
  state when the payment will be made the agreement
  has no validity.
• Suppose ten, twenty or even thirty years from now
  the payment hasnt been made. The agreement
  hasnt been broken because the agreement did not
  specify when the payment was to be made.
• Cash flow diagrams are so valuable in problem
  solving because they show how much money is paid
  or received and when the transfer takes place.

7
What About Inflation?

• The value, economic worth, or purchasing power of
  money changes over time during periods of
  inflation or deflation. We will consider these
  effects later.
• Example 2.2 on p. 32
• Examine closely the two cash flow profiles given
  in Table 2.1. Both alternatives involve an
  investment of 100,000 in ventures that last for
  4 years.

8
Example 2.2 on p. 32

• Alternative A involves an investment in a
  minicomputer by a consulting engineer who is
  planning on providing computerized design
  capability for clients. Since the engineer
  anticipates that competition will develop very
  quickly if the plan proves to be successful, a
  declining revenue profile is anticipated.
• Alternative B involves an investment in a land
  development venture by a group of individuals.
  Different parcels of land are to be sold over a
  4-year period. The land is anticipated to
  increase in value. There will be differences in
  the sizes of parcels to be sold. Consequently, an
  increasing revenue profile is anticipated.

9
Example 2.2 on p. 32
10
Example 2.2 on p. 32

• The consulting engineer has available funds
  sufficient to undertake either investment, but
  not both.
• The cash flows shown are cash flows after taxes
  and other expenses have been deducted.
• Both investments result in 160,000 being
  received over the 4-year period hence, a net
  cash flow of 60,000 occurs in both cases.
•  

11
Example 2.2 on p. 32

• Which would you prefer?
• If you prefer Alternative B, then you are not
  acting in a manner consistent with the concept
  that money has a time value.
• The 60,000 difference at the end of the first
  year is worth more than the 60,000 difference at
  the end of the fourth year.
• Likewise, the 20,000 difference at the end of
  the second year is worth more than the 20,000
  difference at the end of the third year.
• One of the goals of this book is to allow a
  person to evaluate alternatives such as these on
  the basis of sound time value of money principles
  and techniques rather than intuitive arguments.

12
Example 2.3
13
Example 2.3
14
Example 2.3

• Which looks better, C or D?
• C Why? You get your money back faster.
• Why is it important to get your money back
  faster?
• You could invest it somewhere else making more
  money.
• Less risk - the shorter the time period the more
  likely you will get you money back

15
2.2 Cash Flow Diagrams

• What is a cash flow diagram?
• A cash flow diagram has the time on the
  horizontal axis and dollars on the vertical axis.
  Revenues are generally represented by upward
  pointing arrows and costs are downward pointing
  arrows.

1000
1
2
3
4
5
231
231
231
231
231
16
2.2 Cash Flow Diagrams

• Why use a cash flow diagram?
• It is easy to understand - A cash flow diagram
  presents a clear, concise, and unambiguous
  description of the amount and timing of all cash
  flows associated with an economic analysis.
• Usually, a well-drawn cash flow diagram can be
  readily understood by all parties to an economic
  transaction regardless of whether they have had
  any formal training in economic analysis.
• It is a visual tool that allows different
  alternatives to be quickly compared.
• It helps identify significant cash flow patterns
  which might exist within an economic transaction.

17
2.3 Interest Calculations

• How do we calculate the difference between the
  Present value of money P, and the Future value of
  money Fn?
• Fn P In
• Where In accumulated interest in borrowing and
  lending transactions and is a function of P, n,
  and the annual interest rate i.
• How is the annual interest rate defined?
• The annual interest rate is defined as the change
  in value for 1 over a 1-year period.
• There are 2 approaches to calculating In
• Simple interest
• Compound interest.

18
2.3.a - Simple Interest
2.3.a - Simple Interest - assumes that In is a
linear function of time Simple interest is only
use on very short term financial transactions if
at all. The formula for simple interest is In
P i n Where P present principal sum I
Interest earned i interest rate n number of
periods.
19
2.3 Interest Calculations
For example How much simple interest would
accrue on a loan of 1000. The interest rate was
10 per year (not compounded) and the loan was
outstanding for six months. Then, I P i n
I 1000 ( 0.1/ yr.) (6 mon. / 12 mon./yr. )
50 (Note the change in units from months to
years.) The amount to be repaid after six months
is then F (6 months) P P i n P ( 1
i n) F (6 months) 1000 (1.05) 1050
20
2.3.b Compound Interest

• Compound interest occurs when interest earned is
  added to the principal and the sum becomes the
  new principal. (thus interest is earned on
  interest)
• Principle changes every time it is compounded

21
2.3.b Compound Interest

• Suppose that in the previous example the interest
  rate was compounded quarterly (every three
  months). Then what would the future amount be
  after six months?
• At three months
• F1 P P i n P (1 i n )
• F1 1000 1 (0.1) (3 mon. / 12 mon./yr.)
  1000 (1 (.1)(.25))
• F1 1000 1 (.1)(.25)
• F1 1000 (1.025) 1025
• Now, this amount becomes the principal for the
  second quarter (three months) and F2 P P i n
  P (1 i n ) but P is now F1 and not the
  original P
• F2 F1 (1 i n ) 1025 (1.025) 1050.625.

22
2.3.b Compound Interest

• The additional 62.5 cents is from interest earned
  in period 2 on interest that was compounded at
  the end of period 1.
• After 2 periods (6 months), the principle is
• F P I1 I2 1050.625
• Over long periods of time compounding can have a
  dramatic effect on the future value. This effect
  is illustrated in Table 2.2 on page 38.
• The formula for the future value of a present sum
  with compound interest is
• F P ( 1 i )n
• (Note that the number of years is now an
  exponential rather than just multiplied with the
  other values.)

23
2.3.b Compound InterestDeriving The Future Value
Equation

• We have a present value P and want to know its
  future value F at an interest rate i.

P
F1
F2
I1
1
2
F1 P I1 P Pin P(1 i) given that n
1 F1 becomes the P for the next period, now what
is F2? F2 F1 I2 F1 Pin where Pin
F1i(1) and n 1, now factor out F1 F2 F1
F1i(1) F1(1 i) where F1 P(1 i), so
substituting for F1 F2 P(1 i)2 , so what
would the equation be after 3 periods? F3 P(1
i)3 , thus for n periods Fn P(1 i)n
24
2.3.b Compound Interest - Example

• Now suppose P 1000, i 10 per year compounded
  annually and n 30 years.
• What is the future value?
• Fn P(1 i)n
• F 1000 (1.1) 30 17,449.40 .
• If only simple interest was used then,
• F 1000 ( 1 (0.1)(30) ) 4,000.00 .
• The difference between 17,449 and 4,000 (13,449)
  is the interest earned on prior years interest
  (because of the compounding).

25
2.4 Single Sums of Money

• How do we calculate the future value of a present
  sum of money P?
• F P (1 i )n
• Where I is expressed as a decimal. Or use
• F P(F P, i, n)
• Where i is expressed as a percentage amount.
• The factor (1 i)n is called the Single Payment
  Compounded Amount Factor. This quantity can be
  found in Appendix A for various values of i and
  n.
• Think about this formula as moving money through
  time. This equation tells you how to take P
  dollars now and move it into the future.
• Conversely, if the equation is solved for F, then
  money can be moved from a future value to a
  present value, thus if we multiply by (1 i )-n
  then we can isolate P
• P F (1 i )-n

26
2.4 Single Sums of Money - Example

• Lets suppose that after graduation, that you
  want to deposit a single amount of money that
  will make you a millionaire when you retire. How
  much do you need to deposit?
• How long do you plan to work? Lets say 40 years,
  so n 40
• Say you earn 10
• P 1,000,000 (1 .1 )-40 22,094.93
• Most of this course is doing calculations related
  to moving money through time

27
Example Problem 2.3, p.72
How long does it take a deposit in a 4 compound
interest fund to double in value?   Looking
Appendix A  (FP,4,n) 2.00  17 18 between 17 and 18 interest periods
28
2.4 Single Sums of Money

• How do we calculate the present value of a future
  sum of money F?
• P F ( 1 i )-n
• Or use
• P F(PF, i, n)
• The factor ( 1 i ) -n is called the Single
  Payment Present Worth Factor. See examples 2.6
  and 2.7 on pages 40
• How do we calculate the future value of a present
  sum of money P?
• F P ( 1 i )n
• Or use
• F P(FP, i, n)

29
Example Problem 2.6, p.72
(a) If a fund pays 12 compounded annually, what
single deposit now will accumulate 12,000 at the
end of the tenth year?   P F(PF, i, n)
12,000(PF,12,10) 12,000/(1.12)10
3,863.68   (b) If the fund pays 6 compounded
annually, what single deposit is required now in
order to accumulate 6000 at the end of the tenth
year?   P 6000(PF,6,10) 6000/(1.06)10
3350.37
30
2.5 Series of Cash Flows

• If the series has no pattern then each cash flow
  in the series can be considered as a single sum
  and treated according to the methods presented in
  the previous section.

400
300
200
200
300
31
2.5 Series of Cash Flows

• For example, see figure 2.5 on page 41. There are
  five cash flows and the value of these is desired
  at the present time or at the end of period zero.
  
• The present value of each flow is calculated
  using the formula for single sums of money.
• These present values are than added together to
  obtain the present value of the series.

32
2.5 Series of Cash Flows
When more than one cash flow occurs it is
considered to be a series of cash flows. How do
we calculate the present value of a series of
cash flows? P A1(1 i) -1 A2(1 i) -2
An-1(1 i) -(n-1) An(1 i)-n Or where At
magnitude of cash flown at the end of time
period t. How do we calculate the future value
of a cash flow?
33
Example Problem 2.9, p.73
At the end of 8 years, the balance in the fund
is  F8 1000(FP,8,8) 3000(FP,8,6)
4000(FP,8,4)   1000(1.8509) 3000(1.5869)
4000(1.3605)   12,053.60 Principle 1000
3000 4000 8000 Interest 8000 - 12,053.60
4,053.60 4,053.60 is withdrawn and deposited in
the 10 fund it accumulates in 4 years to a
value of  F 4053.60(FP,10,4)
4053.60(1.4641) 5934.88 In the original fund,
the balance in 4 more years will be F
8000(FP,8,4) 8000(1.3605) 10,884.00
34
Special Structure
If the series has a special structure then
formulas are developed to expedite the
calculations. These types of series are of
particular interest Uniform Series (section
2.5.1) Gradient Series (section 2.5.2) Geometric
Series (section 2.5.3)
35
2.5.1 Uniform Series

• What is a Uniform Cash Flow?
• A cash flow series is uniform if
• Each element in the series is for the same amount
  
• The time interval is the same for each cash flow
• The 1st cash flow occurs at the end of the 1st
  period.
• Since the Uniform Cash Flow is used so much in
  the financial world, special formulas have been
  developed to make the calculations easier.

36
2.5.1 Uniform Series
Suppose you were saving for retirement, saving a
fixed amount every year for n years. Say you want
to know what F is.
F
A1
A2
A3
A4
A5
An-1
An
1
2
3
4
5
n-1
n
P
What is the future value of that last term? Add A
n-1 to An, but how much is A n-1 worth at the
last payment (moving it 1 period) (1) F An
An-1 (1 i)1 An-2 (1 i)2 A5 (1 i)n-5
An (1 i) n-1
37
2.5.1 Uniform Series
(1) F An An-1 (1 i)1 An-2 (1 i)2
A5 (1 i)n-5 A1 (1 i) n-1 To get this
equation into a more useful form, multiply the
equation by (1 i) and subtract the equations (1
i) F An An-1 (1 i)1 An-2 (1 i)2
A5 (1 i)n-5 A1 (1 i) n-1 (1 i) (2) (1
i) F An (1 i) An-1 (1 i)2 An-2 (1 i)3
A1 (1 i)n
38
2.5.1 Uniform Series
Now subtract Equation 1 and Equation 2 (1)
F An An-1 (1 i)1 An-2 (1 i)2 A5
(1 i)n-5 A1 (1 i) n-1 (2) (1 i) F An (1
i) An-1 (1 i)2 An-2 (1 i)3 An (1
i)n iF - An A1 (1 i)n (since A is the same
every time, we can drop the subscript) iF A1
(1 i)n - An (factor the A and divide through by
i) F A (1 i)n - 1/i (uniform series
compound factor)
39
2.5.1 Uniform Series
To find the present worth of a future value, use
the following equations
(PA i, n) is referred to as the uniform series,
present worth factor To find the magnitude of a
cash flow from a present amount, use the
following equations (which are the reciprocal of
the equations above)
(AP i, n) is referred to as the capital
recovery factor This equation is often known as
the "loan equation".
40
2.5.1 Uniform Series

• A positive value for A indicates a receipt and a
  negative value indicates a payment.
• The interest rate is i and it must be the
  effective rate for the time period between the
  payments.
• If the cash flow occurs monthly then the interest
  rate must be a monthly rate.
• n is the number of cash flow elements and also
  the number of interest periods.
• The derivation of the above formula assumes that
  each cash flow occurs at the end of a period and
  the present time is the beginning of period one.
• Therefore, the first cash flow in the series
  occurs one period later then the present time
  (see figure 2.6 on page 44 for an illustration).

41
2.5.1 Uniform Series
To find the future worth of a uniform series, use
the following equations
(FA i, n) is referred to as the uniform series,
future worth factor To find the magnitude of a
cash flow from a future amount, use the following
equations (which are the reciprocal of the
equations above)
(AF i, n) is referred to as the sinking fund
factor - used to determine the size of the
deposit one should sink into a fund in order to
accumulate a desired future amount.
42
2.5.1 Uniform Series - Example
Suppose that you make 40,000 when you graduate
and save 10 every year for retirement. So, you
save 4,000 per year for 40 years. How much will
you have in the account after 40 years assuming
that you make 10 interest? F 4000(1.1)40
1/.1 17,770,370 This problem can be solved
using tables starting on p.411 On p.424, look at
the 10 table. The table does not have n 40 ,
but has n 36 and n 42
43
Example - Problem 2.11, p.73
Five deposits of 500 each are made at t 1, 2,
3, 4, 5 into a fund paying 6 compounded per
period. (a) How much will be accumulated in the
fund at t 5?  F5 500(FA,6,5) (look in
Appendix A, 6 Table on p.420 under the uniform
series To find F given A where n 5  F5
500(5.6371) 2,818.55 (b) How much will be
accumulated in the fund at t 10? F10
2818.55(FP,6,5) 2818.55(1.3382) 3771.78
44
2.5.2 Gradient Series of Cash Flows

• A gradient series occurs when each payment in a
  series differs from the previous payment by a
  constant amount , G.
• The series can be represented by the sum of a
  uniform series and a gradient series. (See Figure
  2.10, p.48)
• The gradient series is defined to have the first
  positive cash flow occur at the end of the second
  time period
• The size of the cash flow occurring at the end of
  period t is
• At (t 1)G
• Examples
• Increased maintenance every year
• Increased bonus every year

45
2.5.2 Gradient Series of Cash Flows
A1(n-1)G
A1(n-2)G
A12G
A1G
A1
0
1
2
3
n-1
n
P P1 P2
Figure 2.10 p.48
46
2.5.2 Gradient Series of Cash Flows
A1
A1
A1
A1
A1
0
1
2
3
n-1
n
(n-1)G
(n-2)G
P1
2G
G
0
1
2
3
n-1
n
Figure 2.10 p.48
P2
47
2.5.2 Gradient Series of Cash Flows
To find the present worth of a gradient series,
use the following equations
(PG i, n) is referred to as the gradient
series, present worth factor To find the uniform
series equivalent to the gradient series, use the
following equations
(AG i, n) is referred to as the gradient to
uniform series conversion factor
48
2.5.2 Gradient Series of Cash Flows

• When a series does not start at zero but each
  payment differs from the previous one by G it is
  called a composite series ( see figure 2.11 ).
• A composite series can be partitioned into a
  uniform series plus a gradient series to aid in
  finding its present or future value. ( see
  example 2.17 page 51 ).
• Translation to use the tables in the Appendix,
  you must first separate the Uniform Series from
  the gradient series.

49
Example Problem 2.15, p.73

• John borrows 15000 at 18 compounded annually
  he pays off the loan over a 5 year period with
  annual payments. Each successive payment is 700
  greater than the previous payment. How much was
  the first payment?
• 15,000 A(PA,18,5) 700(PG,18,5)
• For A(PA,18,5) look in Appendix A, 18 Table on
  p.427 under the uniform series To find P given
  A where n 5
• For (PG,18,5) look in Appendix A, 18 Table on
  p.427 under the gradient series To find P given
  G where n 5
• 15,000 A(3.1272) 700(5.2312)
• 15000 - 700(5.2312) A(3.1272)
• 11338.16 / 3.1272 A
• 3625.66 A

50
2.5.3 Geometric Series of Cash Flows

• A geometric series occurs when the size of a cash
  flow increases (or decreases) by a fixed
  percentage from one time period to the next
• If j denotes the percentage change in the size of
  a cash flow from one period to the next, then
  size of the tth flow can be expressed as
• At At-1 ( 1 j ).
• If the first payment is designated as A1 then
  from page 52.
• At A1 ( 1 j )t-1 for t 2 , .. , n
  Equation 2.33.p.52.
• The geometric series is used to represent the
  growth (positive j) or decay (negative j) of
  costs and revenues undergoing annual percentage
  changes.
• Example labor costs or gasoline prices
  increasing 10 per year

51
2.5.3 Geometric Series of Cash Flows

• The present value of a geometric series of cash
  flows is obtained from the following equations .

i not equal to j
i j
i not equal to j j 0

• Where (PA1, i, j, n) is the geometric series,
  present worth factor

52
2.5.3 Geometric Series of Cash Flows

• For the case of j greater than or equal to 0 and
  i not equal to j, the relationship between P and
  A can be expressed in terms of compound interest
  factors

i not equal to j j 0
53
2.5.3 Geometric Series of Cash Flows

• The future worth equivalent of the geometric
  series is obtained by

i not equal to j
i j
i not equal to j j 0

• Where (FA1, i, j, n) is the geometric series,
  future worth factor

54
2.5.3 Geometric Series of Cash Flows

• What is the difference between a gradient series
  and a geometric series?
• A gradient series differs by a constant amount,
  whereas a geometric differs by (1j).
• Thus, there is no need to split a geometric
  series like you do a composite series.

55
Example - Problem 2.19, P.73
John borrows 15,000 at 18 compounded annually,
he pays off the loan over a 5-year period with
annual payments. Each successive payment is 10
greater than the previous payment. How much was
the first payment? P A1 (P A1 i, j, n) we
could use this if we had a table for i 18, but
since we do not have a table for 18, we are
forced to use
56
Example - Problem 2.19, P.73
John borrows 15,000 at 18 compounded annually,
he pays off the loan over a 5-year period with
annual payments. Each successive payment is 10
greater than the previous payment. How much was
the first payment? P A1 (P A1 i, j, n) we
could use this if we had a table for i 18, but
since we do not have a table for 18, we are
forced to use P A1 1 - ( 1 j )n ( 1 i
)-n / i - j 15,000 A11 - (1.10)5 (1
.18) -5/.18-.10 15,000 A1 1 -
(1.61051)(.43711)/.08 15,0000 A1 .2960/.8
A1 3.700 15,000 A1 (3.7004) A1 4053.62
57
2.6 Multiple Compounding Periods in a Year

• What is the difference between the nominal
  interest rate and the effective interest rate?
• The nominal rate occurs at the same interval as
  the payment, while the Effective interest rate
  does not.
• How do we solve these problems?
• Period Interest rate approach
• Effective interest rate approach

58
2.6 Multiple Compounding Periods in a YearPeriod
Interest Rate
Now the interest period and the compounding
period are the same (monthly), so the factors in
the back of the book can be applied directly
59
2.6 Multiple Compounding Periods in a
YearEffective Interest rate Approach

• If interest is compounded more than once, say m
  times, in a period of interest then the effective
  interest rate for that period is
• ieff ( 1 r / m )m - 1
• Where r is called the nominal rate and m is the
  number of times interest is compounded in the
  period for which i is desired.
• You might pay once a year, but compounded monthly
  or pay monthly and compounded daily.
• Either way you have payments and compounding
  occurring at different intervals.

60
2.6 Multiple Compounding Periods in a Year

• All previous formulas are derived on the fact
  that each time there is a payment, there is a
  compounding.
• Now we could have more frequent (or less
  frequent) compounding that payment.
• For example, suppose the interest is 12 per year
  compounded monthly. Also, suppose that the annual
  effective rate is needed.
• The nominal annual rate is 12 or 0.12. Interest
  is compounded 12 times in a Year. From the above
  equation the effective annual rate is
• i ( 1 0.12 / 12 )12 - 1 0.1268 or 12.68 .
• Consider the same situation but suppose the
  effective rate for a quarter is desired then
• i ( 1 0.03 / 3 )3 - 1 0.0303 or 3.03 .

61
2.6 Multiple Compounding Periods in a
YearExamples

• 12 per year compounded semiannually
• Nominal annual interest rate 12/year/6 months
• Period interest rate 6/6 month/6 month
• Effective annual interest rate (1 .12/2)2 1
  .1236 or 12.36
• 12 per year compounded monthly
• Nominal annual interest rate 12/year/months
• Period interest rate 1/month/month
• Effective annual interest rate (1 .12/12)12
  1 .1268 or 12.68

62
2.6 Multiple Compounding Periods in a
YearExample from Test 1 2001
A local bank will lend Katy Conner 1000 on a
2-year car loan as follows Money to pay for car
1000 Two years interest rate at 7 2 (.07)
(1000) 140 24 monthly payments (1000140)/24
47.50 The first payment must be made in 30
days. What is the nominal annual interest rate
the bank is receiving? P A(PA, i, n) 1000
47.50 (PA, i, 24) 21.053 (PA, i, 24)
63
2.6 Multiple Compounding Periods in a
YearExample from Test 1 2001
For 1 (PA, i, 24) 21.2434 For 1.5 (PA, i,
24) 20.0304
Now interpolate y ? y0 .01 y1 .015 x
21.053 x0 21.2434 x1 20.0304
Nominal interest rate 12(1.077) 12.92
64
2.7 Continuous Compounding

• As the frequency of compounding in a year
  increases, the effective interest rate increases
• Continuous compounding implies that the time
  interval between compounding of interest is
  reduced to zero.
• This also implies that interest is compounded an
  infinite number of times.

• Where
• n number of years
• m number of interest periods per year
• r nominal annual interest rate

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2.7 Continuous Compounding

• Where (FP, r, n) denotes the continuous
  compounding single sum future worth factor
• The infinity subscript is provided to denote that
  continuous compounding is being used
• The interest tables for continuous compounding
  are given in Appendix B

66
2.7 Continuous Compounding

• The effective interest rate under continuous
  compounding using

• Where (PF, r, n) denotes the continuous
  compounding single sum present worth factor
• The infinity subscript is provided to denote that
  continuous compounding is being used

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2.7 Continuous CompoundingExample 2.24

• If 2000 is invested in a fund that pays interest
  at a rte of 12 compounded continuously, after 5
  years the cumulative amount in the und will
  total
• F P(FP, 12, 5)?
• F 2000(1.8221)
• F 3644.20

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2.7 Continuous CompoundingExample 2.25

• 1000 is deposited each year into an account that
  pays at a rate of 12 compounded continuously.
  Determine both the amount in the account
  immediately after the 10th deposit and the
  present worth equivalent for 10 deposits.
• F A(FA, 12, 10)?
• F 1000(18.1974)
• F 18197.40
• P A(PA, 12, 10)?
• F 1000(5.4810)
• F 5481

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2.7 Continuous Compounding

• For example, suppose the interest rate is 12 per
  year compounded continuously.
• Q..1 What is the effective annual interest rate ?
• Ans. i (annual) e 0.12 - 1 0.1274 12.74 .
• Q..2 What is the effective quarterly interest
  rate ?
• Ans. We must first get the nominal quarterly rate
  and then calculate i using equation 2.45.
• r (quarterly) r (annual) / 4 0.12 / 4 0.03
• i er - 1 e0.03 - 1 0.03045 3.045 .

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2.7 Continuous Compounding

• Formulas for continuous compounding are found in
  Table 2.4 on page 61. The formula to find F given
  A and the formula to find P given A may not be
  correct (depending on which printing of the book
  that you own).
• What correction is needed ?
• To answer this question go back to the equivalent
  formulas for discrete compounding in Table 2.3 on
  page 55 and substitute er - 1 for i in each
  formula.
•  
• 2.7.2 Continuous Flow Skip this section.

71
2.8.1 Equivalence

• Two cash flows are said to be equivalent at a
  specified interest rate i if they both have the
  same present value.
• Consider the following cash flows

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2.8.1 Equivalence

• If P1 P2 then they are equivalent.
• What is the future value at t 4 for P1?
• F1 must equal F2 if they are equivalent.
• F1 P1 (1 i)4
• F2 P2 (1 i)4
• We have already worked problems of this type.

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2.8.2 Indifference

• A decision maker is indifferent between two cash
  flows if they are equivalent (have the same
  present value).
• Find the interest rate in which P1 P2
• P1 100 (1 i) -1 50(1 i) -2 200(1 i)
  -3
• P2 400 (1 i) -3
• Then solve for i.

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2.8.2 Indifference - Problem 2.54 p.77

• A firm buys a new computer that costs 100,000.
  It may either pay cash now or pay 20,000 down
  and 30,000/year for 3 years. If the firm can
  earn 15 on investments, which would you suggest?
• Pcash 100,000
• Ppay 20,000 30,000(PA, 15, 3)
• Ppay 20,000 30,000(2.2832) 88,496
• Prefer payments