Time Value of Money

1
Interest
• Simple Interest
• Interest paid (earned) on only the original amount, or principal, borrowed
(lent).
• Formula
• SI = P0(i)(n)
SI:Simple Interest
P0:Deposit today (t=0)
i: Interest Rate per Period
n: Number of Time Periods
• Compound Interest
Interest paid (earned) on any previous interest earned, as well as on the
principal borrowed (lent).

2
Simple Interest FV and PV Example
• Assume that you deposit $1,000 in an account earning 7% simple interest for 2
years. What is the accumulated interest at the end of the 2nd year?
• SI = P0(i)(n)
= $1,000(0.07)(2)
= $140
• What is the Future Value (FV) of the deposit?
FV = P0 + SI
= $1,000 + $140
= $1,140
• Future Value (FV) is the value at some future time of a present amount of money, or a series of
payments, evaluated at a given interest rate.
• Present Value (PV) is the current value of a future amount of money, or a series of payments,
evaluated at a given interest rate.
• The Present Value is simply the $1,000 you originally deposited. That is the value today!

3
Why TIME?
• Why is TIME such an important element in your decision?
• TIME allows you the opportunity to postpone consumption and earn
INTEREST.
• Which would you prefer – $10,000 today or $10,000 in 5 years?

4
The Timeline
• A timeline is a linear representation of the timing of potential cash
flows.
• Drawing a timeline of the cash flows will help you visualize the
financial problem.

2-5

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The Timeline (cont’d)
• Assume that you lend $20,000 to a friend. You will be repaid in two
payments, one at the end of each year over the next two years.

2-6

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The Timeline (cont’d)
• Assume that you are lending $10,000 today and that the loan will be repaid
in two annual $6,000 payments.

The first cash flow at date 0 (today) is represented as a negative sum because
it is an outflow.
• Timelines can represent cash flows that take place at the end of any time
period.
2-7

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Three Rules of Time Travel
• Financial decisions often require combining cash flows or comparing
values. Three rules govern these processes.

2-8

8
Summary
• Present Value – earlier money on a time line
• Future Value – later money on a time line
• Interest rate – “exchange rate” between earlier money and later
money
• Discount rate
• Cost of capital
• Opportunity cost of capital
• Required return

4.9
Future Values
• Suppose you invest $1000 for one year at 5% per year. What is the
future value in one year?
• Interest = 1000(.05) = 50
• Value in one year = principal + interest = 1000 + 50 = 1050
• Future Value (FV) = 1000(1 + .05) = 1050
• Suppose you leave the money in for another year. How much will
you have two years from now?
• FV = 1000(1.05)(1.05) = 1000(1.05)2 = 1102.50

4.10
Future Values: General Formula
• FV = PV(1 + r)t
• FV = future value
• PV = present value
• r = period interest rate, expressed as a decimal
• t = number of periods
• Future value interest factor = (1 + r)t

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Effects of Compounding
• Simple interest
• Compound interest
• Consider the previous example
• FV with simple interest = 1000 + 50 + 50 = 1100
• FV with compound interest = 1102.50
• The extra 2.50 comes from the interest of .05(50) = 2.50 earned on the first
interest payment

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Future Values – Example 2
• Suppose you invest the $1000 from the previous example for 5 years.
How much would you have?
• FV = 1000(1.05)5 = 1276.28
• The effect of compounding is small for a small number of periods, but
increases as the number of periods increases. (Simple interest would
have a future value of $1250, for a difference of $26.28.)

4.13
Valuation: The One-Period Case
• Mike Tuttle, Keith’s financial adviser, points out that if Keith takes the
first offer, he could invest the $10,000 in the bank at an insured rate of
12 percent. At the end of one year, he would have:
• $10,000 + (.12× $10,000)=$10,000 × 1.12 = $11,200

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Future Values – Example 3
• Suppose you had a relative deposit $10 at 5.5% interest 200 years
ago. How much would the investment be worth today?
• FV = 10(1.055)200 = 447,189.84
• What is the effect of compounding?
• Simple interest = 10 + 200(10)(.055) = 120.55
• Compounding added $446,979.29 to the value of the investment

4.15
Future Value as a General Growth Formula
• Suppose your company expects to increase unit sales of widgets by
15% per year for the next 5 years. If you currently sell 3 million
widgets in one year, how many widgets do you expect to sell in 5
years?
• FV = 3,000,000(1.15)5 = 6,034,072

4.16
Present Values
• How much do I have to invest today to have some amount in the
future?
• FV = PV(1 + r)t
• Rearrange to solve for PV = FV / (1 + r)t
• When we talk about discounting, we mean finding the present value
of some future amount.
• When we talk about the “value” of something, we are talking about
the present value unless we specifically indicate that we want the
future value.

4.17
Present Value – One Period Example
• Suppose you need $10,000 in one year for the down payment on a
new car. If you can earn 7% annually, how much do you need to invest
today?
• PV = 10,000 / (1.07)1 = 9345.79

4.18
Present Values – Example 2
• You want to begin saving for you daughter’s college education and
you estimate that she will need $150,000 in 17 years. If you feel
confident that you can earn 8% per year, how much do you need to
invest today?
• PV = 150,000 / (1.08)17 = 40,540.34

4.19
Present Values – Example 3
• Your parents set up a trust fund for you 10 years ago that is now
worth $19,671.51. If the fund earned 7% per year, how much did your
parents invest?
• PV = 19,671.51 / (1.07)10 = 10,000

4.20
Present Value – Important Relationship I
• For a given interest rate – the longer the time period, the lower the
present value
• What is the present value of $500 to be received in 5 years? 10 years? The
discount rate is 10%
• 5 years: PV = 500 / (1.1)5 = 310.46
• 10 years: PV = 500 / (1.1)10 = 192.77

4.21
Present Value – Important Relationship II
• For a given time period – the higher the interest rate, the smaller the
present value
• What is the present value of $500 received in 5 years if the interest rate is
10%? 15%?
• Rate = 10%: PV = 500 / (1.1)5 = 310.46
• Rate = 15%; PV = 500 / (1.15)5 = 248.58

4.22
23
 Future Value of a Single Amount

• Suppose you invest $100 at 5% interest, compounded annually. At the end of one
year, your investment would be worth:
$100 + .05($100) = $105
or
$100(1 + .05) = $105
• During the second year, you would earn interest on $105. At the end of two years,
your investment would be worth:
$105(1 + .05) = $110.25

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• In General Terms:
FV1 = PV(1 + i)
and
FV2 = FV1(1 + i)
• Substituting PV(1 + i) in the first equation for FV1 in the second equation:
FV2 = PV(1 + i)(1 + i) = PV(1 + i)2
• For (n) Periods:
FVn = PV(1 + i)n
• Example: Invest $1,000 @ 7% for 18 years:
FV18 = $1,000(1.07)18 = $1,000(3.380) = $3,380

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 Present Value of a Single Amount

• Calculating present value (discounting) is simply the inverse of calculating future

value (compounding):

FVn  PV (1  i ) n Compoundin g
FVn  1 
PV   FVn  n 
Discountin g
(1  i ) n  (1  i ) 
 1 
where :  n 
is the PV of $1 interest factor
 (1  i) 
(See Appendix B for calculations)

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 Present Value of a Single Amount
(An Example)
• How much would you be willing to pay today for the right to receive $1,000 five
years from now, given you wish to earn 6% on your investment:

 1 
PV  $1000  5 
 (1.06) 
= $1000(.747)
= $747

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 Future Value of an Annuity

• Ordinary Annuity: A series of consecutive payments or receipts of

equal amount at the end of each period for a specified number of
periods.
• Example: Suppose you invest $100 at the end of each year for the
next 3 years and earn 8% per year on your investments. How much
would you be worth at the end of the 3rd year?

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 T1 T2 T3
$100 $100 $100

Compounds for 0 years:

$100(1.08)0 = $100.00

Compounds for 1 year:

$100(1.08)1 = $108.00

Compounds for 2 years:

$100(1.08)2 = $116.64
______

Future Value of the Annuity $324.64

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 FV3 = $100(1.08)2 + $100(1.08)1 +$100(1.08)0
= $100[(1.08)2 + (1.08)1 + (1.08)0]
= $100[Future value of an annuity of $1
factor for i = 8% and n = 3.]
(See Appendix C)
= $100(3.246)
= $324.60

FV of an annuity of $1 factor in general terms:

(1  i) n  1
(useful when using a non - financial calculator)
i

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 Future Value of an Annuity
(Example)
• If you invest $1,000 at the end of each year for the next 12 years and earn 14%
per year, how much would you have at the end of 12 years?

FV12 = $1000(27.271) given i  14% and n  12

= $27,271

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 Present Value of an Annuity
Suppose you can invest in a project that will return $100 at the end of each year for the next 3 years. How much should
you be willing to invest today, given you wish to earn an 8% annual rate of return on your investment?

T0 T1 T2 T3
$100 $100 $100

Discounted back 1 year:

$100[1/(1.08)1] = $92.59
Discounted back 2 years:
$100[1/(1.08)2] = $85.73
Discounted back 3 years:
$100[1/(1.08)3] = $79.38
PV of the Annuity = $257.70

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PV  $100[1 /(1.08)1 ]  $100[1 /(1.08) 2 ]  $100[1 /(1.08) 3 ]
= $100[1 /(1.08)1  1 /(1.08) 2  1 /(1.08) 3 ]
= $100[Present value of an annuity of $1 factor for i  8% and n  3]
(See Appendix D.)
 $100(2.577)
 $257.70

PV of an annuity of $1 factor in general terms :

1
1
(1  i ) n
(useful with non - financial calculators)
i

33
 Present Value of an Annuity
(An Example)
• Suppose you won a state lottery in the amount of $10,000,000 to be paid in 20
equal annual payments commencing at the end of next year. What is the present
value (ignoring taxes) of this annuity if the discount rate is 9%?

PV = $500,000(9.129) given i  9% and n  20

= $4,564,500

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 Summary of Compounding and
Discounting Equations
• In each of the equations above:
• Future Value of a Single Amount
• Present Value of a Single Amount
• Future Value of an Annuity
• Present Value of an Annuity
there are four variables (interest rate, number of periods, and two
cash flow amounts). Given any three of these variables, you can solve
for the fourth.

35
 A Variety of Problems
• In addition to solving for future value and present value, the
text provides good examples of:
• Solving for the interest rate
• Solving for the number of periods
• Solving for the annuity amount
• Dealing with uneven cash flows
• Amortizing loans

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 Annuity Due
• A series of consecutive payments or receipts of equal amount at the beginning of each
period for a specified number of periods. To analyze an annuity due using the tabular
approach, simply multiply the outcome for an ordinary annuity for the same number of
periods by (1 + i). Note: Throughout the course, assume cash flows occur at the end of
each period, unless explicitly stated otherwise.
• FV and PV of an Annuity Due:

FVn  FV of an ordinary annuity (1  i)

PV  PV of an ordinary annuity (1  i)

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 Perpetuities

• An annuity that continues forever. Letting PP equal the constant

dollar amount per period of a perpetuity:

PP
PV 
i

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Perpetuities
Some Securities last "forever," and generate the equivalent of a perpetual
cash flow.
Clearly, we cannot evaluate these perpetual cash flows in the
conventional manner.
We do however, have formulas which allows us to evaluate these cash
flows.
• A Perpetuity is a series of equal payments that continues forever.

0 1 2 3 4 5 .......... 98 99 100......
└────┴────┴────┴────┴────┴──..........──┴────┴─┘
15 15 15 15 15 .......... 15 Cash Flow per Period
PV15= 15 .......
• The Present Value of a Perpetuity is: Discount Rate

CF
PV =
r

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Perpetuities
Example: A British Government Bond pays 100,000 pounds a year
forever (Consul). The market rate of interest is 8%. How much
would you pay for this bond?
PV = Cash Flow = 100,000 = 1,250,000
of perpetuity r 0.08
How much is the bond worth if the first coupon is payable
immediately?

PV of Bond = PV Immediate Payment Plus Value of Perpetuity

= 1,250,000 + 100,000
= 1,350,000

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Growing Perpetuity
If the cash flow grows at a constant rate, then the
perpetuity is called a growing perpetuity
CF 1
PV of Growing Perpetuity =
r-g
where CF1= Cash flow next year
r = Market rate interest
g = Constant Growth rate
What is the present value of a cash flow stream that pays
$105,000 at the end of this year, and grows at5% per year
forever?
PV of growing = 105,000 = 3,500,000 perpetuity
0.08 -0.05
= 105,000 = 3,500,000
0.03

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A Clarification on Different
Compounding Periods
• We have assumed that we are dealing with compounding
only once a year.
But what happens when the compounding is done more than
annually?
• Given the periodic interest rate, you can use the tables to
find the present value of a single payment, the present
value of a periodic annuity, as well as the future values.
• Example: Suppose you will receive $1,000 per month for 12
months. at an annual (simple) interest rate of 18%,
compounded monthly, what is the present value of this
cash flow?

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 Nonannual Periods
mn
 i
FVn  PV 1  
 m
1
PV  FVn mn
 i
1  m 
m = number of times compounding occurs per year
i = annual stated rate of interest
• Example: Suppose you invest $1000 at an annual rate of 8%
with interest compounded a) annually, b) semi-annually, c)
quarterly, and d) daily. How much would you have at the
end of 4 years?

43
 Nonannual Example Continued

• Annually
• FV4 = $1000(1 + .08/1)(1)(4) = $1000(1.08)4 = $1360
• Semi-Annually
• FV4 = $1000(1 + .08/2)(2)(4) = $1000(1.04)8 = $1369
• Quarterly
• FV4 = $1000(1 + .08/4)(4)(4) = $1000(1.02)16 = $1373
• Daily
• FV4 = $1000(1 + .08/365)(365)(4)
= $1000(1.000219)1460 = $1377

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Thank You

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