Time value

The Time Value of MoneyThe Time Value of Money
Lohithkumar B

What is Time Value?What is Time Value?
We say that money has a time value because that
money can be invested with the expectation of
earning a positive rate of return
In other words, “a dollar received today is worth
more than a dollar to be received tomorrow”
That is because today’s dollar can be invested so that
we have more than one dollar tomorrow
January 30, 2015 © Lohithkumar B

Why TIME?Why TIME?
TIMETIME allows one the opportunity to postpone
consumption and earn INTERESTINTEREST.
NOT having the opportunity to earn interest on
money is called OPPORTUNITY COST.

The Time Value of MoneyThe Time Value of Money
Which would you rather have -- $1,000 today$1,000 today or
$1,000 in 5 years?$1,000 in 5 years?
Obviously, $1,000 today$1,000 today.
Money received sooner rather than later allows
one to use the funds for investment or
consumption purposes. This concept is
referred to as the TIME VALUE OF MONEYTIME VALUE OF MONEY!!

The Terminology of Time ValueThe Terminology of Time Value
Present Value - An amount of money today, or the
current value of a future cash flow
Future Value - An amount of money at some future
time period
Period - A length of time (often a year, but can be a
month, week, day, hour, etc.)
Interest Rate - The compensation paid to a lender (or
saver) for the use of funds expressed as a percentage
for a period (normally expressed as an annual rate)

AbbreviationsAbbreviations
PV - Present value
FV - Future value
Pmt - Per period payment amount
N - Either the total number of cash flows or
the number of a specific period
i - The interest rate per period

TimelinesTimelines
0 1 2 3 4 5
PV FV
Today
A timeline is a graphical device used to clarify the
timing of the cash flows for an investment
Each tick represents one time period

Calculating the Future ValueCalculating the Future Value
Suppose that you have an extra $100 today that you
wish to invest for one year. If you can earn 10% per
year on your investment, how much will you have in
one year?
0 1 2 3 4 5
-100 ?

January 30, 2015 © Lohithkumar B 9
Single Sum - Future & Present ValueSingle Sum - Future & Present Value
Assume can invest PV at interest rate i to receive future sum, FV
Similar reasoning leads to Present Value of a Future sum today.
1 2 30
FV1 = (1+i)PV
FV3 = (1+i)3
PV
PV
FV2 = (1+i)2
PV
1 2 30
PV = FV1/(1+i)
FV1
PV = FV2/(1+i)2
FV2
PV = FV3/(1+i)3
FV3

1
Calculating the Future Value (cont.)Calculating the Future Value (cont.)
Suppose that at the end of year 1 you decide to
extend the investment for a second year. How much
will you have accumulated at the end of year 2?
0 1 2 3 4 5
-110 ?

1
Generalizing the Future ValueGeneralizing the Future Value
Recognizing the pattern that is developing,
we can generalize the future value
calculations as follows:
If you extended the investment for a third
year, you would have:

1
Compound InterestCompound Interest
Note from the example that the future value is
increasing at an increasing rate
In other words, the amount of interest earned each
year is increasing
• Year 1: $10
• Year 2: $11
• Year 3: $12.10
The reason for the increase is that each year you are
earning interest on the interest that was earned in
previous years in addition to the interest on the
original principle amount

1
Compound Interest GraphicallyCompound Interest Graphically
0
500
1000
1500
2000
2500
3000
3500
4000
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Years
FutureValue
5%
10%
15%
20%
3833.76
1636.65
672.75
265.33

1
The Magic of CompoundingThe Magic of Compounding
On Nov. 25, 1626 Peter Minuit, a Dutchman, reportedly
purchased Manhattan from the Indians for $24 worth of beads
and other trinkets. Was this a good deal for the Indians?
This happened about 371 years ago, so if they could earn 5% per
year they would now (in 1997) have:
If they could have earned 10% per year, they would now have:
$54,562,898,811,973,500.00 = 24(1.10)371
$1,743,577,261.65 = 24(1.05)371
That’s about 54,563 Trillion dollars!

1
The Magic of Compounding (cont.)The Magic of Compounding (cont.)
The Wall Street Journal (17 Jan. 92) says that all of New York
city real estate is worth about $324 billion. Of this amount,
Manhattan is about 30%, which is $97.2 billion
At 10%, this is $54,562 trillion! Our U.S. GNP is only around $6
trillion per year. So this amount represents about 9,094 years
worth of the total economic output of the USA!
At 5% it seems the Indians got a bad deal, but if they earned
10% per year, it was the Dutch that got the raw deal
Not only that, but it turns out that the Indians really had no
claim on Manhattan (then called Manahatta). They lived on
Long Island!
As a final insult, the British arrived in the 1660’s and
unceremoniously tossed out the Dutch settlers.

1
Present ValuePresent Value
Discounting is the process of translating a
future value or a set of future cash flows into
a present value.

1
Calculating the Present ValueCalculating the Present Value
So far, we have seen how to calculate the
future value of an investment
But we can turn this around to find the
amount that needs to be invested to achieve
some desired future value:

1
Present Value: An ExamplePresent Value: An Example
Suppose that your five-year old daughter has just
announced her desire to attend college. After some
research, you determine that you will need about
$100,000 on her 18th birthday to pay for four years of
college. If you can earn 8% per year on your
investments, how much do you need to invest today
to achieve your goal?

1
Present Value ExamplePresent Value Example
Joann needs to know how large of a deposit to make
today so that the money will grow to $2,500$2,500 in 55
years. Assume today’s deposit will grow at a
compound rate ofcompound rate of 4% annually.
0 1 2 3 4 55
4%
PV0 $2,500$2,500

2
Present Value Solution
Calculation based on general formula:
PV0 = FVn / (1+i)n
PV0 = $2,500/(1.04)5
= $2,054.81

2
Finding “n” or “i” when one knows
PV and FV
If one invests $2,000 today and has
accumulated $2,676.45 after exactly five years,
what rate of annual compound interest was
earned?

2
Frequency of Compounding
General Formula:
FVn = PVPV00(1 + [i/m])mn
n: Number of Years
m: Compounding Periods per Year
i: Annual Interest Rate
FVn,m: FV at the end of Year n
PVPV00: PV of the Cash Flow today

2
Frequency of Compounding Example
Suppose you deposit $1,000 in an account
that pays 12% interest, compounded
quarterly. How much will be in the account
after eight years if there are no withdrawals?
PV = $1,000
i = 12%/4 = 3% per quarter
n = 8 x 4 = 32 quarters

2
Solution based on formula:Solution based on formula:
FV= PV (1 + i)n
=
1,000(1.03)32
=
2,575.10

2
Annuities
25
Regular or ordinary annuity is a finitefinite set of
sequential cash flows, all with the same value
A, which has a first cash flow that occurs one
period from now.
An annuity due is a finitefinite set of sequential
cash flows, all with the same value A, which
has a first cash flow that is paid immediatelyimmediately.

2
AnnuitiesAnnuities
An annuity is a series of nominally equal payments
equally spaced in time
Annuities are very common:
• Rent
• Mortgage payments
• Car payment
• Pension income
The timeline shows an example of a 5-year, $100
annuity
0 1 2 3 4 5
100 100 100 100 100

2
The Principle of Value AdditivityThe Principle of Value Additivity
How do we find the value (PV or FV) of an
annuity?
First, you must understand the principle of
value additivity:
• The value of any stream of cash flows is equal to
the sum of the values of the components
In other words, if we can move the cash flows
to the same time period we can simply add
them all together to get the total value

2
Present Value of an AnnuityPresent Value of an Annuity
We can use the principle of value additivity to find
the present value of an annuity, by simply summing
the present values of each of the components:

2
Present Value of an Annuity (cont.)Present Value of an Annuity (cont.)
Using the example, and assuming a discount rate of
10% per year, we find that the present value is:
0 1 2 3 4 5
100 100 100 100 100
62.0
968.3
075.1
382.6
490.9
1
379.08

3
Actually, there is no need to take the present
value of each cash flow separately
We can use a closed-form of the PVA equation
instead:

3
We can use this equation to find the present
value of our example annuity as follows:
This equation works for all regular annuities,
regardless of the number of payments

3
The Future Value of an AnnuityThe Future Value of an Annuity
We can also use the principle of value additivity to
find the future value of an annuity, by simply
summing the future values of each of the
components:

3
The Future Value of an Annuity (cont.)The Future Value of an Annuity (cont.)
Using the example, and assuming a discount rate of
10% per year, we find that the future value is:
100 100 100 100 100
0 1 2 3 4 5
146.41
133.10
121.00
110.00
}= 610.51
at year 5

3
Just as we did for the PVA equation, we could
instead use a closed-form of the FVA equation:
This equation works for all regular annuities,
regardless of the number of payments

3
We can use this equation to find the future
value of the example annuity:

3
Annuities DueAnnuities Due
Thus far, the annuities that we have looked at begin
their payments at the end of period 1; these are
referred to as regular annuities
A annuity due is the same as a regular annuity,
except that its cash flows occur at the beginning of
the period rather than at the end
0 1 2 3 4 5
100 100 100 100 100
100 100 100 100 1005-period Annuity Due
5-period Regular Annuity

3
Present Value of an Annuity DuePresent Value of an Annuity Due
We can find the present value of an annuity due in
the same way as we did for a regular annuity, with
one exception
Note from the timeline that, if we ignore the first cash
flow, the annuity due looks just like a four-period
regular annuity
Therefore, we can value an annuity due with:

3
Present Value of an Annuity Due (cont.)Present Value of an Annuity Due (cont.)
Therefore, the present value of our example
annuity due is:
Note that this is higher than the PV of the,
otherwise equivalent, regular annuity

3
Future Value of an Annuity DueFuture Value of an Annuity Due
To calculate the FV of an annuity due, we can
treat it as regular annuity, and then take it
one more period forward:
0 1 2 3 4 5
Pmt Pmt Pmt Pmt Pmt

4
Future Value of an Annuity Due (cont.)Future Value of an Annuity Due (cont.)
The future value of our example annuity is:
Note that this is higher than the future value
of the, otherwise equivalent, regular annuity

4
Deferred AnnuitiesDeferred Annuities
A deferred annuity is the same as any other
annuity, except that its payments do not
begin until some later period
The timeline shows a five-period deferred
annuity
0 1 2 3 4 5
100 100 100 100 100
6 7

4
PV of a Deferred AnnuityPV of a Deferred Annuity
We can find the present value of a deferred annuity
in the same way as any other annuity, with an extra
step required
Before we can do this however, there is an important
rule to understand:
When using the PVA equation, the resulting PV is
always one period before the first payment occurs

4
PV of a Deferred Annuity (cont.)PV of a Deferred Annuity (cont.)
To find the PV of a deferred annuity, we first
find use the PVA equation, and then discount
that result back to period 0
Here we are using a 10% discount rate
0 1 2 3 4 5
0 0 100 100 100 100 100
6 7
PV2 = 379.08
PV0 = 313.29

4
PV of a Deferred Annuity (cont.)PV of a Deferred Annuity (cont.)
Step 1:
Step 2:

4
FV of a Deferred AnnuityFV of a Deferred Annuity
The future value of a deferred annuity is
calculated in exactly the same way as any
other annuity
There are no extra steps at all

4
Uneven Cash FlowsUneven Cash Flows
Very often an investment offers a stream of
cash flows which are not either a lump sum
or an annuity
We can find the present or future value of
such a stream by using the principle of value
additivity

4
Uneven Cash Flows: An Example (1)Uneven Cash Flows: An Example (1)
Assume that an investment offers the following cash
flows. If your required return is 7%, what is the
maximum price that you would pay for this
investment?
0 1 2 3 4 5
100 200 300

4
Uneven Cash Flows: An Example (2)Uneven Cash Flows: An Example (2)
Suppose that you were to deposit the following
amounts in an account paying 5% per year. What
would the balance of the account be at the end of the
third year?
0 1 2 3 4 5
300 500 700

4
Non-annual CompoundingNon-annual Compounding
So far we have assumed that the time period is equal
to a year
However, there is no reason that a time period can’t
be any other length of time
We could assume that interest is earned semi-
annually, quarterly, monthly, daily, or any other
length of time
The only change that must be made is to make sure
that the rate of interest is adjusted to the period
length

5
Non-annual Compounding (cont.)Non-annual Compounding (cont.)
Suppose that you have $1,000 available for
investment. After investigating the local banks, you
have compiled the following table for comparison. In
which bank should you deposit your funds?

5
To solve this problem, you need to determine which
bank will pay you the most interest
In other words, at which bank will you have the
highest future value?
To find out, let’s change our basic FV equation
slightly:
In this version of the equation ‘m’ is the number of
compounding periods per year

5
We can find the FV for each bank as follows:
First National Bank:
Second National Bank:
Third National Bank:
Obviously, you should choose the Third National Bank

5
Perpetuities
Perpetuity is a series of constant payments, A, each period
forever.
1 2 3 4 5 6 7
A
0
A A A A A A
Intuition:
Present Value of a perpetuity is the amount that must invested today at the
interest rate i to yield a payment of A each year without affecting the value
of the initial investment.
PVperpetuity = ∑[A/(1+i)t
] = A ∑[1/(1+i)t
] = A/i
PV1 = A/(1+r)
PV2 = A/(1+r)2
PV3 = A/(1+r)3
PV4 = A/(1+r)4
etc.
etc.

5
PerpetuitiesPerpetuities
A perpetuity is an annuity that has no definite
end, or a stream of cash payments that continues
forever.
A perpetuity is an annuity in which the periodic
payments begin on a fixed date and continue
indefinitely.
Fixed coupon payments on permanently
invested (irredeemable) sums of money are
prime examples of perpetuities.

5
Methods of Calculating Interest
 Simple interest: the practice of charging an
interest rate only to an initial sum (principal
amount).
 Compound interest: the practice of charging
an interest rate to an initial sum and to any
previously accumulated interest that has not
been withdrawn.

5
Simple Interest Formula
( )
where
= Principal amount
= simple interest rate
= number of interest periods
= total amount accumulated at the end of period
F P iP N
P
i
N
F N
= +
$1,000 (0.08)($1,000)(3)
$1,240
F = +
=

5
Compound Interest
Compound interest: the practice of charging
an interest rate to an initial sum and to any
previously accumulated interest that has not
been withdrawn.

5
Compound Interest Formula
1
2
2 1
0:
1: (1 )
2: (1 ) (1 )
: (1 )N
n P
n F P i
n F F i P i
n N F P i
=
= = +
= = + = +
= = +
M

5
Compound Interest
0
$1,000
$1,259.71
1 2
3
3
$1,000(1 0.08)
$1,259.71
F = +
=

6
Amortized loanAmortized loan
Installment loan in which the monthly
payments are applied first toward reducing
the interest balance, and any remaining sum
towards the principal balance. As the loan is
paid off, a progressively larger portion of the
payments goes toward principal and a
progressively smaller portion towards the
interest. Also called amortizing loan.

6
Continuous CompoundingContinuous Compounding
There is no reason why we need to stop increasing
the compounding frequency at daily
We could compound every hour, minute, or second
We can also compound every instant (i.e.,
continuously):
Here, F is the future value, P is the present value, r is
the annual rate of interest, t is the total number of
years, and e is a constant equal to about 2.718

6
Continuous Compounding (cont.)Continuous Compounding (cont.)
Suppose that the Fourth National Bank is offering to
pay 10% per year compounded continuously. What
is the future value of your $1,000 investment?
This is even better than daily compounding
The basic rule of compounding is: The more frequently
interest is compounded, the higher the future value

6
Continuous Compounding (cont.)Continuous Compounding (cont.)
Suppose that the Fourth National Bank is offering to
pay 10% per year compounded continuously. If you
plan to leave the money in the account for 5 years,
what is the future value of your $1,000 investment?

6
Congratulations!
You obviously understand this material.
Now try the next problem.
The Interactive Exercises are found in book.

6
Comparing PV to FVComparing PV to FV
Remember, both quantities must be present
value amounts or both quantities must be
future value amounts in order to be
compared.

6
How to solve a time value of money problem..
The “value four years from today” is a future
value amount.
The “expected cash flows of $100 per year for
four years” refers to an annuity of $100.
Since it is a future value problem and there is
an annuity, you need to solve for a FUTURE
VALUE OF AN ANNUITY.

Time value

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