

Corporate
Finance FIN 622
VU
Lesson
03
TIME VALUE OF
MONEY
Time
Value of Money offers an
overview of the information required to
calculate the future and
present
values
of individual cash flows, ordinary
annuities, due perpetuities and
investments with uneven cash
flows.
TVM is
based on the concept that a
dollar that you have today
is worth more than the
promise or
expectation
that you will receive a
dollar in the future. Money
that you hold today is worth
more because
you
can invest it and earn
interest. After all, you should
receive some compensation
for foregoing
spending.
This
hand out has been divided
into following topics, which
will be explained in detail:
1.
PRESENT VALUE
2.
FUTURE VALUE
3.
ANNUITIES
4.
PERPETUITY
PRESENT
VALUE
The
present value of a future
cash flow is the nominal amount of
money to change hands at
some future
date,
discounted to account for the time
value of money. A given amount of money
is always more
valuable
sooner
than later because this enables
one to take advantage of investment
opportunities.
The
present value of delayed
payoff may be found by
multiplying the payoff by a discount
factor which is
less
than 1. If C1 denotes the expected
payoff at period 1,
then
Present
Value (PV) = discount factor. C1
This
discount factor is the value today of $1
received in the future. It is usually
expressed as the reciprocal
of 1
plus a rate of
return.
Discount
Factor = 1 / 1+r
The
rate of return r is the reward
that investors demand for
accepting delayed
payment.
The
present value formula may be
written as follow:
PV = 1 /
1+r. C1
To
calculate present value, we discount
expected payoffs by the rate of return
offered by equivalent
investment
alternatives in the capital market. This
rate of return is often referred to a the
discount rate,
hurdle
rate or opportunity cost of
capital. If the opportunity cost is 5
percent expected payoff is
$200,000,
the
present value is calculated as
follows:
PV =
200,000 / 1.05 =
$190,476
FUTURE
VALUE
Future
value measures what money is
worth at a specified time in the future
assuming a certain interest
rate.
This
is used in time value of money
calculations.
To
determine future value (FV)
without compounding:
Where
PV
is
the present value or principal,
t
is
the time in years, and r
stands
for the per annum
interest
rate.
To
determine future value when
interest is compounded:
Where
PV
is
the present value, n
is
the number of compounding periods, and
i
stands
for the interest rate
per
period.
The
relationship between i
and
r
is:
Where
X
is
the number of periods in one year. If
interest is compounded annually, X
=
1. If interest is
compounded
semiannually, X
=
2. If interest is compounded quarterly, X
=
4. If interest is compounded
monthly,
X
=
12 and so on. This works for
everything except compounded continuously, which
must be
handled
using exponential.
Similarly, the
relationship between n
and
t
is:
13
Corporate
Finance FIN 622
VU
For
example, what is the future value of 1
money unit in one year,
given 10% interest? The number of
time
periods
is 1, the discount rate is 0.10, the
present value is 1 unit, and
the answer is 1.10 units. Note
that this
does
not mean that the holder of
1.00 unit will automatically
have 1.10 units in one year,
it means that
having
1.00 unit now is the equivalent of having
1.10 units in one
year.
ANNUITY
An
annuity is an equal, annual
series of cash flows. Annuities
may be equal annual
deposits, equal
annual
withdrawals,
equal annual payments, or
equal annual receipts. The
key is equal,
annual cash
flows. Annuities
work
in the following way.
Illustration:
Assume
annual deposits of $100
deposited at end of year
earning 5% interest for
three years.
Year 1:
$100 deposited at end of
year
=
$100.00
Year 2:
$100 × .05 = $5.00 + $100 +
$100 = $205.00
Year 3:
$205 × .05 = $10.25 + $205 +
$100 = $315.25
There
are tables for working
with annuities. Future Value
of Annuity Factors is the table to be
used in
calculating
annuities due. Just look up
the appropriate number of periods, locate the
appropriate interest,
take
the factor found and
multiply it by the amount of the annuity.
For
instance, on the threeyear 5% interest
annuity of $100 per year.
Going down three years,
out to 5%,
the
factor of 3.152 is found.
Multiply that by the annuity of
$100 yields a future value
of $315.20.
The
present value of annuity can
be finding out by the following
formula:
Present
value of annuity = C
[1/r1/r(1+r)t]
The
expression in brackets is the annuity
factor, which is the present value at
discount rate r of an annuity
of $1 paid at the
end of each of t
periods.
PEPETUITY
Perpetuity is a
cash flow without a fixed
time horizon.
For
example if someone were
promised that they would
receive a cash flow of $400
per year until they
died,
that
would be perpetuity. To find the
present value of a perpetuity, simply
take the annual return in
dollars
and
divide it by the appropriate discount
rate.
The
present value of perpetuity
can be finding out by the
following formula:
Present
value of perpetuity=C/r
Where
C is the annual return in dollars and r
is the discount rate.
Illustration:
If
someone were promised a cash
flow of $400 per year
until they died and they could earn 6% on
other
investments
of similar quality, in present
value terms the perpetuity
would be worth
$6,666.67.
Present
value of perpetuity= ($400 /
.06 = $6,666.67)
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