Citibank: basic of corporate finance
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3-16 TIME VALUE OF MONEY
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Summary
In this section, we discussed discrete and continuous compounding
of interest. We looked at examples for calculating the interest
payment on investments for annual and nonannual periods and the
future value of these investments. The formulas for calculating
interest payments and future value are:
Interest payment
I = P (1 + R)
T
- P
I = Interest earned or paid
P = Principal
R = Stated (nominal) interest rate for each
interest paying period
T = Number of interest paying periods
Future value of
investment
discretely
compounded for
annual periods
FV
T
= P (1 + R)
T
FV
T
= Future value after T paying periods
P = Principal
R = Stated (nominal) interest rate for
each interest paying period
T = Number of interest paying periods
Future value of
investment
discretely
compounded for
non-annual
periods
FV
T
= P (1 + R/M)
T x M
FV
T
= Future value after T years
P = Principal
R = Stated (nominal) annual
interest rate
M = Number of compounding periods
within a year
T = Number of years
Future value
of investment
compounded
continuously
FV
T
= P x e
RT
FV
T
= Future value after T periods
P = Principal
e = Base of the natural logarithm
(2.718282)
R = Stated (nominal) annual interest
rate for each period
T = Number of periods
You can use these formulas or your financial calculator to answer the questions in the
Practice Exercise that begins on page 3-17.
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PRACTICE EXERCISE 3.2
Directions: Calculate the answer to each question and write the correct answer in the
blank. Check your solution with the Answer Key on the next page.
7. A deposit of $100 is made into an account on Jan. 1, 1950. If the account has a
balance of $480.10 on Dec. 31, 1989, what is the annual rate of interest earned on
the account?
______%
8. What is the future value of $2,000 after 5 years if the interest rate is 30% p.a.?
$_____________________
9. If you invest $250,000 for one year at 8% compounded semi-annually, what is the
total amount of interest you will receive during the year?
$_____________________
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ANSWER KEY
7. A deposit of $100 is made into an account on Jan. 1, 1950. If the account has a
balance of $480.10 on Dec. 31, 1989, what is the annual rate of interest earned on
the account?
4.0%
Use the future value formula and solve for R.
FV
T
= P (1 + R)
T
$480.10 = $100 (1 + R)
40
$4.8010 = (1 + R)
40
Divide both sides by $100
($4.8010)
1/40
= (1 + R) Raise each side of the equation by the exponent 1/40
1.0400 = 1 + R
0.04 = R
R = 4.0%
8. What is the future value of $2,000 after 5 years if the interest rate is 30% p.a.?
$7,425.86
FV
T
= P (1 + R)
T
FV
5
= $2,000 (1 + 0.30)
5
FV
5
= $7,425.86
9. If you invest $250,000 for one year at 8% compounded semi-annually, what is the
total amount of interest you will receive during the year?
$20,400
I = P (1 + R/M)
T x M
- P
I = $250,000 (1 + 0.08/2)
1 x 2
- $250,000
(If the annual rate is 8%, the semi-annual rate is 4% and the number of periods is 2)
I = $250,000 (1.04)
2
- $250,000
I = $250,000 (1.0816) - $250,000
I = $270,400 - $250,000
I = $20,400
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PRACTICE EXERCISE 3.2
(Continued)
10. If the interest in question 9 is compounded quarterly, how much interest will you
receive during the year?
$_____________________
11. If you borrow $200,000 for three years at a 10% annual rate, compounded
quarterly, but payable at the end of the loan, how much will you have to repay at the
end of the three years?
$_____________________
12. What is the future value of $1,000 after 10 years, with an annual rate of 12%
compounded continuously?
$_____________________
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ANSWER KEY
10. If the interest in question 9 is compounded quarterly, how much interest will you
receive during the year?
$20,608.04
I = P (1 + R)
T
- P
I = $250,000 (1 + 0.02)
4
- $250,000
I = $250,000 (1.082432) - $250,000
I = $270,608.04 - $250,000
I = $20,608.04
11. If you borrow $200,000 for three years at a 10% annual rate, compounded
quarterly, but payable at the end of the loan, how much will you have to repay at the
end of the three years?
$268,977.76
FV
T
= P (1 + R/M)
T x M
FV
3
= P (1 + 0.10/4)
3 x 4
FV
3
= $200,000 (1 + 0.025)
12
FV
3
= $200,000 (1.34488)
FV
3
= $268,977.76
12. What is the future value of $1,000 after 10 years, with an annual rate of 12%
compounded continuously?
$3,320.12
FV
T
= P x e
R x T
FV
10
= $1,000 x e
0.12 x 10
FV
10
= $1,000 x (2.7183)
1.2
FV
10
= $1,000 x (3.32012)
FV
10
= $3,320.12
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PRESENT VALUE
Discounting –
reversal of
compounding
Discounting is the process we use to equate a future cash flow to its
present value. The calculation is the reversal of the compounding
process and also may be done discretely or continuously.
To understand the discounting process, let's begin with the future
value equation.
FV
T
= PV (1 + R)
T
Notice that there is one slight difference in this future value equation
— the variable P (principal) has been replaced by a PV variable. The
PV stands for present value and refers to the value of the investment
at the present time. To solve for PV, we divide both sides of the
equation by the interest factor (1 + R)
T
. The resulting present value
equation is:
PV =
FV
T
________
(1 + R)
T
Present value
formula
The present value equation is often expressed in another format.
PV = FV
T
[1/(1 + R)]
T
Where:
PV = Present value of the investment (value today)
FV
T
= Future value of the investment at time T
T = Time in years until maturity
R = Annual discount rate
Present value
interest factor
The term 1 / (1 + R)
T
is the present value interest factor at R
percent for T periods. FV
T
is the future value of the investment at
time T.
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Opportunity
cost of money
Notice that we now refer to R as the "discount" rate. The discount
rate represents the opportunity cost of money, which is the rate
of return that could have been earned on the best alternative
investment. In finance, analysts often use the concept of opportunity
cost to evaluate investment alternatives. The logic is to estimate the
next best return scenario if a different investment decision had been
made. Applying this concept to discount rates, analysts say that, given
approximately equal time frames and levels of risk, the appropriate
rate to discount cash flows is the rate of return that represents the next
best alternative in the list of investment choices. The discount rate or
opportunity cost is also called the investor's required rate of return.
Discount rate = Opportunity cost = Investor's required rate of return
Present value terminology is similar to the future value terminology
we studied in the last section. You can use most financial calculators
for quick and easy present value calculations.
Discrete Discounting
A present value interest factor is the reciprocal of the future value
interest factor. The value of the present value interest factor
decreases as the time period or interest rate increases. Let's apply
the present value formula to a few examples.
Example
Suppose that you expect to receive four future cash flows of $1,000:
1) In one year at a discount rate of 10%
2) In two years at a discount rate of 10%
3) In two years at a discount rate of 12%
4) In three years at a discount rate of 10%
What is the present value of each of these $1,000 cash flows?
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Time Discount Rate Present Value
1 year 10% PV = FV
T
[ 1 / (1 + R) ]
T
PV = $1,000 [ 1 / (1 + 0.10) ]
1
PV = $1,000 [0.90909]
PV = $909.09
2 years 10% PV = FV
T
[ 1 / (1 + R) ]
T
PV = $1,000 [ 1 / (1 + 0.10) ]
2
PV = $1,000 [ 0.82645 ]
PV = $826.45
2 years 12% PV = FV
T
[ 1 / (1 + R) ]
T
PV = $1,000 [ 1 / (1 + 0.12) ]
2
PV = $1,000 [0.79719]
PV = $797.19
3 years 10% PV = FV
T
[ 1 / (1 + R) ]
T
PV = $1,000 [ 1 / (1 + 0.10) ]
3
PV = $1,000 [0.75131]
PV = $751.31
Look at the three cash flows that are discounted at 10% p.a. You can
see that the present value of those three cash flows decreases as the
length of time increases. Also, for the two cash flows expected in two
years, the cash flow discounted at an annual rate of 10% has a greater
present value ($826.45) than the cash flow discounted at an annual
rate of 12% ($797.19).
Present value
versus future
value
If an investor has the opportunity to invest funds today at the same
rate as the discount rate, it doesn't matter if the cash flow is received
today for the present value amount, or later for the future value
amount. For example, the $1,000 cash flow expected in three years,
given an annual discount rate of 10%, equals $751.31 today. An
investor with opportunities to invest at 10% compounded annually
would be indifferent between receiving $751.31 today or $1,000 in
three years, all other factors being equal.
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Zero-coupon
bond
The zero-coupon bond concept has a very practical application in the
valuing of securities. For example, it is a security that pays no
interest; it is priced at a discount from the face value to give the
investor a specific rate of return. The face value is the amount the
issuer will pay at the time of maturity. The issuer prices the bond at a
discount so that investors receive a return on their investments.
Example
Suppose that a company is issuing zero-coupon bonds to raise funds
for its operation. The bonds have a face value of $10,000 and a
maturity of one year. What would an investor requiring a 9% return
be willing to pay for one of the bonds today? Let's apply
our present value formula with FV
T
= $10,000, T = 1 year, and
R = 0.09.
PV = FV
T
[ 1 / (1 + R) ]
T
PV = $10,000 [ 1 / (1 + 0.09) ]
1
PV = $10,000 [ 0.917431 ]
PV = $9,174.31
An investor requiring a 9% return on investment would pay
$9,174.31 for one of the bonds. As an alternative, s/he could invest
the $9,174.31 in an account earning 9% annually; at the end of one
year, the account would be worth $10,000. As you can see,
discounting to present value and compounding to future value are
simply inverse calculations.
Continuous Discounting
Reverse of
continuous
compounding
We can also reverse the calculation of continuous compounding to
calculate a present value that results from continuous discounting.
Recall that the formula for continuous compounding is:
FV
T
= PV x e
RT
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Formula for
continuous
discounting
To discount continuously, we rearrange the continuous compounding
formula and use present value (PV) in the place of principal (P).
Solving for PV, we arrive at the continuous discounting formula.
PV = FV
T
[ 1 / (e
RT
)]
Where:
PV = Present value of the investment (value today)
FV
T
= Future value of the investment at time T
e = Base of the natural logarithm (2.718282)
T = Time in years until maturity
R = Annual discount rate (opportunity cost of money)
As you can see, the present value interest factor for continuous
discounting 1 / (e
RT
) is the reciprocal of the future value interest factor
for continuous compounding e
RT
. A more common notation for this
present value interest factor is e
-RT
; the formula is:
PV = FV
T
[ e
-RT
]
Example
As an example, let's calculate the present value of a $1,500 cash
flow to be received in 4 years, discounted continually at an annual
rate of 10%.
PV = FV
T
[ e
-RT
]
PV = $1,500 [ 2.718282
-0.10 x 4
]
PV = $1,500 [ 0.6703200]
PV = $1,005.48
Use the the formula for discrete discounting (PV = FV
T
[1/(1 + R)
T
]) or continuous
discounting (PV = FV
T
[ e
-RT
]) or your financial calculator to solve the discounting
problems in the Practice Exercise that follows.
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