Citibank: basic of corporate finance
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3-36 TIME VALUE OF MONEY
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Example
To illustrate, we will calculate the present value of an investment
with expected payments of $250 at the end of Year One, $500 at
the end of Year Two, $400 at the end of Year Four, and $100 at the
end of Year Five. The investor's required rate of return is 8%.
Year Payment x PVIF = Present Value
5 $100 x 1/(1.08)
5
= 68.06
4 $400 x 1/(1.08)
4
= 294.01
3 $000 x 1/(1.08)
3
= 0.00
2 $500 x 1/(1.08)
2
= 428.67
1 $250 x 1/(1.08)
1
= 231.48
PV = $1,022.22
Figure 3.7: Present Value of Unequal Discounted Cash Flows
Using a
financial
calculator
The present value of this investment is $1,022.22. Another approach
to solving this problem is to use a financial calculator that allows
the input of individual cash flows. The process is to input each cash
flow and the appropriate discount rate. Then, push the present value
key to derive the present value of the investment as $1,022.22.
Many calculators can handle this type of calculation. The logic
is to input each payment for the period and the interest rate the
investment is earning. Then push the future value key to derive the
future value.
You have completed the section on calculating the present value and future value of annuities,
perpetuities, and uneven payment streams. Please check your understanding of these
calculations by completing the Practice Exercise that begins on page 3-37. Check your
answers with the Answer Key.
Following the Practice Exercise is a summary of the concepts we have presented in this unit.
Please read the summary, then answer the questions in Progress Check 3. It is important that
you master these basic concepts, as they form the foundation for understanding the material
in succeeding units of this course. If you answer any of the questions incorrectly, we
recommend that you reread the appropriate section.
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PRACTICE EXERCISE 3.4
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.
18. A security is structured to pay cash flows of $100 per year for ten years. If an
investor's required rate of return is 5%, what price would the investor be willing to
pay for the security?
$_______________________
19. You have agreed to pay five installments of $100 for a television, the first payment
being due immediately. If the discount rate is 10%, what is the equivalent cash price
for the television?
$_______________________
20. PQR Inc. sets up a fund into which it will make annual payments that will
accumulate at 5% per annum. The fund is designed to repay a bond issue at the end
of 25 years. If the annual payment into the fund is fixed at $5,000,000, how large is
the bond issue?
$_______________________
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ANSWER KEY
18. A security is structured to pay cash flows of $100 per year for ten years. If an
investor's required rate of return is 5%, what price would the investor be willing to
pay for the security?
$772
Building the table and adding the present Using a financial calculator:
value interest factors for the ten cash Number of payments = 10
flows. Interest rate = 5%
PV
a
= $100 (7.7217) Payment = $100
PV
a
= $772.17 Press the present value key and
you should get $772.17
19. You have agreed to pay five installments of $100 for a television, the first payment
being due immediately. If the discount rate is 10%, what is the equivalent cash price
for the television?
$417
With the formula, calculate the present Using a financial calculator:
value of the last four payments and add Number of payments = 4
the total to the deposit. Interest rate = 10%
PV
a
= $100 (3.1699) Payment = $100
PV
a
= $317 Press the present value key and
Cash Price = $317 + $100 = $417 you should get $316.99. Then
add the deposit of $100 to get
a cash price of $417.
20. PQR Inc. sets up a fund into which it will make annual payments that will
accumulate at 5% per annum. The fund is designed to repay a bond issue at the end
of 25 years. If the annual payment into the fund is fixed at $5,000,000, how large is
the bond issue?
$238,635,494
Using a financial calculator:
Number of payments = 25
Interest rate = 5%
Payment = $5,000,000
Press the future value key and you should get $238,635,494.
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PRACTICE EXERCISE 3.4
(Continued)
21. You are offered a contract that will pay $50.00 per year indefinitely. If your
discount rate is 10%, what would you be willing to pay for the offered contract?
$_______________________
22. If your required return is 10%, what would you be willing to pay for a perpetuity
cash flow of $5.00 per year?
$_______________________
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ANSWER KEY
21. You are offered a contract that will pay $50.00 per year indefinitely. If your
discount rate is 10%, what would you be willing to pay for the offered contract?
$500.00
PV
p
= A x (1/R)
PV
p
= $50.00 (1 / 0.10)
PV
p
= $500.00
22. If your required return is 10%, what would you be willing to pay for a perpetuity
cash flow of $5.00 per year?
$50.00
PV
p
= A x (1/R)
PV
p
= $5.00 (1 / 0.10)
PV
p
= $50.00
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UNIT SUMMARY
In this unit, we focused on the changes in the value of money over
time.
Future value —
simple /
compound
interest
The future value of money at the end of a defined period of time
results from adding the amount of interest that may be earned on an
investment to the principal amount.
• Simple interest is paid on the principal at the end of the payment
period at a defined rate.
• Interest also may be compounded, which means that interest is
reinvested after each payment period and earns interest in each
subsequent period. Interest may be compounded at the end of
discrete annual or non-annual periods, or continuously throughout
the life of the investment.
Present value –
discount
Sometimes we want to know the current value of a cash flow that will
occur at some future date. Discrete or continuous discounting is the
process for equating a future cash flow with its present value. The
discount rate may be referred to as the opportunity cost of an
investment or the investor's required rate of return.
Annuity
An annuity is a security that involves a series of equal payments
made at regular time intervals. To derive the future value of an
annuity, each payment is compounded for the remaining life of the
security. A payment made at the end of the first year of a ten-year
annuity is compounded for nine years, the second payment is
compounded for eight years, etc. All compounded payments added
together equal the future value.
Calculating the present value of an annuity is the reverse process of
compounding. The payment due at the end of the tenth year is
discounted over the ten year period, the ninth payment is discounted
over nine years, etc. The present values of all payments added together
equal the present value of the annuity.
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Perpetuity
A perpetuity is a special type of annuity with an infinite number of
payment periods. The future value of a perpetuity is infinite — and
of little consequence. The present value is easy to calculate and
provides the investor with information to facilitate a comparison of
investment prices.
The present and future values of investments with payments of uneven
size or spacing also may be calculated by totaling the compounded or
discounted values of the cash flows.
You can see that the present / future value of an investment depends
on the discount / interest rate, the number of payments, the length of
time until maturity, and the amount of each cash flow. By
manipulating these factors, we can discover the future value of an
interest-bearing investment or the discounted present value of future
cash flows.
You have completed Unit Three: Time Value of Money. Before proceeding to Unit Four:
Valuing Financial Assets, please answer the following questions. They will check your
comprehension of the concepts presented in this unit. If you answer any questions
incorrectly, we suggest that you review the appropriate text.
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4 PROGRESS CHECK 3
Directions: Select the one best answer for each question, unless otherwise indicated.
Check your responses with the Answer Key on the next page.
1. ABC Bank receives a loan payment of $1,000 principal and interest from a
customer. The bank loans the $1,000 to another customer for one year at the rate of
6.25% per annum. The second customer will pay $1,062.50 to the bank. This
amount represents the:
_____a) loan made to the second customer discounted to its present value on the
maturity date.
_____b) compounded value of the first customer's payment after it was loaned to
customer two.
_____c) future value of the loan payment received from the first customer
calculated on a simple interest basis.
_____d) simple interest future value of the amount of the loan made to the first
customer.
2. In the future value formula FV
T
= P (1 + R/M)
T x M
, if:
_____a) M equals two, interest is compounded semi-annually for each year in
the period.
_____b) T equals one and M equals twelve, interest is compounded once during
each twelve month period.
_____c) T equals three, interest is compounded every four months.
_____d) T equals five and M equals one, simple interest is calculated once each
year for five years.
3. In the formula FV
T
= P x e
RT
, e
RT
represents:
_____a) the future value interest factor for a discrete number of compounding
periods.
_____b) the future value interest factor for an infinitely large number of
compounding periods.
_____c) simple interest calculated on a continuous basis for a specified period
of time.
_____d) interest factor compounded for a discrete period (e) at a certain rate (R)
for a specific amount of time (T).
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3-44 TIME VALUE OF MONEY
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ANSWER KEY
1. ABC Bank receives a loan payment of $1,000 principal and interest from a
customer. The bank loans the $1,000 to another customer for one year at the rate of
6.25% per annum. The second customer will pay $1,062.50 to the bank. This
amount represents the:
c) future value of the loan payment received from the first customer
calculated on a simple interest basis.
2. In the future value formula FV
T
= P (1 + R/M)
T x M
; if:
a) M equals two, interest is compounded semi-annually for each year in
the period.
3. In the formula FV
T
= P x e
RT
, e
RT
represents:
b) the future value interest factor for an infinitely large number of
compounding periods.
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PROGRESS CHECK
(Continued)
4. If you borrow $150,000 for five years at an 11.5% annual rate, compounded
monthly, payable at the end of the loan, how much will you have to pay at the end of
five years?
$_____________________
5. To calculate the present value of a bond portfolio, we apply a discount rate that
represents the (select two):
_____a) rate of return on the next best alternative investment.
_____b) reciprocal value of an interest rate on a comparable investment.
_____c) investor's required rate of return.
_____d) investor's return on the bond portfolio.
_____e) risk factor associated with the bond portfolio.
6. XYZ Company is issuing zero-coupon bonds to fund its expansion plans. The bonds
have a face value of $100,000 and a maturity of five years. How much would an
investor requiring a 10% return be willing to pay today for one of
these bonds?
$_____________________
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