TIME VALUE OF MONEY
Future Value versus Present Value
Suppose a firm has an opportunity to spend
$15,000 today on some investment that will
produce $17,000 spread out over the next five
years as follows:
Year Cash flow
1 $3,000
2 $5,000
3 $4,000
4 $3,000
5 $2,000
Is this a wise investment?
Future Value versus Present Value (cont.)
To make the right investment decision, managers
need to compare the cash flows at a single point in
time.
Figure 5.1 Time Line
Future Value versus Present Value (cont.)
When making investment decisions, managers usually
calculate present value.
Figure 5.2 Compounding and Discounting
Basic Patterns of Cash Flow
The three basic patterns of cash flows include:
• A single amount: A lump sum amount either held
currently or expected at some future date.
• An annuity: A level periodic stream of cash flow.
• A mixed stream: A stream of unequal periodic
cash flows.
Future Value of a Single Amount
• Future value is the value at a given future date of
an amount placed on deposit today and earning
interest at a specified rate. Found by applying
compound interest over a specified period of time.
• Compound interest is interest that is earned on a
given deposit and has become part of the principal
at the end of a specified period.
• Principal is the amount of money on which interest
is paid.
Future Value of a Single Amount: The
Equation for Future Value
• We use the following notation for the various
inputs:
– FVn = future value at the end of period n
– PV = initial principal, or present value
– r = annual rate of interest paid. (Note: On financial
calculators, I is typically used to represent this rate.)
– n = number of periods (typically years) that the money is
left on deposit
• The general equation for the future value at the end
of period n is
FVn = PV (1 + r)n
Future Value of a Single Amount: The
Equation for Future Value
Jane Farber places $800 in a savings account paying 6%
interest compounded annually. She wants to know how much
money will be in the account at the end of five years.
FV5 = $800 (1 + 0.06)5 = $800 (1.33823) = $1,070.58
This analysis can be depicted on a time line as follows:
Figure 5.4
Future Value Relationship
Present Value of a Single Amount
• Present value is the current dollar value of a
future amount—the amount of money that would
have to be invested today at a given interest rate
over a specified period to equal the future amount.
• It is based on the idea that a dollar today is worth
more than a dollar tomorrow.
• Discounting cash flows is the process of finding
present values; the inverse of compounding
interest.
• The discount rate is often also referred to as the
opportunity cost, the discount rate, the required
return, or the cost of capital.
Present Value of a Single Amount: The
Equation for Present Value
The present value, PV, of some future amount, FVn, to
be received n periods from now, assuming an interest
rate (or opportunity cost) of r, is calculated as
follows:
Present Value of a Single Amount: The
Equation for Future Value
Pam Valenti wishes to find the present value of
$1,700 that will be received 8 years from now. Pa
m’s opportunity cost is 8%.
PV = $1,700/(1 + 0.08)8 = $1,700/1.85093 = $918.46
This analysis can be depicted on a time line as
follows:
Figure 5.5
Present Value Relationship
Annuities
An annuity is a stream of equal periodic cash flows,
over a specified time period. These cash flows can be
inflows of returns earned on investments or outflows
of funds invested to earn future returns.
– An ordinary (deferred) annuity is an annuity for which
the cash flow occurs at the end of each period
– An annuity due is an annuity for which the cash flow
occurs at the beginning of each period.
– An annuity due will always be greater than an otherwise
equivalent ordinary annuity because interest will
compound for an additional period.
Personal Finance Example
Fran Abrams is choosing which of two annuities to
receive. Both are 5-year $1,000 annuities; annuity A is
an ordinary annuity, and annuity B is an annuity due.
Fran has listed the cash flows for both annuities as
shown in Table 5.1 on the following slide.
Note that the amount of both annuities total $5,000.
Table 5.1 Comparison of Ordinary Annuity and
Annuity Due Cash Flows ($1,000, 5 Years)
Finding the Future Value of an Ordinary
Annuity
• You can calculate the future value of an ordinary
annuity that pays an annual cash flow equal to CF
by using the following equation:
• As before, in this equation r represents the interest
rate and n represents the number of payments in
the annuity (or equivalently, the number of years
over which the annuity is spread).
Personal Finance Example
Fran Abrams wishes to determine how much money she
will have at the end of 5 years if he chooses annuity A, the
ordinary annuity and it earns 7% annually. Annuity A is
depicted graphically below:
This analysis can be depicted on a time line as follows:
Finding the Present Value of an Ordinary
Annuity
• You can calculate the present value of an ordinary
annuity that pays an annual cash flow equal to CF
by using the following equation:
• As before, in this equation r represents the interest
rate and n represents the number of payments in
the annuity (or equivalently, the number of years
over which the annuity is spread).
Finding the Present Value of an Ordinary
Annuity (cont.)
Braden Company, a small producer of plastic toys, wants
to determine the most it should pay to purchase a
particular annuity. The annuity consists of cash flows of
$700 at the end of each year for 5 years. The firm requires
the annuity to provide a minimum return of 8%.
This situation can be depicted on a time line as follows:
Finding the Future Value of an Annuity
Due
• You can calculate the present value of an annuity
due that pays an annual cash flow equal to CF by
using the following equation:
• As before, in this equation r represents the interest
rate and n represents the number of payments in
the annuity (or equivalently, the number of years
over which the annuity is spread).
Personal Finance Example (cont.)
Future value of ordinary Future value of annuity
annuity due
$5,705.74 $6,153.29
The future value of an annuity due is always higher
than the future value of an ordinary annuity.
Finding the Present Value of an Annuity
Due
• You can calculate the present value of an ordinary
annuity that pays an annual cash flow equal to CF
by using the following equation:
• As before, in this equation r represents the interest
rate and n represents the number of payments in
the annuity (or equivalently, the number of years
over which the annuity is spread).
Finding the Present Value of a Perpetuity
• A perpetuity is an annuity with an infinite
life, providing continual annual cash flow.
• If a perpetuity pays an annual cash flow of
CF, starting one year from now, the present
value of the cash flow stream is
PV = CF ÷ r
Personal Finance Example
Ross Clark wishes to endow a chair in finance at his
alma mater. The university indicated that it requires
$200,000 per year to support the chair, and the
endowment would earn 10% per year. To determine
the amount Ross must give the university to fund the
chair, we must determine the present value of a
$200,000 perpetuity discounted at 10%.
PV = $200,000 ÷ 0.10 = $2,000,000
Future Value of a Mixed Stream
Shrell Industries, a cabinet manufacturer, expects to
receive the following mixed stream of cash flows over
the next 5 years from one of its small customers.
Future Value of a Mixed Stream
If the firm expects to earn at least 8% on its
investments, how much will it accumulate by the end
of year 5 if it immediately invests these cash flows
when they are received?
This situation is depicted on the following time line.
Present Value of a Mixed Stream
Frey Company, a shoe manufacturer, has been
offered an opportunity to receive the following mixed
stream of cash flows over the next 5 years.
Present Value of a Mixed Stream
If the firm must earn at least 9% on its investments,
what is the most it should pay for this opportunity?
This situation is depicted on the following time line.
Compounding Interest More Frequently
Than Annually
• Compounding more frequently than once a year
results in a higher effective interest rate because
you are earning on interest on interest more
frequently.
• As a result, the effective interest rate is greater
than the nominal (annual) interest rate.
• Furthermore, the effective rate of interest will
increase the more frequently interest is
compounded.
Compounding Interest More Frequently
Than Annually (cont.)
A general equation for compounding more frequently
than annually
Recalculate the example for the Fred Moreno example
assuming (1) semiannual compounding and (2)
quarterly compounding.
Continuous Compounding
• Continuous compounding involves the
compounding of interest an infinite number of times
per year at intervals of microseconds.
• A general equation for continuous compounding
where e is the exponential function.
Personal Finance Example
Find the value at the end of 2 years (n = 2) of Fred
Moreno’s $100 deposit (PV = $100) in an account
paying 8% annual interest (r = 0.08) compounded
continuously.
FV2 (continuous compounding) = $100 e0.08 2
= $100 2.71830.16
= $100 1.1735 = $117.35
Nominal and Effective Annual Rates of
Interest
• The nominal (stated) annual rate is the
contractual annual rate of interest charged by a
lender or promised by a borrower.
• The effective (true) annual rate (EAR) is the
annual rate of interest actually paid or earned.
• In general, the effective rate > nominal rate
whenever compounding occurs more than once per
year
Personal Finance Example
Fred Moreno wishes to find the effective annual rate
associated with an 8% nominal annual rate (r = 0.08)
when interest is compounded (1) annually (m = 1);
(2) semiannually (m = 2); and (3) quarterly (m = 4).
Special Applications of Time Value: Deposits
Needed to Accumulate a Future Sum
The following equation calculates the annual cash payment (CF)
that we’d have to save to achieve a future value (FVn):
Suppose you want to buy a house 5 years from now, and you
estimate that an initial down payment of $30,000 will be
required at that time. To accumulate the $30,000, you will wish
to make equal annual end-of-year deposits into an account
paying annual interest of 6 percent.
Special Applications of Time Value: Loan
Amortization
• Loan amortization is the determination of the
equal periodic loan payments necessary to provide
a lender with a specified interest return and to
repay the loan principal over a specified period.
• The loan amortization process involves finding the
future payments, over the term of the loan, whose
present value at the loan interest rate equals the
amount of initial principal borrowed.
• A loan amortization schedule is a schedule of
equal payments to repay a loan. It shows the
allocation of each loan payment to interest and
principal.
Special Applications of Time Value: Loan
Amortization (cont.)
• The following equation calculates the equal periodic loan
payments (CF) necessary to provide a lender with a specified
interest return and to repay the loan principal (PV) over a
specified period:
• Say you borrow $6,000 at 10 percent and agree to make
equal annual end-of-year payments over 4 years. To find the
size of the payments, the lender determines the amount of a
4-year annuity discounted at 10 percent that has a present
value of $6,000.
Table 5.6 Loan Amortization Schedule
($6,000 Principal, 10% Interest, 4-Year
Repayment Period)
Special Applications of Time Value:
Finding Interest or Growth Rates
• It is often necessary to calculate the compound
annual interest or growth rate (that is, the annual
rate of change in values) of a series of cash flows.
• The following equation is used to find the interest
rate (or growth rate) representing the increase in
value of some investment between two time
periods.
Personal Finance Example
Ray Noble purchased an investment four years ago
for $1,250. Now it is worth $1,520. What compound
annual rate of return has Ray earned on this
investment? Plugging the appropriate values into
Equation 5.20, we have:
r = ($1,520 ÷ $1,250)(1/4) – 1 = 0.0501 = 5.01% per
year
The GNP doubled between 1970 and 1990 (20 years).
Approximate the average growth
rate of the GNP.
Ans: ?
Growing Perpetuity
• Calculation of Growing Perpetuity: A stream of
cash flows that grows at a constant rate forever is
known as growing perpetuity.
• Assuming that the discount rate is 7% per annum,
how much would you pay to receive $50, growing
at 5%, annually, forever?
Examples with Multiple Annuities
• I want to withdraw $5,000 a year for the next five
years and $8,000 a year for the following five
years. I can earn 10.5 percent. How much do I
have to invest today? Ans : $36,889.61
• Let’s look at this problem in a different way. I will
invest $36,889.61 today in an asset that will earn
10.5 percent. I plan to withdraw $5,000 a year for
the next five years. How much can I withdraw each
year for the next 5 years? We know the answer is
$8,000 but how would we calculate it? Ans $8000
Retirement Problem
• You are planning to save for the college education
of your children. They are two years apart; one will
begin college in 15 years and the other will begin
college in 17 years. Assume both will be on the four
year plan. You estimate each child’s education will
cost $23,000 per year, payable at the beginning of
each school year. The annual interest rate is 6.5
percent. How much must you deposit each year in
an account to fund your children’s education? You
will make your last deposit when your oldest child
enters college.
Retirement problem
• You are trying to plan for retirement on 10 years.
You currently have $200,000 in a bond account and
$400,000 in a stock account. You plan to add
$10,000 per year at the end of each of the next 10
years to your bond account. The stock account
earns 12.5% and the bond account earns 8.5%.
When you retire, you plan to withdraw an equal
amount for the next 20 years (at the end of each
year) and have nothing left. Additionally, when you
retire you will transfer your money to an account
that earns 7.25%. How much can you withdraw
each year?
EAR and APR
L. Shark is willing to lend you $10,000 for three
months. At the end of six months, L. Shark requires you
to repay the $10,000, plus 50%.
•a) What is the length of the compounding period?
•b) What is the rate of interest per compounding
period?
•c) What is the annual percentage rate associated with
L. Shark's lending activities?
•d) What is the effective annual rate of interest
associated with L. Shark's lending activities?
• a) Six months
• b) 50%
• c) APR = 50% x 2 = 100%
• d) EAR = (1 + 0.50)2 - 1 = 125%
• The Consistent Savings and Loan is designing a new
account that pays interest quarterly. They wish to
pay, effectively, 16% per year on this account.
Consistent desires to advertise the annual
percentage rate on this new account, instead of the
effective rate, since its competitors state their
interest on an annualized basis. What is the APR
that corresponds to an effective rate of 16% for this
new account?
• EAR = 16%
• APR = 1.160.25 this takes the fourth root of 1 +
EAR
• i = 3.78%
• APR = 3.78% x 4 = 15.121%
Finding N in annuity
• A credit card charges an annual rate of 14%
compounded monthly. This month’s bill is $6000.
The minimum payment is $125. Suppose I keep
paying $125 each month. How long will it take to
pay off the bill? What is the total interest paid
during that period?
• A car dealer offers either 0% financing for 3 years
or a $3000 rebate on a $25000 car and assuming
that you could find a loan for 6% for 36 months
which deal gives you a smaller payment?
• Suppose you wish to retire forty years from today.
You determine that you need $50,000 per year
once you retire, with the first retirement funds
withdrawn one year from the day you retire. You
estimate that you will earn 6% per year on your
retirement funds and that you will need funds up to
and including your 25th birthday after retirement.
• a) How much must you deposit in an account today
so that you have enough funds for retirement?
• b) How much must you deposit each year in an
account, starting one year from today, so that you
have enough funds for retirement?
