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Chapter 18
Financial Mathematics
Learning Objectives
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Explain why a dollar today is worth more
than a dollar in the future.
Define future value and present value.
Explain the difference between an ordinary
annuity and an annuity due.
Calculate the future value and present value
of an annuity.
Decisions for Financial Managers
1. Where should we invest our funds?
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i.e. What types of equipment, buildings, services,
programs, or other opportunities should we
invest?
2. How will we finance our investment needs?
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Will we use debt, equity, or both?
Investment and Financing
• Investment decisions involve expending funds
today while expecting to realize returns in the
future
• Financing decisions involve the receipt of
funds today in return for a promise to make
payments in the future
• Therefore, manager’s major task is evaluating
the relative attractiveness of alternative
investment and financing opportunities
Time Value of Money
• Concept of time value money:
– Payment that is made or received in the first year
has a greater value than an identical payment made
or received in the tenth year
– “A dollar today is worth more than a dollar
tomorrow.”
• The idea that the value of money changes over
time is absolutely critical to investment and
financing decisions.
Time Value of Money
• If we accept that a dollar today is worth more
than a dollar received in the future, how can
we determine what that difference is?
– We must determine the cost for money
– e.g. the interest rate associated with borrowing, the
opportunity cost of investment, etc.
• Interest rate - the price for the commodity
called money (e.g. 10% per year)
Present Value & Future Value
• If the interest rate is 10%, a dollar received 1
year from now is only worth $0.9091 in
today’s dollar.
– This concept is called “Present Value”
• Conversely, if the interest rate is 10%, $0.9091
today will be worth exactly $1 one year from
now.
– This concept is called “Future Value”
Simple vs. Compound Interest
• Simple interest - interest is only calculated on
the principal
• Compound interest - interest is calculated on
the principal and the interest.
– Compound interest is the basis for calculating
future values over multiple time periods.
Future Value – Single Sum
• Example – nursing home may want to invest
$100,000 today in a fund to be used in 2 years for
replacement
• Question – what sum of money would be available 2
years from now?
• Solution – set up time graph using:
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–
–
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# of periods during which compounding occurs (n)
Present value of future sum (p)
Future value (f) of present sum
Interest rate per period (i)
Future Value, cont.
• Knowing values of any three variables can
solve for the fourth
• Time line - helps conceptualize the problem
and identify the known values
• Value 0 – represents present time (today),
Values 1 and 2 represents Year 1 and Year 2
Figure 18–1 Future Value of Present
Sum
Future Value, cont.
• We can solve for future value through
substitution of following formula:
f = p  f(i,n)
• Factor (i, n) – future value of $1 invested today
for n periods at i rate of interest per period:
f = $100,00  f(10%,2) or
f = $100,000  1.210 or
f = $121,000
Present Value – Single Sum
• Same table of values used
• Formula:
p = f  p(i,n)
• Factor p (i, n) represents present value of $1
received in n periods at interest rate of i
Present Value, cont.
• Example – assume that an HMO has $100,000
debt obligation due in 2 years
• Question – how much money to set aside today to
meet the obligation at the expected yield of 12%?
• Calculation to solve the problem:
p = f  p(i,n) or
p = $100,000  p(12%,2) or
p = $100,000  .7972 or
p = $79,720
Figure 18–3 Present Value of a Future
Sum
Present and Future Value
• Future and present value tables are reciprocals
of each other as shown in the formula:
f(i,n) = 1/p(i,n)
• Only one of the two tables is necessary to
solve present or future value problem
Annuities
• Often, there is more than one payment or receipt
• If each payment or receipt is constant per time period, there is
an annuity situation
• Basic formula:
F = R  F(i,n)
• Consider example:
– F – future value of invested annuity deposits at the end of
period n
– R – periodic deposits that are constant for each period
($50,000) invested for 3 years at 8%
– Deposits are made at the end of each period => ordinary
annuity
Figure 18–5 Determining the Future
Value of an Annuity
Future Value of Annuity, cont.
To solve the problem:
F = R  F(i,n) or
F = $50,000  F(8%,3) or
F = $50,000  3.2464 or
F = $162,320
Alternative Annuity Calculation
Values in annuity future value table could be
determined through addition for a single sum
of future values:
Future Annuity Formula
In general, a future value annuity factor can be
expressed as follows:
F(i,n) = f(i,1) + f(i,2) +…+ f(i,n - 1) + 1.0
Annuity and Single Sum
• A financial mathematics problem may be part annuity
and part single sum
• In this case, time graph is helpful
• Consider example:
– Hospital has sinking fund payment requirement for last 10
years of a bond’s life
– $45M must be available at the end to retire debt
– $5M available today
– Time line – 20 years before debt retirement
– Investment yield – 10% per year
– Question – what annual deposit is required?
Figure 18–6 Determining the Annual
Sinking Fund Deposit
Annuity Sinking Fund, cont.
• Future value:
f = $5,000,000  f(10%,20) or
f = $33,637,500
f = $5,000,000  6.7275 or
• Final solution:
F - f = R  F(i,n) or
$11,362,500 = R  F(10%,10) or
$11,362,500 = R  15.9374 or
R = $712,946
Present Value of Annuity
• Procedure analogous to future value of annuity
• Payments at the end of the period
• General equation:
P = R  P(i,n)
• Factor P (i, n) - present value of $1 received at
the end of each period for n periods when i is
the rate of interest
Present Value Annuity Problem
Consider example:
– Purchase of an older hospital
– Obligation – payments of $100,000 per year for 4
years to vested employees
– Discount rate – 12%
– Question - what the present value of this obligation
is so that it can be subtracted from the negotiated
purchase price?
Figure 18–8 Present Value of the
Pension Obligation
Present Value of Pension Obligation,
cont.
To solve the problem:
P = $100,000 x P(12%,4) or
P = $100,000 x 3.0373
P = $303,730
Alternative Annuity Present Value
Calculation
• Present-value annuity problems can be thought of as a
series of individual single-sum problems
• The present-value annuity factor P(i,n) is the sum of
the individual single-sum present values:
Present Annuity Formula
The present-value annuity factor [P(i,n)] can be
expressed as follows:
P(i,n) = p(i,1) + p(i,2) +... + p(i,n)
Lease Liability Example
• In most business situations, ordinary annuity
problems do not arise
• Classic exception – lease with front-end payments
• Example:
– Computer lease for 5 years
– Quarterly payments of $1,000 due at the beginning of each
quarter => annuity due
– Discount rate – 16% per year
– Question – what is the present value of the lease liability?
Figure 18–9 Present Value of a Lease
Liability
Lease Liability Example, cont.
The present value of the lease liability can be
calculated as follows:
P = $1,000 + $1,000  P(4%,19) or
P = $1,000 + $1,000  13.1339 or
P = $14,134