Time Value of Money Chapter 5 © 2003 South-Western/Thomson Learning.
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Transcript of Time Value of Money Chapter 5 © 2003 South-Western/Thomson Learning.
Time Value of Money
Chapter 5
© 2003 South-Western/Thomson Learning
2
Time is Money
$100 in your hand today is worth more than $100 in one year Money earns interest
• The higher the interest, the faster your money grows
3
Time is Money
Present Value The amount that must be deposited today to
have a future sum at a certain interest rate The discounted value of a sum is its present
value
4
Outline of Approach
Deal with four different types of problems Amount
• Present value• Future value
Annuity• Present value• Future value
5
Outline of Approach
Mathematics For each type of problem an equation will be
presented Time lines
Graphic portrayal of a time value problem
0 1 2
• Helps with complicated problems
6
Amount Problems—Future Value The future value (FV) of an amount
How much a sum of money placed at interest (k) will grow into in some period of time
• If the time period is one year• FV1 = PV + kPV or FV1 = PV(1+k)
• If the time period is two years• FV2 = FV1 + kFV1 or FV2 = PV(1+k)2
• If the time period is generalized to n years• FVn = PV(1+k) n
7
Amount Problems—Future Value
The (1 + k)n depends on Size of k and n
• Can develop a table depicting different values of n and k and the proper value of (1 + k)n
• Can then use a more convenient formula• FVn = PV [FVFk,n]
These values can be looked up in an interest
factor table.
8
Other Issues
Problem-Solving Techniques Three of four variables are given
• We solve for the fourth
The Opportunity Cost Rate The opportunity cost of a resource is the
benefit that would have been available from its next best use
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Financial Calculators
Work directly with equations How to use a typical financial calculator in time
value Five time value keys
• Use either four or five keys
Some calculators distinguish between inflows and outflows
• If a PV is entered as positive the computed FV is negative
10
The Expression for the Present Value of an Amount
The future and present values factors are reciprocals Either equation can be used to solve any amount
problems
n
n
n n
Interest Factor
FV PV 1+k
Solve for PV
1PV = FV
1 k
k,nk,n
1FVF
PVF Solving for k or n involves
searching a table.
11
Annuity Problems
Annuity A finite series of equal payments separated
by equal time intervals• Ordinary annuities
• Payments occur at the end of the time periods
• Annuity due• Payments occur at the beginning of the time periods
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The Future Value of an Annuity—Developing a Formula
Future value of an annuity The sum, at its end, of all payments and all
interest if each payment is deposited when received
13
The Future Value of an Annuity—Developing a Formula
Thus, for a 3-year annuity, the formula is
FVFAk,n
0 1 2
0 1 2 n -1
n
nn i
ni=1
FVA = PMT 1+k PMT 1+k PMT 1+k
Generalizing the Expression:
FVA = PMT 1+k PMT 1+k PMT 1+k PMT 1+k
which can be written more conveniently as:
FVA PMT 1+k
Factoring PMT outside the summation, we
n
n i
ni=1
obtain:
FVA PMT 1+k
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The Future Value of an Annuity—Solving Problems
There are four variables in the future value of an annuity equation The future value of the annuity itself The payment The interest rate The number of periods
• Helps to draw a time line
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The Sinking Fund Problem
Companies borrow money by issuing bonds for lengthy time periods No repayment of principal is made during the
bonds’ lives• Principal is repaid at maturity in a lump sum
• A sinking fund provides cash to pay off a bond’s principal at maturity
• Problem is to determine the periodic deposit to have the needed amount at the bond’s maturity—a future value of an annuity problem
16
Compound Interest and Non-Annual Compounding
Compounding Earning interest on interest
Compounding periods Interest is usually compounded annually,
semiannually, quarterly or monthly• Interest rates are quoted by stating the nominal
rate followed by the compounding period
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The Effective Annual Rate
Effective annual rate (EAR) The annually compounded rate that pays the
same interest as a lower rate compounded more frequently
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The Effective Annual Rate
EAR can be calculated for any compounding period using the following formula:
m
nominalEAR - 1k
1 m
Effect of more frequent compounding is greater at higher interest rates
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The Effective Annual Rate
The APR and EAR Annual percentage rate (APR)
• Is actually the nominal rate and is less than the EAR
Compounding Periods and the Time Value Formulas Time periods must be compounding periods Interest rate must be the rate for a single
compounding period• For instance, with a quarterly compounding period the
knominal must be divided by 4 and the n must be multiplied by 4
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The Present Value of an Annuity—Developing a Formula
Present value of an annuity Sum of all of the annuity’s payments
• Easier to develop a formula than to do all the calculations individually
2 3
1 2 3
1 2 n
PMT PMT PMTPVA =
1+k 1+k 1+k
which can also be written as:
PVA = PMT 1+k PMT 1+k PMT 1+k
Generalized for any number of periods:
PVA = PMT 1+k PMT 1+k PMT 1+k
Factoring PMT and using summation, we o
n
i
i=1
btain:
PVA PMT 1+k
PVFAk,n
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The Present Value of an Annuity—Solving Problems
There are four variables in the present value of an annuity equation The present value of the annuity itself The payment The interest rate The number of periods
• Problem usually presents 3 of the 4 variables
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Spreadsheet Solutions
Time value problems can be solved on a spreadsheet such as Microsoft Excel™ or Lotus 1-2-3™
To solve for: FV use =FV(k, n, PMT, PV) PV use =PV(k, n, PMT, FV) K use =RATE(n, PMT, PV, FV) N use =NPER(k, PMT, PV, FV) PMT use =PMT(k, n, PV, FV)
Select the function for the unknown
variable, place the known variables in
the proper order within the
parentheses, and input 0 for the
unknown variable.
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Spreadsheet Solutions
Complications Interest rates in entered as decimals, not
percentages Of the three cash variables (FV, PMT or PV)
• One is always zero• The other two must be of the opposite sign
• Reflects inflows (+) versus outflows (-)
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Amortized Loans
An amortized loan’s principal is paid off regularly over its life Generally structured so that a constant
payment is made periodically• Represents the present value of an annuity
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Loan Amortization Schedules
Detail the interest and principal in each loan payment
Show the beginning and ending balances of unpaid principal for each period
Need to know Loan amount (PVA) Payment (PMT) Periodic interest rate (k)
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Mortgage Loans
Mortgage loans (AKA: mortgages) Loans used to buy real estate
Often the largest single financial transaction in an average person’s life Typically an amortized loan over 30 years
• During the early years of the mortgage nearly all the payment goes toward paying interest
• This reverses toward the end of the mortgage
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Mortgage Loans
Implications of mortgage payment pattern Early mortgage payments provide a large tax
savings which reduces the effective cost of a loan
Halfway through a mortgage’s life half of the loan has not been paid off
Long-term loans like mortgages result in large total interest amounts over the life of the loan
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The Annuity Due
In an annuity due payments occur at the beginning of each period
The future value of an annuity due Because each payment is received one
period earlier• It spends one period longer in the bank earning
interest
n -1
n
n k,n
FVAd = PMT + PMT 1+k PMT 1+k 1 k
which written with the interest factor becomes:
FVAd PMT FVFA 1 k
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The Annuity Due
The present value of an annuity due Formula
k,nPVAd PMT PVFA 1 k
Recognizing types of annuity problems Always represent a stream of equal payments Always involve some kind of a transaction at one
end of the stream of payments• End of stream—future value of an annuity• Beginning of stream—present value of an annuity
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Perpetuities
A perpetuity is a stream of regular payments that goes on forever An infinite annuity
Future value of a perpetuity Makes no sense because there is no end point
Present value of a perpetuity A diminishing series of numbers
• Each payment’s present value is smaller than the one before
p
PMTPV
k
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Continuous Compounding
Compounding periods can be shorter than a day As the time periods become infinitesimally
short, interest is said to be compounded continuously
To determine the future value of a continuously compounded value:
knnFV PV e
32
Multipart Problems
Time value problems are often combined due to complex nature of real situations A time line portrayal can be critical to
keeping things straight
33
Uneven Streams and Imbedded Annuities Many real world problems have sequences of
uneven cash flows These are NOT annuities
• For example, if you were asked to determine the present value of the following stream of cash flows
$100 $200 $300
Must discount each cash flow individually Not really a problem when attempting to determine either a
present or future value• Becomes a problem when attempting to determine an interest rate
34
Calculator Solutions for Uneven Streams
Financial calculators and spreadsheets have the ability to handle uneven streams with a limited number of payments
Generally programmed to find the present value of the streams or the k that will equate a present value to the stream
35
Imbedded Annuities
Sometimes uneven streams of cash flows will have annuities embedded within them We can use the annuity formula to calculate
the present or future value of that portion of the problem