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Learning Objectives• Discuss the role of time value in finance and the use
of computational aids used to simplify its application.
• Understand the concept of future value, its calculation
for a single amount, and the effects of compounding
interest more frequently than annually.
• Find the future value of an ordinary annuity and an
annuity due and compare these two types of annuities.
• Understand the concept of present value, its
calculation for a single amount, and its relationship to
future value.
Learning Objectives
• Calculate the present value of a mixed stream of cash
flows, an annuity, a mixed stream with an embedded
annuity, and a perpetuity.
• Describe the procedures involved in:
– determining deposits to accumulate a future sum,
– loan amortization, and
– finding interest or growth rates
The Role of Time Value in Finance
• Most financial decisions involve costs & benefits that
are spread out over time.
• Time value of money allows comparison of cash flows
from different periods.
Question?
Would it be better for a company to invest $100,000 in a product that would return a total of $200,000 in one year, or one that would return
$500,000 after two years?
Answer!
It depends on the interest rate!
The Role of Time Value in Finance
• Most financial decisions involve costs & benefits that
are spread out over time.
• Time value of money allows comparison of cash flows
from different periods.
Basic Concepts
• Future Value: compounding or growth over time
• Present Value: discounting to today’s value
• Single cash flows & series of cash flows can be
considered
• Time lines are used to illustrate these relationships
Computational Aids
• Use the Equations
• Use the Financial Tables
• Use Financial Calculators
• Use Spreadsheets
Simple Interest
• Year 1: 5% of $100 = $5 + $100 = $105
• Year 2: 5% of $100 = $5 + $105 = $110
• Year 3: 5% of $100 = $5 + $110 = $115
• Year 4: 5% of $100 = $5 + $115 = $120
• Year 5: 5% of $100 = $5 + $120 = $125
With simple interest, you don’t earn interest on interest.
Compound Interest
• Year 1: 5% of $100.00 = $5.00 + $100.00 = $105.00
• Year 2: 5% of $105.00 = $5.25 + $105.00 = $110.25
• Year 3: 5% of $110.25 = $5 .51+ $110.25 = $115.76
• Year 4: 5% of $115.76 = $5.79 + $115.76 = $121.55
• Year 5: 5% of $121.55 = $6.08 + $121.55 = $127.63
With compound interest, a depositor earns interest on interest!
Time Value Terms
• PV0 = present value or beginning amount
• k = interest rate
• FVn = future value at end of “n” periods
• n = number of compounding periods
• A = an annuity (series of equal payments or
receipts)
Four Basic Models
• FVn = PV0(1+k)n = PV(FVIFk,n)
• PV0 = FVn[1/(1+k)n] = FV(PVIFk,n)
• FVAn = A (1+k)n - 1 = A(FVIFAk,n) k
• PVA0 = A 1 - [1/(1+k)n] = A(PVIFAk,n)
kFV: future valuePV: present valueIF: interest factorA: annuity
Future Value Example
You deposit $2,000 today at 6%
interest. How much will you have in 5
years?
$2,000 x (1.06)5 = $2,000 x FVIF6%,5
$2,000 x 1.3382 = $2,676.40
Algebraically and Using FVIF Tables
Future Value Example
You deposit $2,000 today at 6%
interest. How much will you have in 5
years?
Using Excel
PV 2,000$ k 6.00%n 5FV? $2,676
Excel Function
=FV (interest, periods, pmt, PV)
=FV (.06, 5, , 2000)
Compounding 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.
有效利率
Compounding More Frequently than Annually
• For example, what would be the difference in future
value if I deposit $100 for 5 years and earn 12%
annual interest compounded (a) annually, (b)
semiannually, (c) quarterly, an (d) monthly?
Annually: 100 x (1 + .12)5 = $176.23
Semiannually: 100 x (1 + .06)10 = $179.09
Quarterly: 100 x (1 + .03)20 = $180.61
Monthly: 100 x (1 + .01)60 = $181.67FVn=PV0×(1+k/m)m×n
Compounding More Frequently than Annually
Annually SemiAnnually Quarterly Monthly
PV 100.00$ 100.00$ 100.00$ 100.00$
k 12.0% 0.06 0.03 0.01
n 5 10 20 60
FV $176.23 $179.08 $180.61 $181.67
On Excel
Continuous Compounding• With continuous compounding the number of
compounding periods per year approaches infinity.
• Through the use of calculus, the equation thus
becomes:
FVn (continuous compounding) = PV x (ekxn)
where “e” has a value of 2.7183.
• Continuing with the previous example, find the Future
value of the $100 deposit after 5 years if interest is
compounded continuously.
kn
m
nmn ePV
m
kPVFV
00 )1(
Continuous Compounding• With continuous compounding the number of
compounding periods per year approaches infinity.
• Through the use of calculus, the equation thus
becomes:
FVn (continuous compounding) = PV x (ekxn)
where “e” has a value of 2.7183.
FVn = 100 x (2.7183).12x5 = $182.22
Nominal & Effective Rates• The nominal interest rate is the stated or contractual rate of
interest charged by a lender or promised by a borrower.
• The effective interest rate is the rate actually paid or earned.
• In general, the effective rate > nominal rate whenever
compounding occurs more than once per year
EAR = (1 + k/m) m -11
1)/1(
n nmmkEAR
Nominal & Effective Rates
• For example, what is the effective rate of interest on
your credit card if the nominal rate is 18% per year,
compounded monthly?
EAR = (1 + .18/12) 12 -1
EAR = 19.56%
Present Value• Present value is the current dollar value of a future
amount of money.
• It is based on the idea that a dollar today is worth
more than a dollar tomorrow.
• It is the amount today that must be invested at a given
rate to reach a future amount.
• It is also known as discounting, the reverse of
compounding.
• The discount rate is often also referred to as the
opportunity cost, the discount rate, the required return,
and the cost of capital.
Present Value Example
How much must you deposit today in order to
have $2,000 in 5 years if you can earn 6%
interest on your deposit?
$2,000 x [1/(1.06)5] = $2,000 x PVIF6%,5
$2,000 x 0.74758 = $1,494.52
Algebraically and Using PVIF Tables
Present Value Example
How much must you deposit today in order to
have $2,000 in 5 years if you can earn 6%
interest on your deposit?
FV 2,000$ k 6.00%n 5PV? $1,495
Excel Function
=PV (interest, periods, pmt, FV)
=PV (.06, 5, , 2000)
Using Excel
Annuities• Annuities are equally-spaced cash flows of equal size.
• Annuities can be either inflows or outflows.
• An ordinary (deferred) annuity has cash flows that occur at
the end of each period.
• An annuity due has cash flows that occur at the beginning
of each period.
• The future value of an annuity due will always be greater
than the future value of an otherwise equivalent ordinary
annuity because interest will compound for an additional
period.
Future Value of an Ordinary Annuity
• Annuity = Equal Annual Series of Cash Flows
• Example: How much will your deposits grow to if you
deposit $100 at the end of each year at 5% interest for
three years.
FVA = 100(FVIFA,5%,3) = $315.25
Using the FVIFA Tables
0 1 2 3
100 100 100
100X1.05=105
100X(1.05)2=110.25
Future Value of an Ordinary Annuity
• Annuity = Equal Annual Series of Cash Flows
• Example: How much will your deposits grow to if you
deposit $100 at the end of each year at 5% interest for
three years.
Using Excel
PMT 100$ k 5.0%n 3FV? 315.25$
Excel Function
=FV (interest, periods, pmt, PV)
=FV (.05, 3,100, )
Future Value of an Annuity Due
• Annuity = Equal Annual Series of Cash Flows
• Example: How much will your deposits grow to if you deposit $100 at the
beginning of each year at 5% interest for three years.
FVA = 100(FVIFA,5%,3)(1+k) = $330.96
Using the FVIFA Tables
FVA = 100(3.152)(1.05) = $330.96
Future Value of an Annuity Due
• Annuity = Equal Annual Series of Cash Flows
• Example: How much will your deposits grow to if you
deposit $100 at the beginning of each year at 5%
interest for three years.
Using Excel
Excel Function
=FV (interest, periods, pmt, PV)
=FV (.05, 3,100, )x(1.05)
=315.25*(1.05)
PMT 100.00$ k 5.00%n 3FV $315.25FVA? 331.01$
Present Value of an Ordinary Annuity
• Annuity = Equal Annual Series of Cash Flows
• Example: How much could you borrow if you could
afford annual payments of $2,000 (which includes
both principal and interest) at the end of each year for
three years at 10% interest?
PVA = 2,000(PVIFA,10%,3) = $4,973.70
Using PVIFA Tables
2000 2000 20002000÷1.1
2000÷(1.1)2
2000÷(1.1)3
Present Value of an Ordinary Annuity
• Annuity = Equal Annual Series of Cash Flows
• Example: How much could you borrow if you could
afford annual payments of $2,000 (which includes
both principal and interest) at the end of each year for
three years at 10% interest?
Using Excel
PMT 2,000$ I 10.0%n 3PV? $4,973.70
Excel Function
=PV (interest, periods, pmt, FV)
=PV (.10, 3, 2000, )
Present Value of a Mixed Stream
• A mixed stream of cash flows reflects no particular
pattern
• Find the present value of the following mixed stream
assuming a required return of 9%.
Using Tables
Year Cash Flow PVIF9%,N PV
1 400 0.917 366.80$
2 800 0.842 673.60$
3 500 0.772 386.00$
4 400 0.708 283.20$
5 300 0.650 195.00$
PV 1,904.60$
Present Value of a Mixed Stream
• A mixed stream of cash flows reflects no particular
pattern
• Find the present value of the following mixed stream
assuming a required return of 9%.
Using EXCEL
Year Cash Flow
1 400
2 800
3 500
4 400
5 300
NPV $1,904.76
Excel Function
=NPV (interest, cells containing CFs)
=NPV (.09,B3:B7)
Present Value of a Perpetuity
• A perpetuity is a special kind of annuity.
• With a perpetuity, the periodic annuity or cash flow
stream continues forever.
PV = Annuity/k
• For example, how much would I have to deposit today in
order to withdraw $1,000 each year forever if I can earn
8% on my deposit?
PV = $1,000/.08 = $12,500
…1000 1000 1000
………………
Determining Interest or Growth Rates
• At times, it may be desirable to determine the
compound interest rate or growth rate implied by a
series of cash flows.
• For example, you invested $1,000 in a mutual fund in
1994 which grew as shown in the table below?1994 1,000$ 1995 1,127 1996 1,158 1997 2,345 1998 3,985 1999 4,677 2000 5,525
It is first important to notethat although there are 7
years show, there are only 6time periods between the
initial deposit and the final value.
Determining Interest or Growth Rates
• At times, it may be desirable to determine the
compound interest rate or growth rate implied by a
series of cash flows.
• For example, you invested $1,000 in a mutual fund in
1994 which grew as shown in the table below?1994 1,000$ 1995 1,127 1996 1,158 1997 2,345 1998 3,985 1999 4,677 2000 5,525
Thus, $1,000 is the presentvalue, $5,525 is the futurevalue, and 6 is the numberof periods. Using Excel,
we get:
Determining Interest or Growth Rates
• At times, it may be desirable to determine the
compound interest rate or growth rate implied by a
series of cash flows.
• For example, you invested $1,000 in a mutual fund in
1994 which grew as shown in the table below?1994 1,000$ 1995 1,127 1996 1,158 1997 2,345 1998 3,985 1999 4,677 2000 5,525
PV 1,000$ FV 5,525$ n 6k? 33.0%
Determining Interest or Growth Rates
• At times, it may be desirable to determine the
compound interest rate or growth rate implied by a
series of cash flows.
• For example, you invested $1,000 in a mutual fund in
1994 which grew as shown in the table below?1994 1,000$ 1995 1,127 1996 1,158 1997 2,345 1998 3,985 1999 4,677 2000 5,525
Excel Function
=Rate(periods, pmt, PV, FV)
=Rate(6, ,1000, 5525)