principles of corporate finance session 10 unit ii: time value of money
TRANSCRIPT
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Principles of Corporate Finance
Session 10
Unit II: Time Value of Money
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TIME allows you the opportunity to postpone consumption and earn
INTEREST.
Why TIME?Why TIME?
Why is TIME such an important element in your decision?
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Types of InterestTypes of Interest
• Compound InterestInterest paid (earned) on any previous
interest earned, as well as on the principal borrowed (lent).
• Simple Interest
Interest paid (earned) on only the original amount, or principal, borrowed (lent).
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Simple Interest FormulaSimple Interest Formula
Formula SI = P0(i)(n)
SI: Simple Interest
P0: Deposit today (t=0)
i: Interest Rate per Period
n: Number of Time Periods
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• SI = P0(i)(n)= $1,000(0.07)(2)= $140
Simple Interest ExampleSimple Interest Example
• Assume that you deposit $1,000 in an account earning 7% simple interest for 2 years. What is the accumulated interest at the end of the 2nd year?
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FV = P0 + SI = $1,000 + $140= $1,140
• Future Value is the value at some future time of a present amount of money, or a series of payments, evaluated at a given interest rate.
Simple Interest (FV)Simple Interest (FV)
• What is the Future Value (FV) of the deposit?
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The Present Value is simply the $1,000 you originally deposited. That is the value today!
• Present Value is the current value of a future amount of money, or a series of payments, evaluated at a given interest rate.
Simple Interest (PV)Simple Interest (PV)
• What is the Present Value (PV) of the previous problem?
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0
5000
10000
15000
20000
1st Year 10thYear
20thYear
30thYear
Future Value of a Single $1,000 Deposit
10% SimpleInterest
7% CompoundInterest
10% CompoundInterest
Why Compound Interest?Why Compound Interest?
Fu
ture
Val
ue
(U.S
. Dol
lars
)
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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.
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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!
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Principles of Corporate Finance
Session 11 & 12
Unit II: Time Value of Money
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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?
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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.
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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
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Computational Aids
• Use the Equations
• Use the Financial Tables
• Use Financial Calculators
• Use Spreadsheets
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Computational Aids
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Computational Aids
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Computational Aids
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Computational Aids
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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)
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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)
k
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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
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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)
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Future Value Example A Graphic View of Future Value
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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.
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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 = $181.67
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Compounding More Frequently than Annually
Annually SemiAnnually Quarterly Monthly
PV 100.00$ 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
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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.
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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
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Principles of Corporate Finance
Session 12 & 13
Unit II: Time Value of Money
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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 -1
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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%
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We will use the “Rule-of-72”.
Double Your Money!!!Double Your Money!!!
Quick! How long does it take to double $5,000 at a compound rate of 12%
per year (approx.)?
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Approx. Years to Double = 72 / i%
72 / 12% = 6 Years[Actual Time is 6.12 Years]
The “Rule-of-72”The “Rule-of-72”
Quick! How long does it take to double $5,000 at a compound rate of 12%
per year (approx.)?
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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.
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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
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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
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Present Value Example A Graphic View of Present Value
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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.
• An annuity due will always be greater than an
otherwise equivalent ordinary annuity because interest
will compound for an additional period.
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Annuities
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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
Year 1 $100 deposited at end of year = $100.00
Year 2 $100 x .05 = $5.00 + $100 + $100 = $205.00
Year 3 $205 x .05 = $10.25 + $205 + $100 = $315.25
Using the FVIFA Tables
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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 (.06, 5,100, )
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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
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Future Value of an Annuity Due
• Annuity = Equal Annual Series of Cash Flows
• Example: How much will your deposits grow to if you
deposit $100 at the beginning of each year at 5%
interest for three years.
Using Excel
Excel Function
=FV (interest, periods, pmt, PV)
=FV (.06, 5,100, )
=315.25*(1.05)
PMT 100.00$ k 5.00%n 3FV $315.25FVA? 331.01$
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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
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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? $4,973.70
Excel Function
=PV (interest, periods, pmt, FV)
=PV (.10, 3, 2000, )
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Present Value of a Mixed Stream
• A mixed stream of cash flows reflects no particular
pattern
• Find the present value of the following mixed stream
assuming a required return of 9%.
Using Tables
Year Cash Flow PVIF9%,N PV
1 400 0.917 366.80$
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$
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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)
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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