Principles of Corporate Finance
Session 10
Unit II: Time Value of Money
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?
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).
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
• 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?
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?
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?
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
)
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!
Principles of Corporate Finance
Session 11 & 12
Unit II: Time Value of Money
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
Computational Aids
Computational Aids
Computational Aids
Computational Aids
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)
k
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)
Future Value Example A Graphic View of Future Value
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.
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 = $181.67
Compounding More Frequently than Annually
Annually SemiAnnually Quarterly Monthly
PV 100.00$ 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.
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
Principles of Corporate Finance
Session 12 & 13
Unit II: Time Value of Money
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
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%
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.)?
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.)?
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
Present Value Example A Graphic View of Present Value
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.
Annuities
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
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, )
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 $100 at the beginning of each year at 5%
interest for three years.
Using Excel
Excel Function
=FV (interest, periods, pmt, PV)
=FV (.06, 5,100, )
=315.25*(1.05)
PMT 100.00$ 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
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