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The Time Value of MoneyTranslating Cash Flows Forward and Backward in Time
Future Value
TrCFV )1(0
• Money invested earns interest and interest reinvested earns more interest
• The power of compounding
Future Value Problems
TrCFV )1(0 Solve for any variable, given the other three
• FV: How much will I have in the future?
• C0: How much do I need to invest now?
• r: What rate of return do I need to earn?• T: How long will it take me to reach my goal?
Present Value
TT
r
CPV
)1(
• Discounting future cash flows at the “opportunity cost” (cost of capital, discount rate, minimum acceptable return)
• A dollar tomorrow is worth less than a dollar today
Present Values can be Added
TT
TT
CFr
CFr
CFr
CF
r
CF
r
CF
r
CFCFP
)1(
1...
)1(
1
1
1
)1(...
)1()1(
2210
221
0
• Cash flows further out are discounted more• Discount factors are like prices (exchange
rates)
Net Present Value (NPV)
Best criterion for corporate investment:
Invest if NPV > 0
NPV with a Single, Initial Investment Outlay
T
tt
t
r
CCNPV
10 )1(
• C0 = initial investment outlay
• Ct = project cash flow in period t
• r = discount rate (shareholders’ opp. cost)• T = project termination period
Calculating PV of a Stream (Beware)
• Calculator assumes first CF you give it occurs now (Time 0)
• Excel assumes first CF you give it occurs one year from now (Time 1)
Implications of NPV > 0
01 )1(
Cr
CT
tt
t
• Project benefits exceed cost (in PV terms)• Project is worth more than it costs• Project market value exceeds book value• Project adds to shareholder value
NPV More Generally
T
tt
t
r
CNPV
0 )1(• Treat inflows as +, outflows as –• NPV = PV of all cash flows• Investment may occur throughout project life
Internal Rate of Return
01 )1(
CIRR
CT
tt
t
• IRR sets value of benefits = investment• IRR sets NPV = 0• IRR is the rate of return company expects on
investment C0
NPV > 0 Implies IRR > r
01 )1(
0 Cr
CNPV
T
tt
t
• If NPV > 0, IRR must exceed r• Investing when NPV > 0 implies company expects
to earn more than shareholders’ opportunity cost• Equivalent: Invest when NPV > 0 or when
IRR > r
Different Compounding Periods
m
m
APREAR
1)1(
• m = # of compounding periods in a year• APR = actual rate x m (APR is annualized)• EAR = the annually compounded rate that
gives the same proceeds as APR compounded m times
Semiannual Compounding
1025.12
10.1
2
• m = 2• APR = 10%• EAR = 10.25%
Quarterly Compounding
1038.14
10.1
4
• m = 4• APR = 10%• EAR = 10.38%
Monthly Compounding
1047.112
10.1
12
• m = 12• APR = 10%• EAR = 10.47%
Daily Compounding
10516.1365
10.1
365
• m = 365• APR = 10%• EAR = 10.516%
Continuous Compounding
10517.1
m as 1
10.
e
em
APR APRm
• m = • APR = 10%• EAR = 10.517%
Working with Annuities
Annuities
• All cash flows are the same, so we can factor out the constant payment C and calculate the sum of the discount factors
T
T
rrrC
r
C
r
C
r
CPV
)1(
1...
)1(
1
1
1
)1(...
)1(1
2
2
Special Case: Perpetuity
• If all the cash flows are the same each period forever, the sum of the discount factors converges to 1/r
r
C
rrrCPV
...
)1(
1
)1(
1
1
132
Perpetuity Example
• Let C = $100 and r = .05
• $100 per year forever at 5% is worth:
200005.
100PV
Other Perpetuity Examples
• British Consol Bonds
• Canadian Pacific 4% Perpetual Bonds
• Endowments– How much can I withdraw annually without
invading principal?
Growing Perpetuity
• Suppose the initial payment C grows at a constant rate g per period (where g < r)
• This growing stream still has a finite present value:
gr
C
r
gC
r
gC
r
CPV
...)1(
)1(
)1(
)1(
)1( 3
2
2
Growing Perpetuity Example
• Suppose the initial payment is $100 and that this grows at 3% per year while the discount rate is 5%
• The value of this growing perpetuity is:
000,50$03.05.
100
gr
CP
Other Growing Perpetuity Examples
• Stock price = present value of growing dividend stream (see Class #5)
• M&A: How to estimate terminal value– How fast do earnings grow after the end of the
analysis period?
Finite Annuity=Difference Between Two Perpetuities
C C C C C C C C
0 1 2 3 4 5 6 7 8
C C C C
4)1(
1
rr
CPV
r
CPV
4)1(
11
rr
Cdifference
Annuity Example
• What’s the value of a 4-year annuity with annual payments of $40,000 per year (@5%)?
838,141)05.1(
11
05.
000,40
)1(
11
4
4
rr
CP
Oops, Tuition Payments Due at Beginning of Year
)05.1(838,141930,148
)05.1(
11
05.
11000,40
)1(
11
11
3
1
TrrCP
Other Annuity Applications
• Lottery winnings
• Lease & loan contracts
• Home mortgages (constant monthly payments)
• Retirement savings
• Retirement income
Home Mortgages
• 30-year fixed rate mortgage: 360 equal monthly payments
• Most of early payments goes toward interest; principal repayment gradually accelerates
• At any point: outstanding balance = present value of remaining payments
More Annuity Problems
Saving, Retirement Planning, Evaluating Loans and
Investments