Present and Future Value Translating cash flows forward and backward through time
Mar 31, 2015
Present and Future Value
Translating cash flows forward and backward through time
Future Value
TrPFV )1( • Money invested earns interest
and interest reinvested earns more interest
• The power of compounding
Future Value Problems
TrPFV )1( Solve for any variable, given the other three
• FV: How much will I have in the future?• P: 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
Tr
FVP
)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)
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)
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%
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
CP
)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
rrrCP
...
)1(
1
)1(
1
1
132
Perpetuity Example
• Let C = $100 and r = .05
• $100 per year forever at 5% is worth:
200005.
100P
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
CP
...)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:
000,5$03.05.
100
gr
CP
Other Growing Perpetuity Examples
• Stock price = present value of growing dividend stream (see Class #7)
• 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
CP
r
CP
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
• Retirement savings/ 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
Net Present Value (NPV)
• Best criterion for corporate investment:
• Invest if NPV > 0
NPV with a Single, Initial Investment Outlay
Ir
CNPV
T
tt
t
1 )1(
• I = initial investment outlay
• Ct = project cash flow in period t
• r = discount rate (shareholders’ opp. cost)• T = project termination period
Implications of NPV > 0
Ir
CT
tt
t
1 )1(
• Project benefits exceed cost (in PV terms)• Project is worth more than it costs• Project market value exceeds book value• Project adds 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
IIRR
CT
tt
t
1 )1(• IRR sets value of benefits = investment• IRR sets NPV = 0• IRR is the rate of return company expects on
investment I
NPV > 0 Implies IRR > r
Ir
CNPV
T
tt
t
1 )1(
0
• If NPV > 0, IRR must exceed r• Investing when NPV > 0 implies company
expects to earn more than shareholder’ opp. Cost
• Equivalent: Invest when NPV > 0 or when IRR>I