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