This chapter discusses how to calculate the present value, future value, internal rate of return, and modified internal rate of return of a cash flow stream. Understanding how (and when) to use these formulas is essential to your success as a financial manager! Formulas and examples are included with these notes.
Numbers are rounded to 4 decimal places in tables and formula. However, the actual (nonrounded) numbers are used in the calculations.
Time Value of Money Concepts

TimeLine Conventions

$1 received today (cash inflow)

$1 paid in five years (cash outflow)

$1 received at the end of the third year

$1 received at the beginning of the third year

Fouryear annuity of $1 per year, first cash flow received at t = 1 (ordinary annuity)

Fouryear annuity of $1 per year, first cash flow received at t = 0 (annuity due)

Fouryear annuity of $1 per year, first cash flow received at t = 2 (deferred annuity)


0

1

2

3

4

5

A.

$1






B.






$1

C.




$1



D.



$1




E.


$1

$1

$1

$1


F.

$1

$1

$1

$1



G.



$1

$1

$1

$1


Notation
C_{0} = cash flow at time 0
C = cash flow (used when all cash flows are the same)
r = discount rate or interest rate
t = time period (e.g., t = 4), or number of years (e.g., t years in the future)
m = number of compounding periods per year (e.g., with monthly compounding, m = 12)
g = growth rate in cash flow

Annual Compounding, Single Payments

Future value of $1 as of time 1. Interest rate = 5%.

0

1

2

3

4

5

$1

$0

$0

$0

$0

$0

Formula: C_{0} (1 + r)^{t} = $1(1.05)^{1} = $1.0500
Financial Calculator: N = 1, I/Y = 5, PV = 1, PMT = 0, FV = Answer
Note on financial calculators – The calculator inputs described above are for a Texas Instruments BAII Plus calculator. (Many other financial calculators require similar inputs.)
Notice that you enter a 1 as the PV and the solution is +1.05. Here is the intuition: deposit $1 in the bank (negative cash flow), withdraw $1.05 in one year (positive cash flow). If you had entered +1 as the PV, the solution would be –1.05.

Future value of $1, as of time 5. Interest rate = 5%.

0

1

2

3

4

5

$1

$0

$0

$0

$0

$0

Formula: C_{0} (1 + r)^{t} = $1(1.05)^{5} = $1.2763
Financial Calculator: N = 5, I/Y = 5, PV = 1, PMT = 0, FV = Answer

Present value of $1, received at time 1. Discount rate = 5%.

0

1

2

3

4

5

$0

$1

$0

$0

$0

$0

Formula: C_{1} / (1 + r)^{t} = $1/(1.05)^{1} = $0.9524
Financial Calculator: N = 1, I/Y = 5, PV = Answer, PMT = 0, FV = 1

Present value of $1, received at time 5. Discount rate = 5%.

0

1

2

3

4

5

$0

$0

$0

$0

$0

$1

Formula: C_{5} / (1 + r)^{t} = $1/(1.05)^{5} = $0.7835
Financial Calculator: N = 5, I/Y = 5, PV = Answer, PMT = 0, FV = 1

Compounding periods less than one year

Future value of $1, as of time 5. Interest rate = 5%, compounded “m” times per year.

0

1

2

3

4

5

$1

$0

$0

$0

$0

$0

General (noncontinuous) formula: C_{0} (1 + r/m)^{tm}
Continuous compounding formula: C_{0} e^{rt}
Note: “e” = 2.718281828
Semiannual compounding: $1(1 + (0.05/2))^{(5)(2)} = $1.280085
Monthly compounding: $1(1 + (0.05/12))^{(5)(12)} = $1.283359
Daily compounding: $1(1 + (0.05/365))^{(5)(365)} = $1.284003
Continuous compounding: $1 e^{(5)(0.05)} = $1.284025
Note: Some may use 360 days as the length of one year, other may take into account leap years (366 days every four years). The effects of these changes (from a 365day year) are extremely small.
Financial Calculator (for semiannual): N = 10, I/Y = 5/2, PV = 1, PMT = 0, FV = Answer
Financial Calculator (for monthly): N = 60, I/Y = 5/12, PV = 1, PMT = 0, FV = Answer
Financial Calculator (for daily): N = 1825, I/Y = 5/365, PV = 1, PMT = 0, FV = Answer

Present value of $1, received at time 5. Discount rate = 5%, compounded m times per year.

0

1

2

3

4

5

$0

$0

$0

$0

$0

$1

General (noncontinuous) formula: C_{5} / (1 + r/m)^{tm}
Continuous compounding formula: C_{5} / e^{rt}
Semiannual compounding: $1 / [1 + (0.05/2)]^{(5)(2)} = $0.781198
Monthly compounding: $1 / [1 + (0.05/12)]^{(5)(12)} = $0.779205
Daily compounding: $1 / [1 + (0.05/365)]^{(5)(365)} = $0.778814
Continuous compounding: $1 / e^{(5)(0.05)} = $0.778801
Financial Calculator (for semiannual): N = 10, I/Y = 5/2, PV = Answer, PMT = 0, FV = 1
Financial Calculator (for monthly): N = 60, I/Y = 5/12, PV = Answer, PMT = 0, FV = 1
Financial Calculator (for daily): N = 1825, I/Y = 5/365, PV = Answer, PMT = 0, FV = 1

Constant Finite Annuities

Fouryear annuity of $1 per year, first cash flow received at t = 1. Interest and discount rate = 5%.

0

1

2

3

4

5

$0

$1

$1

$1

$1

$0

Standard formula for the future value of a finite annuity = C [(1 + r)^{t} – 1] / r
This standard formula for the future value of a finite annuity gives a value as of the last period of the annuity (time 4 in this example). The 1.05 is raised to the fourth power because there are 4 payments in the annuity.

Value as of time 4 = $1 [(1.05^{4} – 1) / 0.05] =

$4.3101

You can calculate the value of the cash flows at other points in time by multiplying or dividing by 1+r, where r (the interest and discount rate) is 5% in this example.
For instance, assume you want to know the value of the above cash flow stream at t = 6. Time 6 is two years after time 4. To calculate, use the standard formula to determine the value at t = 4, then multiply by 1.05^{2} to determine the value at t = 6. (Use the second power because you are calculating the value two years after time 4.) The solution is:

Value as of time 6 = $1 [(1.05^{4} – 1) / 0.05] 1.05^{2} =

$4.7519

As a second example, assume that you want to know the value of the above cash flow stream at t = 1. Time 1 is three years before time 4. To calculate, use the standard formula to determine the value at t = 4, then divide by 1.05^{3} to determine the value at t = 1. (Use the third power because you are calculating the value three years before time 4.) The solution is:

Value as of time 1 = $1 [(1.05^{4} – 1) / 0.05] / 1.05^{3} =

$3.7232

Therefore, “multiply” when you want to determine the value at a later date, “divide” when you want to determine the value at an earlier date.
Some more examples:

Value as of time 0 = $1 [(1.05^{4} – 1) / 0.05] / 1.05^{4} =

$3.5460

Value as of time 1 = $1 [(1.05^{4} – 1) / 0.05] / 1.05^{3} =

$3.7232

Value as of time 2 = $1 [(1.05^{4} – 1) / 0.05] / 1.05^{2} =

$3.9094

Value as of time 3 = $1 [(1.05^{4} – 1) / 0.05] / 1.05^{1} =

$4.1049

Value as of time 4 = $1 [(1.05^{4} – 1) / 0.05] =

$4.3101

Value as of time 5 = $1 [(1.05^{4} – 1) / 0.05] 1.05^{1} =

$4.5256

Value as of time 6 = $1 [(1.05^{4} – 1) / 0.05] 1.05^{2} =

$4.7519

Value as of time 7 = $1 [(1.05^{4} – 1) / 0.05] 1.05^{3} =

$4.9895

Financial Calculator (time 0): N = 4, I/Y = 5, PV = 0, PMT = 1, FV = Answer. Divide answer by 1.05^{4}.
Financial Calculator (time 1): N = 4, I/Y = 5, PV = 0, PMT = 1, FV = Answer. Divide answer by 1.05^{3}.
Financial Calculator (time 2): N = 4, I/Y = 5, PV = 0, PMT = 1, FV = Answer. Divide answer by 1.05^{2}.
Financial Calculator (time 3): N = 4, I/Y = 5, PV = 0, PMT = 1, FV = Answer. Divide answer by 1.05^{1}.
Financial Calculator (time 4): N = 4, I/Y = 5, PV = 0, PMT = 1, FV = Answer.
Financial Calculator (time 5): N = 4, I/Y = 5, PV = 0, PMT = 1, FV = Answer. Multiply answer by 1.05^{1}.
Financial Calculator (time 6): N = 4, I/Y = 5, PV = 0, PMT = 1, FV = Answer. Multiply answer by 1.05^{2}
Financial Calculator (time 7): N = 4, I/Y = 5, PV = 0, PMT = 1, FV = Answer. Multiply answer by 1.05^{3}
You can also use the formula for the present value of a finite annuity to calculate the value of a cash flow stream at different points in time.
Standard formula for the present value of a finite annuity = C { [1 – (1 / (1 + r))^{t}] / r}
The standard formula gives a value one period before the first payment of the annuity (time 0 in this example). The 1.05 is raised to the fourth power because there are 4 payments in the annuity.

Value as of time 0 = $1 { [1 – (1/1.05)^{4}] / (0.05) } =

$3.5460

As before, you can calculate the value at other points in time by multiplying or dividing by 1+r, (1.05 in this example).
Two examples:

Value as of time 1 = $1 { [1 – (1/1.05)^{4}] / (0.05) } 1.05^{1} =

$3.7232


Value as of time 6 = $1 { [1 – (1/1.05)^{4}] / (0.05) } 1.05^{6} =

$4.7519

Financial Calculator (time 0): N = 4, I/Y = 5, PV = Answer, PMT = 1, FV = 0
Financial Calculator (time 1): N = 4, I/Y = 5, PV = Answer, PMT = 1, FV = 0. Multiply answer by 1.05^{1}.
Financial Calculator (time 6): N = 4, I/Y = 5, PV = Answer, PMT = 1, FV = 0. Multiply answer by 1.05^{6}.

Threeyear annuity of $1 per year, first cash flow received at t = 0. Interest and discount rate = 5%.

0

1

2

3

4

5

$1

$1

$1

$0

$0

$0

The standard future value annuity formula gives a value as of the last year of the annuity (year 2 in this example). This is a threeyear annuity. Therefore, 1.05 is raised to the third power in the formula.

Value as of time 2 = $1 [(1.05^{3} – 1) / 0.05] =

$3.1525

The value at other points in time can be calculated by multiplying or dividing by 1.05, raised to the appropriate power.

Value as of time 0 = $1 [(1.05^{3} – 1) / 0.05] / 1.05^{2} =

$2.8594

Value as of time 4 = $1 [(1.05^{3} – 1) / 0.05] 1.05^{2} =

$3.4756

The standard present value annuity formula gives a value one period before the first payment of the annuity. Therefore, the formula will give you a value at t = 1. You need to multiply by 1 + r to get the value by t = 0.

Value as of time 0 = $1 { [1 – (1/1.05)^{3}] / (0.05) } 1.05^{1} =

$2.8594

The values at time 2 and 4:

Value as of time 2 = $1 { [1 – (1/1.05)^{3}] / (0.05) } 1.05^{3} =

$3.1525

Value as of time 4 = $1 { [1 – (1/1.05)^{3}] / (0.05) } 1.05^{5} =

$3.4756


Fiveyear annuity of $1 per year, first cash flow received at t = 3. Interest and discount rate = 5%.

0

1

2

3

4

5

6

7

8

$0

$0

$0

$1

$1

$1

$1

$1

$0

The standard future value annuity formula gives the value as of the last year of the annuity (t = 7 in this example). This is a fiveyear annuity. Therefore, 1.05 is raised to the fifth power.

Value as of time 7 = $1 [(1.05^{5} – 1) / 0.05] =

$5.5256

Values at different points in time using the future value annuity formula. A couple of examples

Value as of time 0 = $1 [(1.05^{5} – 1) / 0.05] / 1.05^{7} =

$3.9270

Value as of time 4 = $1 [(1.05^{5} – 1) / 0.05] / 1.05^{3} =

$4.7732

Value as of time 8 = $1 [(1.05^{5} – 1) / 0.05] 1.05^{1} =

$5.8019

The standard present value annuity formula gives the value as of the year before the first payment of the annuity (t = 2 in this example).

Value as of time 2 = $1 { [1 – (1/1.05)^{5}] / (0.05) } =

$4.3295

Values at different points in time using the present value annuity formula. A couple of examples:

Value as of time 0 = $1 { [1 – (1/1.05)^{5}] / (0.05) } / 1.05^{2} =

$3.9270

Value as of time 4 = $1 { [1 – (1/1.05)^{5}] / (0.05) } 1.05^{2} =

$4.7732

Value as of time 8 = $1 { [1 – (1/1.05)^{5}] / (0.05) } 1.05^{6} =

$5.8019


Growing Finite Annuities

Fouryear growing annuity, growing at 10% per year. First cash flow (equal to $1) received at t = 1. Interest and discount rate = 5%.

0

1

2

3

4

5

$0

$1

$1.1

$1.21

$1.331

$0

Standard formula for the present value of a finite growing annuity (for when r is not equal to g) =
C_{first} [1 – [(1 + g) / (1 + r)]^{t} ] / (r – g). This formula gives the value one period before the first payment (t = 0 in this example).
C_{first} is the first cash flow of the annuity. In this above example, C_{first} = C_{1} = $1.

Value as of time 0 = $1 { [1 – (1.10/1.05)^{4}] / (0.05 – 0.10) }

$4.0904

Values at different points in time using the present value growing annuity formula. Multiply or divide by 1+r (raised to the appropriate power) to determine the value at other points in time.

Value as of time 2 = $1 { [1 – (1.10/1.05)^{4}] / (0.05 – 0.10) } 1.05^{2}

$4.5096

Value as of time 4 = $1 { [1 – (1.10/1.05)^{4}] / (0.05 – 0.10) } 1.05^{4}

$4.9719

Value as of time 5 = $1 { [1 – (1.10/1.05)^{4}] / (0.05 – 0.10) } 1.05^{5}

$5.2205


Fouryear growing annuity, growing at 10% per year. First cash flow (equal to $1) received at t = 3. Interest and discount rate = 5%.

0

1

2

3

4

5

6

7

$0

$0

$0

$1

$1.1

$1.21

$1.331

$0

The standard formula for the present value of a growing annuity gives you a value at time 2 (one period before the first payment). This is a 4year annuity. Therefore, 1.05 is raised to the 4^{th} power.

Value as of time 2 = $1 { [1 – (1.10/1.05)^{4}] / (0.05 – 0.10) }

$4.0904

Values at different points in time using the present value growing annuity formula. A few examples:

Value as of time 0 = $1 { [1 – (1.10/1.05)^{4}] / (0.05 – 0.10) } /1.05^{2}

$3.7101

Value as of time 5 = $1 { [1 – (1.10/1.05)^{4}] / (0.05 – 0.10) } 1.05^{3}

$4.7351


Perpetual constant annuities

Perpetual constant annuity of $1 (cash flows start at time 1, interest and discount rate = 5%)

0

1

2

3

4

5


$0

$1

$1

$1

$1

$1

$1

The standard formula for the present value of a perpetual constant annuity is C / r. The formula gives you the value one period before the first payment.

Value as of time 0 = $1 / 0.05

$20.0000

Values at different points in time using the present value perpetual constant annuity formula

Value as of time 1 = ($1 / 0.05) 1.05^{1}

$21.0000

Value as of time 4 = ($1 / 0.05) 1.05^{4}

$24.3101


Perpetual constant annuity of $1 (cash flows start at time 0, interest and discount rate = 5%)

0

1

2

3

4

5


$1

$1

$1

$1

$1

$1

$1

The standard formula for the present value of a perpetual constant annuity gives you the value one period before the first payment (t = 1 in this example). Therefore, you need to multiply by 1.05 to get the value at t = 0.

Value as of time 0 = ($1 / 0.05) 1.05

$21.0000

Values at different points in time using the present value perpetual annuity formula. Two examples:

Value as of time 1 = ($1 / 0.05) 1.05^{2}

$22.0500

Value as of time 4 = ($1 / 0.05) 1.05^{5}

$25.5256


Perpetual constant annuity of $1 (cash flows start at time 5, interest and discount rate = 5%)

0

1

2

3

4

5


$0

$0

$0

$0

$0

$1

$1

The standard formula for the present value of a perpetual constant annuity gives you the value one period before the first payment (t = 4 in this example).

Value as of time 4 = ($1 / 0.05)

$20.0000

Values at different points in time using the present value perpetual annuity formula. Some examples:

Value as of time 0 = ($1 / 0.05) / 1.05^{4}

$16.4540

Value as of time 5 = ($1 / 0.05) 1.05^{1}

$21.0000


Perpetual growing annuity, growing at 3% per year (first cash flow, received at time 1, equals $1, interest and discount rate = 5%)

0

1

2

3


$0

$1

$1.03

$1.0609

3% more

The standard formula for the present value of a perpetual growing annuity is C_{first} / (r – g). The formula gives you the value one period before the first payment.

Value as of time 0 = $1 / (0.05 – 0.03)

$50.0000

Values at different points in time using the present value perpetual growing annuity formula. Some examples:

Value as of time 1 = [$1 / (0.05 – 0.03)] 1.05^{1}

$52.5000

Value as of time 4 = [$1 / (0.05 – 0.03)] 1.05^{4}

$60.7753


Perpetual growing annuity, growing at 3% per year (first cash flow, received at time 11, equals $1, interest discount rate = 5%)

10

11

12

13


$0

$1

$1.03

$1.0609

3% more

The standard formula for the present value of a perpetual growing annuity gives you the value one period before the first payment (t = 10 in this example).

Value as of time 10 = [$1 / (0.05 – 0.03)]

$50.0000

Values at different points in time using the present value perpetual growing annuity formula. A few examples:

Value as of time 0 = [$1 / (0.05 – 0.03)] / 1.05^{10}

$30.6957

Value as of time 9 = [$1 / (0.05 – 0.03)] / 1.05^{1}

$47.6190

Value as of time 11 = [$1 / (0.05 – 0.03)] 1.05^{1}

$52.5000


Two growthrate example: $1 at time 1, 10% growth rate until time 4, 3% growth rate after time 4 (in perpetuity). Use a 5% discount rate

0

1

2

3

4

5


$0

$1

$1.1

$1.21

$1.331

$1.3709

3% more

Value the first four payments using the finite growing annuity formula. Value the payments starting at time 5 using the perpetual growing annuity formula
The first four payments

Value as of time 0 = $1 { [1 – (1.10/1.05)^{4}] / (0.05 – 0.10) }

$4.0904

The payments starting at time 5

Value as of time 0 = {[($1) (1.1^{3}) (1.03)] / [(0.05 – 0.03)]} / 1.05^{4}

$56.3934

Solution = $4.0904 + $56.3934 = $60.4837
Note: [($1) (1.1^{3}) (1.03)] = $1.3709 = the payment at t = 5

Two growthrate example: $1 at time 0, 10% growth rate until time 12, 3% growth rate after time 12 (in perpetuity). Use a 5% discount rate

0

1

2

. . .

11

12

13


$1

$1.1

$1.21


$2.8531

$3.1384

$3.2326

3% more

Value the first thirteen payments (t = 0 to t = 12) using the finite growing annuity formula. Value the payments starting at time 13 using the perpetual growing annuity formula
The first thirteen payments

Value as of time 0 = $1 { [1 – (1.10/1.05)^{13}] / (0.05 – 0.10) } 1.05^{1}

$17.4471

The payments starting at time 13

Value as of time 0 = {[($1) (1.1^{12}) (1.03)] / [(0.05 – 0.03)]} / 1.05^{12}

$90.0011

Solution = $17.4471 + $90.0011 = $107.4482
Note: [($1) (1.1^{12}) (1.03)] = $3.2326 = the payment at t = 13

Two growthrate example: $1 at time 3, 10% growth rate until time 20, 3% growth rate after time 20 (in perpetuity). Use a 5% discount rate

0

1

2

3

4

. . .

20

21


$0

$0

$0

$1

$1.1


$5.0545

$5.2061

3% more

Value the first eighteen payments (t = 3 to t = 20) using the finite growing annuity formula. Value the payments starting at time 21 using the perpetual growing annuity formula
The first eighteen payments

Value as of time 0 = $1 { [1 – (1.10/1.05)^{18}] / (0.05 – 0.10) } / 1.05^{2}

$23.7689

The payments starting at time 21

Value as of time 0 = {[($1) (1.1^{17}) (1.03)] / [(0.05 – 0.03)]} / 1.05^{20}

$98.1063

Solution = $23.7689 + $98.1063 = $121.8752
Note: [($1) (1.1^{17}) (1.03)] = $5.2061 = the payment at t = 21

Internal rate of return (IRR)
Definition: the IRR = the discount rate that causes the sum of the present values of all cash flows to equal zero.
IRR calculation examples

Two cash flows

0

1

2

3

4

5

$1

$0

$0

$0

$0

$2

$1 + $2 / (1 + r)^{5} = $0
r = ($2 / $1)^{(1/5)} – 1 = 14.8698% = IRR
Financial Calculator: N = 5, I/Y = Answer, PV = 1, PMT = 0, FV = 2

0

1

2

3

4

5

$1

$0

$0

$0

$0

$2

$1 + $2 / (1 + r)^{5} = $0
r = ($2 / $1)^{(1/5)} – 1 = 14.8698% = IRR
Financial Calculator: N = 5, I/Y = Answer, PV = 1, PMT = 0, FV = 2

Perpetual constant annuities

0

1

2

3

4


$10

$1

$1

$1

$1

$1

$10 + $1 / r = $0
r = $1 / $10 = 10% = IRR

Perpetual growing annuities

0

1

2

3

4


$10

$1

$1.03

$1.0609

$1.0927

3% more

$10 + $1 / (r – 3%) = $0
r = ($1 / $10) + 3% = 13% = IRR

Other cash flow patterns – solve by your calculator or computer. Example – finite annuity

0

1

2

3

4

5

$3

$1

$1

$1

$1

$1

$3 + $1 { [1 – (1/(1+r))^{5}] / r } = $0
r = 19.8577% = IRR
Financial Calculator: N = 5, I/Y = Answer, PV = 3, PMT = 1, FV = 0

Modified Internal Rate of Return (MIRR)

Step one: Using the discount rate, take a PV (to time zero) of the negative cash flows

Step two: Using the interest rate, take a FV (to time t, where t is the time of the last cash flow) of the positive cash flows

Step three: Calculate the IRR of the two cash flows calculated in the first two steps

Using an interest and discount rate = 5%, what is the MIRR of the following cash flow stream?

0

1

2

3

4

5

$3

$1

$1

$1

$1

$1

Step 1: PV of negative cash flows (at time 0) = $3
Step 2: FV of positive cash flows (at time 5) = $5.5256
Step 3: IRR = ($5.5256 / $3)^{(1/5)} – 1 = 12.9932% = MIRR

Using an interest and discount rate = 5%, what is the MIRR of the following cash flow stream?

0

1

2

3

4

5

+$3

$1

$1

$1

$1

$1

Step 1: PV of negative cash flows (at time 0) = $4.3295
Step 2: FV of positive cash flows (at time 5) = $3.8288
Step 3: IRR = ($3.8288 / $4.3295)^{(1/5)} – 1 = 2.4277% = MIRR

Using an interest and discount rate = 5%, what is the MIRR of the following cash flow stream?

0

1

2

3

4

5

+$3

$0

$0

$1

$1

$1

Step 1: PV of negative cash flows (at time 0) = $1.6865
Step 2: FV of positive cash flows (at time 5) = $4.8288
Step 3: IRR = ($4.8288 / $1.6865)^{(1/5)} – 1 = 23.4154% = MIRR

Application – loan amortization schedules
A 30year home loan has an annual interest rate of 8%. Interest is compounded monthly. What is the monthly payment on a fully amortizing, level payment loan for $100,000? $733.7646
Use this payment to filling in the following loan amortization table for the home loan described above.

Month

Beginning Balance

Total Payment

Interest Payment

Principal Payment

Ending Balance

0





$100,000.000

1

$100,000.000

$733.7646

$666.6667

$67.0979

$99,932.9021

2

$99,932.9021

$733.7646

$666.2193

$67.5452

$99,865.3569

3

$99,865.3569

$733.7646

$665.7690

$67.9955

$99,797.3613

4

$99,797.3613

$733.7646

$665.3157

$68.4488

$99,728.9125

5

$99,728.9125

$733.7646

$664.8594

$68.9052

$99,660.0073

The loan balance will be $0 after the 360^{th} payment.
