Internal Rate of Return (IRR) and
Net Present Value (NPV)

Net present value (NPV): the sum of the present values of all

cash inflows minus the sum of the present values of all

cash outflows.

The internal rate of return (IRR): (1) the discount rate that

equates the sum of the present values of all cash inflows

to the sum of the present values of all cash outflows;

(2) the discount rate that sets the net present value

equal to zero.

The internal rate of return measures the investment yield.

IRR and NPV

Example: Yield on a single receipt.

An investor can purchase a vacant lot for $28,371 and expects to sell it for $50,000 in 5 years. What is the expected IRR for this investment?

EMBED Equation.2

EMBED Equation.2

d = 12%

IRR and NPV
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One payment per year

PV = -$ 28,371

FV = $ 50,000

FV in 5 years

Solve for IRR

IRR and NPV

Example: NPV for a single receipt.

An investor can purchase a vacant lot for $28,371 and expects to sell it for $50,000 in 5 years. What is the expected NPV for this investment if the investor discounts future cash flows at 15%?

EMBED Equation.2

EMBED Equation.2

NPV = -$28,371 + $24,858.84 = - $3,512.16

IRR and NPV
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One payment per year

$50,000 future value

Discount rate = 15%

FV in 5 years

Compute present value

Subtract $28,371

IRR and NPV

Example: Yield on an Ordinary Annuity

An investor has the opportunity to invest in real estate costing $28,371 today. The investment will provide $445.66 at the end of each month for the next 8 years. What is the (annual) IRR (compounded monthly) for this investment?

EMBED Equation.2

IRR and NPV
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Monthly compounding

PV = - $28,371

Monthly pmt = $445.66

96 months

Compute IRR

IRR and NPV

Example: NPV for an Ordinary Annuity

An investor has the opportunity to invest in real estate costing $28,371 today. The investment will provide $445.66 at the end of each month for the next 8 years. What is the NPV for this investment if the investor discounts future cash flows monthly at a 10% annual rate?

EMBED Equation.2

NPV = - $28,371 + $29,369.66 = $998.66

IRR and NPV
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Monthly payments

Monthly pmt = $445.66

Annual discount rate = 10%

96 monthly payments

Compute PV

Subtract $28,371

IRR and NPV

Example: What is the IRR for an investment that costs $96,000 today and pays $1028.61 at the end of the month for the next 60 months and then pays an additional $97,662.97 at the end of the 60th month?

EMBED Equation.2

d/12 = 1.0921% ; d = 13.10%

IRR and NPV
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Monthly payments

PV = -$96,000

Monthly pmt = $1,028.61

FV = $97,662.97

60 months

Compute yield (IRR)

IRR and NPV

Example: NPV for an ordinary annuity with an addition lump

sum receipt at the end of the investment term.

What is the NPV for an investment that costs $96,000 today and pays $1028.61 at the end of the month for the next 60 months and then pays an additional $97,662.97 at the end of the 60th month if the investor discounts expected future cash flows monthly at the annual rate of 13.1047%?

EMBED Equation.2

NPV = - $ 96,000 + $ 96,000 = $ 0

IRR and NPV
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Monthly payments

Monthly pmt = $1,028.61

FV = $97,662.97

60 months of payments

Discount rate = 13.1047%

Compute PV

Subtract $96,000

IRR and NPV

Example: IRR for uneven cash flows.

What is the IRR for an investment that costs $100,000 today and pays $20,000 one year from today; $35,000 two years from today; and $75,000 three years from today?

EMBED Equation.2

IRR and NPV
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One payment per year

Initial CF = - $100,000

1st CF = $ 20,000

2nd CF = $ 35,000

3rd CF = $ 75,000

Compute yield (IRR)

IRR and NPV

Example: NPV for uneven cash flows.

What is the NPV for an investment that costs $10,000 today, $8,000 one year from today, $5,000 two years from today and pays $15,000 three years from today and $25,000 four years from today if future cash flows are discounted at 10%?

EMBED Equation.2

NPV = -$10,000 - $7,272.73 - $4,132.23 + $11,269.72 + $17,075.34

= $ 6,940.10

IRR and NPV
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One payment per year

Initial CF = - $ 10,000

1st CF = - $ 8,000

2nd CF = - $ 5,000

3rd CF = $ 15,000

4th CF = $ 25,000

Discount rate = 10%

Compute net present value

IRR and NPV

Example: IRR for grouped cash flows.

Compute the IRR for an investment that costs $92,725.60 today and is expected to pay $10,000 at the end of the year for the next three years; $15,000 at the end of years 4 and 5; and $100,000 at the end of year 6.

EMBED Equation.2

d = 12%

IRR and NPV
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One payment per year

Initial CF = - $ 92,725.60

1st grouped CF = $ 10,000

Occurs three times

2nd grouped CF = $ 15,000

Occurs twice

3rd CF = $ 100,000 (once)

Compute the yield (IRR)

IRR and NPV

Example: NPV for grouped cash flows.

Compute the NPV for an investment that costs $98,000 today and is expected to pay $791.38 at the end of each month for 12 months; $850.73 at the end of each month for the following 12 months; $914.54 at the end of each month for the following 11 months and a balloon payment of $107,491.18 at the end of month 36 if the investor discounts future cash flows monthly at a 13% annual rate.

NPV = - $554.17 = - $98,000 +

EMBED Equation.2

IRR and NPV
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Monthly payments

Initial CF = - $98,000

1st grouped CF = $791.38

Occurs 12 times

2nd grouped CF = $850.73

Occurs 12 times

3rd grouped CF = $914.54

Occurs 11 times

4th CF = $107,491.18 (once)

Discount rate = 13%

Compute net present value

Net present value (NPV):

the sum of the present values of all

cash inflows minus the sum of the present values of all

cash outflows.

The

internal rate of return (IRR):

(1) the discount rate that

equates the sum of the present values of all cas

h inflows

to the sum of the present values of all cash outflows;

(2) the discount rate that sets the

net present value

equal to zero.

The

internal rate of return

measures the investment yield.

Example: Yield on a single receipt.

An investor can purchase a vacant lot for $28,371 and expects

to sell it for $50,000 in 5 years. What is the expected

IRR

for

this investment?

d = 12%

PV

FV

d

n

=

+

1

1

(

)

$28

,

$50

,

(

)

371

000

1

1

5

=

+

d

50000

CLEAR ALL

P/YR

+ / -

PV

28371

1

FV

5

N

I/YR

Example:

NPV

for a single receipt.

An investor can purchase a vacant lot for $28,371 and expects to sell it for

$50,000 in 5 years. What is the expected

NPV

for this investment if the

investor discounts future cash flows at 15%?

NPV = -$28,371 + $24,858.84 =

- $3,512.16

NPV

PV

FV

d

n

=

-

+

+

1

1

(

)

NPV

=

-

+

+

$28

,

$50

,

(

.

)

371

000

1

1

0

15

5

CLEAR ALL

1

P/YR

50,000

FV

15

I/YR

5

N

PV

+/-

-

28,371

=

Example: Yield on an Ordinary Annuity

An investor has the opportunity to invest in real estate costing $28,371

today. The investment will provide $445.66 at the end of each month for the

next 8 years. What is the (annual)

IRR

(compounded monthly) for this

investment?

PV

PMT

d

k

d

d

d

t

t

nk

t

t

=

+

=

+

=

=

=

=

1

1

371

445

66

1

1

12

12

0

9167%;

11

0%

1

1

96

(

)

$28

,

.

(

)

.

.

12

CLEAR ALL

P/YR

28,371

+/-

PV

445.66

PMT

8

I/YR

x P/YR

Example:

NPV

for an Ordinary Annuity

An investor has the opportunity to invest in real estate costing

$28,371 today. The investment will provide $445.66 at the end of

each month for the next 8 years. What is the

NPV

for this

investment if the investor discounts future cash flows monthly at a

10% annual rate?

NPV = - $28,371 + $29,369.66 = $998.66

NPV

t

t

=

-

+

+

=

$28

,

.

(

.

)

371

445

66

1

1

0

10

12

1

96

CLEAR ALL

12

P/YR

445.66

PMT

10

I/YR

8

PV

+/-

-

28,371

=

x P/YR

Example: What is the

IRR

for an investment that costs $96,000 today and

pays $1028.61 at the end of the month for the next 60 months and then pays

an additional $97,662.97 at the end of the 60th month?

d/12 =

1.0921% ; d = 13.10%

PV

PMT

d

k

FV

d

k

d

d

t

t

nk

nk

t

t

=

+

+

+

=

+

+

+

=

=

1

1

1

000

028

61

1

1

12

662

97

1

12

1

1

60

60

(

)

(

)

$96

,

$1

,

.

(

)

$97

,

.

(

)

CLEAR ALL

12

P/YR

96,000

+/-

PV

1,028.61

PMT

97,662.97

FV

5

I/YR

x P/YR

Example:

NPV

for an ordinary annuity with an addition lump

sum receipt at the end of the investment term.

What is the

NPV

for an investment that costs $96,000 today

and pays $1028.61 at the end of the month for the next 60

months and then pays an additional $97,662.97 at the end of

the 60th month if the investor discounts expected future cash

flows monthly at the annual rate of 13.1047%?

NPV = - $ 96,000 + $ 96,000 = $ 0

NPV

PV

PMT

d

k

FV

d

k

NPV

t

t

nk

nk

t

t

=

-

+

+

+

+

=

-

+

+

+

+

=

=

1

1

1

000

028

61

1

1

0

131047

12

662

97

1

0

131047

12

1

1

60

60

(

)

(

)

$96

,

$1

,

.

(

.

)

$97

,

.

(

.

)

CLEAR ALL

12

P/YR

1,028.61

PMT

97,662.97

FV

5

13.1047

I/YR

PV

+/-

-

96,000

=

x P/YR

Example:

IRR

for uneven cash flows.

What is the

IRR

for an investment that costs $100,000 today

and pays $20,000 one year from today; $35,000 two years from

today; and $75,000 three years from today?

$100

,

$20

,

(

)

$35

,

(

)

$75

,

(

)

.

000

000

1

000

1

000

1

11

59%

2

3

=

+

+

+

+

+

=

d

d

d

d

CLEAR ALL

1

100,000

+/-

CFj

20,000

CFj

35,000

CFj

75,000

CFj

P/YR

IRR/YR

Example:

NPV

for uneven cash flows.

What is the

NPV

for an investment that

costs

$10,000 today,

$8,000 one year from today, $5,000 two years from today and

pays

$15,000 three years from today and $25,000 four years

from today if future cash flows are discounted at 10%?

NPV = -$10,000 - $7,272.73 - $4,132.23 + $11,269.72 + $17,075.34

= $ 6,940.10

NPV

=

-

-

-

+

+

$10

,

$8

,

.

$5

,

.

$15

,

.

$25

,

.

000

000

1

1

000

1

1

000

1

1

000

1

1

2

3

4

CLEAR ALL

1

P/YR

10,000

+/-

+/-

+/-

8,000

CFj

CFj

CFj

CFj

CFj

5,000

15,000

25,000

10

I/YR

NPV

Example:

IRR

for grouped cash flows.

Compute the

IRR

for an investment that costs $92,725.60

today and is expected to pay $10,000 at the end of the year for

the next three years; $15,000 at the end of years 4 and 5; and

$100,000 at the end of year 6.

d = 12%

$92

,

.

$10

,

(

)

$15

,

(

)

$100

,

(

)

725

60

000

1

000

1

000

1

1

3

4

5

6

=

+

+

+

+

+

=

=

d

d

d

t

t

t

t

CLEAR ALL

1

P/YR

92,725.60

+/-

CFj

CFj

CFj

10,000

3

15,000

2

100,000

CFj

IRR/YR

N j

N j

Example:

NPV

for grouped cash flows.

Compute the

NPV

for an investment that costs $98,000 today and is

expected to pay $791.38 at the end of each month for 12 months; $850.73 at

the end of each month for the following 12 months; $914.54 at the end of

each month for the following 11 months and a balloon payment of

$107,491.18 at the end of month 36 if the investor discounts future cash

flows monthly at a 13% annual rate.

NPV =

- $554.17

= - $98,000 +

$791

.

(

.

)

$850

.

(

.

)

$914

.

(

.

)

$107

,

.

(

.

)

38

1

1

0

13

12

73

1

1

0

13

12

54

1

1

0

13

12

491

18

1

0

13

12

1

12

13

24

25

35

36

+

+

+

+

+

+

+

=

=

=

t

t

t

t

t

t

CLEAR ALL

12

P/YR

98,000

+/-

CFj

CFj

CFj

CFj

CFj

I/YR

NPV

791.38

12

850.73

12

914.54

11

107,491.18

13

N j

N j

N j