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?
1
PV  FV
(1  d )n
1
$28,371  $50,000
(1  d ) 5
d = 12%
IRR and NPV
HP 10B Keystrokes
CLEAR ALL
1
P/YR
28371
+/-
50000
5
N
I/YR
FV
PV
Clears registers
One payment per year
PV = -$ 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%?
1
NPV   PV  FV
(1  d )n
1
NPV  $28,371  $50,000
5
(1  0.15)
NPV = -$28,371 + $24,858.84 = - $3,512.16
IRR and NPV
HP 10B Keystrokes
CLEAR ALL
1
P/YR
50,000 FV
15
I/YR
5
N
PV
-
+/28,371
=
Clears registers
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?
nk
1
PV  PMT 
d t
t 1 (1 
)
k
96
1
$28,371  445.66
d t
t 1 (1 
)
12
d
 0.9167%; d  110%
.
12
IRR and NPV
HP 10B Keystrokes
+/-
Clears registers
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?
96
1
NPV   $28,371  445.66 
010
. t
t 1
(1 
)
12
NPV = - $28,371 + $29,369.66 = $998.66
IRR and NPV
HP 10B Keystrokes
x P/YR
+/-
-
=
Clears registers
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?
nk
1
FV
PV  PMT 

d
d nk
t 1 (1  ) t
(1  )
k
k
60
1
$97,662.97
$96,000  $1,028.61

d t
d 60
t 1 (1 
) (1  )
12
12
d/12 = 1.0921% ; d = 13.10%
IRR and NPV
HP 10B Keystrokes
+/-
Clears registers
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%?
nk
1
FV

d t
d
t 1 (1 
) (1  )nk
k
k
60
1
$97,662.97
NPV  $96,000  $1,028.61

0131047
.
0131047
.
t
t 1 (1 
) (1 
)60
12
12
NPV   PV  PMT 
NPV = - $ 96,000 + $ 96,000 = $ 0
IRR and NPV
HP 10B Keystrokes
Clears registers
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?
$20,000 $35,000 $75,000
$100,000 


2
(1  d )
(1  d )
(1  d )3
d  1159%
.
IRR and NPV
HP 10B Keystokes
+/-
Clears registers
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%?
$8,000 $5,000 $15,000 $25,000
NPV  $10,000 



2
3
11
.
11
.
11
.
11
.4
NPV = -$10,000 - $7,272.73 - $4,132.23 + $11,269.72 + $17,075.34
= $ 6,940.10
IRR and NPV
HP 10B Keystrokes
+/+/+/-
Clear registers
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.
3
$10,000 5 $15,000 $100,000
$92,725.60  


t
t
6
(
1

d
)
(
1

d
)
(
1

d
)
t 1
t 4
d = 12%
IRR and NPV
HP 10B Keystrokes
+/-
Clears registers
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 +
12
24
35
1
1
1
$107,49118
.
$791.38
 $850.73
 $914.54 

013
.
013
.
013
.
013
. 36
t 1 (1 
t 13 (1 
t  25 (1 
)t
)t
)t (1 
)
12
12
12
12
IRR and NPV
HP 10B Keystrokes
+/-
Clear registers
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