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

Net present value (NPV): the sum of the present values of allcash inflows minus the sum of the present values of allcash outflows.

The internal rate of return (IRR): (1) the discount rate thatequates the sum of the present values of all cash inflowsto the sum of the present values of all cash outflows;(2) the discount rate that sets the net present valueequal to zero.

The internal rate of return measures the investment yield.

IRR and NPV

E x a m p l e : Y i e l d o n a s i n g l e r e c e i p t .

A n i n v e s t o r c a n p u r c h a s e a v a c a n t l o t f o r $ 2 8 , 3 7 1 a n d e x p e c t st o s e l l i t f o r $ 5 0 , 0 0 0 i n 5 y e a r s . W h a t i s t h e e x p e c t e d I R R f o rt h i s i n v e s t m e n t ?

d = 1 2 %

P V F Vd n

1

1( )

$ 2 8 , $ 5 0 ,( )

3 7 1 0 0 01

1 5

d

IRR and NPVHP 10B Keystrokes

50000

CLEAR ALL

P/YR

+ / - PV28371

1

FV

5 N

I/YR

Clears registers

One payment per year

PV = -$ 28,371

FV = $ 50,000

FV in 5 years

Solve for IRR

IRR and NPVE x a m p l e : N P V f o r a s i n g l e r e c e i p t .

A n i n v e s t o r c a n p u r c h a s e a v a c a n t l o t f o r $ 2 8 , 3 7 1 a n d e x p e c t s t o s e l l i t f o r$ 5 0 , 0 0 0 i n 5 y e a r s . W h a t i s t h e e x p e c t e d N P V f o r t h i s i n v e s t m e n t i f t h ei n v e s t o r d i s c o u n t s f u t u r e c a s h f l o w s a t 1 5 % ?

N P V = - $ 2 8 , 3 7 1 + $ 2 4 , 8 5 8 . 8 4 = - $ 3 , 5 1 2 . 1 6

N P V P V F Vd n

1

1( )

N P V

$ 2 8 , $ 5 0 ,( . )

3 7 1 0 0 01

1 0 1 5 5

IRR and NPVHP 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 $28,371today. The investment will provide $445.66 at the end of each month for thenext 8 years. What is the (annual) IRR (compounded monthly) for thisinvestment?

PV PMTdk

d

dd

tt

nk

tt

1

1

371 445661

112

1209167%; 110%

1

1

96

( )

$28, .( )

. .

IRR and NPVHP 10B Keystrokes

12

CLEAR ALL

P/YR

28,371 +/- PV

445.66 PMT

8

I/YR

x P/YR

Clears registers

Monthly compounding

PV = - $28,371

Monthly pmt = $445.66

96 months

Compute IRR

IRR and NPVE x a m p l e : N P V f o r a n O r d i n a r y A n n u i t y

A n i n v e s t o r h a s t h e o p p o r t u n i t y t o i n v e s t i n r e a l e s t a t e c o s t i n g$ 2 8 , 3 7 1 t o d a y . T h e i n v e s t m e n t w i l l p r o v i d e $ 4 4 5 . 6 6 a t t h e e n d o fe a c h m o n t h f o r t h e n e x t 8 y e a r s . W h a t i s t h e N P V f o r t h i si n v e s t m e n t i f t h e i n v e s t o r d i s c o u n t s f u t u r e c a s h f l o w s m o n t h l y a t a1 0 % a n n u a l r a t e ?

N P V = - $ 2 8 , 3 7 1 + $ 2 9 , 3 6 9 . 6 6 = $ 9 9 8 . 6 6

N P Vtt

$ 2 8 , .(

.)

3 7 1 4 4 5 6 61

10 1 0

1 21

9 6

IRR and NPVHP 10B Keystrokes

CLEAR ALL

12 P/YR

445.66 PMT

10 I/YR

8

PV +/-

- 28,371 =

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

d/12 = 1.0921% ; d = 13.10%

PV PMTdk

FVdk

d d

tt

nk

nk

tt

1

1 1

000 028 611

112

662 97

112

1

1

60

60

( ) ( )

$96, $1, .( )

$97, .

( )

IRR and NPVHP 10B Keystrokes

CLEAR ALL

12 P/YR

96,000 +/- PV

1,028.61 PMT

97,662.97 FV

5

I/YR

x P/YR

Clears registers

Monthly payments

PV = -$96,000

Monthly pmt = $1,028.61

FV = $97,662.97

60 months

Compute yield (IRR)

IRR and NPVE x a m p l e : N P V f o r a n o r d i n a r y a n n u i t y w i t h a n a d d i t i o n l u m p

s u m r e c e i p t a t t h e e n d o f t h e i n v e s t m e n t t e r m .

W h a t i s t h e N P V f o r a n i n v e s t m e n t t h a t c o s t s $ 9 6 , 0 0 0 t o d a ya n d p a y s $ 1 0 2 8 . 6 1 a t t h e e n d o f t h e m o n t h f o r t h e n e x t 6 0m o n t h s a n d t h e n p a y s a n a d d i t i o n a l $ 9 7 , 6 6 2 . 9 7 a t t h e e n d o ft h e 6 0 t h m o n t h i f t h e i n v e s t o r d i s c o u n t s e x p e c t e d f u t u r e c a s hf l o w s m o n t h l y a t t h e a n n u a l r a t e o f 1 3 . 1 0 4 7 % ?

N P V = - $ 9 6 , 0 0 0 + $ 9 6 , 0 0 0 = $ 0

N P V P V P M Tdk

F Vdk

N P V

tt

n k

n k

tt

1

1 1

0 0 0 0 2 8 6 11

10 1 3 1 0 4 7

1 2

6 6 2 9 7

10 1 3 1 0 4 7

1 2

1

1

6 0

6 0

( ) ( )

$ 9 6 , $ 1 , .(

.)

$ 9 7 , .

(.

)

IRR and NPVHP 10B Keystrokes

CLEAR ALL

12 P/YR

1,028.61 PMT

97,662.97 FV

5

13.1047 I/YR

PV +/-

- 96,000 =

x P/YR

Clears registersMonthly paymentsMonthly pmt = $1,028.61FV = $97,662.9760 months of paymentsDiscount rate = 13.1047%Compute PV Subtract $96,000

IRR and NPV

E x a m p l e : I R R f o r u n e v e n c a s h f l o w s .

W h a t i s t h e I R R f o r a n i n v e s t m e n t t h a t c o s t s $ 1 0 0 , 0 0 0 t o d a ya n d p a y s $ 2 0 , 0 0 0 o n e y e a r f r o m t o d a y ; $ 3 5 , 0 0 0 t w o y e a r s f r o mt o d a y ; a n d $ 7 5 , 0 0 0 t h r e e y e a r s f r o m t o d a y ?

$ 1 0 0 ,$ 2 0 ,

( )

$ 3 5 ,

( )

$ 7 5 ,

( )

.

0 0 00 0 0

1

0 0 0

1

0 0 0

1

1 1 5 9 %

2 3

d d d

d

IRR and NPVHP 10B Keystokes

CLEAR ALL

1

100,000 +/- CFj

20,000 CFj

35,000 CFj

75,000 CFj

P/YR

IRR/YR

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

E x a m p l e : N P V f o r u n e v e n c a s h f l o w s .

W h a t i s t h e N P V f o r a n i n v e s t m e n t t h a t c o s t s $ 1 0 , 0 0 0 t o d a y ,$ 8 , 0 0 0 o n e y e a r f r o m t o d a y , $ 5 , 0 0 0 t w o y e a r s f r o m t o d a y a n dp a y s $ 1 5 , 0 0 0 t h r e e y e a r s f r o m t o d a y a n d $ 2 5 , 0 0 0 f o u r y e a r sf r o m t o d a y i f f u t u r e c a s h f l o w s a r e d i s c o u n t e d a t 1 0 % ?

N P V = - $ 1 0 , 0 0 0 - $ 7 , 2 7 2 . 7 3 - $ 4 , 1 3 2 . 2 3 + $ 1 1 , 2 6 9 . 7 2 + $ 1 7 , 0 7 5 . 3 4

= $ 6 , 9 4 0 . 1 0

N P V $ 1 0 ,$ 8 ,

.

$ 5 ,

.

$ 1 5 ,

.

$ 2 5 ,

.0 0 0

0 0 0

1 1

0 0 0

1 1

0 0 0

1 1

0 0 0

1 12 3 4

IRR and NPVHP 10B Keystrokes

CLEAR ALL

1 P/YR

10,000 +/-

+/-

+/-

8,000

CFj

CFj

CFj

CFj

CFj

5,000

15,000

25,000

10 I/YR

NPV

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

E x a m p l e : I R R f o r g r o u p e d c a s h f l o w s .

C o m p u t e t h e I R R f o r a n i n v e s t m e n t t h a t c o s t s $ 9 2 , 7 2 5 . 6 0t o d a y a n d i s e x p e c t e d t o p a y $ 1 0 , 0 0 0 a t t h e e n d o f t h e y e a r f o rt h e n e x t t h r e e y e a r s ; $ 1 5 , 0 0 0 a t t h e e n d o f y e a r s 4 a n d 5 ; a n d$ 1 0 0 , 0 0 0 a t t h e e n d o f y e a r 6 .

d = 1 2 %

$ 9 2 , .$ 1 0 ,

( )

$ 1 5 ,

( )

$ 1 0 0 ,

( )7 2 5 6 0

0 0 0

1

0 0 0

1

0 0 0

11

3

4

5

6

d d dtt

tt

IRR and NPVHP 10B Keystrokes

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

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

E x a m p l e : N P V f o r g r o u p e d c a s h f l o w s .

C o m p u t e t h e N P V f o r a n i n v e s t m e n t t h a t c o s t s $ 9 8 , 0 0 0 t o d a y a n d i se x p e c t e d t o p a y $ 7 9 1 . 3 8 a t t h e e n d o f e a c h m o n t h f o r 1 2 m o n t h s ; $ 8 5 0 . 7 3 a tt h e e n d o f e a c h m o n t h f o r t h e f o l l o w i n g 1 2 m o n t h s ; $ 9 1 4 . 5 4 a t t h e e n d o fe a c h m o n t h f o r t h e f o l l o w i n g 1 1 m o n t h s a n d a b a l l o o n p a y m e n t o f$ 1 0 7 , 4 9 1 . 1 8 a t t h e e n d o f m o n t h 3 6 i f t h e i n v e s t o r d i s c o u n t s f u t u r e c a s hf l o w s m o n t h l y a t a 1 3 % a n n u a l r a t e .

N P V = - $ 5 5 4 . 1 7 = - $ 9 8 , 0 0 0 +

$ 7 9 1 .(

.)

$ 8 5 0 .(

.)

$ 9 1 4 .(

.)

$ 1 0 7 , .

(.

)3 8

1

10 1 3

1 2

7 31

10 1 3

1 2

5 41

10 1 3

1 2

4 9 1 1 8

10 1 3

1 21

1 2

1 3

2 4

2 5

3 5

3 6

tt tt tt

IRR and NPVHP 10B Keystrokes

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

Clear registersMonthly paymentsInitial CF = - $98,0001st grouped CF = $791.38Occurs 12 times2nd grouped CF = $850.73Occurs 12 times3rd grouped CF = $914.54Occurs 11 times4th CF = $107,491.18 (once)Discount rate = 13%Compute net present value