What is an Internal Rate of Return (IRR)?

Internal Rate of Return: It is a rate of return earn on investing per rupees in investment periods. With the help of IRR long term investment decision are taken. It is a discount rate that makes NPV equals to zero. When an IRR is high than hurdle rate that project will be accepted.

Formula for Internal Rate of Return (IRR):

NPV (0) = - C₀+C ₁/ (1+IRR) +C _t/ (1+IRR) ^t

Where,

C₀= Initial Cash Outflow

C ₁ = First future value

IRR = Internal Rate of Return

Single Cash Flow:

Example: You invest Rs.10000 and it pay out single cash flow of Rs.14500 in 4 year. You have to find out the IRR (Internal Rate of Return)?

Solution:

PV = 10,000

FV = 14,500

N = 4

IRR = (FV/PV) ^1/n-1

= (14500/10000) ^¼-1

= 9.73%

Example:Find out the IRR of project A and the cost of capital is 10%.

Year	Project A
0	-10000
1	-6000
2	10000
3	9000

 Solution:

Year	Project A	Discount rate	Present Value
0	-10000	-	-10000
1	-6000	1.12	-5357.14
2	10000	1.25	8000
3	9000	1.40	6428.57
NPV			-928.57

IRR = lower rate+ (NPV at lower rate/ (NPV at lower rate-NPV at higher rate ) )*(higher rate-lower rate)

= 8+ (687.87/ (687.87-(928.57))) * (12-8)

= 8-1.72

= 6.28

IRR<r so,the project cannot accepted.

Multiple Cash Flows:

If a project has a normal cash flows that is -+++ or --+++ or  ++-- consist these sign of cash flows that project have only one IRR but if project has a non-normal cash flows that is -+++-++ or -+++-- consist these sign of cash flows that is known as multiple IRR where project have more than one IRR.

Example: Suppose the initial investment is Rs.1, 000 and it was expected that the future cash inflow and outflows of that investment are Rs.6000, -Rs.11000 and Rs.6000 for 3 years compounded annually. Find out the IRR?

Solution:

NPV = - C₀+C ₁/ (1+IRR) +C _t/ (1+IRR) ^t

Year	Cash flows	Discount rate 200%	Present Value
0	-1000	-	-1000
1	6000	3	2000
2	-11000	9	-1222.22
3	6000	27	222.22
NPV			0

Year	Cash flows	Discount rate 100%	Present Value
0	-1000	-	-1000
1	6000	2	3000
2	-11000	4	-2750
3	6000	8	750
NPV			0

There are 2 rates where NPV is near to 100% and 200%.
There are two IRR rates and we cannot choose anyone rate for project A. In that situation we can accept project on the basis of NPV.

In excel IRR can easily calculated:
=IRR(value,[guess])
=IRR(-1000,6000,-11000,6000,8%)

You can specify cells which contain values e.g A1:A5. similarly you can do with guess percentage e.g B1 which can contain 8% or 10% which ever single percentage you write into it.

There are 2 internal rate of return of an investment which is not correct to analyze the project to accept or reject. So, in that case second technique is used to calculate internal rate of return that is MIRR.

MIRR (Modified Rate of Return): It is modified version of internal rate of return this is used when there is multiple cash flows in investment and it is also used where cash inflow and inflow exist in an investment.

Formula of MIRR:

MIRR = ⁿ √ {FV (Positive cash flow, @ cost of capital) / -PV (Negative cash flow, @ finance rate)} -1

Whereas,

FV = Future Value

PV = Present Value

N= number of years

Example: An initial investment of Rs.200, 000 and the cash flows for 4 years are as follows:

·         Rs. 60,000

·         Rs. 70,000

·         Rs. (40,000)

·         Rs. 80,000

Assuming the cost of capital and finance rate are 8% and 10% respectively.

Solution:

Positive Cash Flow (FV)

1.	60,000*(1+0.1) 3	79,860
	70,000*(1+0.1) 2	84,700
	80,000*(1+0.1)0	80,000
		244560

Negative Cash Flow (PV)

1.	200,000/ (1+0.08) 0	200,000
	40,000/(1+0.08) 3	31753.28
		231753.29

MIRR = ⁿ √ {Future Value (Positive cash flow, @ cost of capital) / -Present value (Negative cash flow, @ finance rate)} ^1/n-1

= ⁴ √ (244560 /231753.29) -1

= 1.35%