Question

You are considering the following two mutually exclusive projects.

YEAR             PROJECT (A)         PROJECT (B)   

    0                  -$35,000                  -$35,000

    1                     22,000                     13,000

    2                     20,000                     21,000

    3                     13,000                     22,000

What is the internal rate of return of PROJECT A?

Answer 1

IRR

IRR is the rate at which NPV of the project become zero, Computing the internal rate of return (IRR) for a possible investment is time-consuming and inexact. IRR calculations must be performed via guesses, , and trial and error. Essentially, an IRR calculation begins with two random guesses at possible values and ends with either a validation or rejection. If rejected, new guesses are necessary.

Here,

0 = PV of cash inflow - PV of cash outflow

We can solve through this trail and error method

For calculating IRR, we need two NPV with their discount rate. One of them is positive and one of them negative. And discount rate used for those NPVs are one of them lower than IRR and one of them Higher than IRR

The first step is to make guesses at the possible values for lower discount rate and higher discount rate to determine

For correcting Guess just use fake payback period and check the closest value of the result in the Present value of annuity $ 1 factor table’s 3 year period row and find the correspondent rate on top of that row. Use this rate as apprpx.IRR and take two rate, one is lower than this rate and one is higher than this rate

Fake payback period = initial investment / average CF

Fake payback period = 35000 / (22000 + 20000 + 13000) / 3

Fake payback period = 35000 / 18333 = 1.909

Look In the table and find the closest value of 1.909 and its rate on top of the column

Closest value = 1.8955      aand its correspondent rate is 27%

So, pretend that 27% is the IRR take two rate, one is below 27% and one is higher than 27% . I have taken here 20% and 30%

Use 20% discount rate and find the NPV of the project. It should be positive.

NPV @ 20%

Year	CF	PV factor $1 @ 20%	CF * PV factor $1 @ 20%
0	- $ 35000	(1 / 1+20%)0 = 1	-35000
1	22000	(1 / 1+20%)1 = 0.8333	18332.6
2	20000	(1 / 1+20%)2 = 0.6944	13888
3	13000	(1 / 1+20%)3 = 0.5787	7523.15

NPV = PV of CF - initial investment

NPV = 18332.6 + 13888 + 7523.15 - 35000 = 4743.75 (positive NPV)

One of thing that if we get a positive NPV, the discount rate used here be lower than IRR

Use 30% discount rate and find the NPV of the project. It should be positive.

NPV @ 30%

Year	CF	PV factor $1 @ 20%	CF * PV factor $1 @ 20%
0	- $ 35000	(1 / 1+30%)0 = 1	-35000
1	22000	(1 / 1+30%)1 = 0.769	16318
2	20000	(1 / 1+30%)2 = 0.592	11840
3	13000	(1 / 1+30%)3 = 0.455	5917.15

NPV = PV of CF - initial investment

NPV = 16318 + 11840 + 5917.15 - 35000 = -325 (negative NPV)

One of thing that if we get a negative NPV, the discount rate used here be higher than IRR

So we can conclude that the IRR is between 20% and 30%

***Next use a formula to find correct IRR

IRR = Lower discount rate + (NPV @ Lower discount rate / Difference between two NPV ) * ( Higher discount rate - Lower discount rate)

Here,

Lower discount rate = 20%

NPV @ Lower discount rate = 4743.75

Difference between two NPV = NPV @ Lower discount rate - NPV @ Higher discount rate

Difference between two NPV = 4743.75 - (- 325) = 5068.59

NPV @ Higher discount rate = - 325

Higher discount rat = 30%

Put the values to the formula

IRR = 20% + ( 4743.75 / 5068.59 ) * ( 30% - 20%)

IRR = 20% + 0.9359 * 10

IRR = 20% + 9.36 = 29.36%

The correct IRR is = 29.36%