Question

Hello! Can you show me how to this problem by hand? No Excel please!

An organization is considering the purchase of new machines to automatically conduct some quality control tasks. The machines are expected to save production cost which is given below as Annual Income. The machines are expected to be part of the production process for 5 years. The company has a minimum attractive rate of return (MARR) of 5.4%. The following data is available for these new machines:

Alternative, Initial Cost, Annual Income, Annual Operation and Maintenance costs, Salvage value

A   $26,000 $9,200 $875 $5,500

B $62,000 $17,100 $1,320 $7,900

C $35,000 $13,200 $1,250 $6,700

Using incremental IRR analysis, find the best alternative.

Answer 1

An organization is considering the purchase of new machines to automatically conduct some quality control tasks. The machines are expected to save production cost which is given below as Annual Income. The machines are expected to be part of the production process for 5 years. The company has a minimum attractive rate of return (MARR) of 5.4%. The following data is available for these new machines:

Alternative, Initial Cost, Annual Income, Annual Operation and Maintenance costs, Salvage value

A   $26,000 $9,200 $875 $5,500

B $62,000 $17,100 $1,320 $7,900

C $35,000 $13,200 $1,250 $6,700

.

Using incremental IRR analysis, find the best alternative.

.

IRR(Internal rate of return)

IRR is the rate at which the NPV of a project become zero, in other word net initial investment is equal to PV of cash flow.

Criteria with IRR

*A project IRR is above the MARR, the project is acceptable.

.

Calculation of 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.

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 the Guess, just use fake payback period and check the closest value of the result in the Present value of annuity $ 1 factor table’s use full life of the assets year (n) 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

So first calculate Fake payback period

Fake payback period = initial investment / average CF

.

.

Alternative = A,

Initial Cost, = $26000

Annual Income, = $9200

Annual Operation and Maintenance costs, = $875

Salvage value = $5,500

Net operating cash flow = 9200 - 875 = 8325

.

Fake payback period = 26000 / 8325 = 3.123

Look In the table’s n=5 period row and find the closest value of 3.123 and its rate on top of that column

Closest value = 3.127      aand its correspondent rate is   18%

So, pretend that 18% is the approx. IRR, take two rate, one is below 18% and one is higher than 18% . I have taken here 5.4% and 22%

one is below 18% = for this, I take the company MARR of 5.4%

Use 5.4% discount rate and find the NPV of the project. It should be positive.

NPV @ 5.4%

Year	CF	PV factor $1 @ 5.4%	PV of cash flow
0	- $ 26000	(1 / 1+5.4%)0 = 1	- $26000
1 to 5 year operating cash flow	8325	(1 / 1+5.4%)^5GT = 4.28202	35647.81
Terminal cash flow	5500	(1 / 1+5.4%)^5 = 0.768771	4228.24

NPV = PV of CF - initial investment

NPV = 35647.81 + 4228.24 - 26000 = 13876 (positive NPV)

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

Use 22% discount rate and find the NPV of the project. It should be negative.

NPV @ 22%

Year	CF	PV factor $1 @ 22%	PV of cash flow
0	- $ 26000	(1 / 1+22%)0 = 1	- $26000
1 to 5 year operating cash flow	8325	(1 / 1+22%)^5GT = 2.8636	23840
Terminal cash flow	5500	(1 / 1+22%)^5 = 0.36999925	2035

NPV = PV of CF - initial investment

NPV = 23840 + 2035 - 26000 = - 125 (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 actual correct IRR is between 5.4% and 22%

***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)

Where,

Lower discount rate = 5.4%

NPV @ Lower discount rate = 13876

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

Difference between two NPV = 13876 - (-125 ) = 14001

NPV @ Higher discount rate = -125

Higher discount rat = 22%

Put the values in to the formula

IRR = 5.4% + ( 13876 / 125 ) * ( 22% - 5.4%)

IRR = 5.4% + ( 0.9912 * 16.6)

IRR = 5.4% + 16.45 = 21.80%

The correct IRR is =   21.80%

.

Alternative = B,

Initial Cost, = $62000

Annual Income, = $17100

Annual Operation and Maintenance costs, = $1320

Salvage value = $7900

Net operating cash flow = 17100 - 1320 = 15780

.

Fake payback period = 62000 / 15780= 3.929

Look In the table’s n=5 period row and find the closest value of 3.929 and its rate on top of that column

Closest value = 3.889      and its correspondent rate is   9%

So, pretend that 9% is the approx. IRR, take two rate, one is below 9% and one is higher than 9% . I have taken here 5.4% and 12%

one is below 9% = for this, I take the company MARR of 5.4%

Use 5.4% discount rate and find the NPV of the project. It should be positive.

NPV @ 5.4%

Year	CF	PV factor $1 @ 5.4%	PV of cash flow
0	- $ 62000	(1 / 1+5.4%)0 = 1	- $62000
1 to 5 year operating cash flow	15780	(1 / 1+5.4%)^5GT = 4.28202	67570.275
Terminal cash flow	7900	(1 / 1+5.4%)^5 = 0.768771	6073.29

NPV = PV of CF - initial investment

NPV = 73643.56 - 62000 = 11643.56 (positive NPV)

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

Use 12% discount rate and find the NPV of the project. It should be negative.

NPV @ 212%

Year	CF	PV factor $1 @ 12%	PV of cash flow
0	- $ 62000	1	- $62000
1 to 5 year operating cash flow	15780	3.6047	56883.36
Terminal cash flow	7900	0.5674	4482.67

NPV = PV of CF - initial investment

NPV = 61366 - 62000 = - 634 (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 actual correct IRR is between 5.4% and 12%

***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)

Where,

Lower discount rate = 5.4%

NPV @ Lower discount rate = 11643.56

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

Difference between two NPV = 11643.56 - (-634 ) = 12277.52

NPV @ Higher discount rate = -634

Higher discount rat = 12%

Put the values in to the formula

IRR = 5.4% + ( 11643.56 / 12277.52) * ( 12% - 5.4%)

IRR = 5.4% + ( 0.94 * 6.6)

IRR = 5.4% + 6.20 = 11.60%

The correct IRR is =   11.60%

.

Alternative = C,

Initial Cost, = $35000

Annual Income, = $13200

Annual Operation and Maintenance costs, = $1250

Salvage value = $6700

Net operating cash flow = 13200- 1250= 11950

.

Fake payback period = 35000 / 11950= 2.9288

Look In the table’s n=5 period row and find the closest value of 2.928 and its rate on top of that column

Closest value = 2.925      aand its correspondent rate is   21%

So, pretend that 21% is the approx. IRR, take two rate, one is below 21% and one is higher than 21% . I have taken here 5.4% and 25%

one is below 21% = for this, I take the company MARR of 5.4%

Use 5.4% discount rate and find the NPV of the project. It should be positive.

NPV @ 5.4%

Year	CF	PV factor $1 @ 5.4%	PV of cash flow
0	- $ 35000	(1 / 1+5.4%)0 = 1	- $35000
1 to 5 year operating cash flow	11950	(1 / 1+5.4%)^5GT = 4.28202	51170
Terminal cash flow	6700	(1 / 1+5.4%)^5 = 0.768771	5150.76

NPV = PV of CF - initial investment

NPV = 56320.90 - 35000 = 21320.9 (positive NPV)

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

Use 25% discount rate and find the NPV of the project. It should be negative.

NPV @ 25%

Year	CF	PV factor $1 @ 25%	PV of cash flow
0	- $ 35000	1	- $26000
1 to 5 year operating cash flow	11950	2.68928	32136.89
Terminal cash flow	6700	0.32768	2195.45

NPV = PV of CF - initial investment

NPV = 34332.35 - 35000 = -667.65 (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 actual correct IRR is between 5.4% and 25%

***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)

Where,

Lower discount rate = 5.4%

NPV @ Lower discount rate = 21320.9

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

Difference between two NPV = 21320.9 - (-667.65 ) = 21988.55

NPV @ Higher discount rate = -667.65

Higher discount rat = 25%

Put the values in to the formula

IRR = 5.4% + ( 21320.9 / 21988.55 ) * ( 25% - 5.4%)

IRR = 5.4% + ( 0.96 * 19.6)

IRR = 5.4% + 18.81= 24.2%

.

The correct IRR is =   24.2%

.

**Using incremental IRR analysis, find the best alternative.

The best alternative is Alternative C, because its IRR is more higher than MARR of alternative.