Question

Use the following information to answer the next two questions.

Consider the after-tax cash flows below:

Year	0	1	2	3	4
Cash Flow	-$75,000	$5,100	-$1,100	$498,000	-$272,000

The required rate of return is 12 percent.

Find an internal rate of return (IRR) for these cash flows.

Should you use the IRR you calculated in the previous question to decide whether the project is acceptable? Explain

show work please

Answer 1

There are negative cash flows in the middle year as well as towards the end. It is apparent that the project will have 2 IRRs with wide range.

IRR is the rate at which the sum of PV of all cash flows (Inflows & Outflows) = 0, or the discounting rate at which NPV = 0

Lets use trial and error and interpolation to get the desired value.

If IRR = 20%,

NPV = -75000/ (1+20%)^0 + 5100/(1+20%)^1 -1100/(1+20%)^2 + 498000/(1+20%)^3 - 272000/(1+20%)^4

= -75000 + 4250 - 764 + 288194 - 131173

= 85,508

If IRR = 40%,

NPV = -75000/ (1+40%)^0 + 5100/(1+40%)^1 -1100/(1+40%)^2 + 498000/(1+40%)^3 - 272000/(1+40%)^4

= -75000 + 3643 - 561 + 181487 - 70804

= 38765

If IRR = 60%,

NPV = -75000/ (1+60%)^0 + 5100/(1+60%)^1 -1100/(1+60%)^2 + 498000/(1+60%)^3 - 272000/(1+60%)^4

= -75000 + 3188 - 430 + 121,582 - 41504

= 7836

We are nearing to NPV = 0

If IRR = 70%,

NPV = -75000/ (1+70%)^0 + 5100/(1+70%)^1 -1100/(1+70%)^2 + 498000/(1+70%)^3 - 272000/(1+70%)^4

= -75000 + 3000 - 381 + 101364 - 32567

= -3584

Now IRR lies between 60% & 70%

If IRR = 65%,

NPV = -75000/ (1+65%)^0 + 5100/(1+65%)^1 -1100/(1+65%)^2 + 498000/(1+65%)^3 - 272000/(1+65%)^4

= -75000 + 3091 - 404 + 110861 - 36697

= 1850

Now IRR lies between 65% & 70%

We will now use interpolation to get the value

The interpolation formula looks like this:

y- y1 = ((y2-y1) / (x2-x1))* (x-x1)

IRR - 65 = ((70-65) /(-3850 - 1850)) *( 0-1850)

= 65 + (5/ -5700)*(-1850)

= 65 + 1.702

= 66.70%

For the other IRR, if we start with 0%

IRR = 0%

NPV = -75000/ (1+0%)^0 + 5100/(1+0%)^1 -1100/(1+0%)^2 + 498000/(1+0%)^3 - 272000/(1+0%)^4

= -75000 + 5100 - 1100 + 49800 - 272000

= 155000

We need to now move further towards negative value,

If IRR = -40%

NPV = -75000/ (1-40%)^0 + 5100/(1-40%)^1 -1100/(1-40%)^2 + 498000/(1-40%)^3 - 272000/(1-40%)^4

= -75000 + 8500 - 3056 + 2305556 - 2098765

= 13,725

If IRR = -50%

NPV = -75000/ (1-50%)^0 + 5100/(1-50%)^1 -1100/(1-50%)^2 + 498000/(1-50%)^3 - 272000/(1-50%)^4

= -75000 +10200 - 4400 + 3984000 - 4352000

= -437200

So IRR lies between - 40% & -50%

If IRR = -45%

NPV = -75000/ (1-45%)^0 + 5100/(1-45%)^1 -1100/(1-45%)^2 + 498000/(1-45%)^3 - 272000/(1-45%)^4

= -75000 +9273 - 3636 + 2993238 - 2972475

= - 48600

So IRR lies between - 40% & -45%

Lets now use interpolation to find the value

y- y1 = ((y2-y1) / (x2-x1))* (x-x1)

IRR - (-45) = ((-40-(-45)) /(13725- (-48600)) *( 0- (-13725)

= -45 + (5/185835) x 13725

= -45 + 1.307

= - 43.69% (This could be further approximated with few more iterations)

2. Clearly a negative IRR is what a business never looks at. And the positive IRR too looks unrealistic. A better way to find the value is to use MIRR method.

MIRR = (ΣFVCF / ΣPVCF) ^(1/n) -1

ΣFVCF = Sum of Future Value of all positive cash flows

FVCF = Cash Flow x ((1+ Investment Rate)^(Term of project - Year of CF))

ΣPVCF = Cash Flow / ((1+ Discount Rate)^(Year of CF))

ΣFVCF (Positive Cash flow)

= 5100 x (1+12%)^(4-1) + 498000 x (1+12%)^(4-3)

= 7165 + 557760

= 564925

ΣPVCF (Negative CF)

= 75000/ (1+12%)^0 + 1100/ (1+12%)^2 + 272000 / (1+12%)^4

= 75000 +877 +172861

= 248738

MIRR = (564925 / 248738)^(1/4) -1

= 2.2711^(1/4) - 1

= 1.2276 -1

= .2276

MIRR = 22.76%

In the above, discount rate and reinvestment rate are both taken at 12% (Required rate of return)

Since MIRR is greater than 12%, the project should be accepted