1. Find Rate of Return

1. The problem statement, all variables and given/known data

Alternative:
immediate investment: $130,000 first-year expenditure:$6,000
the increase \$2,000 and 36,000 annually.
The economic life of the project is forecast to be 12 years
Determine the rate of return of this Plan?

2. Relevant equations

PW = -130,000 + 36,000(P/A, i%, N) – G(A/G, i%, N) - 6000(P/A, i%, N)

3. The attempt at a solution

At i = 10 %
PW = -130,000 + 36,000(P/A, 10%, 12) – G(A/G, 10%, 12) - 6000(P/A, 10%, 12)

Is the PW equation right?

2. 0xCMD, I'm not entirely familiar with all of your notation, so I apologize in advance if I overlook something important in your info.

A project's IRR (aka "rate of return", your question's objective) is that rate which creates a zero NPV, when all the cash flows are discounted by such rate.

Unless told otherwise, questions of this type usually have you assume the CFs all occur at the end of each year. Bad news: Solving for the single rate r which creates the requisite -0- NPV comes down to trial-and-error, if there are several cash flows (as in this case). Good news: Programs such as Excel can blaze through the T-and-E iterations in a blink.

Using Excel as the example (but use your favorite weapon of choice), lay out the cash flows and let the IRR function (or its Goal Seek tool, e.g.) do the high-speed heavy lifting for you.

It appears that what you've done to this point is to find the project's NPV (or "NW"), using a discount rate of 12%, a worthy objective in its own right. But what the question is asking requires you to fiddle with that discount rate until you land on a NPV of zero. Hint: the project you've described has an IRR significantly north of that 12%.

Best of luck!

3. ok thanks

I used trial and error:
PW(18%) = -130,000 - 2,000(A/G, 18%, 12) + 36,000(P/A, 18%, 12)
PW(18%) = -130,000 - 2,000(3.1936) + 36,000(4.4941)
PW(18%) = 230.642

PW(20%) = -130,000 - 2,000(A/G, 20%, 12) + 36,000(P/A, 20%, 12)
PW(20%) = -130,000 - 2,000(3.0739) + 36,000(4.1925)
PW(20%) = -5,112.33

Using Linear Interpolation:

i=18% --> 230.642
x% --> 0
i=20% --> -5,112.33

(18%-x%)/(18%-20%) = (230.642-0)/(230.642-(-5,122.33))
(0.18-x%)/(-0.02) = 0.061984
0.18 - x% = 0.00124
0.18 + 0.00124= x%
x% = 0.18124 or 18.12 %

I'm not sure if the equation for the PW is right based on the cash flows described in the question.

4. Yep, your PVs of Project A at 18% and at 20% are good.

And you correctly reasoned that the actual IRR must lie between 18 and 20, hence your approximation via linear interpolation.

Just for grins, though, keep in mind that the relationship between PV and IRR isn't linear, and so a linear interpolation might miss it by more than your desired accuracy degree. Figure the PV at 19%, and then interpolate using 18 and 19% as your range.

It comes down to how much accuracy is appropriate for the situation--your call. But with the 18-19 interpolation, you'll have an idea of how sensitive the answer is to further fine-tuning. If the answer didn't move much, you'll probably be comfortable calling it a day without any further trial-and-error.

Cheers!