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This is a question in an online finance lesson, and I know what they want for the answer. But I'm not at all a novice at financial matters, so in my mind, I would do this differently.

The question is an example for calculating Net Present Value:

You are considering buying out your boss when he retires at year end. You will need to put up 100k at that time, and your profits will be 20k per year. If your goal is a 6% annual return for 8 years, should you do it?

They want you to calculate the Present Value of the inflows with the 6% return (124,196), and compare with PV of outflow (100k) and draw the conclusion that yes, you should go ahead because NPV is positive.

But in my mind the 6% desired return applies to the 100k initial investment, and not the inflows.

  • So I would calculate Future Value of that, i.e. in 8 years, if the business in fact returned 6% annually, I should have $159,384.81.
  • Then I would ask, ok, will the 20k per year inflows achieve that desired outcome? This would again be a future value calc, not a present value one. Assuming the 6% can be used again for this part, that would be $197,949.36
  • Even with 0%, the FV of the cash inflows would be 160k

So again, yes, the investment would meet the objective. But my intuitive approach is very different than that from the example.

Am I wrong? Why?

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If you are comparing the values of two income streams, it doesn’t matter if you compare them today or in the future - the larger one will be the larger one at all points in time.

So your approach is equivalent and will answer the question the same way.

There is however a reason to use ‘today’ values - if you want to know the amount of the difference, not just which one is larger. You would typically care about the exact value stream difference at the moment of decision taking, not at some arbitrary point in the future.

  • Thanks for that, it definitely helps to know that either taking everything to PV, or taking everything to FV is equivalent. – Sandra Dec 9 '18 at 15:07
  • "the larger one will be the larger one at all points in time." This is ambiguous. Presumably, you mean "given a point in time, the value of the larger one at that point in time is larger than the value of the smaller one at that point in time", but it could be taken as "the value of the larger one at any point in time is larger than the value of the smaller one at any point in time (even if the two points of time aren't the same)". – Acccumulation Dec 9 '18 at 19:10
  • "if you want to know the amount of the difference" The amount is the same regardless of whether it's present value or future value, it's just denominated in different units. – Acccumulation Dec 9 '18 at 19:11
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If you just want to determine which of two amounts is larger, then you can use any units to compare them in. If X is larger in inches then Y is in inches, then clearly X is larger in centimeters than Y is in centimeters. Present dollars and future dollars are basically different units, and the discount rate is the conversion factor. You have to convert everything to the same units to compare them. The standard way of doing that is to keep the $100k in present dollars and convert everything else to present dollars. You could convert the $100k to future dollars, and convert all the inflows to that as well. But that would be more work for the $100k, and you would still have to convert all (except maybe the last) of the inflows to a different time (for instance, the dollars you get 4 years from now have to be shifted to be 8-years-from-now dollars).

Your "Even with 0%, the FV of the cash inflows would be 160k" argument for simplifying the calculation applies with present value as well: you can take the 160k, discount it by 1.06^8 and get 100.3k. Thus, even if you were to discount all the inflows as if they were coming 8 years from now, their PV would still be more than the 100k. So your intuitive approach with the FV isn't any less math than a similar argument with PV.

PV is generally used because there is only one present, so anything put in terms of PV will be comparable to anything else put in PV, but if you put numbers in FV, you have to have the same future for all the different numbers. Furthermore, PV numbers are more meaningful. It might be useful, depending on your political persuasion, to give the cost of Social Security payouts through 2040 in 2040 dollars, but it's more honest to give them in present dollars.

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$124,196 is the value of the loan a lender would provide expecting to receive $20,000 p.a. with 6% interest over 8 years.

If the lender invested $124,196 at 6% the future value would be

$124,196 (1 + 0.06)^8 = $197,950

If you invested $100,000 at 6% the future value would be

$100,000 (1 + 0.06)^8 = $159,385

So it's better to receive the lender's future value, which corresponds to buying out your boss.

Edit

Another way to consider the situation is to see what rate the proposition involves.

With

principal, s = 100,000
payments,  d =  20,000
no. years, n = 8

Net present values

enter image description here

Solve s = (d - d (1 + r)^-n)/r for r

∴ r = 11.81451 %

Likewise net future values

enter image description here

Solve s (1 + r)^n = (d ((1 + r)^n - 1))/r for r

again r = 11.81451 %

It's the same result either way. Both show that the proposition is better than the desired 6%.

  • Chris, I follow the math. The problem I have with the setup, and it may be Sandra's issue too, is that it's worded as if I give you $100K for 8 payments of 20K. But, what of the $100K itself? The business value isn't zero in 8 years, but that seems to be the assumption. – JoeTaxpayer Dec 8 '18 at 17:19
  • If you invest $100K and receive $20k p.a. for 8 years your annual rate of return is 11.815%. – Chris Degnen Dec 8 '18 at 17:34
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    Yes. But, doesn’t this assume no further cash flows, and zero value left for the business? I’m ok with that, but it’s why these problems didn’t make much sense to me in school. Just ask me the PV of N cash flows at x%. – JoeTaxpayer Dec 8 '18 at 17:42
  • You are also using the 6% rate as if it's the current lending rate or investment rate. But the only thing we know from the problem is that we would like to make 6% from this buy-out. So to me, the only place the 6% can be applied is to the 100k. Applying it to the 20k payments makes no sens to me. – Sandra Dec 9 '18 at 15:13
  • I have added an example with the future value calculation. Perhaps the problem might be clearer to you if - when looking at the future value of the business - rather than use the desired 6% you use the actual 11.81451%. Then the future value of the principal will equal the sum of the future values of the payments. – Chris Degnen Dec 9 '18 at 18:18

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