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December 13th, 2010, 05:15 PM   #1
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gain/loss ratios?

Hi there,

I am looking for a general way to measure the, for want of a better term, 'efficiency' of some input given some gain (or loss) resulting from the input. I am assuming that there is some brach of economics mathematics that deals with this stuff.

Take a scenario in which different investments can be made in something. One type of investment costs $2 and this investment would increase that something’s value by $10. Another type of investment would cost $5 and this would also increase that something’s value by $10.

In this case, the first investment is a better, more 'efficient' option than the second since less is invested with the same return. This can be represented with 10/2 > 10/5.

Similarly, if one type of investment of $10 resulted in a $100 increase and another type of investment of $10 resulted in a $200 increase, then the latter option is better, and 200/10 > 100/10

So in maximising efficiency we simply look to maximise the numerator and minimise the denominator.

When potential loss is taken into account, the numerator can become negative. But I am not sure how to capture this precisely.

Using the inverse of the above examples, one type of investment costs $2 and results in a loss of $10. The other type costs $5 and would also result in a loss of $10. In this case, the former is better, since both losses are equal but only $2 was spent on the first and $5 on the second. Accordingly, -10/2 < -10/5, so the better option has a numerically lower measure.

With the other example, $10 is invested in both cases and the first investment results in a loss of $100 and the second results in a loss of $200. The first of these is a better option, since the loss is less. But -100/10 > -200/10, so the better option has a numerically higher measure.

thanks
- Simon
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December 17th, 2010, 10:25 AM   #2
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Re: gain/loss ratios?

The apparent contradiction you're bumping heads with is similar to the contest between Net Present Value and IRR, for the title of Best Investment-Ranking Metric.

E.g., is it better to invest 10 to earn 20, or 1,000 to earn 1,200? The former wins on rate-of-return (100% vs. 20%), while the latter comes out on top in terms of total dollars of profit (200 vs. 10). Personally, I'll take a 200 profit over 10 any day, and the concept underlying my preference is analogous to one of NPV's advantages over IRR. (Of course, if the investments are divisible and not mutually exclusive, the optimal play is to first put 10 into the 100% deal, and then my remaining 990 goes into the 20% deal.)

So using my numbers with your ratio-based approach, while it's true that 20/10 beats 1,200/1,000 from a ratio standpoint, it's also true that (1,200 - 1,000) is preferable to (20 - 10)in terms of how many beers you can buy, no?

In fact, you're sensing that very same idea when you discuss your loss-generating examples. You've noted that it's preferable to invest 2 to lose 10, than to invest 5 to lose 10. That's true on a 'magnitude of dollars' basis: Losing 12, counting the investment, is better than losing 15.

But interestingly, the bet-2-and-lose-10 is actually an inferior play, from a rate-of-return viewpoint. That's a loss of 5 for every 1 invested, whereas the latter only lost on a 2-for-1 basis. Put another way, had the latter investment been as bad as the former, it would have lost 25 instead of just 10.

The takeaway is that in the context you're working in, you may need to finesse it a bit and decide whether it's total dollars, or rate of return, that matters most. And further it may be that one's more important than the other within a certain range, whereas the other metric is the better yardstick within a certain other range.
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December 18th, 2010, 06:20 AM   #3
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Re: gain/loss ratios?

cheers for that, it was quite helpful.
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