# losing 10% return need gain off 11.11% to original capital: What is the theory called

Losing 10% means you have to gain 11.11% to get back to the original capital.

Losing 20% means you have to gain 25% to get back to original capital.

Losing 50% means you have to gain 100% to get back to original capital.

I'm looking for the name of this theory.

It is called the multiplicative inverse in mathematics meaning that if you start with P and then have 0.9P left (10% loss), your residual capital must increase by a factor X so that your capital is P again. So, X times (0.9P) = P, leading to X = 1/0.9 = 1.1111111...., the multiplicative inverse of what happened previously. You can think of this as a 11.11% gain if you like (it is mathematically just a tad larger) for all practical purposes unless you are applying it to a large sum such as the US National Debt in which case more digits would be needed for accuracy.

What all this is called officially in financial circles, I do not know but I suspect what it is described as around the office coffeepot at brokerages is

Average yield is for selling to suckers, of which there is one born every minute.

As in your example, in the first year, my fund went down 50%, while in the second year, it went up 50% and so the average of these two annual returns was 0%. And yet, my investment in the fund, which was \$10K at the beginning of the first year, went down to \$5K at the end of the first year (50% loss) and then up to \$7.5K at the end of the second year (50% gain). But, I (who just fell off the turnip truck) am perfectly happy because my broker has told me that my average annual return over these two years is 0%; and so my investment is worth the same as it was at the beginning of the two-year period. As an example of this phenomenon, consider the following quote from this question

He suggested setting up a mutual fund with American Funds and that would probably return 7% over the long run. He went through charts/data and such, but all I could pull in was yearly percentages since that's all I know. I understood any ups and downs and how they're all a part of the end goal on hitting a 'good' average.

What you really want to compare is not the average yield (obtained as the arithmetic average of all the annual (or quarterly or monthly or weekly or daily yields) but rather the Compounded Annual Growth Rate (CAGR) which is calculated as follows:

If the investment value changed by factors of X1, X2, ...., Xn over n years resulting in your investment having value P(X1)(X2)...(Xn) after n years, then the

The CAGR is {the n-th root of the product (X1)(X2)...(Xn)} minus 1

The CAGR is a standard notion (with variations such as CQGR, CMGR, CWGR, CDGR) that is used in financial circles even if it not revealed to suckers (I meant to say, customers). The mathematically inclined will recognize the "n-th root of (X1)(X2)...(Xn)" as the geometric mean of the numbers X1, X2, ... Xn.

In contrast, the "average yield" as quoted by brokers to customers is more likely to be the arithmetic average of the yields

Average Yield is {(the sum of X1, X2, ... Xn) divided by n} minus 1