siamond wrote:LadyGeek wrote:siamond- I understand the need for geometric math, but I'm having some difficulty to help. Can you please show the first few steps to derive that "little bit of algebra"?

Sure. Let's start from the formula that clearly represents a geometric way to subtracting E from R.

(1+R)/(1+E) - 1

Which is the same as:

(1+R)/(1+E) - (1+E)/(1+E)

Which is the same as:

((1+R) - (1+E)) / (1+E)

Which is the same as:

(R-E) / (1+E)

Makes sense?

PS. to be honest, the corresponding formula in Simba did give me pause the first time I saw it. Took me a minute or two to realize that it was the same as the more intuitive formula I would have used!

Now it does, thank you. I missed two things:

- The formula for subtraction is also called Relative Return / Excess Return (Geometric). It took some thinking to realize the same formula can be used for any geometric difference (subtraction, which is a ratio in geometric terms) - not just the benchmark shown in the example. Once I realized that, I could understand what you were up to.

- Multiplying by 1, which is substituting the "- 1" with "-(1+E)/(1+E)" to derive the equation.

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Since this thread is now on a new page, Kevin M's top-level overview on the exponential part is repeated here. It's helpful to read this first, then siamond's part. CAGR is a very important concept, as it's used for apples-to-apples performance comparisons between two investments.

Kevin M wrote:To add a bit more to the geometric math subtraction of ER ...

The return values are annualized return, also referred to as compound annual growth rate, or CAGR. This is the value, r, that you can plug into this formula to get cumulative growth over N years:

(1 + r) ^ N

Since we are compounding r over N years, we must subtract the expense ratio, E, from the gross return, R, in a way that also works with compounding. This is because the expense is subtracted each year, reducing the net amount you have to continue compounding in subsequent years.

Note that if you plug in reasonable values, the difference between R - E and (1+R)/(1+E) - 1 is quite small. Try it with R = 2% and E = 0.2%, in which case R - E = 1.800%, while (1+R)/(1+E)-1 = 1.796%, so they both round to 1.80%. Or try it with R=5% and E=0.2%, and you get 4.80% vs. 4.79%--again, not much difference.

Note the similarity to the difference between the exact and approximate formulas for adjusting returns for inflation. The approximate formula is r = R - I, where r is real return, R is nominal return, and I is inflation rate. The exact formula is r = (1+R)/(1+I) - 1. Again, plugging in some sample numbers, I'll use R = 5% and I = 2%.

Approximate: r = R - I = 5% - 2% = 3%.

Exact: r = (1+R)/(1+I) - 1 = 1.05/1.02 - 1 = 2.94%.

Pretty close, so most people are going to be OK with the approximate value, but it's not exactly correct, and when compounded over many years could make a relevant difference.

Kevin