# A math question

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...I'm tired, and don't want to think.

A hypothetical question. Say you have x initial amount amount to be invested, and y years to invest. Assume you have a total growth rate of, say g% (over y years, not per year). What distribution equation/curve of the per-year-growth gives the maximum output after y years? Does (g/y)% growth per year gives the best output? Or it is some bell curve?

If a generic equation is hard, please use an example of some example numbers. \$1 million initial amount, invested for 20 years, a total growth of  400% of the initial amount over 20 years. I'm trying make some mental models for the portfolio allocation.

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...I'm tired, and don't want to think.

A hypothetical question. Say you have x initial amount amount to be invested, and y years to invest. Assume you have a total growth rate of, say g% (over y years, not per year). What distribution equation/curve of the per-year-growth gives the maximum output after y years? Does (g/y)% growth per year gives the best output? Or it is some bell curve?

If a generic equation is hard, please use an example of some example numbers. \$1 million initial amount, invested for 20 years, a total growth of  400% of the initial amount over 20 years. I'm trying make some mental models for the portfolio allocation.

I don’t understand what you mean by “the best output”. You provided number of years and total return as givens. So, what are you trying to optimize?

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Maybe you are saying if you have 400 percentage points of return to allocate over a given number of years, what allocation would provide the highest total return. If you are asking that then the answer is to earn equal returns each year. Anytime you have a given amount of something and want to divide it into a fixed number of pieces and multiply those pieces together to get the highest total product, you should divide into equal sized pieces. Example: a 5x5 square has a larger area than a 6x4 rectangle or any other rectangle with adjacent sides totaling 10 units. The mathematical principle is called the “arithmetic mean/geometric mean inequality”.

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Using smaller numbers for illustration purpose:

Say in 5 years, it is supposed to produce 100% increment in the stock price.

For a starting position of 100K, the growth table is as follows:

Case 1:

year 0: 100K

Year 1: Price change: 0%, ending balance: 100K

Year 2: Price change: 0%, ending balance: 100K

Year 3: Price change: 0%, ending balance: 100K

Year 4: Price change: 0%, ending balance: 100K

Year 5: Growth: 100%, ending balance: 200K

Case 2:

year 0: 100K

Year 1: Price change: -50%, ending balance: 50K

Year 2: Price change: -50%, ending balance: 25K

Year 3: Price change: -50%, ending balance: 12.5K

Year 4: Price change: -50%, ending balance: 6.25K

Year 5: Price change: 300%, ending balance: 18.75K

For both the cases, the 5 year cumulative % price change was 100% (case1: 0+0+0+0+100 = 100%, case2: (-50)+(-50)+(-50)+(50)+(300) = 100%). But the 5th year end balance is vastly different between both the cases.

I'm trying to find the case where the %price change every year that gives the highest \$ amount after the end of the period.

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Using smaller numbers for illustration purpose:

Say in 5 years, it is supposed to produce 100% increment in the stock price.

For a starting position of 100K, the growth table is as follows:

Case 1:

year 0: 100K

Year 1: Price change: 0%, ending balance: 100K

Year 2: Price change: 0%, ending balance: 100K

Year 3: Price change: 0%, ending balance: 100K

Year 4: Price change: 0%, ending balance: 100K

Year 5: Growth: 100%, ending balance: 200K

Case 2:

year 0: 100K

Year 1: Price change: -50%, ending balance: 50K

Year 2: Price change: -50%, ending balance: 25K

Year 3: Price change: -50%, ending balance: 12.5K

Year 4: Price change: -50%, ending balance: 6.25K

Year 5: Price change: 300%, ending balance: 18.75K

For both the cases, the 5 year cumulative % price change was 100% (case1: 0+0+0+0+100 = 100%, case2: (-50)+(-50)+(-50)+(50)+(300) = 100%). But the 5th year end balance is vastly different between both the cases.

I'm trying to find the case where the %price change every year that gives the highest \$ amount after the end of the period.

Ok. In your example the optimal annual return is 20%. This is 100% divided by 5 years. No other distribution of returns will exceed what you will get with 20% per year.

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Using smaller numbers for illustration purpose:

Say in 5 years, it is supposed to produce 100% increment in the stock price.

For a starting position of 100K, the growth table is as follows:

Case 1:

year 0: 100K

Year 1: Price change: 0%, ending balance: 100K

Year 2: Price change: 0%, ending balance: 100K

Year 3: Price change: 0%, ending balance: 100K

Year 4: Price change: 0%, ending balance: 100K

Year 5: Growth: 100%, ending balance: 200K

Case 2:

year 0: 100K

Year 1: Price change: -50%, ending balance: 50K

Year 2: Price change: -50%, ending balance: 25K

Year 3: Price change: -50%, ending balance: 12.5K

Year 4: Price change: -50%, ending balance: 6.25K

Year 5: Price change: 300%, ending balance: 18.75K

For both the cases, the 5 year cumulative % price change was 100% (case1: 0+0+0+0+100 = 100%, case2: (-50)+(-50)+(-50)+(50)+(300) = 100%). But the 5th year end balance is vastly different between both the cases.

I'm trying to find the case where the %price change every year that gives the highest \$ amount after the end of the period.

Ok. In your example the optimal annual return is 20%. This is 100% divided by 5 years. No other distribution of returns will exceed what you will get with 20% per year.

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The question explores the difference between the arithmetic and geometric means.

The compound annual return (geometric) equals the arithmetic mean only when the return is the same every year. Otherwise the geometric return (as a function of total end value) is always lower than the arithmetic mean. The mathematical proof lies in log work but you can use the general rule that the difference between the two means will be a function of the standard deviation (volatility of returns over time) squared. This can be useful math if your compensation is tied to the arithmetic return and can 'work' despite a high water mark feature.

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