Aug

3

Say you start with equity amount A and lose x percent every day (or year, doesn't matter). After number of days N1, your equity reached to B. From that day, you start to gain x percent every day and after number of days N2, your equity gets back to A.

Surprisingly to me, the difference between N1 and N2 almost does not depend on x. The second surprise to me is that the difference is not very big at all. It's not surprising though that N2 (on the gain side) is larger than N1 (on the loss side). And it depend on A and B, but not by a whole lot.

The same applies when you inter-change words "gain" and "lose" in the first paragraph.

Example: 
A=100000, B=50000, then N2-N1 is about 0.7, to be precise:
when x=0.01 (1%), N2-N1=0.6931529570463937
when x=0.2 (20%), N2-N1=0.69550029741854
A=100000, B=10000, then N2-N1 is about 2.3, to be precise:
when x=0.01 (1%), N2-N1=2.3026042820666532
when x=0.2 (20%), N2-N1=2.3104019779971665
A=100000, B=1000, then N2-N1 is about 6.9, to be precise:
when x=0.01 (1%), N2-N1=6.907812846199931
when x=0.2 (20%), N2-N1=6.931205933991492
Interchanging "gain" and "lose", and N1 and N2
A=1000, B=2000, then N2-N1 is about 0.7, to be precise:
when x=0.01 (1%), N2-N1=0.6931529570463937
when x=0.2 (20%), N2-N1=0.69550029741854
A=1000, B=10000, then N2-N1 is about 2.3, to be precise:
when x=0.01 (1%), N2-N1=2.3026042820666532
when x=0.2 (20%), N2-N1=2.3104019779971665
A=1000, B=100000, then N2-N1 is about 6.9, to be precise:
when x=0.01 (1%), N2-N1=6.907812846199931
when x=0.2 (20%), N2-N1=6.931205933991492

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