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Say there is a mutual fund and its annual return in the past 3 years i.e. 2017, 2018, 2019 were -10%, +20%, and -10% respectively.

A and B invest $100 in this fund at different time points. A invests in 2017 and B begins investing in 2018. The return at the end of each year was:

2017

A invested $100 so return is $90 (=100*0.9)

B invested $0 so return is $0

2018

Return for A is 90*1.2 = $108

Return for B is 100*1.2 = $120

2019

Return for A is 108*0.9 = $97.20

Return for B is 120*0.9 = $108

Thus at the end of 2019 A has a return of $97.20 and B has a return of $108. Though A was invested for 3 years, he lost money while B made money.

Now my doubt is, it was A's investment during the down time in 2017 that enabled the fund to provide a 20% return in 2018. How can A be at a loss and B gets all the gains? Did I miss something in my calculations?

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Your calculations are correct. Keeping it simple...

On the surface, one might think that because A made 20% in one year and lost 10% in two years that he should be at break even because -10% +20% -10% equals zero. Unfortunately, that's not how it works. A's nominal gains and losses were -$10.00 + $18.00 - $10.80 for a total loss of $2.80

B's winning year was twice as much in percent as his losing year so he should obviously do better than A. His nominal gains and losses were +$20 and -$12 for a total gain of $8

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