Performance of multiple accounts does not add up

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Example (adapted from Renditen addieren sich nicht wie erwartet (TTWROR) - #16 by chirlu):

  • There are two securities (assume one in each of your accounts), C and S.
  • During the first month, C gains +100%, during the second, it falls to –50% (in all, ±0).
  • S falls to –50% during the first month and gains +100% during the second (in all, ±0).
  • Initially, you invest $100 into C and $50 into S (total $150). After one month, you have $225 (corresponding to +50%) and change your investment to have $50 in C and $175 in S. After the second month, you have $375 (corresponding to +150% as compared to the initial $150).
  • Note that +150% is not between ±0 and ±0.
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