AppliedNumeric answerFree solution

An index rises 10%10\% then falls 10%10\%. Where does a 2×2\times daily ETF finish, relative to its start, as a percentage?

Answer with a number in %. Fractions, powers and expressions like 23/6 or C(52,5) are read correctly in practice.

Show the answer and worked solution

Answer

−4%-4\%

A daily-reset 2×2\times product delivers twice each day’s return, so you compound 1.201.20 then 0.800.80: 1.20×0.80=0.961.20 \times 0.80 = 0.96, a loss of 4%4\%. The index itself finishes at 1.10×0.90=0.991.10 \times 0.90 = 0.99, down 1%1\%, so twice the index return would have been −2%-2\%. The extra 2%2\% is volatility drag, and it grows with the square of the leverage and with realised volatility. That is why these products track their daily multiple faithfully and their multi-day one not at all.

Worked solution

  1. Formula
    ∏t(1+2rt)\prod_t (1 + 2r_t)
  2. Substitute
    =(1+0.20)(1−0.20)= (1 + 0.20)(1 - 0.20)
  3. Solve
    =1.20×0.80=0.96= 1.20 \times 0.80 = 0.96
  4. Answer
    −4%-4\%

Sanity check. The index is down 1%, so twice that is 2%. The extra 2% is volatility drag.

Takeaway: A leveraged ETF delivers the daily multiple, not the cumulative one.

Answer it in practice – your answer is marked and recorded.

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An index rises 10% then falls 10%. · QuantMax