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If I have daily prices for $N$ stocks, how do I calculate the Sharpe ratio for an equal volatility weight portfolio?

On each day, I have calculated log returns as: $$ r_{t}^{s} = \ln{price_{t}\over{price_{t-1}}} $$

I have then measured the standard deviation of each this series in a 60-day window, and normalised this so each series has approximate deviation of 1.

Question 1:

Is the portfolio daily return equal to the arithmetic mean of the individual stocks $s$, assuming I will allocate on a equal volatility basis before trading:

$$ \text{Portfolio Log Return}_t = \frac{1}{N} \sum_s r_t^s $$

Question 2:

If I take the portfolio return from above, and use these figures to create a Sharpe ratio

$$ \text{Sharpe ratio} = \frac{\sqrt(252)*\text{Arithmetic Average of LogReturns}}{\text{Std Deviation of LogReturns}} $$

  • Does this make any sense?
  • What's the relationship of a Sharpe ratio calculated in this way to the Sharpe ratios published in journals and fund brochures?
  • Can a Sharpe calculated in this way be converted back to 'realized returns'?

I have calculated this value for a period 1993-1997 for the 500 most traded stocks, weighted an equal volatility basis, and got a value of approximately 7, which is definitely not right.

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  • $\begingroup$ The return on a portfolio (assuming certain weights and a certain rebalancing policy) is generally not equal to the mean of daily returns. What are your weights and what is you rebalancing frequency (weekly, monthly, quarterly, ...)? What do you mean by equal volatility? $\endgroup$
    – nbbo2
    Commented Aug 31, 2021 at 17:46
  • $\begingroup$ I'm rebalancing daily, based a 60 day moving average of the volatility of log returns, such that balancing factor would lead to standard deviation of log-returns of 1 for every stock. $\endgroup$
    – cjm2671
    Commented Aug 31, 2021 at 19:10
  • $\begingroup$ Daily rebalancing (with no transaction costs taken into account) generally leads to overestimates of rates of return because of microstructure effect such as bid-ask bounce. In real life such rates of return are generally not achievable. $\endgroup$
    – nbbo2
    Commented Aug 31, 2021 at 20:01

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