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You are not accounting for all the factors in your calculation.
Consider that you have a $200,000 account.
Case 1 - Invest $20,000
In this case you invest only $20,000 of the account and achieve a 20%
return on that investment = $4,000. Assume that the standard
deviation of those returns is 15% or $3,000.
Assume you invest the remaining $180,000 at the risk-free rate of 3%
for a $5,400 return with zero standard deviation. Thus, the Sharpe
Ratio of the portfolio is:
Sharpe = (Return - Risk_free_return) / Standard_deviation
= (4000 + 5400 - 3% of 200000) / 3000
= 3400 / 3000 = 1.13
Case 2 - Invest $200,000
In this case you invest all $200,000 in the same investment and
achieve the same 20% return on that investment = $40,000. Assume that
the standard deviation of those returns is again 15% or $30,000.
Sharpe = (Return - Risk_free_return) / Standard_deviation
= (40000 - 3% of 200000) / 3000
= 34000 / 30000 = 1.13
So we see that the Sharpe Ratio of the overall portfolio is THE SAME
IN EITHER CASE!
That is one of the advantages of using the Sharpe Ratio - it measures
the return-to-risk ratio. In both of the above cases, the excess
return (return in excess of the risk-free rate) was proportional to
the risk, so the Sharpe Ratio of the resulting portfolio is
independent of what percentage of the portfolio you invest.
It measures the Sharpe Ratio of the investment itself and is
independent of how much money you invest in it.
If the investment is a trading system, for example, the Sharpe Ratio
measures the performance of the trading system, independent of the
trade-size. (Caveat: if trade size gets very large - near the
so-called "Vince optimal_F point" - the relationship ceases to be
linear and the Sharpe Ratio begins to decrease as the trade size
increases, [but only an idiot would use trades that large...])
In your case, you should calculate the return each day on the amount
at risk each day and you will find that you will get the right
answers.
Bob Fulks
At 3:02 PM +0200 8/10/02, Bengtsson, Mats wrote:
>From this list I have learned to trust the Sharpe-value. The
>calculation behind it seems fair, but yet, when looking at the
>results I get usig it compared to Kelly value and other figures, I
>realise I have to change the portfolio calculation.
>
>The problem seems to lie in the assumptions behind my own portfolio
>calculations, and thus I need advise on how to adjust the
>calculations. If i do the calculations on only one stock everything
>is fine. What is happening is that on some shorter days there are a
>lot less trade than on longer days, and on some days by pure chance
>some less traded stocks are not traded at all. The portfolio
>calculation translates this to a risk using the Sharpe value, which
>is fair, but looking at it I see that the risk is overestimated in
>strategies using fewer buys than strategies using many buys.
>
>My guess is that this can be remedied by using equity per stock for
>each day in the calculation instead of just equity as is done now. My
>question is if I am right, and if not what else to do.
>
>Below is an explanation of the calculation at what I consider as the
>wring assumption.
>
>Date Equity Number of stocks Equity per stock
>990119 280000 303 924
>990121 260000 281 925
>990122 283000 305 928
>990123 261000 281 929
>
>As can se above, when calculating Sharpe ration based on equity
>change from day to day, the value will go up and down (-20000 first
>day, +23000 next day, minus 21000 thirs day, ...). This is how the
>calculation is done today, giving a low sharpe ratio in the end. If I
>instead would calculate the risk based on changes in equity per stock
>included in the measure, I would get a much smoother curve.
>
>It can be argued that the risk is higher since the trades include
>stocks that by some reason do not trade each and every day, thus the
>Sharpe ratio shall swing more. I can accept that as the truth, but I
>prefer to view that as a micro-level calculation which is to be
>considered and remedied when creating the trading systems and trading
>them, not a macro level assumption that shall result in such swings
>in the portfolio Sharpe calculation as it does. Instinctively I feel
>that using the equity per stock should be a much fairer result than
>equity that is travelling up and down just because some stocks do not
>trade same days as others (this will always be the case when trading
>international markets, some markets will have holidays when others do
>not).
>
>I would very much appreciate any help and input on this subjet. It is
>from this list I have learned to trust the Sharpe ratio, now I would
>like to have a little help in applying it better to Portfolio level
>calculations than I do today.
>
>--- Mats ---
>
>
>
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