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At 10:52 AM -0600 4/8/01, caw wrote:
>Bob, I think I disagree with your comments (below) and could I ask
>you to help me better understand. I'm having difficulty with your
>example in which you state that the doubling of exposure by the use
>of leverage also doubles the standard deviation.
The Sharpe Ratio is:
Sharpe = (Return - Risk_free_return) / Standard_deviation
Using an example:
In the first case, assume you have $10,000 and you invest in a stock
that appreciates 25% in a year for a $2500 profit. Assume the
annualized standard deviation of, for example, weekly returns over
the years was $2000 or 20% of the initial $10,000 investment.
Sharpe = (Return - Risk_free_return) / Standard_deviation
Sharpe = (25% - 5%) / 20% = 1.0
In the second case, you double the exposure with 100% leverage so you
borrow $10,000 from your broker and invest your $10,000 plus the
borrowed $10,000 at the same time as above, for a total exposure of
$20,000. The stock still gains 25% with a 20% standard deviation of
returns so you made 25% of $20,000 which is $5000. The standard
deviation was 20% of $20,000 or $4000.
You pay back the $10,000 loan plus, say, 5% interest ($500) so you
total profit was:
Profit = $5000 - $500 = $4500 which is 45% of your initial position.
The standard deviation of $4000 is now 40% of you initial position so:
Sharpe = (45% - 5%) / 40% = 1.0
>Given the same performance of two portfolios with differing SDs, how
>would a third portfolio identical to the first except for the
>application of leverage to cause its SD to be equal to that of the
>second portfolio prove your example?
I think I understand the question but am not sure.
Assume two investment vehicles with the following performance:
Vehicle 1: Return = 25%, Standard_deviation = 20%, Sharpe = 1.0
Vehicle 2: Return = 25%, Standard_deviation = 10%, Sharpe = 2.0
Now assume a portfolio using vehicle 2 with 100% leverage:
Portfolio 3: Return = 45%, Standard_deviation = 20%, Sharpe = 2.0
So now the standard deviation of Portfolio 3 is the same as the
Vehicle 1 with almost twice the return.
There are now other possibilities:
Assume portfolio 4 uses Vehicle 1 100% short plus Vehicle 2 200% long
(using 100% leverage).
Now we have a net return of 45% - 25% = 20% and a calculated standard
deviation of 20% - 20% = 0%. You would actually get 0% standard
deviation only if Vehicle 1 and Vehicle 2 are 100% correlated such
that the daily gains and losses track each other. Then the
fluctuations of the short position will exactly cancel those of the
long position so we end up with no fluctuations and a standard
deviation of 0% for the portfolio. This would then be an infinite
Sharpe Ratio.
It is impossible to have 100% correlation so the standard deviation
of the portfolio will be greater than zero and the Sharpe Ratio will
be some finite number - possibly well above 2 or 3.
This is what a hedge fund tries to do. With very high correlations
between the long and short position, the risk is low so it is safe to
extend the leverage to higher and higher values. This is what Long
Term Capital Management was doing with their Nobel-Prize-winning
staff. It works well until the correlation starts to fall apart as it
did in their case...
Bob Fulks
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