Daily Return of a Leveraged / Shorted Asset

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Daily Return of a Leveraged / Shorted Asset

David St John
Dear All,

In the literature, it seems to be popular / standard to use the percentage
change:
d(t) = x(t)-x(t-1) / x(t-1)
To define the 'return' of an asset being held with position s(t) as:
r(t) = ln(1+s(t)d(t))

This is already problematic, even if s(t) takes on values of only 1, -1, 0,
since you could be short on a day when d(t)>1.  It's especially problematic
when s(t) is allowed to take on any real (possibly bounded, possibly
normalized) value corresponding to a more or less leveraged / cautious
position.

So, is there some reason why the measure:
r(t) = ln(1+s(t)d(t))
Is preferable to the more obvious, never undefined (for nonzero prices):
s(t)ln(x(t)/x(t-1))
???

Thanks,
-David

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Re: Daily Return of a Leveraged / Shorted Asset

Patrick Burns-2
I'm not following your notation, so I don't
really understand your question.  But I have
one comment that might help.

When you short an asset, you are really reversing
time in terms of returns.  What we normally think
of as time t-1 is really the "buy time" and time t
is the "sell time".



Patrick Burns
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(home of "The R Inferno" and "A Guide for the Unwilling S User")

David St John wrote:

> Dear All,
>
> In the literature, it seems to be popular / standard to use the percentage
> change:
> d(t) = x(t)-x(t-1) / x(t-1)
> To define the 'return' of an asset being held with position s(t) as:
> r(t) = ln(1+s(t)d(t))
>
> This is already problematic, even if s(t) takes on values of only 1, -1, 0,
> since you could be short on a day when d(t)>1.  It's especially problematic
> when s(t) is allowed to take on any real (possibly bounded, possibly
> normalized) value corresponding to a more or less leveraged / cautious
> position.
>
> So, is there some reason why the measure:
> r(t) = ln(1+s(t)d(t))
> Is preferable to the more obvious, never undefined (for nonzero prices):
> s(t)ln(x(t)/x(t-1))
> ???
>
> Thanks,
> -David
>
> [[alternative HTML version deleted]]
>
> _______________________________________________
> [hidden email] mailing list
> https://stat.ethz.ch/mailman/listinfo/r-sig-finance
> -- Subscriber-posting only.
> -- If you want to post, subscribe first.
>
>

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