Why Sharp Ratio is not affected when we add cash or use leverage?
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Let’s say you start out with your Sharpe ratio:
(Rp – Rf )/(Risk p)
Now you want to lever it up three times. But you have to borrow money to pay for your leverage, and you borrow at the risk-free rate, Rf. So in the numerator, you have to subtract one Rf for each of the the two “extra returns” you’re adding to your base-case return through leverage:
[3(Rp) – 2(Rf)] – Rf
This simplifies to 3(Rp – Rf).
Meanwhile, the denominator just increases 3 times for 3 times the risk. So the new Sharpe ratio is:
3(Rp – Rf) / 3(Risk p)
The 3’s cancel out, and the ratio remains unaffected by the addition of leverage.
You can apply the same reasoning to adding cash if you consider cash to be inverse leverage. In other words, going to 50% cash is like levering by 1/2. And instead of paying Rf on the funds you borrow, you’re earning Rf on the cash you hold. Just to walk through it:
Your numerator is [0.5(Rp) + 0.5(Rf)] – Rf
That simplifies to 0.5(Rp – Rf)
Your denominator is 0.5(Risk p)
And the Sharpe ratio remains unchanged by the addition of cash.