Impact of cash or leverage on Sharpe and Information Ratio?

I just cant wrap my head around why the Sharpe ratio is unaffected by leverage and why information ratio is?

I understand the impact of change in weights on the two ratios, and that both numerator and denominator change by the same factor but I cannot understand the impact of leverage.

Can anyone please explain? Thanks!

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Hey Steve, I’ve got the exact same question; this notion seems quite tricky for one reason: CFA tells us Information Ratio is affected by leverage and cash. However it underlines the fact that with an allocation to the benchmark portfolio the result blended portfolio will have the same Information Ratio than the original one AS THE NUMERATOR AND DENOMINATOR EVOLVE IN SAME PROPORTION. Inconsistence: leverage and cash allocation has impact on Information Ratio, but benchmark portfolio allocation does not. In my view, the active return evolve when there is more leverage in comparison with the benchmark and the volatility of the difference between the portfolio risk & benchmark portfolio evolves in same proportion hm? So same story than with Sharpe ratio right? (namely: leverage & cash --> no effect). --> I suppose I’m wrong somewhere and intellectual help would be much appreciated Many thanks in advance for your consideration guys

*****, I wake up this morning with the answer in my head,

Some CFA Saints highlighted me the path to the understanding of this intellectual issue during night, I will write you this answer in few days to let the suspense make you enjoying even more the coming answer.

Hi, Alex

Would really appreciate your answer.

AlexTe you made me laugh, thanks for the comic relief :slight_smile:

Wow guys sorry for the really late reply; I thought I was the only one with some questions about it since there had not been any activity on the thread for months; Benchmark move => delta numerator & delta denominator is similar = no changes, for a 100% investment in benchmark & active portfolio The trick is that with the information ratio, you have to think about what it really measures: it is a measure of one unit of active return for one unit of active risk : conceptually, it will never change with more or less of the active or benchmark portfolio, this ratio will always stay the same with playing with the value of this two (10% active + 90% benchmark IR (information ratio) = 90% active + 10% benchmark IR) ; as you will increase the numerator & denominator in same proportion; there is no impact in increasing the benchmark portfolio; What clarifies it is imagine the case when this benchmark portfolio is the risk free rate; then you’ve got a Sharpe Ratio and the same mathematical concept applies to the sharpe ratio without “surprising us” Leverage or cash => delta numerator & delta denominator can be different in size = changes, for a X% investment in cash / us of leverage & (100% - X% cash / + x % leverage) What surprises us is when we then underline than leverage & cash will impact the ratio despise what we previously explained; and now a proper example is needed: let’s imagine your benchmark portfolio (50% of total portfolio) performed -5% and your active portfolio (10% of total portfolio) -8%; if you had massive cash reserve (40% of total portfolio) with +0.10 % performance , you increased the active return but decreased the active risk (cash = more stable right) If you had leveraged the portfolio with 50% of the active portfolio (-8% performance) + 50% benchmark portfolio (-5% performance) + 40% cash (+0.10% performance) you would have: less active return and more active risk (active portfolio = less stable right) = shitty ratio , IR decreases. And that is basically it guys

Here is an explanation why, at least, Sharpe ratio isn’t affect by leverage/cash.

Background

Sharpe ratio = rp - rf / std p

and Rp = Wp * Rp + (1-Wp) * Rf

…and Rf is risk-free and zero std. dev

Lets break it down - step by step

  1. [Wp * Rp + (1-Wp) * Rf - Rf] / [Wp * std p] (Note that i substituted in the Rp formula into the Sharpe formula)

  2. [Wp * Rp + Rf - WpRf - Rf] / [Wp * std p] (I simply multiplied Rf with 1 and - Wp from point 1)

  3. [Wp * Rp - WpRf] / [Wp * std p] (Note that we had +Rf and -Rf which I removed)

  4. Wp(Rp - Rf) / [Wp * std p] (Remove Wp from top and bottom)

  5. Rp - Rf / std p

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Anyone have a simpler explanation here? Magician? My quick thought was that as you increase leverage, you increase the portfolio standard deviation…and as you deleverage (shift more to cash), you decrease the portfolio standard deviation. Hence the Sharpe Ratio is not impacted by the addition of cash or leverage? I think that needs some refinement but starting with that idea conceptually. Thank you.

I was reading this earlier and I believe the numerator and denominator move in the same proportion, so the ratio (Sharpe) stays the same…try a numerical example to prove this!
Ash

Hello I have a question!

In your equation 1, you have a sharpe ratio which is Rp-rf/std.p * Wp. Where does this Wp come from in Denominator?

When you invest in the risky portfolio that has volatility of \sigma_{risky} (with a weight of w_{risky} ) and in the risk-free asset that has a volatility of \sigma_{rf} = 0 (with a weight of 1 - w_{risky} ), the standard deviation of the combination is w_{risky} \times \sigma_{risky} .

\sigma^2_p = w^2_{risky} \times \sigma^2_{risky} + w^2_{rf} \times \sigma^2_{rf} + 2 \times w_{rf} \times w_{risky} \times \rho_{rf,risky} \times \sigma_{rf} \times \sigma_{risky}

\sigma^2_p = w^2_{risky} \times \sigma^2_{risky} + w^2_{rf} \times (0)^2 + 2\times w_{rf}\times w_{risky}\times \rho_{rf,risky} \times (0)\times \sigma_{risky}

\sigma^2_p = w^2_{risky} \times\sigma^2_{risky}

\sigma_p = w_{risky} \times\sigma_{risky}