FRM Fun 19: Sharpe ratio against volatility

[[email protected]]

Hi David,

I hope you are well.

On an interview I was asked what is the sensitivity of the Sharpe's ratio to the volatility,

The interviewer told it is almost zero...

Could you give a hint please?

Many thanks,
Indira

 

[David Harper CFA FRM]

Hi Indira,

Thanks, I hope you are well too. That is a provocative interview question, never heard it before.
Do you mind if we pose it as an FRM Fun question? i.e., I award you a star and will also to the best given answer (stars enter into our weekly drawing). The reason is, i think it's a clever application of an FRM Foundations assignment/concept (or even a T9. current), albeit a tough on-the-spot interview question!

P1 preferred. FRM Fun Question (for star credit). Either Indira's question above, or my paraphrase-restatement:

• Under what conditions is the Sharpe ratio insensitive (invariant) to [portfolio] volatility, if any? Is the interviewer correct, incorrect or merely imprecise?

 

[[email protected]]

Hi David,

Glad to hear your prompt feedback,

Sure, we can post it on the FRM Fun questions,

Thanks for the star from you,

Have a good week-end,
Indira

 

[ShaktiRathore]

from CAPM,
E(Rp)=Rf+β*(Rm-Rf)
Or E(Rp)-Rf= β*(Rm-Rf)
Or (E(Rp)-Rf)/σ(p)= β*(Rm-Rf)/ σ(p)
Or S.Rp.= β*(Rm-Rf)/ σ(p)
Or S.R.p= β*(Rm-Rf)/ σ(p)
Now since, σ(p)= β* σ(m)
implies S.Rp.= (Rm-Rf)/ σ(m)=S.R.m

Now measuring sensitivity of SR to portfolio volatility,
d(S.Rp)/d σ(p)= d//d σ(p)[ (Rm-Rf)/ σ(m)]
or (Rm-Rf)*d//d σ(p)[1/ σ(m)]+ (1/σ(m))*d//d σ(p)[ (Rm-Rf)]
or -(Rm-Rf)*(1/ β)*(1/ σ(m)^2)+ (1/σ(m))* dRm/d σ(p)
or –[(Rm-Rf)/ σ(m)]*(1/ β)*(1/ σ(m))+ (1/σ(m))* dRm/d σ(p)
or (1/σ(m))*[ –[(Rm-Rf)/ σ(m)]*(1/ β)+ dRm/d σ(p)]
or (1/ β *σ(m))*[ –[(Rm-Rf)/ σ(m)]+ dRm/d σ(m)]…E
for very small changes in volatility of Rm we assume linear realtion between Rm and volatility which implies,
Rm=Rf-k* σ(m) (ignoring higher order terms)
dRm/d σ(m)=-k
Rm-Rf=-k* σ(m)
Putting these in equation E,
d(S.Rp)/d σ(p)= (1/ β *σ(m))*[ –[-k* σ(m)/ σ(m)]+k]
d(S.Rp)/d σ(p)= (1/ β *σ(m))*[ –[-k]+k]=0 which is the required result..
Assuming market provides returns in linear proportion to volatility for small changes in volatility, x unit of return for 1 unit change in market volatility, this will provide sharpe ratio which is insensitive to volatility of portfolio. So assuming linear relation between market returns and market volatility and ignoring higher order terms for changes in volatility we find that sensitivity of Sharpe ratio is almost zero to changes in volatility.

thanks

 

[[email protected]]

Hi, ShaktiRathore,

Many thanks for this demonstration.

• Just a small note why do you define β=σ(p)/σ(m)?

I thought β=cov(p,m)/σ²(m)=ρ(p,m)σ(p)/σ(m). [ρ is correlation]

• Finally do you understand the economic sense of this conclusion?

Best Wishes,
Indira

 

[ShaktiRathore]

• According to single index model, the security volatility is given by,

σ(i)^2= β^2* σ(m)^2+e^2 where β^2* σ(m)^2 is systematic risk and e^2 is firm specific risk
now for a portfolio of n securities the firm specific risk diversifies away leaving ep=0=>ep^2=0
so we have. σ(p)^2= β^2* σ(m)^2+ep^2 =β^2* σ(m)^2
or σ(p)=β* σ(m) .....(please refer to any book for index model)

• your second question as what economic sense does this conclusion makes is nothing that investors demand returns for the risk they undertake according to risk aversion theory. So greater risk they are willing to take for greater return. Lower risk for lower return like investing in T-bills provides low risk and lower return while investing in securities has high risk and higher returns.So in a portfolio SR= (E(Rp)-Rf)/σ(p) so if σ(p) changes by small +d σ(p) than the E(Rp) also increases(why as cited above) by same almost same percentage thereby making overall SR insensitive to change in σ(p) and if σ(p) changes by small -d σ(p) than the E(Rp) also decreases by same almost same percentage. This makes the overall sharpe ratio approx. the same and insensitive to small changes in volatility of portfolio. So finally the economic rational is that investors are risk averse and expect more returns for taking more risks or that securities the more risk the more returns they provide.

thanks

 

[David Harper CFA FRM]

As Shakti shows (nice! ) in his first few lines, the Sharpe ratio of a (perfectly) diversified portfolio is equal to the Sharpe ratio of the market portfolio. This is a key feature of the capital market line (CML): unlike the SML, all points (portfolios) on the CML, which includes the Market Portfolio, are efficient with identical Sharpe ratios which correspond to the slope of the CML:

So, I would re-phrase the original statement into something like the following:

• If the portfolio remains efficient (on the CML), then its expected (ex ante) Sharpe ratio is already optimal and will remain invariant to changes in volatility.
  I add the expected (ex ante) to omit alpha and remind its an expectation rather than a realized Sharpe. I personally think the interviewer's initial assertion appears to be incorrect because we tend to think not in expected theoretical, but realized terms.

I think a way to look at this is with the CAPM:

• If E[excess portfolio return] = beta (p, M)*ERP, where beta(p,M) = correlation(p,M)*volatility(p)/volatility(M), then:
• E[excess portfolio return] = correlation(p,M)*volatility(p)/volatility(M)*ERP; but the (perfectly) diversified portfolio will maintain a correlation of 1.0 (an interesting property of correlation; it holds up even in the leverage), such that consistent with Shakti's final analysis:
• E[excess portfolio return | rho = 1.0] = 1.0 rho*volatility(p)/volatility(M)*ERP; i.e., E[return | diversified portfolio] is a linear function of volatility (p)