Treynor vs sharpe

[sumeetg]

Hi​

​

Just came across this question​

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The treynor and sharpe ratios will :​

a) give identical rankings when the assets have identical correlations with the market.​

b) give identical rankings when the assets have identical standard deviations.​

c) give identical rankings when the same minimum acceptable return is chosen for the calculations.​

d) always give identical rankings.​

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According to me , none of the condition in itself is complete. but the answer is (a).​

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Can someone please explain this?​

 

[ShaktiRathore]

hi understand it like this i hope this is the logic you would appreciate ans would see how its true,
betas of assets are,
betai=Cov(i,m)/sigmam^2
betaj=Cov(j,m)/sigmam^2
=>Cov(i,m)/Cov(j,m)=betai/betaj ---1
identical correlation with the market means
corr(i,m)=Cov(i,m)/sigmai*sigmam
corr(j,m)=Cov(j,m)/sigmaj*sigmam
=>Cov(i,m)/sigmai=Cov(j,m)/sigmaj
Cov(i,m)/Cov(j,m)=sigmai/sigmaj ...2
from 1 and 2,
betai/betaj =sigmai/sigmaj
trynor of i/trynor of j=betaj/betai
sharpe of i/sharpeof j= sigmaj/sigmai
from above its clear that trynor of i/trynor of j=sharpe of i/sharpeof j thus the if trynor of i>trynor of j=>sharpe of i> sharpeof j and vise versa thus treynor and sharpe ratio will produce identical rankings.
option b there is no standard deviation in trynor so no question of these
option c no Mar and same risk free rate is used in both the measures so it would not affct ranknings
thanks

 

[sumeetg]

Thanks Shakti...It is clear now...I had misinterpreted the question.

 

[David Harper CFA FRM]

ShaktiRathore interesting proof! Doesn't your proof also show that, technically, for a correlation < zero, the assertion is not mathematically true (if "ranking" extends into the negative Treynor outcomes!) because the beta can then be negative (while the sigmas cannot) such that, with negative correlation, the Treynor will be negative; i.e., a high-ranking Sharpe ratio equates to a low-ranking Treynor. For example, assume correlation = -0.30, Rf = 1.0%, market vol = 20%. If portfolio return = 11% with port vol = 10%, Sharpe = 1.0 and Treynor = -0.667 (which in my small dataset gives a high rank for Sharpe but a low "rank" for Treynor).

I admit it's a sort of a cheap shot (as we tend to presume positive asset betas while negative betas arise from short positions), but this begs the question: how should negative Treynor be treated in a ranking; e.g., is absolute value of Treynor appropriate?

 

[k.simpson]

That's an interesting point David. Taking a five-minute break from question solving, perhaps indulge some thought-experimenting it may be useful discussion (post-exam?) for someone.

Suppose we had the following setup for two instruments (hypothetical numbers which hopefully are internally consistent):

sigma_1 = sigma_2 = .3
sigma_m = .2
rho_1 = -rho_2 = 1
both give same expected returns above r_f of 3%

Then T1 = 0.02 and T2 = -0.02. Using the absolute value argument that David presents, they are equally attractive to add to a diversified portfolio (which is when we'd use Treynor to evaluate). They also have the same Sharpe ranking. But, if there was an instrument like #2 that give diversification benefits (-ve relationship to a portfolio that has positive correlation to the market), wouldn't people buy it up, which may reduce its returns and lower its attractiveness relative to instrument 1 in a risk-reward tradeoff?

I feel (i.e. no mathematical/empirical proof at the moment ) like a Treynor ratio caused by a negative beta and positive excess return that is at top of the rankings would not be "stable" in the long run. Nice, but not something to count on, if markets react rationally. Does that make sense to anyone?

Not sure how the argument would work for portfolios though. Who can consistently play against the markets and win? But my five-minute experiment is over!

 

[ShaktiRathore]

Hi,
Yes #2 will definitely lower the returns due to negative beta but due to diversification its also lowering the risk of the portfolio so that determining reward to risk is ambiguous. In terms of reward to risk the attractiveness of the instrument can be different sometimes more attractive and sometimes less but saying that Treynor ratio would not be stable in long run on the basis of returns alone is not feasible in fact what investors yearns for are an optimal reward to risk and not just return. The negative beta portfolio might survive in the long run but who knows!!
Davids Question of whether that if we can use absolute value for negative Treynor, because if for P1: beta=-1 and sigma=beta^2*sigma(mkt)^2+sigma(e)^2, sigma1=-1^2*.1^2+.01^2=.01+.0001=.0101 and for P2:beta=1 and sigma2=1^2*.1^2+.01^2=.0101 also and compare the two they are identical in Sharpe and treynor when compared to a benchmark when absolute treynor value is taken for P1 but not when negative treynor value is taken so i think its more appropriate to take absolute value of treynor. Because beta whether negative or positive shall yield the same stdDev and thus the same sharpe ratios for the two portfolios with opposite sign treynor ratios, because the numerator is identical in both the ratios.
And the question above uses assumption of identical correlation that is of the same magnitude and signs so that signs of betas cancels and thus we get identical rankings for trynor and sharpe. And i think we would not consider opposite signs of betas.
thanks

 

[southeuro]

I DO like Shakti's proof, but I don't understand this conceptually.

if sharpe's denominator includes both beta and idiosyncratic risk factors, per choice a if assets have identical correlations to the market (so let's say beta=1 for the assets) then how can both sharpe's and treynor's denominators be the same? Treynor's will be 1, and sharpe's will be x + 1 where x is the idiosyncratic risk. To me, ONLY if eliminate the idiosynratic risk (highly doubtful) can we both have same ratios. Where am I thinking wrongly?

thanks

 

[David Harper CFA FRM]

Hi @southeuro They won't be the same, that's clearly true; e.g., if Rf = 4.0%, sigma(M) = 10%, sigma(p) = 20%, corr(M,p) = 0.30, and portfolio return is 10%, the portfolio excess returns = 10% - 4% = 6.0%, beta = 0.30 *20%/10% = 0.60 and Treynor = 0.06 /0.60 = 0.10 and Sharpe = 0.06 / 0.20 = 0.30.

In fact, it's interesting, they are never the same except when the denominator is the same (as the numerators are already the same) such that:

• beta(i,M) = sigma(i), or
• corr(i,M)*sigma(i)/sigma(M) = sigma(i), which requires:
• corr(i,M)/sigma(M) = 1.0 or corr(i,M)=sigma(M); so that's the special case when Treynor = Sharpe

for example, to revise the above example, if we reduce the correlation, corr(M,p) to 0.10 to match the market volatility of 10%, then both Treynor and Sharpe = 0.30. Then, if we vary only portfolio volatility, they will remain equal but change!

But that's not the question nor @ShaktiRathore 's proof, the question says they will give identical rankings, which appears to be true in the non-negative cases (and Shakti's proof only tries to show rank). Thanks,

 

[ShaktiRathore]

For negative cases this might also hold true.My logic is:
trynor of i/trynor of j=betaj/betai as both betas are negative say -a and -b a,b>0
trynor of i/trynor of j=-a/-b=> trynor of i<trynor of j as a>b
sharpe of i/sharpeof j= sigmaj/sigmai= betaj/betai= -a/-b=> sharpe of i<sharpeof j
As DavidHarper pointed out sigma cant be negative but negative Sharpe ratio is possible.So we can write sharpe of i/sharpeof=-a/-b such that above relation holds if not sigmaj/sigmai=-a/-b.
Thanks

 

[Delo]

From the formula perspective it is easy to learn and see the difference between Treynor vs sharpe. But I am still struggling to warp my head around conceptual interpretation of beta and std dev. for portfolio.

Imagine Beta and Std like a 2X2 matrix.
- When does (in which market condition) is Beta high and std dev low for a portfolio?
- Like wise for other combinations.

Appreciate any help !

 

[ShaktiRathore]

Hi
You can interpret relation b/w beta b and std dev sd of potfolio,
Let d be sd of mkt and r be correlation b/w mkt and portfolio,
b=r*(sd/d) so your 4*4 matrix leds to 4 cases
1)b high and sd low, this can happen when r is high which indicates when mkt and portfolio follow almost same returns provided there is not much change in d i.e. Mkt std dev. This case do indicate adverse mkt conditions when correlation r gpes high,also when pprtfolio closely resembled the index.
2)b is high and sd is high, this seems logical from above relation as sd increase b also increase provifed other params d and r do not change much as under normal mkt cinditions
3)b is low and sd is low ,this also seems ligical as sd and b are in direct correlation so high sd indicattes high b under normal mkt conditions except when mkts are highly volatile
4)b is low and sd is high, when r is very low or d is high this can happen during volatile mkts and when portfolio corr r is very low with the mkt e.g. portfolii is> 90% invested in safe instruments
Coclusion of above analysis:1 and 4 are possile in volatile mkts while 2 and 3 arepossible in normal markets.
Thanks