Increased unsystematic risk lowers both the Sharpe and Treynor ratios

A

archived_user

New member
Joined
Jun 18, 2026
Messages
0
Reaction score
0
The statement included in CFAI volume 6, page 174 - “…increased nonsystematic risk will lower the Sharpe ratio but leave the Treynor measure unaffected” seems to be wrong.
Just take a look at how these measures are computed.
Beta = (std dev portfolio return/std dev market return)*correlation of portfolio return, market return.
Sharpe ratio = (return on portfolio - risk free return)/std dev portfolio return.
Treynor ratio = (return on portfolio - risk free return)/beta portfolio
Thus, it is certainly true that increased nonsystematic risk will increase total risk, which is measured by the standard deviation of the portfolio. The Sharpe ratio will now be lower, as stated in the CFAI text.
But this increase in standard deviation will also be reflected in a higher beta measure, since a higher standard deviation will be used in the numerator to compute beta, thus raising the value of the numerator while keeping the denominator unchanged.
And this higher beta will necessarily mean a lower Treynor measure.
Any comments?
 
hmmmm naw G
they right son
you are correct with your formula but the assumption is that
nonsystematic risk is not rewarded by higher returns so your numerator is unchanged and the beta is the same
 
Std dev of returns = Beta + Non-systematic risk
Increased non-systematic risk ===> Increased Std dev
===> No effect on beta
So higher nonsystematic risk increases the denominator of the Sharpe Ratio, which decreases the actual value of the Sharpe ratio, but has no effect on the Treynor ratio
 
Sorrry, but portfolio beta does not stay the same when the Std dev of the portfolio return increases.
Again, check out the formula for beta. It is the ratio of the Std dev of portfolio return to the Std dev of the market return, multiplied by the correlation between the portfolio return and the market return.
So, when the Std Dev of the portfolio increases, ceteris paribus, beta must increase.
And the numerator in the Treynor ratio stays the same, but its denominator - the beta - increases. So, the Treynor measure goes down.
 
Dude I think you’re misunderstanding the concept…standard deviation measures total risk, which is the sum of systematic and nonsystematic risk. The two components drive total risk, not the other way around.
 
I’m just saying that the beta coefficient is a function of 3 different inputs, one of them being the standard deviation (the total risk) of the asset or portfolio.
The other two inputs to beta are the standard deviation of the market index and the correlation of the portfolio or asset with the market index.
If a portfolio increases its unsystematic risk and thus its total risk (measured by Std deviation) then its beta will also increase - unless this is offset by a decrease in the correlation of the portfolio with the market index.
All this is implied by the definition of beta in footnote 5, page 229, of CFAI volume 6.
 
take it for what its worth for the next 13 days. after that get a doctorate in this…
 
Jose
You misunderstood the whole concept of systematic vs. unsystematic risk.
When unsystematic (company-specific) risk increases, it does not affect beta which concerns only systematic risk. That a company’s stock is volatile because of incompetent management does not affect its beta, only its stddev.
This is level I stuff.
 
elcfa,
Standard deviation is a measure of total risk (systematic and unsystematic risk).
Beta of portfolio is equal to (portfolio standard deviation)*(correlation of portfolio return and market return/market standard deviation).
Thus the total risk of the portfolio (its standard deviation) affects its beta.
QED
 
You’re right Jose but for the exam in 13 days you’d be better off putting down the “incorrect” CFAI answer if you get a question on this.
 
When unsystematic risk increases, the correlation between portf return and mkt return will decreases. Think about it: if the stock varies more independent of market (thus unsystematic risk increases), its correlation with market can’t stay the same. It does not make sense. The correlation decreases
It will result in beta remaining constant.
All of this is level I stuff.
 
“When unsystematic risk increases, the correlation between portf return and mkt return will decrease…”.
Not true.
You cannot conclude anything about a portf’s correlation with the mkt returns from the increase in that portf’s std dev. They’re different concepts.
Correlation depends on the direction of returns. If the portf’s volatility increases by rising more as the market rises and dropping more as the market plummets, then its correlation with the market will increase.
So correlation may increase, decrease or stay the same, depending on the circumstances.
Anyway, I agree that “putting down the “incorrect” CFAI answer if you get a question on this…” is the best option on exam day!
 
Jose G. Wrote:
——————————————————-
> “When unsystematic risk increases, the correlation
> between portf return and mkt return will
> decrease…”.
>
> Not true.
>
>
> So correlation may increase, decrease or stay the
> same, depending on the circumstances.
>
Sight!! It is amazing to see that one still has this kind of arguments at level III forum !!!
Good luck with your exam, anyway.
 
If unsystematic risk were to increase, then you are right the s.d. will increase, BUT the correlation between the asset and the market will decrease, which results in the beta remaining unchanged. However, since the sharpe ratio depends only on the s.d. of the asset, then the sharpe will decrease while the treynor will remain the same.
Think about what unsystematic risk is, it’s the portion of the deviations UNEXPLAINED by the the market. So if market goes up by 5%, the asset goes down by 10%, that is unsystematic risk since the downwards move certainly isn’t due to the market. The market goes up by 5%, the asset goes up by 10%, that’s systematic risk (the asset moves with the market but moves more than the market). Therefore, the higher the unsystematic risk, the lower the correlations between an asset’s returns and and the the markets returns.
Recall, again that the question talks about unsystematic risk… Not total risk…
Trust me, it works. You are making Markowitz turn in his (future) grave.
 
Correlation decreases because of the commensurate increase in unsystematic risk, which the market portfolio has none of. Thus BETA stays the same.
 
“When unsystematic risk increases, the correlation between portf return and mkt return will decreases”.
This is the key question. Does correlation necessarily decrease - and do it in the precise amount necessary to keep beta constant?.
Let’s plug some figures to try to see the conditions under which an increase in portfolio std dev would not impact beta.
Assumptions:
covariance portfolio, market = 50
Std dev portfolio = 10%
Std dev market = 10%
correlation portfolio, market = covariance portf, mkt/(std dev portf*std dev mkt) = 50/100 = .5
Beta of portfolio= correlation portf, mkt/(std dev portf*std dev mkt) = .5*(10/10) = .5
Now let’s suppose that, because of unsystematic risk, the portfolio std dev increases to 20%.
What happens to beta?
It depends. If the mkt std dev AND the covariance don´t change, then beta will stay the same:
correlation portf, mkt/(std dev portf*std dev mkt) = 50/200 = .25
and
Beta of portfolio = .25*(20/10) = .5
But what if covariance does change? Let’s say it decreases because now “…the stock varies more independent of market (thus unsystematic risk increases)…”?
In that case beta decreases also. If covariance goes down to 40, beta will now be .4.
And if - for some reason not foreseen in the CAPM theory - covariance increases to say 60, then beta will be .6.
So the effect on beta hinges on what will happen to the covariance of the portfolio return to the market return, when portfolio standard deviation rises because of increased unsystematic risk.
 
dude if you really feel you are onto something send in a paper to the financial journal AFTER THE EXAM
in the meantime take the CFAI and our word for it
increased nonsystematic risk will lower the Sharpe ratio but leave the Treynor measure unaffected
 
You are thinking about it wrong
Given the SAME level of covariance but now a higher measure of standard deviation, what can you conclude about the nature of risk in an asset?
Higher risk, but same amount of beta
 
You must log in or register to reply here.
Back
Top