Treynor Ratio and Sharpe Ratio

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deriv108

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If portfolio A has a higher Sharpe Ratio but a lower Treynor ratio than portfolio B, then
1) Portfolio A has a higher systematic risk: true or false
2) Portfolio A is more diversified: true or false
3) Portfolio A has achieved a positive return for taking more risk: true or false
 
1. True, I made a hypothetical situation:
Portfolio A Sharpe: (15-10)/10 = .5
Portfolio B Sharpe: (12-10/10 = .2
Portfolio A Treynor: (15-10)/15 =.33
Portfolio B Treynor: (12-10)/5 = .4
Portfolio A needs to have more systematic risk in order for Treynor ratio to fall below B’s.
2. False, Portfolio A manager took on a massive amount of unsystematic risk. This could be wrong though, Schweser has some weird explanation on risk relative to total wealth in one of the practice exam books.
3. True, Portfolio A manager took on a massive amount of unsystematic risk to generate the higher return
 
treynor = rp-rf/beta
sharpe = rp-rf / stddev
A has higher sharpe
A has lower treynor
sharpe A > Sharpe B could happen bcos
=============================
1. (RpA > RpB and StdDevA <= StdDevB ) – A is more diversified.
OR
2. (RpA=RpB) and StdDevA < StdDevB ) – A is more diversified
OR
3. If RpA < RpB then StdDev A must be << StdDev B – again A is more diversified.
treynor A < Treynor B
================
1. RpA > RpB and BetaA > BetaB (more systematic risk for A)
or
2. RpA = RpB and Beta A > Beta B (more systematic risk for A)
or
3. RpA < RpB then beta A >> beta B …
so higher systematic risk for A.
more diversified
there is a particular scenario where A could achieve lower return than B.
(Not sure about the 3rd part of the question).
 
1. True
2. True
3. False
Since not much info is given besides a comparisons of returns ratio, I assumed Rp and Rf are the same for both portfolios.
Summary:-
B has lower systematic risk
A has lower total risk.
A has higher systematic risk + lower unsystematic risk (mostly diversified away).
Therefore 2 is True. And Q3 follow through with False given B has higher total risk (mostly unsystematic risk).
 
I made a new example, the old one seemed too sloppy:
RF Rate = 3
Portfolio A return = 25
Portfolio B return = 20
Portfolio A SD = 30
Portfolio B SD = 25
Portfolio A Sharpe = (25-3)/30 = .73
Portfolio B Sharpe = (20-3)/25 = .68
Portfolio A Beta = 1.7
Portfolio B Beta = 1.2
Portfolio A Treynor = (25-3)/1.7 = 12.94
Portfolio B Treynor = (20-3)/1.2 = 14.17
So, 1 is true (unless I’m assuming something in all these calcs that I shouldn’t be). If I lower Portfolio A’s beta to 1.2, the Treynor measure will actually be higher. The only way to force it down is to increase beta.
 
Great Question keep them coming:
If my logic is way off please correct me 12:10am >6 hours of sleep every day in the past 10
1) Portfolio A has a higher systematic risk: true or false
True
2) Portfolio A is more diversified: true or false
True
3) Portfolio A has achieved a positive return for taking more risk: true or false
False
I’m thinking the positive return is being generated by increasing Beta, if the Sharp is lower then they must have a lower std dev, going back to L2 on Treynor Black,
Var a= Beta^2 X Var of Mkt + Var ea
Port A and B both produce returns of 15, RFR= 10 (15-10= 5)
Portfolio A Sharpe: 5/10 = .5
Portfolio B Sharpe: 5/15 = .33333
Portfolio A Treynor: 5/1.5 =3.3333
Portfolio B Treynor: 5/1 = 5
 
1. false (it may be so, we do not know if higher systematic risk)
2. true (always need one ratio with beta and one with stdev, cannot be judged using just one ratio)
3. false (this sentence is unclear, more risk than what?)
 
example:
beta same
stdev B much higher than A
return B slightly higher than A
good night
 
V6, page 174
However, it is pos- sible for the Sharpe ratio and M2 to identify a manager as not skillful, although the ex post alpha and the Treynor measure come to the opposite conclusion. This outcome is most likely to occur in instances where the manager takes on a large amount of nonsystematic risk in the account relative to the account’s systematic risk.
this is B portfolio, high non-systematic, worse diversification, therefore 2. is True.
 
True lower Treynor – higher systematic risk
True – higher Sharpe , lower total risk , so more diversified
Depends – you are generally rewarded only for systematic risk (arguably) which is the case here( lower Treynor) , but the wording says “more risk” without clarification. risk could be more systematic or less systematic but much more unsystematic , hence question is not clear
 
true
true
cannot possibly be determined from the info given, thus false
 
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