How does Si > Sp * Cor(i,p) really hold?

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lemonkit

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The text says.. to determine whether adding a new asset i to a current portfolio p would be benefical or not, we can use this relationship.
RULE 1: If Si > Sp * Cor(i,p), then the new asset i has benefit to be added into the portfolio p. The Sharpe Ratio of final portfolio will INCREASE.
RULE 2: If Si = Sp * Cor(i,p), then the new asset i provides no benefit and no harm to be added to the portfolio p, provided Cor(i,p) = Si/Sp, because adding the new asset i will leave the final portfolio sharpe ratio UNCHANGED.
RULE 3: If Si < Sp * Cor(i,p), then the new asset i will harm or DECREASE the sharpe ratio of the portfolio p, if it is added.
where Si = sharpe ratio of asset i, Sp is sharpe ratio of current portfolio p, Cor (i,p) is return correlation between asset i and portfolio p.
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OK, let me put it in practice to reconcile this relationship. Assume Risk Free Rate=3%
Asset i: RETURN=12%, SD=35%, then Sharpe Ratio i=0.2571
Current portfolio: RETURN=10%, SD=25%, then Sharpe Ratio p=0.2800
Assume Weight i=50%, Weight p=50%
According to RULE 2, if my Cor(i,p)=0.2571/0.2800=0.9184, then I expect that after I add asset i to the current portfolio, the final portfolio’s sharpe ratio remain UNCHANGED, at least not worsen than before.
Then, I calculated the final portfolio return and SD after the mix with asset i, as:-
R(final portfolio) = 50%x12%+50%x10%=11%
SD(final portfolio) = 50%^2*35%^2+50%^2*25%^2 + 2*50%*50%*(0.9184)*35%*25%=29.40%
Sharpe Ratio(final portfolio) = (11%-3%)/29.40%=0.2721
So, my question is why is the Sharpe Ratio of final portfolio=0.2721 which however is not same as Sharpe Ratio p= 0.2800, as contraditary to what was mentioned by RULE 2 provided Cor(i,p)=0.9184?
Thanks for help in advance.
 
I confirmed your #’s - frustrating. Does not look like this formula ties into the standard calc of portfolio variance.
Seems like it is more a rule of thumb that doesn’t work at the extremes.
 
I confirmed your #’s - frustrating. Does not look like this formula ties into the standard calc of portfolio variance.
Seems like it is more a rule of thumb that doesn’t work at the extremes.
 
I confirmed your #’s - frustrating. Does not look like this formula ties into the standard calc of portfolio variance.
Seems like it is more a rule of thumb that doesn’t work at the extremes.
 
Huh, don’t know why I got a triple post….
I retract the above. The rule oly tells you if a Sharpe improvement is possible. It doesn’t tell you at what weights. Your assumption of a 50/50 split is what gives you the lower Sharpe ratio. If you try a 25/75 split, you will get a higher Sharpe ratio.
 
Hi ftwcfa, thanks for your reply first.
To this topic, what you meant is that the RULE only means the the reduction of sharpe rate is POSSIBLE, but not guaranteed?
Of course I also felt something strange when I looked at this relationship. This is because i knew that the sharpe ratio of a portfolio is also subject to different combinations of weights of its underlying assets. That’s why it triggered me to put it into maths work for reconcilations.
So, I cannot convince myself in the future to use this relationship to judge whether adding a new asset is appropriate or not, because this relationship cannot guarantee 100% correct but only meant POSSIBLE.
Anyway, for the examination only, this is OK that I would force myself to memorise!!!
Thanks and your view is very reasonable.
 
If the rule holds, it says that at some weight, an increase in Sharpe will happen if you add the asset. This does not mean that at all weights, an increase in Sharpe will happen.
If the rule says an increase in Sharpe is not possible, it means that there is not weight at which you could add the asset class where there would be an increase in the Sharpe.
Hope that clarifies. Try changing the 50/50 assumption in your calculations to 25/75 and then 75/25. You may see it then.
 
good explanation ftwcfa… this formula is more insightful than it seems at first glance…
 
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