# returns-based style analysis (RBSA)

Hi,

i am confused by the item “Equity Portfolio Management - Sonera”

The betas given do not sum to 1 , whereas one can read in the curriculum :

“Returns-based style analysis involves a constraint that the coefficients or betas on the indices are nonnegative and sum to 1. That constraint permits us to interpret a beta as the portfolio’s proportional exposure to the particular style (or asset class) represented by the index”

I am missing something?

If the coefficients (betas) sum up to 1 - e.g. 0.2, 0.3, 0.5 - then you can say 20%, 30% and 50% of the returns are attributable to the 3 components respectively. if they do not sum up to 1 – then you cannot make such a proportional prediction. (is what I think that means).

ok, but if the coefficients are :

Small-cap value index, beta =1.05

Small-cap growth index, beta =0.48

Large-cap value index, beta =0.96

Large-cap growth index, beta =0.37

Does it make sense to say that for this manager , the “growth index betas” sum to 0.37+0.48 = 0.85 over a total of 0.37+0.48 +0.96+1.05=2.86 , hence 0.85/2.86 = 30% of the returns are attributable to “growth” components ?

I think that is precisely what they are saying IS not right. What contributed more to the 30% was it the higher proportion of Small Cap Growth (31%) vs. 27% of the Large Cap growth? Or was it something else? Is the real value add from the Growht portion really 30% is also contestable.

Thanks for the answer but i still have the feeling of not having understood the conclusions that can (or can’t) be drawn from the coefficients when they do not sum to 1.

In the actual question , the information given for manager B is :

Small-cap value index, beta =0.98

Small-cap growth index, beta =0.43

Large-cap value index, beta =1.1

Large-cap growth index, beta =0.39

The correction then states: “Manager B’s investment style is consistent with a value investment style, with a higher beta for the two value indices—the small-cap value index and the large-cap value index”

It looks like they translate higher beta into the conclusion than more returns are attributable to Value styles , even though the coefficients do not sum to 1.

Do you agree with the conclusion given in the answer?

On a side note, i do not understand the 31% figure and 27% figure in your last post, can you please elaborate on those numbers?

31% = 0.48 / (0.48+1.05)

likewise for 27%… (0.37/(0.37+0.96))

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I do not know if the beta provided in the problem is the average beta of the stocks in that index, or if it is the “coefficient” after regression of the “index” used.

I also do not have access to the Sonera problem you are referring to, since that is only available to current candidates.

In all problems shown in the text - the sum of the coefficients of the indexes after regression for the RBSA sum up to 1.

Unfortunately, “beta” is used to mean too many things in this context.

There are the betas that you’ve cited above: how fast the returns of each style change compared to the changes in returns on that style’s index.

Those are not the betas that sum to one.

There are betas that represent the coefficients on each style in explaining the returns on the portfolio: this portfolio’s returns are closest to 20% large-cap value plus 10% 10-year T-Bonds plus 30% small-cap growth plus 40% mid-cap value.

These betas sum to one.

Ok thanks to the two of you.

I misinterprated the two different kinds of beta

My pleasure.

Hi,

I am a little confuse that based on CFAI

“…(the factor weights are also known as styles weights or Sharpe style weights). We expect the portfolio to move 0.75 times whatever happens to large-cap stocks (holding everything else constant) and 0.25 times whatever happens to small-cap stocks (holding everything else constant)”

It sound the same as the Beta you cited " how fast the returns of each style change compared to the changes in returns on that style’s index."

Why " this portfolio’s returns are closest to 20% large-cap value plus 10% 10-year T-Bonds plus 30% small-cap growth plus 40% mid-cap value."?