Portfolio Construction

[ckyeh]

Dear David:

In bionic turtle's tutorial 2010-8-a-Investment Page 5, you mentioned this equation:
E{Rn} = 1+(iF) + βn• μB+ βn• ΔfB + αn
Could you explained the meaning of “ 1 “ ? Why we need to put “ 1 ” into this equation?

And one more question, in the Grinold’s book “ Active portfolio management” chapter 2 page 14, the regression is

Rp(t) = αp +βp• rM(t)+ εp(t) (2.2)

Then Grinold break the excess return on that portfolio into a market component and a residual component

Rp= βp• rM+ Θp (2.3)

Where is the Alpha“α”? Why Alpha“α” disappear in equation 2.3?
Thanks for your help!!!


CKyeh

 

[David Harper CFA FRM]

Hi CKyeh,

The first is a keen observation. I am certain i have bored certain customers with the relevance of compound frequency, and here it is again! ... the reason is simply that Grinold et al (you can see in the Appendix) have defined returns as SIMPLE RETURNS; e.g., $10 growing to $11 in Grinold is given as 11/10 = 1.1. Rather than, say Hull's use of continuous which would give LN(11/10) = 9.53%
... I think it's a nice trick b/c without the "1+" we might assume or not know, but the "1+" is a reminder that we are referring to simple returns!
... and here is something else: it is natural for them to use simple returns in this book, because log returns are time additive but do not add across the portfolio. However, simple returns (while not time additive "horizontally") do add "vertically" so, it seems to me, this is more natural when the task is attribution

The second is the difference between ex ante (what we expect beforehand) and ex post (the realized truth which will not lie on a perfect line)

In equation (2.2), alpha is the "realized" alpha; i.e., after the returns are regressed against the common factor (in this case, the excesss market return) alpha is the regression intercept: whatever cannot be explained by factor exposure

Equation (2.3) is an ex ante EXPECTED RETURN, where under equilibrium, the E[alpha] is zero. To have a non-zero alpha here would violate the assumption that return can only be a function of beta exposure to the factor.
... a maybe deeper way to think about this: if ex ante the E[alpha] were non-zero, then you could get excess return (above Rf) but without any risk (factor exposure), so you would arbitrage that away

BTW, this ex ante/ex post pair is analogous to equations that took me a while to understand in Ch 7 APT:

APT Chapter 7, Eq 7.1 has a specific return term (u)
but Eq 7.2 omits the specific return (u)

and it is the same idea here in APT (which after all, generalizes from CAPM)

Eq 7.2: ex ante (expected) APT excess return = exposure1 * factor_2 + exposure_2 *factor_2 + ... exposure_n * factor_n
i.e., we don't expect an intercept, or if we do, we figure the average error is zero anyhow

Eq 7.3: an ex post (realized) "after-the regression" APT where
excess return = exposure1 * factor_2 + exposure_2 *factor_2 + ... exposure_n * factor_n PLUS (+) the specific return

(... but it's only an analogy, right? the alpha in CAPM connotes luck/skill, something manager-specific, whereas the u in APT is instrument specific. But, still they are both regression intercepts)

Hope that helps, great points you have raised!

David

 

[ckyeh]

Dear David:

Thanks !!!
It helps a lot !
I appreciated it.

CKyeh