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The portfolio beta in the conventional sense is simply the sum of weighted beta coefficients for each holding in the portfolio.

Is it the same for portfolio up and down beta, where I can simply take the weighted up betas of each holding and sum them up to find the portfolio up beta? And I can do the same to find portfolio down beta? I’m unsure if there’s something stopping me from doing this. enter image description here

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    $\begingroup$ Run two linear regressions, one on those days where the market is up, and one where it is down. ("Up" and "Down" may be defined relative to some $\theta$ not necessarily zero.) $\endgroup$
    – shabbychef
    Nov 3, 2021 at 5:02
  • $\begingroup$ Maybe I wrote my question in a confusing way but I know how to compute upside beta. I’ve revised my question above. $\endgroup$ Nov 3, 2021 at 14:45
  • $\begingroup$ Ah, I see. Yes, when there is a single market used to determine the betas, you should just take the weighted average of the up-betas and that of the down-betas to get your portfolio up-beta and down-beta. $\endgroup$
    – shabbychef
    Nov 3, 2021 at 18:35

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Below, I describe three cases:

  1. The standard $$\beta=Cov(r_p,r_m)/Var(r_m)$$
  2. The case of a (up-)sided beta with arbitrary market return threshold $\theta$, $$\beta^+_m+(\theta)= Cov(r_p,r_m|r_m>\theta)/Var(r_m|r_m>\theta)$$
  3. The case where we condition on your portfolio instead of the market, $$\beta^+_p(\theta)=Cov(r_p,r_m|r_p>\theta)/Var(r_m|r_p>\theta)$$

The standard case:

Assume a portfolio of $n$ assets with weights $w_1+\ldots+w_n=1$. We collect the weights into vector $w$ and the individual asset returns into vector $r$, i.e. $r_p=w^Tr$. Each asset has $\beta_i=Cov(r_i,r_m)/Var(r_m)$, and we collect all betas into vector $b$. Given the definition of $\beta$, the beta of your portfolio $$ \begin{align} \beta&\equiv\frac{Cov(r_p,r_m)}{Var(r_m)}\\ &=\frac{Cov(w^Tr,r_m)}{Var(r_m)}\\ &=\frac{Cov(w_1r_1+\ldots w_nr_n,r_m)}{Var(r_m)}\\ &=\frac{w_1Cov(r_1,r_m)+\ldots+w_nCov(r_n,r_m)}{Var(r_m)}\\ &=w_1\beta_1+\ldots+w_n\beta_n\\ &=w^Tb \end{align} $$ and thus, as you have written, $\beta_p=\sum_i w_i\beta_i$.

Upside beta conditioned on the market

Let us rewrite the covariance as $$ \begin{align} \beta_m^+(\theta)&\equiv \frac{Cov(r_p,r_m|r_m>\theta)}{Var(r_m|r_m>\theta)}\\ &= \frac{E((r_p-E(r_p|r_m>\theta))(r_m-E(r_m|r_m>\theta))|r_m>\theta)}{Var(r_m|r_m>\theta)}\\ &=\frac{E(r_pr_m|r_m>\theta)-E(r_p|r_m>\theta)E(r_m|r_m>\theta)}{Var(r_m|r_m>\theta)}\\ &=\frac{E((w_1r_1+\ldots+w_nr_n)r_m|r_m>\theta)-E((w_1r_1+\ldots+w_nr_n)|r_m>\theta)E(r_m|r_m>\theta)}{Var(r_m|r_m>\theta)}\\ &=\frac{\sum_i w_iE(r_ir_m|r_m>\theta)-\sum_i w_iE(r_i|r_m>\theta)E(r_m|r_m>\theta)}{Var(r_m|r_m>\theta)}\\ &=\sum w_i\frac{E(r_ir_m|r_m>\theta)-E(r_i|r_m>\theta)E(r_m|r_m>\theta)}{Var(r_m|r_m>\theta)}\\ &=\sum_i w_i \beta_i^+(\theta) \end{align} $$ .. as you have guessed. Same holds for the downisde beta. Intuitively, you simply "split" your dataset into two sections: One where the market is below its time series average return, and one where it is above. From there, things are additive again.

Case 3: Conditioning on $r_p$

For $\beta_p^+(\theta)$, the single asset sided betas cannot be used (you cannot aggregate $\beta^+_i(\theta)$, but you can still condition on your portfolio being above or below a certain threshold,

$$ \beta_p^+(\theta)=\sum_i w_i Cov(r_ir_m|r_p>\theta)/Var(r_m|r_p>\theta) $$

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  • $\begingroup$ Focusing on up beta, can I still think of up beta as the sum of weighted betas if I use theta equal to the mean excess market return as is done in the definition above? To be clearer, I have a three stock portfolio with 1/3 in each stock. I want to know if up beta can be computed as simply the weighted sum of each up beta for each stock, and then I can see how up beta changes given different weights. $\endgroup$ Nov 3, 2021 at 15:34
  • $\begingroup$ So effectively I’m asking if your answer holds for theta equal to not only zero, as I’m your answer, but also for theta equal to other constants such as mean market excess return (as in the definition above). $\endgroup$ Nov 3, 2021 at 15:44
  • $\begingroup$ I updated the answer, hope that helps? $\endgroup$ Nov 4, 2021 at 10:24
  • $\begingroup$ Great answer thank you $\endgroup$ Nov 4, 2021 at 20:38

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