When Portfolio Up and Down Betas Can Be Aggregated
Summary
The document explains when a portfolio’s conditional beta can be calculated from the conditional betas of its holdings. For ordinary beta, covariance is linear in portfolio returns, so the portfolio beta equals the sum of each asset’s beta multiplied by its portfolio weight. The same reasoning applies to upside or downside beta when the conditioning event is defined by whether the market return is above or below a threshold: the portfolio and each asset use the same subset of observations and the same conditional market variance.
The aggregation rule does not apply directly when the subset is defined by whether the portfolio return itself is above or below a threshold. In that case, the asset-level covariances must be calculated using observations selected by the portfolio’s condition. The explanation is algebraic and assumes fixed portfolio weights; it does not discuss estimation error, changing weights, or empirical performance.
Key ideas
- Ordinary portfolio beta is the weighted sum of the component asset betas when weights are fixed.
- Market-conditioned upside beta aggregates by weights when each asset and the portfolio use the same market-return condition.
- The same aggregation logic applies to downside beta conditioned on market returns.
- Betas conditioned on the portfolio’s own return cannot generally be assembled from independently conditioned asset betas.
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Full text
# Can you still sum the weighted up betas to find portfolio up beta, or not?
# Can you still sum the weighted up betas to find portfolio up beta, or not?
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.
## Answer by Kermittfrog (score 2, accepted)
https://quant.stackexchange.com/a/68636
Below, I describe three cases:
- The standard $$\beta=Cov(r_p,r_m)/Var(r_m)$$
- The case of a (up-)sided beta with arbitrary market return threshold $\theta$, $$\beta^+_m+(\theta)= Cov(r_p,r_m|r_m>\theta)/Var(r_m|r_m>\theta)$$
- 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,
$$ \beta_p^+(\theta)=\sum_i w_i Cov(r_ir_m|r_p>\theta)/Var(r_m|r_p>\theta) $$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.