> [!tldr] Effective Number of Stocks > Suppose we have a portfolio of $n$ stocks. If we hold all of them in equal size and they have equal volatility, then we surely have a diversified portfolio with $n$ stocks. > > But if 90% of the GMV is in one stock, then our portfolio will behave like that stock, offering little diversification. > > The **effective number of stocks** provides a quantified measure of diversification. - Note that "diversification" means different things for different investment strategies, so the definition of the effective number also changes. Let the portfolio have weights $\mathbf{w}$, (non-idiosyncratic) returns $\mathbf{r}$, and suppose $\mathbf{r}$ follows some family with location and scale parameters (i.e. idio volatilities) $\pmb{\mu}$ and matrix $\Sigma$. ### Directional Betting on Drift Suppose we want to bet on the drift $\mu$, then the portfolio has variance $\mathrm{Var}=\mathbf{w}^{T}\Sigma \mathbf{w}$. In an idealized world with an equal-weighed portfolio with the same total market cap (WLOG assume $\| \mathbf{w} \|_{1}=1$), and independent returns that have homoscedastic variance $\bar{\sigma}^{2}:= \frac{1}{n}\sum_{i}\Sigma_{ii}$, the variance would have been $\widetilde{\mathrm{Var}}= \frac{\mathbf{1}^{T}\tilde{\Sigma} \mathbf{1}}{\| w \|_{1}^{2} } =\frac{\bar{\sigma}^{2}}{n},$where $\tilde{\Sigma}:=\mathrm{diag}(\bar{\sigma},\dots \bar{\sigma})$ and $\mathbf{1}=(1,\dots,1)^{T}$. Therefore, we can define the effective diversification as the ratio of the two: $\frac{\widetilde{\mathrm{Var}}}{\mathrm{Var}}=\frac{\bar{\sigma}^{2} / n}{\mathbf{w}^{T}\Sigma \mathbf{w}}.$ Lastly, to convert back to the scale of number of stocks, recall that variance of diversified portfolio $\propto n$, simply multiply by $n$ to get $\text{effective no. of stocks}:= \frac{\bar{\sigma}^{2}}{\mathbf{w}^{T}\Sigma \mathbf{w}}.$ Note that in this case, - An equal-sized portfolio $\mathbf{w}=\mathbf{1} / n$ does not give ENS of $n$, consistent with the fact that if one stock has $100\times$ the average volatility, then the whole (equal-weighed) portfolio will be dominated by its noise. - On the other hand, a homoscedastic noise $\Sigma=\mathrm{diag}(\bar{\sigma})$ gives ENS $=1 / \| \mathbf{w} \|_{2}^{2}$. ### Directional Betting on Idiosyncrasy *In the context of factor-hedged directional betting, we care less about the size of each stock's (idiosyncratic) vol*: both the expected returns and portfolio vol scales linearly in the stock's weight in the portfolio, so high/low vol itself does not change sharpe ratio too much. Suppose we hedge out the non-idio risks, and we are directionally correct about the idio returns with constant probability $p$, then the expect returns due to (correctly predicting) the idiosyncrasy is $\mathbb{E}[\text{returns}]= (2p-1)| \mathbf{w} |\cdot \mathbb{E}[| \mathbf{r} |],$where the $| \cdot |$ is taken element-wise. If we further assume that then $\mathbb{E}[\text{returns}]\propto | \mathbf{w} |\cdot \pmb{\sigma}=\| \mathbf{w} \| _{1,\Sigma}$where and $\| v \|_{1,\Sigma}:= \| \Sigma v\|_{1}$ is the $l_{1}$ norm weighted by $\Sigma$. - Here we assume the idio volatilities are independent. The variance is $\mathrm{Var}(\text{returns})=\mathbf{w}^{T}\Sigma \mathbf{w}=\| \mathbf{w} \|_{2,\Sigma}^{2} ,$where $\| v \|_{2,\Sigma}=\sqrt{ v^{T}\Sigma v }$ is the $l_{2}$ norm weighted by $\Sigma$. Therefore, the [[Sharpe Ratio]] (aka information ratio in this case) is $\text{Sharpe} \propto \sqrt{ \frac{\| w \|_{1,\Sigma} }{\| \mathbf{w} \| _{2,\Sigma}^{2}} }=: \sqrt{ \text{effective number of stocks} }.$ Therefore, > [!definition|*] Effective number of stocks > The **effective number of stocks** of a portfolio is > $\frac{| \mathbf{w} |^{T} \Sigma\mathbf{1}}{\mathbf{w}^{T}\Sigma \mathbf{w} },$where $\mathbf{1}=(1,\dots,1)$. - It's easy to see that it matches our intuition: with equal weights $\mathbf{w}=\mathbf{1} / n$ and homoscedastic idiosyncrasies, the above reduces to $n$. - In particular, this reduction does not depend on $\Sigma$, consistent with the comment at the start of the section.