Given a portfolio of stocks, with weights $w$, we want to limit our exposure to a number of risks. ## Factor Risk Limits Following the result from factor models, > [!embed] > ![[Factor Models#^5d1660]] Now the [[Pearson's R2|$R^{2}$]] of the above regression measures the amount of contribution factors have toward the portfolio's variance, which equals $1-\frac{\mathrm{RSS}}{\mathrm{TSS}}=1-\left(\frac{\mathbb{E}[\text{idio returns}] / \text{Info. ratio}}{\mathbb{E}{\text{[total returns]}} / \text{Sharpe}} \right)^{2} \approx 1-\left( \frac{\text{Sharpe}}{\text{Info. ratio}} \right)^{2},$assuming $\text{idio returns} \approx \text{total returns}$ (as in the case where the investor do not have factor-level edge). Therefore, $\frac{\text{Sharpe}}{\text{Info. ratio}}=\sqrt{ 1-R^{2} } < 1.$i.e. idiosyncratic edge is always diluted by the factors, which brings no edge by assumption and extra risk exposure. In addition, the more variance factors contribute to the portfolio (a high $R^{2}$), the worse the ratio is. In practice, we do not seek 100% idiosyncrasy because: - It's costly (in terms of trading costs) to always maintain perfect hedges. - The assumption of 0 factor edge, and that idio returns are fully independent of factor returns are not realistic, e.g. when the manager is good at sizing. - Since the factor models are not 100% correct, we are wrong anyways. ### Market Exposure Limit In addition, the macroeconomic factors (e.g. country) have a small positive Sharpe, so it might be worth it to keep some of them. It's simple to allocate exposure between idio and market factors: since the two have uncorrelated returns, we can simply decompose our portfolio into two sub-portfolios, one with purely idio exposures, and one with purely factor exposures. Now we can use simple rules for portfolio allocation, e.g. [[Mean-Variance Optimal Portfolio|MVO]] to find the proportion of GMV that should be allocated to the market exposure: In the case of the MVO, the weight can be written as a function of volatilities and Sharpe ratios of the two sub-portfolios, as they are uncorrelated, because of [[Mean-Variance Optimal Portfolio#^0cfff3|this result]]: $\frac{\mathrm{GMV}_{\text{market}}}{\mathrm{GMV}} = \frac{\frac{\text{Sharpe}_{\text{market}}}{\sigma_{\text{market}}}}{\frac{\text{Sharpe}_{\text{market}}}{\sigma_{\text{market}}}+\frac{\text{Sharpe}_{\text{idio}}}{\sigma_{\text{idio}}}}.$ ### Other Single-Factor Limits ## Single Stock Limits