CVaR optimisation (minimum expected shortfall)

用通俗的话解释

Minimum variance treats a gain and a loss alike. CVaR optimisation looks only at the bad tail: on the 5% worst days (or the level you choose), how much did the portfolio lose on average? It picks the weights that make that number smallest, so tickers that crash together are avoided even if their ordinary volatility is low.

计算方法

Over the daily scenarios, CVaR is VaR plus the average loss beyond VaR. Rockafellar and Uryasev showed it can be minimised directly; the optimiser uses projected subgradient steps over the history.

CVaR_α(w) = min_z { z + (1 ÷ ((1 − α)T)) Σₜ max(0, −wᵀrₜ − z) }
minimise CVaR_α(w)  subject to  Σwᵢ = 1, 0 ≤ wᵢ ≤ w_max

如何使用

• When crash days matter most to you: drawdown-sensitive money, leveraged accounts.
• Set the level under Settings (95% by default; 97.5% looks further into the tail).
• Use three or more years of data so the tail has enough days.

得出结论

• The study's notes show the average loss on the worst days for the chosen mix.
• Results close to minimum variance mean the tails are roughly normal.

注意事项

• Few tail days means a noisy estimate; the weights can jump between studies.
• It protects against the tails that happened, not against new kinds of crisis.

一点历史

R. Tyrrell Rockafellar and Stanislav Uryasev, "Optimization of Conditional Value-at-Risk" (Journal of Risk, 2000). CVaR (expected shortfall) is now the regulatory measure of bank trading risk.