Portfolio construction
CVaR optimisation (minimum expected shortfall)
The mix with the smallest average loss on its worst days, measured on the real history rather than a bell curve.
The idea in plain words
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.
How it is calculated
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_maxHow to use it
- 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.
Drawing conclusions
- 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.
Example
Running example (used in every portfolio guide): US500, GER40, gold (XAUUSD) and EURUSD, two years of daily data. Expected returns 8%, 7%, 7% and 1% a year; volatilities 18%, 20%, 15% and 7%. US500 and GER40 move together (correlation about 0.7); gold hardly moves with the indices (about 0.04). CVaR optimisation: 7% US500, 14% gold, 79% EURUSD. Average loss on the worst 5% of days 0.90% (equal weights: 1.31%). Compared with minimum variance it holds more gold (14% against 6%), which had milder bad days than its volatility suggests.
The example's inputs are illustrative; the weights and figures are what our optimiser returns for them.
Watch out for
- 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.
A little history
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.
Educational content only. No portfolio model knows the future: every one of them works on estimates from the past, and nothing here is investment advice.