Portfolio construction
Optimal f (Ralph Vince)
For each ticker, the fraction of capital that would have grown its past daily results the most, measured against its worst day.
The idea in plain words
Ralph Vince adapted Kelly to trading results that are not a simple win or loss. For each ticker he looks at the actual history and asks: which fraction f of your capital, risked against the worst loss, would have compounded the history to the most money (the terminal wealth relative, TWR)? The fractions together give the weights.
How it is calculated
For each ticker: worst = its largest daily loss. For f from 0.001 to 0.999, TWR(f) = Π(1 + f × r ÷ worst). Optimal f maximises TWR; the capital fraction is f ÷ worst. Long, or short when the mean is negative and shorts are allowed. The total is capped at the maximum leverage.
TWR(f) = Πₜ (1 + f × rₜ ÷ |worst|)
f* = argmax TWR(f)
weight = f* ÷ |worst|How to use it
- To see how aggressive the history alone would justify being.
- Like Kelly, use a fraction of it in practice.
Drawing conclusions
- Optimal f is driven by the worst day: one extreme loss in the history shrinks the size.
- The note shows the leverage before the cap.
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). The f values are 0.08, 0.06, 0.10 and 0.02; divided by each ticker's worst day they add up to 9.4× leverage. Capped at 2×: 53% US500, 38% GER40, 67% gold, 43% EURUSD. Return 11.9%, volatility 20.4%.
The example's inputs are illustrative; the weights and figures are what our optimiser returns for them.
Watch out for
- Extremely aggressive: optimal f is the edge of ruin if the future has a worse day than the past.
- Each ticker is sized on its own; correlations between them are ignored.
A little history
Ralph Vince, Portfolio Management Formulas (1990) and The Mathematics of Money Management (1992).
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.