Why an Average Active Portfolio May Not Beat the Market on Sharpe
Summary
The document examines a claim that active portfolios collectively have lower risk-adjusted returns than a passive market portfolio. It presents a numerical example with a market holding and an independent active holding, then constructs two funds with offsetting active positions. Each fund’s Sharpe ratio is calculated, and their simple average appears higher than the market’s Sharpe ratio.
The replies challenge that conclusion on two related grounds. First, a simple average gives equal weight to funds of different sizes; a size-weighted average in the example falls just below the market Sharpe ratio. Second, when the funds are considered together, their active positions cancel, leaving the aggregate market exposure unchanged. The discussion therefore highlights how the averaging method and unit of comparison affect conclusions about active management. The example is illustrative rather than a general proof: it assumes particular expected returns, risks, and correlations, and does not include fees in the calculations. It also notes that active trading could affect the composition of the market portfolio through investment and price signals, a possibility outside the static-market example.
Key ideas
- The simple average of fund Sharpe ratios can mislead when fund sizes differ.
- Weighting the example by fund size changes its aggregate risk-adjusted result.
- Offsetting active positions can leave the combined holdings equivalent to the market portfolio.
- Active managers may redistribute performance among themselves without adding aggregate return in a static market.
- The numerical example omits fees and depends on its chosen assumptions.
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Full text
# Given future price probability distribution, what is a strategy that maximizes return? # Given future price probability distribution, what is a strategy that maximizes return? Say I know the price probability distribution, e.g., `lognormal(p,s)`, of a stock `X` at a future time `T` that is perhaps one or two years into the future. `p` is price and `s` is a standard deviation. What should I trade to maximize my expected total return at time `T`? Should I for example roll over short-term stock options, long-term stock options, buy the stock using some amount of leverage, etc? Is there some way or tool that calculates this automatically? How do I need to change this problem or my thinking so that the solution is not to simply use infinite amounts of leverage? -- Edit: I am assuming I have some positive amount of capital C to invest. Available instruments are: - Long and short stocks - Long and short call and put options - Stock futures and swaps - T-Bills, notes and bonds - (If it makes things easier we can ignore warrants and convertibles) ## Answer by Jon Grah (score 1) https://quant.stackexchange.com/a/14054 If you have an accurate 'future price probability distribution' (I will simplify this phrase: prediction), you buy the futures contracts (or any instrument) using leverage in a grid that covers the expected range and trade it aggressively with increased sizes over the range (e.g. martingale). You can also increase the sizes significantly when the 'prediction' is more short term (over a shorter range) as the drawdown is affected by the total range length prices travel from the first trade + all maximum subsequent trades made in that basket. This is an example spreadsheet which simulates how to setup the grid from the first trade, and calculates the drawdowns and points to breakeven at each trade level. While you can technically use stocks, you only get 1:2 leverage in most cases. Futures you can have leverage of up to 1:2000 from the exchange (Eurodollar) and day trading can give you 2x or more from your broker (assuming you close your positions out before daily close). Forex derivatives operate similarly. And you really do not have the manipulation or fraud that you do in stock market, assuming you are with a 'proper' broker.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.