Applying the Kelly Criterion to Correlated Portfolios
The Kelly criterion is a position-sizing framework designed to maximise the long-term growth rate of capital. It links the size of a trade or investment to expected return, risk, and the probability of success. While the basic formula is easy to understand for one wager, applying it to several assets requires more care because returns may move together.
Correlation is especially important in portfolios containing technology shares, cryptocurrency, forex positions, or sector-focused exchange-traded funds. Five apparently different holdings can behave like one trade during a market shock. A portfolio Kelly approach helps account for that overlap, but it relies on realistic assumptions and disciplined risk controls.
What Kelly Is Measuring
For a single opportunity, the classic Kelly formula is often expressed as f = (bp − q) / b. Here, f is the fraction of capital to allocate, b is the net payout relative to the stake, p is the probability of winning, and q is the probability of losing. In market portfolios, the same idea becomes an optimisation problem involving expected returns and a risk or covariance matrix.
For a set of assets, a simplified excess-return version is:
w = Σ⁻¹ μ
In this expression, w represents portfolio weights, μ is the vector of expected returns above the risk-free rate, and Σ⁻¹ is the inverse covariance matrix. The formula increases exposure to assets with attractive expected returns while reducing exposure where volatility and co-movement create excessive risk.
The framework is growth-oriented rather than comfort-oriented. A full-Kelly allocation can produce large drawdowns when estimates are wrong. Investors comparing a long-term portfolio with active speculation may benefit from reading about the investing or trading mindset before deciding whether an aggressive sizing method suits their objectives.
Why Correlation Changes the Answer
Correlation measures how closely two assets have moved historically. A value near +1 means they have tended to rise and fall together, while a value near 0 suggests less consistent co-movement. Negative correlation can provide diversification, although relationships can weaken or reverse during periods of stress.
Suppose a portfolio contains an ASX technology ETF, a global technology fund, Bitcoin, and a technology-heavy US share. Their labels differ, but their economic drivers may overlap. Interest-rate expectations, growth-stock sentiment and liquidity conditions could push all four lower at the same time. Treating each holding as independent would overstate diversification and produce an oversized Kelly allocation.
The covariance matrix captures both individual volatility and the relationship between assets. When correlations are high, the matrix usually reduces the combined allocation compared with simply adding separate single-asset Kelly bets. A near-singular matrix can also make the inverse unstable, meaning a small change in estimated returns may create a very large change in recommended weights.
Compare Sizing Approaches
The following comparison shows why a portfolio method is generally more suitable when holdings share common risk drivers.
| Approach | Main input | Handles correlation? | Typical weakness |
|---|---|---|---|
| Fixed percentage | Chosen portfolio fraction | No | Ignores changing opportunity quality |
| Single-asset Kelly | Win rate and payoff, or return and variance | No | Double-counts exposure across related assets |
| Equal risk contribution | Volatility and covariance | Yes | Does not directly use expected returns |
| Full portfolio Kelly | Expected return and covariance matrix | Yes | Highly sensitive to estimation errors |
| Fractional portfolio Kelly | Full Kelly scaled down | Yes | May sacrifice theoretical growth for smoother outcomes |
A practical comparison should include the cash rate in Australian dollars, trading costs, taxes where relevant, and the timeframe used for return estimates. For an Australian investor, the Reserve Bank of Australia cash-rate environment can affect the opportunity cost of holding cash, while currency movements can materially change the result for unhedged US assets.
Kelly is a model, not a forecast. If expected returns are inflated or volatility is understated, the output can look precise while being economically fragile. A conservative investor may use half-Kelly or quarter-Kelly as a risk budget rather than treating the full result as an instruction to invest.
Build Inputs From Realistic Data
Start by choosing a consistent return frequency. Daily data may suit liquid shares or major currency pairs, while weekly data can reduce noise for a longer-term portfolio. Use total returns where possible, including distributions and relevant price adjustments. For a portfolio containing ASX-listed funds, US shares and crypto, convert returns into the same reporting currency, such as AUD.
Expected returns deserve more scepticism than historical averages often receive. A simple estimate might use a long-term annualised return, a forward valuation assumption, or several scenarios. It is sensible to test optimistic, central and pessimistic estimates instead of relying on one number.
Covariance estimates can be improved with a longer sample, shrinkage methods, or a factor model. A factor model may reveal that several assets are all exposed to global equities, duration, the US dollar or commodity prices. It is also useful to cap individual positions and total exposure to a single theme before applying the Kelly result.
Inputs Worth Checking Before Calculation
- Expected return after fees, spreads and applicable tax effects
- Volatility measured over a period relevant to the investment horizon
- Correlations during both calm markets and sharp sell-offs
- AUD currency exposure for overseas shares, funds and digital assets
- Liquidity, leverage and the ability to exit at a reasonable price
Apply Fractional Kelly in Australia
Fractional Kelly is usually easier to defend in a real portfolio. Applying 25% or 50% of the calculated allocation recognises that return estimates, correlations and volatility are uncertain. This is particularly relevant for crypto markets, where a short price history and changing market structure can make a full-Kelly result unreliable.
Australian investors also need to account for the local regulatory and tax setting. ASIC regulates financial services and markets, while leveraged products such as CFDs can magnify losses quickly. Crypto disposals may create capital gains tax consequences under Australian Taxation Office rules, and frequent trading can have different tax treatment from long-term investing. Professional advice may be appropriate for personal circumstances.
An investor in Brisbane, Perth or Melbourne may hold a mixture of ASX shares, global funds and digital assets through different platforms. Consolidating exposures in a spreadsheet can reveal that several accounts depend on the same US technology factor. After defining a fractional allocation, a learning account can be used to explore educational tools without confusing theoretical portfolio sizing with a guaranteed outcome.
Guardrails That Keep Kelly Practical
- Set a maximum weight for any one asset and investment theme
- Limit portfolio leverage and exclude borrowed funds from aggressive assumptions
- Keep a cash reserve for living costs and unexpected obligations
- Recalculate with lower expected returns and higher correlations
- Use a drawdown limit that triggers a review rather than automatic doubling down
Review and Rebalance Without Overfitting
Kelly weights should change when the underlying assumptions change, not whenever prices move by a small amount. Monthly or quarterly reviews can be more stable than constant re-optimisation. A review should examine expected returns, volatility, correlations, position limits, tax effects and whether the original investment thesis remains valid.
Rebalancing can be based on target bands. For example, an allocation might be reviewed when an asset moves five percentage points away from its intended weight. This reduces unnecessary turnover and helps manage brokerage, bid-ask spreads and potential tax events. Superannuation portfolios may have different liquidity and access constraints, so the same process cannot be copied mechanically into every account.
Backtesting is useful for understanding how a method behaved in earlier conditions, but it cannot prove future performance. Test different starting dates, stressed correlations and delayed rebalancing. For a practical review, investors can review their dashboard alongside a separate record of assumptions, transactions and portfolio-level risk.
Warning Signs That Call for a Reset
- The covariance matrix changes sharply after a market shock
- Several holdings begin responding to the same macroeconomic news
- Portfolio losses exceed the level used when choosing the Kelly fraction
- Expected returns depend on a single forecast or recent price trend
- Trading costs or tax consequences make rebalancing uneconomic
The best use of Kelly is often as a ceiling rather than a target. It can show how uncertainty, correlation and concentration interact, while separate constraints reflect the investor’s capacity to tolerate losses. Before placing any trade, calculate the portfolio with conservative assumptions, apply a fractional scale, and record the resulting weights in an AUD-based risk sheet.