The Optimisation Problem
Portfolio optimisation is formally a constrained mathematical optimisation problem. The objective is to find the set of portfolio weights (w1, w2, ..., wn) that maximises the Sharpe Ratio:
Maximise: Sharpe Ratio = (w'R - Rf) / SQRT(w'Cw)
Subject to:
Sum(wi) = 1 (weights sum to 100%)
wi >= w_min (minimum weight per stock)
wi <= w_max (maximum weight per stock)
Sector_j <= S_max (optional sector concentration limit)
Where:
w = vector of portfolio weights
R = vector of expected returns
Rf = risk-free rate
C = covariance matrix of returns
' = transpose operator
Fintiq solves this problem numerically using a quadratic programming algorithm that runs thousands of simulations to map the full Efficient Frontier, then identifies the Tangency Portfolio (maximum Sharpe) and other key points. The entire computation takes a few seconds — what would have required a Bloomberg terminal and a quant analyst in 2010 is now available free at app.fintiq.uk.
What Data You Need
The good news: Fintiq provides all the necessary data automatically. You do not need to download price series, calculate covariance matrices, or build your own model. Here is what the optimiser uses behind the scenes:
Historical Price Data
Fintiq downloads 3 years of daily closing prices for each stock you enter. This provides 756+ observations per stock, which is sufficient for statistically reliable return, volatility, and correlation estimates.
Expected Return Estimates
By default, Fintiq uses the trailing 3-year annualised total return (price + dividends) for each stock as the expected return input. You can override this with your own estimates if you have a view that differs from the historical trend. For example, if a stock has just been through a period of unusually strong performance driven by a one-off catalyst, you might adjust the expected return downward to reflect more normalised conditions.
The Covariance Matrix
Fintiq calculates the full covariance matrix from the historical price data. This is the computational backbone of the optimisation. The covariance between each pair of stocks captures how much they move together — the foundation of MPT's diversification mathematics.
The Most Common Mistakes in Portfolio Optimisation
1. Too Few Stocks
Meaningful frontier construction requires at least 8-10 stocks. With fewer than 8 stocks, the frontier becomes poorly defined and the "optimal" weights reflect noise in the historical data rather than genuine diversification opportunities. A 4-stock portfolio on the frontier is often nearly the same as a 4-stock equally-weighted portfolio — there is not enough diversity to exploit.
The ideal input size for the Fintiq optimiser is 12-18 stocks from at least 6 different sectors. This provides enough diversity for the mathematical optimisation to find meaningful weight differentials, while keeping the portfolio concentrated enough to reflect genuine stock-picking conviction.
2. Ignoring Constraints
Running the optimiser without weight constraints is a common mistake that produces unreliable results. Unconstrained optimisation exploits noise in the historical data, often recommending 60-80% allocations to a single stock that happened to perform well in the lookback period. This is not investment wisdom — it is data overfitting.
Always set weight constraints. The standard professional approach: minimum 3-5% per stock (to ensure positions are meaningful), maximum 20-25% per stock (to prevent dangerous concentration). Many professional funds also apply sector caps (maximum 30% in any single sector) to prevent hidden sector concentration risk.
3. Sector Concentration
The optimiser does not know what sector each stock is in. It only sees price histories. If you enter 12 UK bank stocks, it will optimise among them — but the resulting portfolio, however "optimal" within that universe, will be dangerously concentrated in UK financial sector risk. Always ensure your input universe spans multiple sectors before running the optimisation.
4. Over-Trading
Re-running the optimiser daily and immediately rebalancing to the new "optimal" weights is a path to excessive transaction costs and tax friction. The optimal weights change frequently as prices shift, but these changes are often within the noise. A practical approach: rebalance when any position drifts more than 5% from its target weight, or quarterly at maximum.
Why You Need 8-10 Stocks Minimum
The mathematical reason for the minimum stock count is rooted in the behaviour of the covariance matrix. With n stocks, the covariance matrix has n*(n-1)/2 unique off-diagonal elements — the pairwise covariances that capture diversification potential.
Number of unique correlations by portfolio size:
5 stocks: 10 unique pairs (limited frontier shape definition)
8 stocks: 28 unique pairs (minimum for meaningful frontier)
10 stocks: 45 unique pairs (recommended minimum)
15 stocks: 105 unique pairs (good definition)
20 stocks: 190 unique pairs (rich frontier, diminishing marginal benefit)
Below 8 stocks: the frontier is poorly constrained and optimal weights
are highly sensitive to small data changes (unstable)
Beyond approximately 20 stocks, the marginal diversification benefit diminishes and the portfolio starts to resemble a broad index. Professional active fund managers typically hold 25-60 stocks to maintain a balance between diversification and conviction.