values) and rewards (negative values). These coefficients can be derived from the pairwise correlations.

The minimization of the QUBO objective function given by expression (\[eq:QUBO\]) should optimize the risk-adjusted returns by the use of the metrics of Sharpe ratio [@sharpe1966mutual] for the $a_i$ parameters (measuring expected excess asset return in the asset volatility units) and correlation between assets (as a measure of diversification) for the $b_{ij}$ coefficients.

The Sharpe ratio is calculated as $(r-r_0)/\sigma$ where $r$ is the asset’s expected annualized return, $r_0$ is the applicable risk-free interest rate and $\sigma$ is asset’s volatility (annualized standard deviation of the asset’s log-returns). The higher fund’s Sharpe ratio, the better fund’s returns have been relative to the risk it has taken on. Volatility can be estimated as historical annualized standard deviation of the Net Asset Value (NAV) per share log-returns. Expected return can be either estimated as historical return on fund investment or derived independently by the analyst/portfolio manager taking into account different considerations about the future fund performance. The asset correlation matrix $\rho_{ij}$ is constructed from the time series of log-returns [@hull2016options].

#### QUBO instances

Instead of using the raw real numbers obtained from financial data for the QUBO coefficients, we 