The general assumption underpinning equation 8, is that the asset returns (of what? Duh! the assets in question) follow a multivariate normal distribution.
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Now, here's a hypothetical problem from which we'll derive the parameters that we'll use in the formulation of a problem matrix for Orion to solve:
Let us postulate that a regulated financial institution is required by law to keep 50% of its capital reserves in liquid risk-free assets like T-bills. The institution is free to invest the entire unregulated half (50%) of its reserves as it wishes. Therefore, the institution's management decide to invest in the stock of Google (NYSE: GOOG) and Apple (NYSE: AAPL). Hence, the management tasks the Chief Risk Officer to calibrate the lowest level of risk (variance) that the institution would be exposed to owing to a 50-50 investment of the unregulated capital in Apple and Google stocks.
The risk manager starts by examining the volatility patterns and the correlation matrix of Google and Apple stock in the following tables that were prepared by his analyst:
Hence, from the data he derives the following parameters:- Annual Standard deviation of Google shares, σ1 = 0.4926
- Annual Standard deviation of Apple shares, σ2 = 0.5508
- Anual Covariance of Apple and Google shares, p12σ1σ2 = 0.1826 (to 4 s.fs.)
To conserve computing resources, he decides to treat the optimization problem as a two asset binary quadratic problem, and thus plugs the values he derived from the tables into equation 8, and gets an expression that looks like this (which we'll call equation 9):
min{0.2427(x1.x1) + 0.3034(x2.x2) + 0.3652(x1.x2)}
The risk manager decides that in binary notation, x1 and x2 = 1 and derives the following binary quadratic portfolio optimization problem for DWave's Orion to minimize:
...Not so fast: The matrix above will not work in Orion because The third column in each row specifies the value to be assigned to the selected Qij element of the matrix. Qij must be an integer value. Therefore, we round-off everything in the third column to 1 significant figure and multiply it by ten to get this matrix (which will work in Orion):
Stay tuned for Part 4 of this installation. It will contain the result of the computation plus an evaluation of Orion
...Not so fast: The matrix above will not work in Orion because The third column in each row specifies the value to be assigned to the selected Qij element of the matrix. Qij must be an integer value. Therefore, we round-off everything in the third column to 1 significant figure and multiply it by ten to get this matrix (which will work in Orion):
Stay tuned for Part 4 of this installation. It will contain the result of the computation plus an evaluation of Orion
