March & Sanchis Investments Ltd.¶

Problem Definition¶

Suppose you are a financial advisor at March & Sanchis Investments Ltd. and your client has €1,000,000 to invest in four different types of assets: stocks, bonds, real estate, and crypto assets, with the Expected Returns and risk (an estimation of the risk per euro invested) expressed in the table below:

Asset Type	Expected Return (%)	Risk (%)
Stocks	10	1.25
Bonds	6	0.5
Real Estate	8	0.7
Crypto Assets	15	2.1

Your task is to help the client select the optimal portfolio of investments, maximizing the Expected Return. Assume that the investments are independent and take into account the following constraints:

Budget: The total investment cannot exceed the budget of the client (€1,000,000)
Personal Preferences: The client wants to invest at least €200,000 in real estate due to personal interests in owning property, and does not want to invest more than €400,000 in crypto assets as she would like to be cautious and limit exposure.
Risk: The overall risk estimation cannot exceed 12% of the expected return

Solution¶

To model this as a continuous linear programming problem, let:

$x_1$ = Amount invested in stocks (€)
$x_2$ = Amount invested in bonds (€)
$x_3$ = Amount invested in real estate (€)
$x_4$ = Amount invested in crypto assets (€)

Then, we want to maximize the expected return, which is given by:

$\max z = 0.1*x_1 + 0.06*x_2 + 0.08*x_3 + 0.15*x_4$ (Expected return in €)

Subject to:

$x_1 + x_2 + x_3 + x_4 = 1,000,000$ (total investment must equal $1,000,000)

$0.0125*x_1 + 0.005*x_2 + 0.007*x_3 + 0.021*x_4 ≤ 0.12*(0.1*x_1 + 0.06*x_2 + 0.08*x_3 + 0.15*x_4)$ (risk limit: the risk of the portfolio, cannot exceed 12% of the total expected return)

$x_3 ≥ 200,000$ (minimum investment in real estate: the client wants to invest at least €200,000 in real estate)

$x_4 ≤ 400,000$ (maximum investment in crypto assets: the client is cautious about the high risk of crypto assets and does not want to invest more than $400,000 in them)