## Example 1

Consider an investor that wants to construct a portfolio from scratch, drawing from a set of 3 stocks. Let us assume that the investor has generated, in some way, the rate of return of each stock under three scenarios: say a pessimistic, a neutral, and an optimistic scenario. The rate of return of each stock under each scenario is given in the table below along with the probability associated with each scenario. The last column reports, for each stock, the expected return computed over the three scenarios.

 Scenario Expected Pessimistic Neutral Optimistic Return Stock A -7% 3% 8% 1.35% Stock B -15% 15% 25% 8.40% Stock C -2% 1% 3% 0.67% Probability 0.33 0.34 0.33

Assuming that the investor has selected a risk measure, the portfolio optimization problem aims at determining the optimal portfolio weights invested in each stock that minimize the portfolio risk given a value of expected return or, conversely, that maximize the portfolio expected return for a specific value of risk.

But, what are the optimal portfolio weights?

Investing all the budget available in the stock that yields the highest expected return, i.e., Stock B, does not consider the portfolio risk. Indeed, portfolio risk can be usually reduced through diversification, i.e., the sub-additivity property of coherent risk measures. Conversely, building a portfolio comprising only the least risky stock does not consider its return (other than the aforementioned sub-additivity property). As a consequence, the optimal portfolio will likely include more than one stock and a decision about the proportion of the budget invested in each of them has to be made. Portfolio optimization aims at providing investors and financial institutions with an effective tool to choose the optimal portfolio composition.

## Example 2

In Example 1 we assumed that the investor is building the portfolio from scratch. Nevertheless, in most of the practical situations, the investor already holds a portfolio of stocks (referred to as the current portfolio) and, due to sudden changes in the market trend, he/she might prefer to rebalance the portfolio composition to possibly reduce losses or better exploit returns growth. In this context, fixed and/or proportional transaction costs (i.e., the cost paid to move from the current portfolio to the new one) play a crucial role as they directly reduce the portfolio return. A fixed transaction cost is a fixed cost paid if a stock is traded, irrespective to the amount negotiated. A proportional transaction cost is a cost that increases with the amount traded of a given stock.

Consider an investor that holds a current portfolio composed in the proportions shown in the table below. Let us assume that the scenarios generated by the investor are those reported in the table shown above in Example 1 and that he/she would like to reduce the investment in Stock C while increasing that in Stock B in order to achieve a higher expected return but without assuming an excessive additional risk.

 Curr. Port. Cost Prop. Fixed Prop. Stock A 7% 10 \$ 0.18% Stock B 3% 8 \$ 0.195% Stock C 90% 15 \$ 0.225%

But, also in this example, what are the optimal portfolio weights?

Selling all the securities of Stock C in the current portfolio and investing all the cash earned in Stock B would probably imply a sharp increase in portfolio risk and the payment of high transaction fees.

Transaction costs and the presence of a current portfolio are real-life features that have to be included in portfolio optimization models. Also in this case, portfolio optimization aims at providing an effective tool to support the decision-maker in determining the optimal portfolio composition.