Our first model is the basic mean-variance Model PFO-3.2.3, which we reproduce here:

We have available, in our data set, estimates of expected returns and the variance-covariance matrix for a universe of eight assets: an index of Italian government bonds with short maturity, some stock indices including the general Italian index, and a cash position in Euro as the risk-free asset. We compute the expected returns for each asset and the variance-covariance matrix through Estimate.gms (see Section 3.2.1 for details).

The output of Estimate.gms is dumped into a GDX container (Estimate.gdx), and the data needed are then loaded into MeanVar.gms by the GDX utility:

Note that the SET and PARAMETERS are first declared and then filled via the load call. In particular, the set Assets is assigned the parameter subset. The latter is defined and filled in Estimate.gms and dumped into the GDX container. VarCov is simply listed, without assignment, since this parameter has the same name as that contained in Estimate.gms.

In the VARIABLES section of the model we declare the vector of holdings of the assets. This vector is restricted to be positive since we are assuming for now that short sales are not allowed. In the model discussed in Section 3.2.2 we remove this restriction. Other variables are the expected return, the variance of the portfolio, and the objective function value.

The portfolio expected return (PortReturn) and the portfolio variance (PortVariance) are defined by the equations:

Different risk attitudes are suitable for different investors. We therefore trace the mean-variance efficient frontier corresponding to portfolios that are optimal for different values of λ. In GAMS, this can be done using a FOR loop: This code segment will solve the model for values of λ ranging from zero to one in increments of 0.1.

The results of the optimization are stored in a file by means of the PUT statement. The data needed to plot the efficient frontier are the optimal portfolio variance (PortVariance.L) and the optimal portfolio return (PortReturn.L). With each mean-variance pair corresponding to a given value of λ, there is an optimal portfolio (x.L) stored in the same file.

An alternative way of writing and presenting the results obtained is through the GDX utility GDXXRW. In this case, data structures have to be declared in order to receive the output produced. As a starting point we define the set of points that will describe the efficient frontier.

To each point there will correspond a minimum variance, an expected return, and an optimal portfolio. Information about the solver status is also stored. All these parameters have to be indexed by p:

The optimization model is solved inside a LOOP statement indexed by p, and the results obtained are cast in the SummaryReport parameter.

The latter is “unloaded” into the GDX container Summary.gdx, and the GDX utility GDXXRW will produce the Excel file MeanVarianceFrontier.xls.

The values of the optimal (expected return, variance) pairs are shown in Figure 3.1. As expected, the portfolios corresponding to high risk aversion (large λ) have the lowest variances (around 0.01), but also the lowest expected returns (around 10%). In contrast, the portfolios for low risk-averse investors have variances greater than 0.5 and expected returns around 23%. In Figure 3.2 we show a set of portfolios for different values of λ. Note that the portfolios with the highest number of stocks correspond to the more risk-averse investors, as these investors diversify their portfolios to reduce their risk.

For λ = 0.1, only the assets EMERGT and NORD_AM (with the highest expected returns) are held, but as λ increases, the model diversifies into up to five assets, as risk is decreased together with the portfolio expected returns. It is typical behavior for the mean-variance model to diversify into a high number of assets when risk aversion is high. Such a portfolio can be impractical and costly to manage in practice, and in Section 3.4 we give an alternative model to address this problem.

An alternative formulation to Model PFO-3.2.3 minimizes the portfolio variance subject to the portfolio expected return reaching an investor-specified target μ (called mu in the GAMS formulation). We reproduce here Model PFO-3.2.2:

The two models are equivalent in the sense that they generate the same efficient frontier by varying lambda or mu, respectively. Of course, the latter model can be made infeasible by specifying a target return outside the range of expected returns given by the underlying asset universe. An example of this alternative formulation is found in Section 3.5.

Let the observation times be denoted τ_t = (τ₀ , τ, . . . , τ_T), where τ₀ = January 1, 1990 and τ_T = January 1, 2000, and let Δ =be the time interval (one month) between observations. The observed prices are first converted to a common currency, the Italian Lira,¹ and stored in the parameter data_ITL(t,i), for each time point t and asset i; the file ExchangeRates.inc contains exchange rates for each observation date.

We first calculate the annualized return of each stock in each period, Ret_ITL(t,i), using the continuous-time formula where P_ti is the asset price called DATA_ITL(t,i) in the GAMS model, using the following statement:

The $-operator serves to exclude references to a non-existing observation prior to the first one available.

The discrete-time version of the formula for calculating returns, might be more familiar; either one could be used.

¹ The data used predate the introduction of the Euro, on January 1, 2002, in most EU countries.

Given these returns, it is straightforward to calculate the expected returns over the entire observation period, and the covariances of returns among pairs of assets. Mean returns μ_i , and covariances of returns σ_ij (see Equation PFO-2.1 and Definition PFO-2.3.6, respectively) are estimated as and Note that, unlike the definitions in PFO, we use the historical time periods to denote scenarios, adopting the simple bootstrapping method of Section PFO-9.4.1 for scenario generation, whereby it is assumed that every historical observation is a likely future scenario. The GAMS implementation is given by: where i1, i2 are aliases of the asset index i.

From the resulting data, MU and Q, we extract the input data for the MeanVar.gms model, ExpectedReturns and VarCov: where SUBSET is the subset of the assets we are interested in.

The whole input data set and the parameters computed are dumped into the GDX container Estimate.gdx. The latter can be used as input for other GAMS models (see previous section) and inspected, through the GAMSIDE application, to check consistency of the input data and results.

The GAMS file MeanVarShort.gms solves the mean-variance model with two types of short sales restrictions: first, only a short position in each individual asset class of up to 20% of the total investment is allowed and second, the total short positions are limited to 50% of the total investments. These restrictions are implemented using negative lower bounds on asset positions in the first case, and accounting for short sales (through Short(i)) for each asset in the second case:

As expected, the resulting portfolios have even better risk-reward characteristics. The resulting efficient frontier, together with the frontier obtained with no short sales, is shown in Figure 3.3. The optimal portfolios are now better diversified than the MeanVar model (see Figure 3.4), when also taking into account the short positions. Of course, when short sales are allowed, we also introduce the risk of getting caught in a “short squeeze.”

We use as a benchmark index the low-risk Euro Cash account, CASH_EU. The return and variance-covariance statistics are calculated as usual, but relative to the excess portfolio return over the benchmark.

The model in Estimate.gms performs these calculations:

The excess returns are calculated with respect to the CASH_EU, which is assumed to be the risk-free asset. The model equations in Sharpe.gms follow directly:

Notice the need to put a strictly positive lower bound on the Variance variable because it is used as an argument of the square root function. In general, nonlinear solvers will not attempt to evaluate any nonlinear expressions until all linear constraints, including bounds, are satisfied.

The optimal Sharpe portfolio is similar to the optimal mean-variance portfolio obtained for λ = 0.2. It consists of about 35% Italian bonds of short maturity, about 20% emerging markets, and about 45% North American stocks. Recall from Theorem PFO-2.3.1 that this is the tangent portfolio.

In Figure 3.5, we show how the efficient frontier changes once risk-free lending is allowed. The new linear efficient frontier dominates the previous one. The investor will split investments between the risk-free asset and the tangent portfolio fund, identified by θ = 1.

The scalar theta is obtained as the ratio between the volatility of the portfolio, and the volatility of the tangency portfolio,

For lambda = 0 only the return of the portfolio is relevant and, therefore, the model will try to maximize the expected return. This of course can be made as large as we like, as the amount of holdings increases by borrowing at the risk-free rate. Therefore, without a limit on the amount borrowed, the model instance with lambda = 0 is unbounded. This is shown in the output file where we report the status of the model after optimization. A value MODELSTAT = 3 means that the model is unbounded.

In Figure 3.6 (left), we show the absolute asset allocations for different values of lambda. Note that the lower the lambda (low risk aversion) the higher the amount borrowed and invested in the tangent portfolio. For example, when lambda = 0.04 the amount borrowed is 4.37, which means that more than four units of capital were borrowed and invested in the tangent portfolio in addition to one unit of own capital. To confirm that the proceeds of the borrowing are allocated in the tangent portfolio, we display in Figure 3.6 (right) the relative asset allocations obtained by normalizing the holdings with the available capital (one plus the borrowing). As expected, the portfolio compositions are the same for all values of the risk aversion parameter.

This version of the mean-variance model (MeanVarMip.gms) is a mixed-integer, nonlinear program, MINLP. For high values of λ, where risk aversion causes diversification, the MeanVarMip model results in portfolios with only three assets. These portfolios are usually dominated by those obtained with the unconstrained MeanVar.gms (see Figure 3.7).

We devote a part of the model in MeanVarMIP.gms to this case. The new equation definitions are: Note that equation ReturnDefWithCost, which defines the expected return of the portfolio, is made up of two components: one where the transaction fee is fixed, and a second where the transaction costs are proportional (see Equation PFO-3.38).

By setting the transaction costs to we obtain the portfolio displayed in Figure 3.8, for different values of λ. On the left, we report the asset allocations relative to x_0(i); on the right the asset allocations relative to x_1(i).

In the third part of the model in MeanVarMIP.gms we implement zero-or-range constraints, which ensure that the asset allocation will either not change at all from its current value, or, if it does change, the changes will stay within a bounded range near the current allocation. Following the model given in Section PFO-3.2.4 we define the buying and selling variables by declaring: The new equations for portfolio revision are: Equation InventoryCon(i) defines, for each asset and for a given initial holding, the total holding, x(i), after selling and purchasing of assets. We split the range constraints in two inequalities, since GAMS does not deal explicitly with inequalities ranges; those are given by UpBuyLimits(i), UpSellLimits(i), LoBuyLimits(i) and LoSellLimits(i). The binary variables, Yb(i) and Ys(i), force the buy and sell variables to be zero or to stay within the prespecified range. Finally, the BuyTurnover constraint takes care of the total turnover. Note that, since the asset allocations must add up to one, the total turnover limits both the buy and the sell amounts.

In Figure 3.9 we show the asset allocation after the revision of the portfolio. The initial holdings are set equal to those of the optimal mean-variance portfolio with λ = 0.5 and no turnover constraints. By comparing Figure 3.9 with Figure 3.2, we note that the portfolio with λ = 0.5 did not change. As expected, when the expected returns and the variance-covariance matrix stay fixed or move little, the revision of the portfolio should have no effect. However, if we change the risk aversion parameter λ to 0.1 or 0.2, the new asset allocation will hit the bounds of the selling and purchasing variables.

The raw data consists of monthly observations over a 10-year period (January 1, 1990 through January 1, 2000) of prices of 46 different asset classes, including a risk-free asset. These are 23 Italian sector stock indices and the general Italian stock index, three Italian bond indices, seven international bond indices, five international stock indices, three international cash rates and three US corporate bond sector indices. The risk-free asset is an Italian cash account. The data file also contains a time series of exchange rates between all relevant currencies and the Italian Lira, which is used as the common numeraire.

This model deals with asset allocation among broad asset classes by an Italian investor. Our investor first has to determine the structure of her portfolio by allocating funds in the major stock indices of the Milano stock exchange, and government bonds of different maturities. She then extends the allocation to international asset classes, namely government bonds, and finally invests in addition in US corporate bonds. We will see how the efficient frontier can be pushed out when diversifying internationally. However, we must note that all returns are measured in the foreign currency (e.g., in USD for the corporate bond market) so that foreign exchange risk is not included in this model. We can either assume that our Italian investor hedges currency exposure with forward contracts, or we can be patient until Section 7.2.3 to incorporate hedging decisions in asset allocation models.

The portfolio selection in InternationalMeanVar.gms is structured as follows:

We trace now the efficient frontiers for each step of our models. In the MeanVar.gms model this was done by changing the weight, lambda, between the portfolio expected return and its variance, but in the InternationalMeanVar.gms model we use the variance minimization Model PFO-3.2.2, and set a target return, MU_TARGET, which varies from zero to the largest expected return among the assets, MAX_MU.

The asset universe set and a number of subsets, corresponding to the different asset subsets used in the models, are declared in the include file AssetUniverse.inc. For example, the subset IT_ALL is given by: The subsets allow us to change dynamically and in a flexible way the universe of assets that participate in each of the six models. This is done through a dynamic set, ACTIVE, which at any point contains the set of active assets: Of course, the basic decision variables and constraints are unchanged from the previous models: Notice that all indexing in the constraints is done using the indices a, a1, and a2, which are aliases of the set ACTIVE. This set will be initialized to different values for the models in Steps 1 to 5. When it is re-initialized, the aliases a, a1, and a2 change their domains as well.

By dynamically changing the contents of ACTIVE, we are effectively changing the domain of the constraint summations. Hence, before solving each of the five models, the following is executed:

The decision variables x(i) are defined over the whole asset universe, but in any specific model only a subset is used. The default in GAMS is to leave the values of x(i) from the previous steps unchanged as it proceeds to solve the next step. This is unfortunate for our purposes, since when we display the vector of optimal investments, it may appear, for instance, that corporate bonds are part of the universe when our model does not include them since the corresponding x(i) may have a non-zero value from the solution of the previous step.

The GAMS option changes this behavior so that variables that do not participate in a specific model have their values reset to 0 so that they do not show up in the solution again.

Finally, in Step 6, we include a liability and essentially use the Italian assets to track the Italian general stock index. This model is very similar in essence to the Sharpe model of Section 3.3, but instead of calculating the excess returns and variance-covariances relative to the general stock index, we now use a different approach: the general stock index is included in the portfolio, but with a fixed “proportion” of -1, indicating that it is now really a liability (or a short position). The remaining part of the model is unchanged, except that the sum of asset proportions - including the liability position -1 - now must sum to 0: In this model, with its relatively large set of assets, the optimal portfolios often contain assets with very small proportions, even less than 1%. This indicates that the model would be a good candidate, in a real-world setting, for the zero-or-range constraints presented in Section 3.4. See also Section 5.3.1 and Section PFO-5.3.3 for an alternative formulation of this problem using scenarios.

In Figure 3.10 we display the mean-variance frontier for each set of assets. Since each set includes the previous one, the mean-variance frontier dominates the ones obtained by optimizing over a smaller asset set.