7.2.1 Asset Diversification and Efficient Frontier

We assume that there are only two assets (namely 1 and 2) in the investment universe, which are denominated in the same currency. It follows that portfolio value V_U is indicated by:

7.1

The weights w₁ and w₂ add up to 1: extreme portfolios can be constructed by investing 100% into either of the two given assets and zero in the other.

Mean-Variance is based on the estimate of portfolio expected returns and standard deviations, which are a combination of the expected returns, standard deviations, and pairwise correlations of any potential security in the portfolio. Expected returns need to be explicitly estimated for any given time horizon T: moving from a short-term representation to a long-term representation typically requires re-estimation of all parameters by using a different length of time series. It is conveniently assumed that the longer the time series, the more “appropriate” the estimation of long-term expected returns, as a way to capture long-term trends in the market variables or mean reversion. Therefore, portfolio expected return is indicated by:

7.2

While the portfolio return is a linear combination of the returns of the underlying assets, weighted by the relative w₁ and w₂ contributions to total portfolio value V_U, standard deviation ρ_U is not a linear measure but a quadratic function of asset volatility. Therefore, given the volatility and the portfolio weight of each asset, portfolio risk is typically indicated by:

7.3

where

7.4

One can therefore express the above equation as follows:

7.5

The correlation coefficient ρ can take a maximum value of +1 (i.e., the two assets are perfectly correlated, that is they move in perfect unison) and a minimum of –1 (i.e., they are perfectly negatively correlated, that is their movement is the opposite of the other). To understand how portfolio risk is affected by the correlation assumptions, we can investigate the extreme cases when correlation is equal to –1, is null, or equal to +1.

If pairwise correlation is equal to +1, the covariance between the two assets will equal the product of the two volatilities: the volatility of the portfolio becomes a linear combination of the volatility of the underlying assets. Hence, plotting the relationship between portfolio returns and portfolio risk on the cartesian plane, for any given combination of asset weights, would lead to a straight line.

7.6

Similarly, where correlation equals –1, one can plot on the cartesian plane a segment that, although monotonic in the expected returns, can solve portfolio returns for two different attainable but equally likely asset allocations, that depend on the portfolio relevance of the exposure of each of the two assets against the other:

7.7

When instead correlation equals 0, then the resulting function is not linear:

7.8

Figure 7.2 shows a numerical example in which the expected returns are respectively and , while the standard deviations are ρ₁ = 8% and ρ₂ = 3%. The plotted line representing the risk/return characteristics of a potential model portfolio invested in the two assets is contained in the space identified by the extreme cases, in which the assets are perfectly correlated and perfectly uncorrelated, for any value of ρ_(1,2) between −1 and +1.

Hence, for any given estimate of ρ_(1,2) and any given portfolio composition (w₁,w₂), one can identify the minimum return portfolio, the maximum return portfolio, and the minimum variance portfolio. As ρ_(1,2) is also an input in the optimization exercise, the objective function would solve for a suitable combination of portfolio weights. When the number of assets is greater than two, as in the three assets case represented in Figure 7.3, then pairwise correlations are indicated by a variance-covariance matrix. Allocation constraints are usually specified for the various asset classes: the problem becomes multi-dimensional and mathematically advanced routines are required to identify the global minimum/maximum of the optimization objective function (e.g., minimization of standard deviation). The best mix of risky assets for every maximum level of expected return is a curve, indicated as the efficient frontier.

7.2.2 The Mean-Variance Model Portfolio

Finding the model portfolio, for a given investor and a given set of constraints, requires us to move along the efficient frontier to attain the desired expected return with the minimum portfolio variance. Therefore, the efficient frontier is the “collection” of all portfolios that optimize the objective function, under the same set of constraints but for different target expected returns. We assume that the universe U of the available securities is made up of any j_th securities. Each security has been associated with an expected return corresponding to the mean return of the historical distribution or a subjective opinion of the portfolio manager, while w_j denotes each security's fair value exposure within the portfolio. As w_j, can equal zero, we can identify a portfolio with the notation U as a set of all securities that an individual can invest or not invest in. We can also assume that σ_j is the volatility of the j_th security, while cov_(i,j) refers to the covariance between the returns of any pair of securities (often asset classes).

The classical case identifying the model portfolio by setting a target return and finding the optimal asset allocation to minimize portfolio variance takes the form of a quadratic programming problem:

7.9

subject to:

1. - the resulting portfolio yields at least a target
   7.10
2. - all portfolio weights sum to 1:
   7.11
3. - short selling is typically not allowed:
   7.12

Varying between the return of the minimum variance portfolio and the return of the maximum variance portfolio indicates the efficient frontier, as in Figure 7.1.

The optimization engine is usually guided by setting constraints on any amount that can be invested in a particular asset class or currency A⊂U so that the exposure in A is no more than a certain b percentage of V_U:

7.13

Once the efficient frontier is built, Robo-Advisors may break it into segments and identify levels of risk and expected return that map onto client profiles as defined in the on-boarding mechanism. Hence, a model portfolio is presented to the investor as a personalized and optimally chosen investment recommendation.

7.2.3 Final Remarks About Mean-Variance

Mean-Variance is the simplest and most convenient optimization case, which limits the representation of real securities to linear products or their equivalent asset class estimates (e.g., indices), deals with a single investment horizon at a time, restricts to the use of expected returns as a measure of future profitability, and plugs in variance as a measure of risk. Although more recent techniques allow enhancing the risk estimate with more refined measures, such as tracking error volatility or expected tail losses, wealth managers would still be asked to optimize portfolios separately for different time horizons, and reduce the conversation about investment ambitions to the expected return instead of discussing the probability of achieving personal goals through the market cycle.

7.3.1 The Equilibrium Market Portfolio

The starting point is the equilibrium market portfolio (denoted by M) which corresponds to the investment decisions of all market participants. Continuous trading enables the market to adjust towards a long-term equilibrium value where the market weights are governed by frictionless price discovery. Equilibrium market weights can be estimated by assessing the capital value of each asset divided by the total capital value of the whole market:

7.14

The initial investment recommendation would be to hold a combination of the risk-free and the market portfolio (e.g., buy each asset in the universe according to the respective capital weights in the market): for any given level of risk no other portfolio can provide higher expected excess return because trading against the market equilibrium would not be valuable. Yet, trying to construct the real market portfolio would be unrealistic, since the number of assets is enormous in the real world. That is why wealth managers might select a representative market index to approximate the initial equilibrium weights of M.

Alternatively, the approach can be initialized by using the CAPM bet to model the expected excess returns of individual asset classes (as in Sharpe (1964) and Lintner (1965)), thus performing reverse optimization to derive the equivalent CAPM weights that represent the initial market equilibrium. The time series of the excess returns of each j_th asset with respect to the risk-free r_f are the initial input for the estimation of the CAPM equilibrium, so that:

7.15

in which,

7.16

7.17

7.18

The variance-covariance matrix of the excess returns of all assets in the universe is indicated by Σ and allows us to derive the vector of the initial portfolio weights by so-called reverse optimization:

7.19

λ indicates the CAPM risk aversion coefficient (Sharpe ratio) that represents the change in the expected return of the investor's portfolio per unit change in portfolio volatility:

7.20

The implied and reverse optimized excess returns indicate the specified market risk premiums, hence a larger λ implies more excess return per unit of risk adding a positive influence on the level of the estimated excess returns. Under the CAPM theory it is assumed that the idiosyncratic risks of the assets are uncorrelated so that risk can be reduced by diversification. Therefore, the coefficient of the linear regression analysis is assumed to be null since the investor holding the market portfolio will be rewarded only for the systemic risk estimated by β_(j,M).

7.3.2 Embedding Professional Views

Wealth managers willing to include their own views in the optimization process would be asked to overwrite the Mean-Variance initial market statistics, thus facing excessive sensitivity on the inputs which would in turn lead to extreme portfolios. Instead, according to Black and Litterman, the views do not directly replace the original historical inputs, but add new information to the exercise as they are combined with the prior expectation of market returns by means of a Bayesian model, thus leading to more stable posterior asset allocations. The information contained in the views can be an absolute or a relative expectation, as investors can express their own beliefs about the performance of an individual asset or the expected performance relative to other investments. Black and Litterman deviate from the assumption of symmetric information (i.e., that all market participants invest or are willing to invest in the same efficient frontier). Robo-Advisors and digital wealth management might believe that they possess superior information compared to the market as a whole and, although accepting asset prices as a starting point to indicate initial optimal weights (prior), they might want their views on short-term dynamics of asset prices to be reflected in the building of the final optimal portfolio (posterior). Their expectations indicate the belief that in the short term, a given asset or set of assets would not converge to the equilibrium but would yield a different return. This belief is uncertain, as it refers to a future state of the world, and it can be described by an expectation and a probability distribution, that is, a view. Therefore, a posterior vector of combined expected excess returns and their related uncertainty is supplied for the final optimization process so that the posterior vector of tilted asset weights is derived: this indicates the optimal portfolio.

The scope of the Black-Litterman approach is to combine the probability density function of initial excess returns with the probability density function of the views, so that the posterior probability density function can be used as an input to the traditional Mean-Variance optimization. The original proposition assumes the distributions are normal, but this assumption could also be relaxed:

7.21

7.22

Q is a column vector made up of r number of rows, where each row element represents the subjective expected excess return attached to a view and Ω is the confidence interval of the views. The Bayes theorem allows us to construct the combined posterior distribution so that if Ω = 0, then all views are certain so that for all assets specified in the views the return is given by the views themselves; if Ω = ∞, then the views are totally uncertain so that by simplifying the equation becomes .

7.3.3 The Black-Litterman Optimal Portfolio

Having derived the probability density function of the posterior distribution of the excess returns, one can optimize the Mean-Variance objective function and estimate the posterior “tilted” weights of the assets that indicate the optimal portfolio. This is achieved by solving the following unconstrained maximization problem:

7.23

Similarly to the Mean-Variance case, wealth managers can estimate the Black-Litterman efficient frontier and break it into segments which identify different levels of portfolio expected return and standard deviation. Thus, map investors' risk/return profiles as indicated by the on-boarding mechanism, leading to a “seemingly” personalized and optimally chosen investment proposal.

7.3.4 Final Remarks on Black-Litterman

The main advantage of Black-Litterman is to facilitate the embedding of explicit professional subjectivity about future return distributions without creating excessive sensitivity by tilting input parameters. However, portfolio construction is still dependent on a very simplified representation of real investment opportunities which does not allow us to move comfortably outside the convenient zone of ETF investing. This can hinder the simulation and risk management of resulting actual portfolios, and disallow a consistent representation of portfolio sensitivities to market scenarios over time.

7.4.1 Final Remarks on Mean-Variance and Mental Accounts

Clearly, this approach is a significant step forward in modelling portfolios for private wealth. Yet, it still suffers from key limitations of Mean-variance and Black-Litterman: fixed income cannot be conveniently modelled, multi-period goals and their representation cannot be featured, and derivatives cannot become part of portfolio construction. Most of all, since Robo-Advisors invite final investors to stay the course towards the long term, we believe that long-term simulation techniques need to be the basis of portfolio construction and digital representation; hence, Probabilistic Scenario Optimization.

7.5.1 The PSO Process

PSO is a step-by-step process of portfolio filtering and ordering according to a probability measurement criterion, as synthesized in Figure 7.4: the end result is the asset allocation that shows the highest probability of achieving an investment goal, while complying with given allocation constraints and risk limits.

Advanced risk management methods are required to deal with the estimation of uncertainty about risk/return realizations of actual financial products. Such estimates involve the generation of tens of thousands of stochastic scenarios about the evolution of the market risk factors, that drive fair value pricing of financial securities. The accuracy and meaningfulness of this process are influenced by the quality of the input data and the methodology adopted to simulate the future states of the world for each class of risk factor. The process can be represented by the following steps:

1. Definition of the optimization problem: selection of the investment universe, indication of the allocation constraints, declaration of the investment risk/return profile to depict the investment goal and the risk limit.
2. Generation of the space of future total returns by simulating real securities over time, conditional on probabilistic scenarios.
3. Exhaustive generation of the quasi-random space of the admissible asset allocations and reduction to the risk-adequate set: filtering of all admissible allocations that fulfil the asset allocation constraints and reduction to the set of risk-adequate portfolios with respect to the investor's risk profile.
4. Probability measurement and portfolio ranking: optimization of the objective function and graphical representation of the resulting asset allocation and its characteristics.
5. Performance measurement and portfolio comparison: investment performance can be tracked over time and the distance to optimality can be measured by computing the residual probability to achieve a target across time.

Exhaustive enumeration techniques are very unrestricted methods and can be applied to any type of investment problem. Hence, Robo-Advisors and traditional wealth managers are provided with a framework that can scale up irrespective of their business focus (e.g., human or unmanned advice), securities universe (e.g., ETF or fixed income), or target clientele (e.g., affluent or UHNW).

7.5.2 The Investor's Risk and Return Profile

The process starts with the elicitation of the risk/return profile of the investor, which is a combination of the client's tolerance for risk as well as their declared ambition. Clearly, while risk tolerance is expressed in terms of a statistical moment or a quantile loss of the distribution of potential returns of the final portfolio, the ambition does not refer to any moment or quantile a priori. Yet, portfolio construction will allow achievement of the desired combination of risks so that the resulting model portfolio maximizes the probability that the return sought is achieved, that is the one corresponding to the equivalent lowest right-tail quantile. Robo-Advisors and digital wealth managers can gain three benefits:

1. encompass both tails of the distribution at once, moving out of the restrictions of the expected return;
2. assess convexity of products and portfolios in the joint assessment of risks and returns;
3. create a much larger set of goals and risk combinations, which operate on multiple time steps at once.

For example, Sironi (2015) reports in Figure 7.5 the case of a hypothetical moderate investor willing to take an opportunity (hence risk) in the short term, but achieving capital protection in the medium or long term (i.e., a balanced risk/return profile). Figure 7.6 illustrates the hypothetical case of a risk tolerant individual.

The method is sensitive to the setting of the risk tolerance profile as well as the ambition level. Having elicited a risk tolerance, investors can assess how strong their ambition appears when compared to existing market conditions and volatility levels. Given any portfolio that complies with the risk limit, the greater the ambition, the lower the probability of achieving the target. Clearly, different portfolios can exhibit a higher probability of achieving the target, yet comply with the risk constraint. The PSO process allows for multi-period verification of the risk limit, constraints, and objective function. Therefore, the time discretization assumption for compliance checks and rebalancing can be customized to fit wealth managers' or clients' preferences.

7.5.3 Generation of Scenarios and Scenario Paths

PSO is based upon the full revaluation of security prices, conditional on Monte Carlo scenarios of the underlying risk factors (e.g., inflation expectations, stock prices, term structures of interest rates, credit spreads). In Figure 7.7, a scenario h is a potential state of the world at a particular time, featuring a defined set of risk factors that take on values which are potentially different from their respective evaluation at the beginning of the holding period. A scenario path H is a sequence over time of scenarios h∈H and models the potential evolution of a set of risk factors at any point t along the investment horizon Γ. The set of all scenario paths H is denoted by S.

The measurable uncertainty about the realization of scenario returns is called risk. Conditional on the state of the risk factors in each future scenario, all securities can be repriced and their cashflows tracked to estimate their total return contribution to any portfolio performance over time.

7.5.4 Stochastic Simulation of Products and Portfolios Over Time

A Monte Carlo simulation can be computed for each of the elements of Ψ_U. This represents the future space of the potential total returns of each portfolio, as a linear combination of the potential total returns of each security conditional on scenarios h∈H∈S and weighted by its portfolio contribution, as in Figure 7.8.

7.5.5 Potential and Admissible Portfolios: Allocation Constraints

The scope of PSO is to investigate the density function of potential returns of all portfolios which are admissible, which means that they comply with a given set of allocation constraints and clients' preferences. V_U,H,t is an initial amount of capital to be invested across a universe of opportunities indicated by U, at time t = 0 and conditional on base scenario path H = 0. Φ_U indicates the space of elements identifying all the unrestricted potential portfolio allocations (i.e., the set of the vectors of the potential allocation weights on each of the j assets in the universe). The elements of Φ_U correspond to the individual percentage weights w_j,0,0 which are constant through scenario paths and time step, since only the fair value of each j security is allowed to change:

7.24

Investments can also be added and money can be withdrawn from existing allocations by modelling potential capital inflows and outflows over time (e.g., income streams, dividends, real estate investments), so that weights can change according to defined rebalancing rules. The amount invested in a particular category A of portfolio U (e.g., asset class, sector, or currency) can be floored (fragmentation limit a) or capped (concentration limit b), in order to avoid over-concentration or under-representation of specific names, sectors, regions, or currencies:

7.25

Ψ_U is the final set of admissible portfolios, that is the subset of the unrestricted potential portfolio compositions of Φ_U complying with the allocation constraints. The number of admissible portfolio allocations can grow exponentially with the number of the assets in U and the minuteness of the investment step size, ranging from a few millions to more than one sextillion. Therefore, techniques based on low discrepancy sequences allow us to alleviate the computational burdens without losing accuracy and meaningfulness. In Sironi (2015) we argued for the non-binding adoption of a lexicographical ordering to generate an ordered series of portfolios in an unambiguous canonical order that is similar, but not restricted, to an alphabetical representation (from which the name is taken). Similarly, the explicit list of the ordered asset allocations that make up Ψ_U can be generated by ordering the compositions in such a way that the order associated to each individual asset allocation in the portfolio is preserved throughout the sequence, and is normalized to the unit interval [0,1]. The low discrepancy sequence methods generate a sequence of draws from such a unit interval, in such a way that every draw is far away from the preceding, that is, clustering is avoided (groups of numbers close to each others), and that the draws are maximally avoiding each other, that is larger gaps are avoided. The resulting uniform distribution in the unit interval is used to derive the samples from the ordered universe of the admissible portfolios, that is the final set Ψ_U.

7.5.6 Adequate Portfolios: Risk Adequacy

Θ_U is the subset of the sampled admissible portfolios in Ψ_U which also comply with the risk limit definition. A risk limit can be imposed as a hard line or as a boundary condition, so that the risk measure (e.g., VaR) of the simulated portfolio is lower than a risk limit or falls within a target bandwidth at preselected time steps, as in Figure 7.9. The risk limit can be idiosyncratic or chosen out of a set of standard profiles that the wealth manager has created ex-ante and underlie the on-boarding risk assessment mechanism.

For a given confidence level 1-α, is a risk-limit function over the investment horizon Γ:

7.26

In the case of a hard limit, the probabilistic risk-limit function states a constraint on ξα_U,S,t which is the α-quantile profile of the investor applicable to portfolio returns R_U,H,t.

7.27

This translates into the following statement: the left quantile of the optimal portfolio at any selected time point t, with confidence interval 1-α, shall be contained within the investor's risk appetite .

Clearly, the risk-limit function can operate on a number of reallocation steps which is smaller than the one contained in the simulation framework. Thus, it can incorporate the form of a single point in time constraint. The optimization exercise will check the risk limit only at specified verification points and leave it open otherwise.

7.5.7 Objective Function: Probability Maximization

The optimization journey started with an initial space Φ_U of quasi-random potential allocations, reduced the dataset to a space of quasi-random Ψ_U admissible compositions, and further reduced the set to the risk-adequate initial allocations Θ_U. Thus, the objective function can now be imposed on the risk/return properties of the portfolios contained in Θ_U. The total return distribution of the investment returns over time which entered the optimization exercise can be plotted on digital tools without any discrepancy between the optimal asset allocation and the operational portfolio. Figure 7.10 shows a Monte Carlo simulation and the overlapping of the ambition line and the risk limit.

The optimization problem can be performed in the multi-period where t∈Γ. Hence, one needs to have a notion of preference that determines at which level of the target function a particular point in time dominates another. This can be achieved by introducing a multi-period weighting scheme K which is a vector of k∈K that allocates a positive weight at each t∈Γ. The weighting scheme is phrased rather generally and need not necessarily integrate to 1, since it might incorporate a normalization of the target function over time as well. Alternatively, one could estimate a more refined indicator of the multi-period probability by computing at the final investment horizon the conditional probability of reaching such a final step given the probability measurement at all previous allocation steps.

The process can be summarized in Table 7.1.

ΦU

Generate all potential portfolios

↓

ΨU

Identify only the admissible portfolios

↓

ΘU

Filter the risk-adequate portfolios

↓

Indicate the optimal goal based portfolio

Objective function: maximize the probability of complying with a minimum, time-dependent ambition or target return , subject to a multi-period weighting scheme K over a time horizon Γ and across all elements in Θ_U, so that:

7.28

in which, is the ambition line expressed as a total return function over the investment horizon Γ:

7.29

Clearly, the multi-period optimization can be turned into a discrete or single point in time optimization by assigning full weight to a discrete set of points or a single point only. However, if the weight is not allocated to a single point only, the construction of the weighting scheme should reflect the nature of the respective variable and its term structure to allow for meaningful results.

The probability measures, such as the probability of beating the investment goal or of falling short of the risk appetite limit, can be plotted for the optimal or any other portfolio (as in Figure 7.11).

Robo-Advisors performance over time can also be investigated to highlight whether the investment goals are still attainable, have been achieved, or are challenged by adverse market movements (as in Figure 7.12).

Robo-Advisors can indicate minimum probability targets for each time step and verify ex-ante and ex-post the compliance of both the strategic and the tactical asset allocations with the ambitions and risk appetite of final investors. This allows us to anticipate the needs for a proactive revision of the asset allocation, in case the investment has performed better than expected, as the market turned in favour of the elected strategy (indicating the possibility of cashing in and entering into a new portfolio to enhance investment returns) or in case the investment has under performed, as the market turned against the elected strategy or might not provide enough drift or volatility to achieve the stated ambition within the investment horizon (indicating the need to revise the asset allocation and optimize the timing of such a decision).

An appealing feature of PSO is that we can operate multiple problems without having to recalibrate the full set of simulation inputs: we can redefine time horizons, time steps, allocation constraints, client ambitions, or risk appetite levels and operate on the same stochastic distribution of the total returns of individual products. This should facilitate the institutionalization of the methodology across advisory networks, well outside the specialized desks of quantitative professionals, hence allowing sell-side institutions to deliver better and more transparent support to buy-side players. Moreover, we can identify an intuitive metric that permits us to compare strategic and tactical asset allocations with a certain level of intuition. Probability is such a measure, such as the probability of beating a financial goal, of yielding a minimum total return, or avoiding a capital loss.

7.5.8 Final Remarks on PSO

PSO is a fairly unrestricted framework based on exhaustive enumeration that can encompass a large variety of optimization exercises in terms of:

1. timeline discretization;
2. definition of the risk limit (e.g., VaR, Worst Case, Tail-Loss);
3. definition of the ambition (e.g., total return, asset value, income stream);
4. definition of the objective function (e.g., maximum probability of achieving a return target, maximum probability of affording an annuity);
5. a different combination of ambitions, risk tolerances, and time discretization.

In particular, the framework is suited to meaningful stress test analysis of operational and model portfolios because it is based on scenario analysis. Economic cycles can be conveniently modelled over time to test the robustness of investment policies and rebalancing assumptions, which can be a relevant input to Gamification exercises.

7.5.9 Conclusions

Goal Based Investing seeks to facilitate the implementation of investment policies which are more transparent and consistent with individual preferences and long-term ambitions. The probability of reaching or falling short of an investment target becomes a mainstream indicator to establish the adequacy of model portfolios with regard to individual ambitions and risk tolerance. Markowitz and his co-authors (Das et al., 2011) have recently innovated beyond the original Mean-Variance and Black-Litterman propositions, by replacing the standard deviation with the shortfall probability measure. Probabilistic Scenario Optimization provides a more flexible and consistent framework for the maximization of goal probabilities in the multi-period, within a risk constrained scenario simulation framework. The adoption of probabilistic scenarios requires thorough understanding of modern risk management techniques, based upon full revaluation methods of actual securities by means of multi-period stochastic simulations. The next chapter discusses scenario analysis in the context of educational Gamification, which is a relevant example of sustaining innovation that can enhance Robo-Advisors' propositions, and facilitate the understanding of the impact of personal investment behaviour with regard to otherwise complex investment decisions.