### Goal Based Investing and Financial Planning

Lately wealth managers have increased their focus on helping their clients achieve long-term financial goals in addition to generating high portfolio returns. Goal Based Investing (GBI) represents a holistic approach to portfolio construction and financial planning, which takes various investment and life-cycle parameters into account. Among other factors this process considers all current assets and liabilities of the investor, a projection of their income and expenses, and a risk profile which may change over time. The asset manager’s main objective is to facilitate the achievement of long-term goals by tracking the goal probability of an investor’s portfolio and rebalancing it appropriately over time.

### InCube’s Goal Based Advisory Solution

Traditional investment processes map investors onto one of several risk profiles with the help of questionnaires. This form of profiling may reveal the investor’s current risk appetite, but it generally does not reflect how well the corresponding investment strategy is aligned with long-term investment goals. InCube’s Digital Goal Based Advisory Solution offers interactive tools for a more targeted way of risk profiling. One can intuitively play with settings for risk and investment goals in a browser-based user interface. Our system computes an optimized wealth projection with various statistics on risk and reward within seconds. Through successive refinement of the parameters this process ultimately results in a bespoke strategic asset allocation for the investor and a plan for rebalancing their portfolio such that the goal achievement probability is maximal and in line with the risk constraints.

### Goal Based Investing vs. Markowitz Portfolios

In this series of blog posts we shall highlight various features of our approach to goal based investing. In this first article we are going to focus on quantitative measures for portfolio risk and utility which are supported by our portfolio optimization framework. We shall show that the choice of these measures can have a significant impact on the optimal portfolio allocation. We start by revisiting the standard approach to portfolio optimization initiated by Markowitz with Modern Portfolio Theory (MPT). Here risk and reward are defined in terms of variance and expectation, respectively. We then introduce different common measures that are more intuitive for the investor such as the expected shortfall and the probability of capital preservation.

### Case Study: Goal Based Investment with Two Assets

In this short study we present optimal portfolio allocations with respect to different risk and utility functions in a simple universe of two assets.

#### Instrument Universe

We consider two assets: the first one is a bond fund with an effective duration of approximately 8 years. The second instrument is a barrier reverse convertible (BRC), which is an exotic derivative. We employ a risk model given by 10’000 asset returns jointly sampled from historic data over three years and adjusted for forward-looking market expectations. We accumulate the returns over the three years under consideration. In the following, all quantities, such as expected returns or variances, are accumulated over the three-year horizon.

#### Statistical Properties of Instrument Universe

On the one hand, the bond fund shows an approximately symmetric distribution of accumulated returns. On the other, the BRC’s accumulated returns appear highly skewed. The respective histograms are depicted in Figures 1 and 2.

The statistical properties of these accumulated returns are listed in the table below. The returns of both assets exhibit similar mean and volatility. Both assets show a very low correlation of 9%. However, the exotic asset evidently shows more skewness.

Bond | BRC | |
---|---|---|

Mean | 3.6% | 3.2% |

Volatility | 9.8% | 10.3% |

Skewness | 0.2 | -4.8 |

#### Minimizing Risk versus Expected Returns

Suppose the investor wishes to maximize expected returns as in MPT. Since the returns of both assets show almost identical means, any convex combination of these assets will have nearly the same expected return. Hence, the focus of the investor will lie on minimizing risk. In the simple case of only two assets we can discretize the weight space and easily enumerate all valid portfolios consisting of two non-negative weights that sum up to one.

Two commonly used functions for quantifying risk are portfolio variance and the conditional value at risk (CVaR), often also referred to as expected shortfall. We use the 95% quantile for computing the CVaR. Figure 3 shows the value of these risk measures of the portfolio with respect to the weight of the BRC. As can be seen, both functions attain their minimum at significantly different locations: The variance is minimized for portfolios with almost equal weight on both assets. The conditional value at risk, however, is minimized for portfolios with approximately 20% weight on the BRC. Hence, the choice of the risk measure can have a substantial impact on the optimization result.

#### Maximizing the Goal Achievement Probability

Now consider an investor whose objective is **capital preservation**, that is, the investor wants to maximize the probability of having positive returns. Again, we can discretize the weight space and plot the value of this utility function for all portfolios:

The function in Figure 4 is evidently not convex. It is therefore unsuitable for conventional optimization frameworks. InCube’s goal based portfolio optimizer combines the power of convex optimization with high performance Monte Carlo simulations implemented on graphical processing units (GPUs). Thus, it can produce good solutions to utility functions with complicated shapes even in high dimensional weight spaces within seconds.

We conclude this case study by optimizing the goal achievement probability and computing efficient frontiers with respect to variance and CVaR. We compare the results of InCube’s GBI with conventional convex optimizers, which optimize for expected returns. When the investor wishes to bound portfolio variance, we compare GBI to classical mean variance optimization (MVO). When the investor limits CVaR, the competitor of GBI is mean CVaR optimization (MCVaRO).

#### Efficient Frontiers: GBI vs Mean-Variance Optimization (MVO)

The optimization results for various bounds on portfolio variance are depicted in Figures 5 and 6. Evidently, GBI achieves significantly higher goal-probabilities. For example, when the investor wants to limit the volatility of the accumulated returns to be bounded from above by 9.5 % of the initial investment, GBI allocates more than 80% into the BRC. However, MVO which maximizes expected returns invests only less than 20% in the BRC. As a result, the probability of capital preservation differs greatly: GBI achieves a probability of over 90 % for this risk bound, whereas MVO yields a goal probability of only approximately 65 %.

#### Efficient Frontiers: GBI vs. Mean-CVaR Optimization (MCVaRO)

We obtain similar results when the investor chooses to limit the portfolio’s CVaR instead. The results are presented in Figures 7 and 8. If, for example, the investor wants to limit the expected shortfall by 25 % of the initial portfolio value, GBI invests approximately 80 % into the BRC. On the other hand, MCVaRO, which maximizes expected returns, concentrates the entire weight into the bond fund for this risk bound. Consequently, GBI reaches a goal probability of over 85 %, whereas the conventional optimizer’s probability stays below 65 %.

#### Curse of Dimensionality

In both cases, GBI outperforms classical optimizers that maximize returns by a significant margin. As the space of admissible portfolios is rather small, the brute force enumeration algorithm succeeds to produce the true global optimum in all cases as expected. Clearly, the computational cost of extensive portfolio enumeration grows exponentially with the number of assets due to the curse of dimensionality. In higher dimensions our optimization framework employs techniques from convex optimization combined with Monte Carlo simulations accelerated using GPU technology.

### Conclusion

We have seen how the choice of the risk measure can have a significant impact on the optimal asset allocation. Therefore, goal-based portfolio optimization frameworks need to model risk constraints in an intuitive way to capture the investor’s true risk preference. InCube’s optimizer can express risk limits in terms of variance, (conditional) value-at-risk, or maximum drawdown. Moreover, it supports varying these limits along the investment horizon, for example, to reduce risk over time.

We have also demonstrated that portfolio allocations can be very far from the actual goal when they are optimized for expected returns. Thus, portfolio optimizers must allow for flexible utility functions which take the entire distribution of portfolio returns into account. InCube’s optimizer supports this with sophisticated numerical methods and GPU computing.

### Outlook

In our next blog, we will study the effect of time-varying risk profiles and expand the simple two asset allocation problem to a realistic portfolio optimization scenario. We shall explain how InCube’s advisory tools can aid in constructing truly goal-based bespoke strategic asset allocations.

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