What Investment Robots Need To Know – Evidence from Panel Survey Data

June 2016

Bernd SchererManaging Director, Deutsche Asset ManagementVisiting professor, WU WienResearch Associate, EDHEC Business School

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AbstractAutomated asset management offerings, so called investment robots (or robo-advisors), assign risky portfolios to individual investors based on investor characteristics such as age, net income or self-assessments of risk aversion. Using new German household panel data, we investigate the key household characteristics that predict financial market participation. This information allows us to assess which set of variables is most needed to model private portfolio decisions. Using heavily cross-validated classification trees, we find that a combination of household balance sheet variables – describing the ability to take risks (e.g. net wealth) – and household personal characteristics – describing the willingness to take risks (e.g. risk aversion) – best explain the cross sectional variation in financial market participation. This is also in line with models of portfolio choice under decreasing relative risk aversion and fixed investment costs. For robo-advisory firms, our results require a more holistic modelling of household characteristics. Including background risks in the form of household leverage does not only make investment sense, but is also the new regulatory reality under MIFID II rules. Robo-advisors are strongly advised to act accordingly.

JEL Classification: G11, C8

EDHEC is one of the top five business schools in France. Its reputation is built on the high quality of its faculty and the privileged relationship with professionals that the school has cultivated since its establishment in 1906. EDHEC Business School has decided to draw on its extensive knowledge of the professional environment and has therefore focused its research on themes that satisfy the needs of professionals.

EDHEC pursues an active research policy in the field of finance. EDHEC-Risk Institute carries out numerous research programmes in the areas of asset allocation and risk management in both the traditional and alternative investment universes.

Copyright © 2016 EDHEC

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1 - My Private Banking (2015) estimates the market for automated investment advice to have reached USD 20 billion by the end of 2015 and to grow to USD 450 billion by 2020. The effect of media attention given seems disproportionate, unless one has strong beliefs about the future of wealth management. The most prominent firms are Betterment and Wealthfront in the US. 2 - See Gollier (2001) for a general discussion on the impact of background risk on portfolio choice.3 - This limits the role of behavioural finance beyond overcoming typical biases in human decision making. 4 - Financial market non-participation, documented early on by Mankiv/Zeldes (1991), is now a stylised fact in empirical finance. It is somewhat puzzling, as it does not sit well with the equity risk premium puzzle (high historical risk premium offered by stocks).

1. Introduction Automated asset management offerings, so called investment robots (robo-advisors1) assign risky portfolios to individual investors based on investment algorithms. These algorithms use investor characteristics such as age, net income or assessments of individual risk aversion to recommend suitable asset allocations. So far, these solutions are deeply rooted in standard textbook asset-only portfolio choices. Outside wealth in the form of real estate or shadow assets (exogenously given and non-tradable assets like human capital) and household liabilities are largely neglected.2 Instead, the focus of robo-advisors is narrowly defined on providing cheap access to diversified beta. Visible (headline) cost savings (via lower fees) dominate the value proposition. The implicit costs of providing inadequate advice (loss in security equivalent) by failing to model differences in household characteristics (as summarised in household balance sheets) are ignored even though they could be many times larger than the anecdotal 50bps savings from ETF investments. Modelling client characteristics does not only make investment sense, but is also a legal requirement (e.g. “Know Your Client” under MIFID II) imposed by regulators under the impression of past mis-selling scandals. We acknowledge that investment advice is by definition normative: what households should do.3 This paper instead uses an empirical setting in trying to understand how the observable heterogeneity in household portfolio allocations is driven by heterogeneity in client characteristics, i.e. we ask: what are households actually doing? This is a difficult empirical exercise. Financial market participation is limited.4 We do not observe allocations into risky assets for the overwhelming majority of households (68% in our sample). Consequently, non-participation results in the loss of data points as we observe investments into risky assets for a mere 32% of households. In addition, observable allocations are bounded (allocations in financial markets are limited between 0 and 1 with many observations sitting at these boundaries) which creates a whole set of methodological problems. We therefore model financial market participation instead, in order to increase the number of usable data points. Implicitly we assume here that variables that drive the asset allocation decision (percentage allocation into a risky portfolio) will also drive the participation decision.

In Section 2, we delve deeper into the above conjecture. We derive a simple asset allocation model for investors with decreasing relative risk aversion (as a function of household equity, i.e. net wealth as a fraction of household assets) under fixed investment costs. We show that asset allocation and participation decisions are jointly made using the same set of variables. In addition, we can model non-participation as a result of risked preferences and fixed costs rather than a form of sinister social discrimination.

Section 3 describes the household characteristics that enter our empirical model for household participation. It adds a set of theoretically and empirically motivated variables to our base set of variables identified in our theoretical model. We use the HFCS panel dataset for German households provided by the ECB to find the (socio) economic drivers of financial market participation. In other words: which household characteristics allow us to correctly classify households into owners (“yes”) and non-owners (“no”) of risky assets:

Why do we believe this is an interesting research question? First, it would help providers of automated asset management platforms to identify the variables driving private investors’ asset allocation decisions. This should provide guidance for building more realistic asset allocation models. Second, unequal financial market participation is relevant for the distributional consequences of monetary policies creating asset price inflation (e.g. via quantitative easing). If richer investors hold more stocks, they would benefit disproportionally from those monetary

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policies.5 Third, it could help asset management product providers tailor their effort and offerings to clients with particular characteristics. Should the sales person arrive with an equity or bond pitchbook to maximise the odds of sales for a given client?

In Section 4, we describe our empirical methodology. Given the little theoretical guidance we have on the functional form of the relationship between financial characteristics and financial market participation, and also that we need highly predictive but interpretable results, we opt for classification trees. We offer a brief description on how to compute and interpret classification trees. Combining finance and methods from data science (cross-validated classification trees) differs markedly from previous (econometric) work that relied on full-sample analysis using a model that relied heavily on linearity instead. We believe that good models need to be able to generalise to new data sets, such that our conclusions remain robust. Cross validation is indispensable in this process.

Section 5 provides more details on our cross validation algorithm (to avoid data mining) and presents the results. We find that a combination of household balance sheet variables – describing the ability to take risks (e.g. net wealth) – and household personal characteristics – describing the willingness to take risks (e.g. risk aversion) – best explain the cross sectional variation in financial market participation. This is in line with our simple model of portfolio choice under decreasing relative risk aversion and fixed investment costs in Section 2. For robo-advisory firms this requires a more holistic modelling of household characteristics.

2. Financial Market Non-Participation and Portfolio ChoiceWe start with the standard one period mean variance model for portfolio choice in frictionless markets. All wealth is assumed to be held in financial assets.6 This results in well-known two fund separation. Each investor optimally holds a linear combination of a risky asset (for example the global market portfolio) and cash. We denote risk aversion by l, expected excess return (expected return in excess of cash after fees) by m and the risk (volatility) of excess returns by s. The demand for risky assets is given by (1)

while the remaining wealth ( ) is invested in cash, i.e. used for (de)leverage. As long as expected excess returns are positive, each investor would at least hold some fraction of his portfolio in risky securities. Financial market non-participation does not occur. In reality however, markets are far from frictionless.

One way to introduce non-participation is to endogenise it as the result of optimising agents. For this, we introduce declining relative risk aversion and fixed costs associated with participating in financial markets (opportunity costs of time spent, complexity costs, information costs). With decreasing relative risk aversion, lower wealth households become increasingly risk averse such that the expected return of these conservative portfolios does not cover the fixed costs of investing. Suppose (local) risk aversion is a function of base case risk aversion - given by - as well as current wealth, and minimum reservation wealth :

(2)

5 - The answer could raise concerns on fairness and put political doubt on European Central Bank independence even though it is economically easy to reconcile with decreasing relative risk aversion?6 - Note that introducing human capital (outside wealth representing background risk) in combination with no short constraints can also lead to non-participation in financial markets. Let q denote the fraction of financial wealth relative to total wealth (financial wealth plus human capital). The sensitivity of changes in the market value of human capital with respect to changes in market returns is given by b (imagine this to represent the regression beta of human capital returns on asset market returns). In this case

Unconstrained optimal portfolio weights become negative if .

This occurs when human capital contains a large beta component and simultaneously assumes a large fraction of total wealth, i.e. the household holds too much equity in its human capital. Combined with a long only constraint, this leads to non-participation (zero portfolio weight).

If household wealth approaches minimum reservation wealth, risk aversion approaches infinity: . Should wealth fall further below minimum reservation wealth, we can view

this as household bankruptcy. We can think of as the level of household liabilities. This allows us to interpret the difference as net household wealth (cushion against the uncertainties in life). Rising levels of household wealth in turn lets risk aversion approach base case risk aversion, i.e. . Equation (2) describes local risk aversion as a function of total wealth relative to total wealth (household balance sheet leverage). Substituting (2) into (1), we arrive at a new expression for optimal portfolio weights:

(3)

This reminds us of portfolio insurance. The optimal investment into a risky asset is defined as the product of an optimal multiplier times a state variable describing the percentage distance towards a portfolio floor.

To model non-participation we now introduce fixed costs. We know that risk averse households would only invest into financial markets if the expected payoff does at least cover the fixed costs associated with investing, . The corresponding condition is

(4) Substitute (3) into (4) and recall that the Sharpe-ratio is defined as . We can now solve for (net) wealth levels associated with positive investments into capital markets.

(5) Equation (5) defines the necessary level of net wealth to start financial market participation. Higher risk aversion and higher fixed costs raise this threshold level, while increasing investment opportunities (higher Sharpe-ratio) lower the required level of net wealth.7

Figure 1: Optimal investment into risky securities with decreasing relative risk aversionContour plots display the allocation into risky securities. We assume m = 0.05,s = 0.2, = 5000 . Portfolio allocations are calculated from (3), while the separation between capital market participation and non-participation is given by the straight line with slope ƒ-1SR2 in Panel (b).

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7 - Investment robots can reduce the number of non-participating households by lowering m (cheap ETF based access to diversified beta) or by reducing (individualised portfolio advice reduces complexity and monitoring costs).

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Demand for risky assets now becomes

Assume a globally diversified portfolio offers a Sharpe ratio of 0.25, risk aversion amounts to 5 and fixed costs sum up to 500 USD. In this case we require a minimum level of net wealth of 40,000 USD to find investing into financial markets worthwhile (its associated fixed costs).

Figure 1 illustrates the impact of net wealth and base line risk aversion on the optimal allocation into risky assets with and without fixed costs. Panel (a) plots allocations without fixed costs. As expected, low net wealth leads to small allocations into risky assets. However, as long as net wealth remains positive, portfolio allocations also assume positive weights. There is always financial market participation. This picture changes in Panel (b), where low net wealth and/or high risk aversion lead to zero allocations, i.e. nonparticipation. The separation between capital market participation and non-participation is given by the straight line with slope .

The above model is useful for two reasons. First, it models both the household ability and the capacity to take risks. As net wealth is likely to be affected by profession, net income, age, cognitive abilities, savings rate, i.e. a variety of variables found to be influential for household decision making in both the empirical and theoretical literature it can be viewed as a proxy for all these additional effects. Hence net wealth spans a variety of balance sheet related client characteristics. This is in stark contrast to virtually all robo-advisors, where balance sheet variables are painfully absent. Second, the model simultaneously derives optimal asset allocations as well as financial market participation decisions. The same variables that drive the participation decision, also drive the investment decision.

3. Data SelectionThe previous section introduced a parsimonious one period model with decreasing relative risk aversion and fixed costs for household portfolio choice. In the chosen setting we modelled non-participation as the result of rational choice by optimising agents, rather than an act of social injustice. Low net wealth and high risk aversion are the main drivers of financial market non-participation. At the minimum a model of stock market participation should look like

(7)

where (…) represents a set of additional – theoretically or empirically motivated – variables that could also help to explain the cross sectional variation in financial market participation. In this section we describe all variables we conjecture entering (7).

We use HFCS household balance sheet data for 3,565 German households.8 Our focus on German data originates in two arguments. First, Germany is the wealthiest country in Europe and is therefore of particular interest for asset management services. Second, and more importantly, the German data set contains self-assessment data for individual risk aversion, while most other data set do not.9

Financial market participation. Financial market participation is a binary (1/0) variable. Households that either invest into equities, bonds or mutual funds are classified as households participating in financial markets. In our HFCS dataset only 32% of all German households are classified as participating. In other words 68%

8 - Eurosystem Household Finance and Consumption Survey (HFCS) as described by European Central Bank (2012). 9 - For ease of replication we use the same data identifiers as described in the HFCS core and non-core variables catalogue provided by ECB (2012 a, b). We collect personal information on risk aversion (“HD1800”), age (“RA0300”), gender (“RA0200”), marital status (“PA0100”), household members and education (“PA0200”) for each household (given by its unique household ID). Financial wealth consists of cash ("HD1110"), savings accounts ("HD1210"), mutual funds ("HD1320g"), bonds ("HD1420") and shares (“HD1510"). For mutual fund holdings we have the additional information whether they are mostly equity ("HD1320a"). Net wealth ("DN3001") is available as already consolidated figure, i.e. we do not need to calculate it. We collect income (“DI2000”) and real estate ownership (“DA1100”, “DA1120”). Using outstanding mortgage (“DL1100”) we can calculate home leverage. Human capital is given as after tax disposable income multiplied with the number of years left until an assumed retirement age of 65.

of all households do not directly participate in the return opportunities of global asset markets.

Net wealth. Higher wealth leads to a larger ability to pay fixed financial market participation costs as discussed in Guiso et al (2003) and suggested in our model. Empirical studies by Vissing/Jorgenssen (2002) confirm that even small entry costs can explain low levels of financial market participation. For investors displaying decreasing relative risk aversion, higher net wealth not only makes the fixed costs associated with investing more affordable, but it also makes risk taking more desirable. Net wealth is defined as the difference between household assets and house hold liabilities. Nonfinancial wealth (human capital) does not enter the net wealth calculation. Net wealth is quantiled into 4 buckets of low, average, high and very high wealth. Each buckets consist of 25% of the available data. The 25%, 50% and 75% quantiles amount to 20, 147.8 and 378.5 thousand Euro.

Risk aversion. The willingness to take on risks (broadly described as risk aversion) is an important driver for the allocation to risky assets. Unfortunately, risk aversion is a latent (unobservable) variable. One can either look for observable and highly correlated client characteristics (answers to ad hoc questions or psychometric questionaires) or derive risk preferences from observable experimental or investment decisions (i.e. backing out risk aversion parameters or utility functions from a decision theoretic framework). The HFCS data provide the self-assessment of household decision makers. We find 37% of German households to self asses themselves as average risk aversion, 59% as high risk aversion, 3% as low risk aversion and 1% as very low risk aversion.

While our model provided us with the conjecture that high net wealth and low risk aversion households are more likely to participate in financial markets the empirical and theoretical literature10 suggest a variety of additional variables is still significant variation in the equity allocations across high net wealth investors in our data set. In addition, both the theoretical and empirical literature provides us with a set of additional variables that might also proof influential for household portfolio choice. We therefore add the following variables to explain household participation.

Income. Higher household income leads to both higher financial wealth (accumulated actual savings) as well as nonfinancial wealth in the form of human capital (discounted future saving). This leads to less leveraged household balance sheets. Wealthier households are likely to be more (financially) educated and suffer from less health risk (less background risk) due to better medical access or insurance cover. Also higher income facilitate to cover fixed costs of financial market investments. All these factors tend to be correlated with higher allocations to risky assets. Hence income might be a proxy for a variety of factors and is therefore one piece of information that is almost always required by robo-advice platforms. Income is calculated as income divided by the square root ofhousehold members and quantiled into 4 buckets. Each buckets consist of 25% of the available data. The 25%, 50% and 75% quantiles amount to 18.4, 30.5 and 50 thousand Euro.

Real Estate. Real Estate ownership is one store of household wealth. High levels of real estate in relation to financial wealth indicate substitution effects, i.e. real estate investments crowd out financial investments as discussed in Frantantoni (1998). This leads to unbalanced portfolios (in terms of sources of risk). While a large fraction of real estate relative to financial wealth implies low levels of financial market investments it would also necessitate aggressive allocations into more risky

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10 - Curcuru et al. (2004) provide a set of variables that could explain heterogeneities portfolios.

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assets within the remaining fraction of financial wealth. We classify real estate into 38% none, 29% very small (1-250), 20% small (250-500), 9% large (500-1000) and 4% very large (>1000).

Ratio of home value to mortgage. Leveraged real estate works as a background risk. Even high levels of net wealth can proof unstable if home values drop and rolling over mortgage debt becomes more expensive. Background risk (additional source of wealth variation) leads ceteris paribus to lower levels of investments into risky assets. We classify house leverage into 71% none 11% small (below 25%), 12% medium (25%to 75%) and 6% high (>75%).

Figure 2: Strength of Association between Participation and Explanatory VariablesWe plot Cramer’s V to measure the strength of association between financial market participation set of explanatory variables. Data source: HFCS data.

Figure 3: Strength of Association between Explanatory VariablesWe plot Cramer’s V to measure the strength of association between all explanatory variables identified. Data source: HFCS data.

Human capital as fraction of total wealth. Households endowed with labour income should – ceteris paribus – display larger equity allocations than households without labour income. First, because higher wealth could translate into lower relative risk aversion and hence larger demand for risky assets, and second as a high present value of labour income necessities investments into risky assets to meaningfully diversify labour income risk that represents a large fraction of total wealth. This result will also depend on the nature of

un-insurable labour market risk as put forward by Heaton/Lucas (2000). Capital market related labour income variations will reduce investments that increase total household risk. We calculate human capital as the present value of future savings (up to retirement) divided total wealth (including human capital). We classify human capital into 42% small (0-25%), 14% medium (25%-50%), 17% high (50%-75%) and 27% high (>75%).

Education. A lack in cognitive abilities might create a psychological barrier to participate in financial markets. Unfamiliarity with a complex subject such as investing also raises costs (measured in time and money) for low skill households and hence leads to lower levels of investments. Grinblatt (2011) shows that cognitive skills lower information costs and hence increase the likelihood of participating in financial markets. Campbell (2006) finds evidence that stock market participation positively correlates with education. Finally lower skills lead to lower wealth accumulation as argued in Hsu (2012). If households also display decreasing relative risk aversion, optimal demand for risky assets will decrease with wealth levels as local risk aversion increases. We classify household education (depending on the education of the household earner) into 1% primary, 8% lower secondary, 52% higher and 38% tertiary.

Age. We already established that investor’s willingness to take risks depends might depend on education levels. Having been trained by the University of life, older people have accumulated experience and investment know how during the course of their lives. We would hence expect that allocations to risky assets increase – ceteris paribus - with age as conjectured by King/Leape (1987). We classify age into 10% under 30, 30% between 30 and 50, 30% between 50 and 65 and 30% above 65.

Marital status. Married couples with outside labour income might enjoy income diversification via relatively uncorrelated income streams. If we view this as a reduction in background risk, it will lead to larger equity allocations. However, this might be counterbalanced by divorce risk as argued in SCHERER (2013). We classify marital status into 35% single households and 65% married households. In order to measure the association between financial market participation and our set of explanatory variables as well as between our explanatory variables we use Cramer's V. It is a scaled measure of association with a maximum value of 1 where unassociated variables result in a Cramer’s V of 0, based on the Chi-square independence test. Suppose we observe n pairs of two variables that are classified into c1 and c2 distinct classes. Cramer’s V is given as

(8)

where χ2 is the chi-square independence test statistic. Cramer’s V is by definition nonnegative.Figure 2 we see that net wealth and risk aversion display the highest measures of association, closely followed by income. This coincides with both our theoretical model as well as with our intuition on the role of income in characterising the financial health of a given household. Figure 3 looks how closely associated our explanatory variables are. Again, we find income strongly associated with measures of wealth (net wealth or real estate).

4. MethodologyOur objective is to classify German households into financial market participating and non-participating households. At the same time we need an interpretable model. The later objective is

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important in the context of modelling advice for investment robots. We therefore need a model combining predictive accuracy as well as human interpretability (what are the main influential predictors?). Our task is complicated by the absence of theoretical guidance on the form of a functional relationship between household characteristics and equity participation. We therefore look for a nonparametric method that can deal with nonlinear relations using categorical covariates with high predictive accuracy. This leads us to classification trees advanced by Breiman et al (1984), which have increasingly found their way into data science applications (finding structures in large datasets). Technical discussions of classification trees can be found in numerous textbooks.11 Instead, we provide a brief intuitive introduction using our sample data.

How can we interpret classification trees? Suppose we want to classify household equity participation using risk aversion and age alone. The results for a binary tree are shown in Figure 4. The whole sample of 3,467 households (remove observations with missing data) is first split into 2,052 households with high risk aversion and 1,415 households with low risk aversions. Risk aversion is the dominant (main) effect as all high risk aversion nods (8 and 9) display a lower fraction of equity participation than average to low risk aversion nods (3, 5 and 6). However within a given risk aversion bucket, higher age leads to increased equity market participation. This is consistent with age related growth in investment experience and net wealth as well as reduced career uncertainty (background risk).

How did we derive this classification trees? Put differently: How did we split the data into a tree like structure? First, we test for independence between any predictors and the response. Stop building the tree further, if the null hypothesis of independence can’t be rejected (this is different to earlier tree method’s that relied on pruning to cut tree sizes. Here tree sizes depend on the level of statistical significance. Select the input variable

Figure 4 – Financial market participation (full sample classification tree)Financial market participation classified by a binary Classification tree (R implementation ctree) for German HFCS household panel data. Binary growing if statistically significant splits can no longer be found (p-value of 0.05).

adjusted p-value corresponding to the independence test). This step determines both the predictor variable used as well as its cutoff value. The purity of nods (more and more households within the same class are allocated to a given nod) - measured by entropy - increases with each significant split. Second, implement a binary split for the selected predictor. Third, continue recursively until no statistically significant split is left available (critical p-value of 0.05).

11 - See Hastie/TishraniI/Friedman (2001) as the classical reference.

How can we make predictions from a tree? At each terminal nod (3, 5, 6, 8, 9) we can calculate both class probabilities (number of households in this nod participating in equities, divided by total number of households in this nod) as well as a classification label attached to all households falling into this nod.

(9)

A household falling into a given nod will be classified as participating in equity markets if the classprobability of its nod exceeds 50%.

(10)

Household ending up wrongly in a given nod will therefore be misclassified. The number of misclassified households will be small if nods are pure. In our example this amounts to the following class probabilities and class assignments in Figure 5.

For some applications, class probabilities are more informative than classifications. Suppose we need a model to decide which households are likely to be more profitable clients than others in order to allocate acquisition costs to the more likely customers. Depending on costs and likely profits of client acquisition a bank might want to contact households displaying a much lower class probability than 50%.

Figure 5 – Classification and associated class probabilitiesFinancial market participation classified by the binary Classification tree for German HFCS household panel data from Figure 4.

How can we evaluate the predictive accuracy of a given classification tree? In a two class classification problem we typically start with a 2x2 confusion matrix that contains the number of correctly classified “yes” and “no” cases (true positives and true negatives) as well as the number of incorrectly classified ”yes” and “no” cases (false positives and false negatives. In our example the confusion matrix looks like Figure 6.

Figure 6 – Predictive AccuracyIn-sample predictive accuracy for the binary Classification tree for German HFCS household panel data from Figure 4.

Our predictions show an accuracy of 74% (# of true positive and true negative classifications divided by # of households). This looks higher than it is. Even without any predictive accuracy (always classifying a household as “no”) we would have obtained an accuracy of 68.7%. While the improvement over the naive case looks small now, is it at least significant? The 95% confidence band for a binomial test (number of successful predictions) is (0.72, 0.75), i.e. our improvement in the accuracy rate is statistically significant at the 95% confidence level.

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How can we improve the tree fitting process? Classification trees are prone to overfitting i.e. learning both the signal and the noise of the training data set. Few observations might be responsible that a given predictor and/or cutoff value has been chosen. A slight variation of the data nearer to the top of a classification tree, then lead to different top level choices that will alter the whole sequence of nods affecting chosen predictors and cutoff values. This will make classification trees instable across samples, i.e. limit their interpretability. Also, applied to a new data set that contains (hopefully) the same signal but certainly new noise, the predictive ability will drop well below the precision displayed within the training data. To limit these dangers we will use cross validation. The empirical section will contain more information on this.

5. Empirical ResultsNonlinear models with large degrees of freedom (many variables, specifications), used in-sample (applied to all available data) are likely to yield overly optimistic classifications. As a consequence, in-sample classification accuracy will be impossible to repeat for newly arriving data (i.e. out-of-sample).

However, this is exactly the purpose of a good predictive model. For our classification tree consider the impact of two parameters to (over-) adjust to a given dataset: maximum tree size and number of variables. In order to build models, that extract the signal in the data rather than adjusting to noise, we apply the following algorithm:

First, we allow an exhaustive set of variable combinations. The number of distinct variable combinations (models) out of n variables can be computed as

This amounts to 511 for n = 9 in our dataset (described in the previous section). Second, for each selection of predictors we run repeated k-fold cross validation. This involves the following steps.

Step 1. Impose a maximum tree size from 1 (stump, i.e. 3 nodes) to 5.

Step 2. Partition the data into k=5 non-overlapping subsets. Take k-1=4 subsets to train a classification tree (in-sample) and apply its predictions to the remaining test data (out-of-sample) set. Calculate and store the kappa statistic as a performance measure for the out-of-sample predictions. The kappa statistic is given as the difference between observed accuracy and expected accuracy (from the marginal of the confusion matrix as in a Chi-square independence test) divided by one minus expected accuracy. It takes a value between -1 and 1 where 0 means no predictive ability. Repeat this k times such that every subset has been used as a test set once (Calculate k=5 classification trees).

Step 3. Repeat this k-fold process 50 times and store the kappa statistics

Step 4. Repeat steps for a maximum tree size ranging from 1 to 5.This results into 511 average kappa values (each calculated from 250 out-of-sample values foreach maximum tree size).

In the third step we choose model specification and tree size by looking for the best estimated (averaged) out-of-sample kappa. Finally once we found model and tree size, we refit the classification tree to the whole data set to arrive at the best classify newly arriving data points.Our results are provided in Figure 7.12

12 - All calculations are performed using ctree() from the R package rparty.

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Figure 7 – Financial Market Participation of German Households (Cross-validated Classification Tree)Final classification tree (fit to full sample) after cross validation. Market participation is classified by a binary Classification tree (R implementation German HFCS household panel data. Binary split. Tree stops growing if statistically significant splits can no longer be found (p-value of 0.05). Repeated 50 times, five-fold cross validation where participating and non-participating households are drawn separately to maintain sample proportions.

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The top variable split in our 4 layer classification tree for predicting financial market participation is net wealth. High net wealth households are more likely to participate in financial markets for almost all other household characteristics. The only exception are highly educated average net wealth households that display low risk aversion. For these the complexity costs of investing in financial markets are low. The lowest participation is displayed by low net worth and high risk aversion households. For higher net wealth households, risk aversion is the next watershed. Low risk aversion leads to higher participation. At a more granular level, higher education also adds further to the likelihood of participation. Interestingly, household with low levels of real estate also invest into financial markets despite higher risk aversion. Here financial assets are less likely to be crowded out by real estate. Our results are consistent with a model of portfolio choice under decreasing relative risk aversion and fixed costs of investment. How robust is this model if we replace financial market participation with either equity or bond market participation instead? Are similar patterns arising?

Figure 8 displays the key variables driving equity market participation. Cross validation results in a much smaller sized tree with only two variables. The result is intuitive and easy to summarise: the likelihood of equity market participation increases with lower risk aversion and higher wealth. Despite running the data through a complex machine learning algorithm, the resulting classification tree confirms the intuition from our model in Section 2. Risk aversion is not the dominant variable but instead interacts with net wealth. High risk aversion households with high net wealth display a larger likelihood for equity participation than low risk aversion households with low net wealth. Independent of risk aversion, higher net wealth will ceteris paribus result in increased equity market participation. This effect is amplified for lower risk aversion households. We do not make a statement of causality as we do not know whether higher equity participation also led to higher net wealth. Relative to the decision tree in Figure 7, risk aversion becomes themost important split for equity investments.

Figure 8 - Equity Market Participation of German Households (Cross-validated Classification Tree)Final classification tree (fit to full sample) after cross validation. Equity market participation is classified by a binary Classification tree (R implementation for German HFCS household panel data. Binary split. Tree stops growing if statistically significant splits can no longer be found (p-value of 0.05). Repeated 50 times, five-fold cross validation where participating and non-participating households are drawn separately to maintain sample proportions.

Figure 9 shows the cross-validated classification tree for bond market participation. Here, net wealth is established as the most important split. Very high net wealth individuals are likely to hold bonds, particular if they are beyond their 50s. This is consistent with bonds as storage of wealth designed to fund future liabilities in the form of household expenditures. High net wealth

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households in particular suffer from a pension gap (the difference between retirement income relative to aspirational levels for consumption habits formed during periods of high income). Neither education nor real estate ownership play a role for these households.

The results confirm the conjecture of our simple asset allocation model for all three forms of household classification. This is remarkable as all classification trees undergo a rigorous cross-validation exercise and are not simply fit in-sample. This also explains the parsimonious structure with a limited number of nods and branches. In-sample trees would deliver a much more granular structure but do not generalise well across subsets they have not been trained on. However, classification with a cross-validated model is a difficult exercise. The improvements over a naive model are not always breathtaking.

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Figure 9 – Bond Ownership of German Households (Cross-validated Classification Tree)Final classification tree (fit to full sample) after cross validation. Bond ownership is classified by a binary Classification tree (R implementation German HFCS household panel data. Binary split. Tree stops growing if statistically significant splits can no longer be found (p-value of 0.05). Repeated 50 times, five-fold cross validation where participating and non-participating households are drawn separately to maintain sample proportions.

6. ConclusionsWhile portfolio advice is, by definition, about “what investors should do” we undertake a mainly empirical approach, i.e. we ask “what do investors actually do?”. Using heavily cross-validated classification trees, we find that a combination of household balance sheet variables – describing the ability to take risks (e.g. net wealth) – and household personal characteristics – describing the willingness to take risks (e.g. risk aversion) – best explain the cross sectional variation in financial market participation. This is in line with a simple model of portfolio choice under decreasing relative risk aversion and fixed investment costs. Our result translates well to both equity and bond market participation. For robo-advisory firms, this requires a more holistic modelling of household characteristics. Including background risks in the form of household leverage does notonly make investment sense, but is also the new regulatory reality under MIFID II rules. Robo-advisors are strongly advised to act accordingly.

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