Scoring and Predicting Risk Preferences

Abstract

This study presents a methodology to determine risk scores of individuals, for a given financial risk preference survey. To this end, we use a regression-based iterative algorithm to determine the weights for survey questions in the scoring process. Next, we generate classification models to classify individuals into risk-averse and risk-seeking categories, using a subset of survey questions. We illustrate the methodology through a sample survey with 656 respondents. We find that the demographic (indirect) questions can be almost as successful as risk-related (direct) questions in predicting risk preference classes of respondents. Using a decision-tree based classification model, we discuss how one can generate actionable business rules based on the findings.

Appendices

Appendix A: Selected Survey Questions

Following are selected direct (risk-related) questions from the survey of the case study, which constitute the corresponding direct (risk-related) attributes.

Q34

Over the long term, typically, investments which are more volatile (i.e., that tend to fluctuate more in value) have greater potential for return (Stocks, for example, have high volatility; whereas government bonds have low volatility). Given this trade-off, what would be the level of volatility you would prefer for your investment?

• a Less than 3%

• b 3% to 5%

• c 5% to 7%

• d 7% to 13%

• e More than 13%

Q32

What is your most important investment priority?

• a I aim to protect my capital; I cannot stand losing money.

• b I am OK with small growth; I cannot take much risk.

• c I aim for an investment that delivers the market return rate.

• d I want higher than market return; I am OK with volatility.

• e Return is the most important for me. I am ready to take high risk for high return.

Q23

Compared to others, how do you rate your willingness to take risk?

• a Very low

• b Low

• c Average

• d High

• e Very high

Q42

What is your most preferred investment strategy?

• a I want my investments to be secure. I also need my investments to provide me with modest income now, or to fund a large expense within the next few years.

• b I want my investments to grow and I am less concerned about income. I am comfortable with moderate market fluctuations.

• c I am more interested in having my investments grow over the long-term. I am comfortable with short-term return volatility.

• d I want long-term aggressive growth and I am willing to accept significant short-term market fluctuations.

Appendix B: ScoringAlgorithm

Following is the mathematical presentation of the developed scoring algorithm:

Sets

${\mathcal{I}}$::

set of respondents (observations, rows) in the sample; i=1,…,I

${\mathcal{J}}$::

set of attributes (questions, columns); j=1,…,J

${\mathcal{V}}$::

set of ordinal values for each attribute; v=1,…,V. For the presented case study, ${\mathcal{V}}= (a,b,c,d,e )$, where a≤b≤c≤d≤e

Inputs

$\mathbf{O}={[o_{ij}]}_{I\times J}$::

matrix of ordinal values of all attributes for all respondents

m _j ::

number of possible ordinal values for attribute j; m _j ≤5 in this study

Internal Variables

$\mathbf{A}={[a_{ij}]}_{I\times J}$::

matrix of numerical (nominal) values of all attributes for all respondents

y _i ::

temporary adjusted risk score for respondent i, to be used in regression

Parameters

E::

threshold on absolute percentage error (falling below this value will terminate the algorithm)

α::

threshold for type-1 error (probability of rejecting a hypothesis when the hypothesis is in fact true)

M::

a very large number

$\mathbf{B}$::

transformation matrix for converting the ordinal input value matrix ${\mathbf{O}}$ into the numerical (nominal) value matrix ${\mathbf{A}}$

Outputs

z _j ::

whether attribute j is to be included in computing the risk score; z _j ∈{0,1}

w _j ::

weight for attribute j; w _j ≥0

β _0j ::

intercept value for attribute j

β _1j ::

slope value for attribute j

Γ _j ::

sign multiplier for attribute j; Γ _j ∈{−1,1}

x _i ::

risk score for respondent i

Functions

$f (v,n ):({\mathcal{V}}, \{2,\ldots,V \})\to [0,3]$: mapping function for an attribute with n possible values, that transforms the ordinal value v collected for that attribute to a nominal value b _v,n−1.

$$f (v,n )=b_{v,n-1}$$

where, for V=5,

$$\mathbf{B}=[b_{vn}]_{V\times(V-1)}=\left [\begin{array}{c@{\quad}c@{\quad}c@{\quad}c}0.00 & 0.00 & 0.00 & 0.00 \\3.00 & 1.50 & 1.00 & 0.75 \\\cdot & 3.00 & 2.00 & 1.50 \\\cdot & \cdot & 3.00 & 2.25 \\\cdot & \cdot & \cdot & 3.00\end{array}\right ]$$

$\mathit{regression}(\mathbf{y},\mathbf{a}')$

solve regression model $\mathbf{y}={\beta}_{0}+{\beta }_{1}\mathbf{a}'+\varepsilon$ for vectors $\mathbf{y}$ and $\mathbf{a}'$

return (p, β ₀,β ₁), where p is the p-value for the regression model

$\mathit{preprocess}()$

// transform ordinal attribute values to nominal values

$a_{ij}={\mathbf{\ }}f (o_{ij},m_{j} ); \forall(i,j)\in\ {\mathcal{I}}\times{\mathcal{J}}$

Iteration-Related Notation

k::

iteration count

N::

number of attributes included in risk score computations at a given iteration

W::

sum of weights for attributes

ε _k ::

absolute error at a given iteration k

e _k ::

absolute percentage error at a given iteration k

$\overline{e}_{k}$::

average absolute percentage error at a given iteration k

$\mathit{ScoringAlgorithm}({\mathbf{O}},\ m_{j} )$

BEGIN

// perform pre-processing to transform ordinal data to nominal data

$\mathit{preprocess}()$

// initialization:

// initially, all attributes are included in scoring,

// with unit weight of 1 and sign multiplier of 1.

// all of the regression intercepts are 0.

z _j =1, w _j =1, Γ _j =1, β _0j =0; $\forall j\in{\mathcal{J}}$

N=∑ _j z _j

// begin with iteration count of 1

k=1

$\mathit{Begin\_Iteration}$

// standardize the weights, so that their sum W will equal to N

W=∑ _j w _j z _j

w _j ←(Nw _j )/W; $\forall j\in{\mathcal{J}}$

// compute the average of the intercepts

${\overline{\beta}}_{0\cdot}={(\sum_{j}{{\beta}_{0j}z_{j}})}/{N}$

// compute/update the risk scores at iteration k,

// which is composed of the average intercept value

// and the sum of weighted values for attributes

$x_{ik}=\overline{\beta}_{0\cdot}+\sum_{j}{\varGamma }_{j}$ w _j a _ij ; $\forall i\in{\mathcal{I}}$

// compute total absolute error

ε _k =∑ _i |x _ik −x _i,k−1|

// correction for the initial error values

if k=1 then

 ε ₀=ε ₁

// termination condition

if ε _k =0 then

 go to $\mathit{Iterations\_Completed}$

// compute absolute percentage error,

// and then its average over the last two iterations

${\overline{x}}_{\cdot k}={\sum_{i}{x_{ik}}}/{I}$

$e_{k}={100{\varepsilon}_{k}}/{{\overline{x}}_{\cdot k}}$

${\overline{e}}_{k}=(e_{k}+e_{k-1})/2$

// if the stopping criterion is satisfied, terminate the algorithm

if ${\overline{e}}_{k}<E$ then

 go to $\mathit{Iterations\_Completed}$

// otherwise, continue with the regression modeling for each attribute j,

// and then go to next iteration

$\forall j\in{\mathcal{J}}$

 // if the attribute is included in the risk score calculation

 if z _j =1 then

  // first remove the attribute value from the incumbent score

  // to eliminate its effect

  $y_{i}=x_{ik}-a_{ij};\ \forall i\in{\mathcal{I}}$

  // then define the vectors for the regression model of that attribute

  $\mathbf{y}= (y_{i} )$; $\mathbf{a}'=(\varGamma _{j} a_{\cdot j})$

  $(p,\ {\beta}_{0},{\beta}_{1} )=\mathit {regression}(\mathbf{y},\mathbf{a}')$

  // if the regression yields a high p value

  // that is greater than the type-1 error,

  // this means that attribute j does not contribute significantly

  // to the risk scores

  if p>α then

   // and the attribute should not be included in risk calculations

   z _j =0

  else

   // else it will be included (will just keep its default value)

   z _j =1

   // and weight for the attribute will be the slope value

   // obtained from the regression

   w _j =β ₁

   // the sign of the slope is important;

   // if it is negative, this should be noted

   if β ₁<0 then

    // record the sign change in the sign multiplier

    Γ _j =−1

   else

    Γ _j =1

// advance the iteration count and begin the next iteration

k++

go to $\mathit{Begin\_Iteration}$

$\mathit{Iterations\_Completed}$

x _i =x _ik

return x _i , z _j ,w _j ,Γ _j , β _0j

END