# Measuring Financial Advice: aligning client elicited and revealed risk

John R.J. Thompson, Longlong Feng, R. Mark Reesor,
Chuck Grace, and Adam Metzler
Department of Mathematics
Wilfrid Laurier University
Waterloo, Ontario, Canada N2L 3C5

## Abstract

Financial advisors use questionnaires and discussions with clients to
determine a suitable portfolio of assets that will allow clients to reach
their investment objectives. Financial institutions assign risk ratings to
each security they offer, and those ratings are used to guide clients and
advisors to choose an investment portfolio risk that suits their stated risk
tolerance. This paper compares client Know Your Client (KYC) profile risk
allocations to their investment portfolio risk selections using a value-at-
risk discrepancy methodology. Value-at-risk is used to measure elicited and
revealed risk to show whether clients are over-risked or under-risked, changes
in KYC risk lead to changes in portfolio configuration, and cash flow affects
a client’s portfolio risk. We demonstrate the effectiveness of value-at-risk
at measuring clients’ elicited and revealed risk on a dataset provided by a
private Canadian financial dealership of over $50,000$ accounts for over
$27,000$ clients and $300$ advisors. By measuring both elicited and revealed
risk using the same measure, we can determine how well a client’s portfolio
aligns with their stated goals. We believe that using value-at-risk to measure
client risk provides valuable insight to advisors to ensure that their
practice is KYC compliant, to better tailor their client portfolios to stated
goals, communicate advice to clients to either align their portfolios to
stated goals or refresh their goals, and to monitor changes to the clients’
risk positions across their practice.

Keywords: Risk measures, Value-at-risk, Portfolio management, Financial
advice, Client-advisor relationship

John R.J. Thompson

Department of Mathematics

Wilfrid Laurier University

Waterloo, Ontario N2L 3C5

<EMAIL_ADDRESS>
Longlong Feng

Department of Mathematics

Wilfrid Laurier University

Waterloo, Ontario N2L 3C5

<EMAIL_ADDRESS>
R. Mark Reesor

Department of Mathematics

Wilfrid Laurier University

Waterloo, Ontario N2L 3C5

<EMAIL_ADDRESS>
Chuck Grace

Department of Finance

Ivey Business School

London, Ontario N6G 0N1

<EMAIL_ADDRESS>
Adam Metzler

Department of Mathematics

Wilfrid Laurier University

Waterloo, Ontario N2L 3C5

<EMAIL_ADDRESS>
##  1 Introduction

In previous research, we find that that Know Your Client (KYC)
information–such as gender, residence region, and marital status–does not
explain client investment behaviours, whereas eight variables for trade and
transaction frequency and volume are more informative [Thompson et al., 2021].
With these findings that trading behaviour (recency and volume patterns) is
linked to actual client risk preference and capacity, our recommendation to
financial regulators and advisors is to use metrics more advanced than those
gleaned from KYC information to understand clients better. The dataset used in
that paper spanned from January 1st 2019 to August 12th 2019, and consisted of
daily KYC information (age, gender, annual income, risk tolerance) and
financial information (account type, total value of that account, proportion
of total value invested into different securities) for $50,980$ accounts of
$27,644$ clients and 311 advisors.

This paper demonstrates how value-at-risk (VaR) can be used as an advanced
metric by financial advisors to manage their clients’ portfolio risks. We show
that VaR provides unique real-time empirical insights that can help
stakeholders visualize gaps between the elicited KYC-derived profile risk and
the actual portfolio risk revealed by trading behaviours. We examine how
clients’ trading behaviour and cash flow impact their portfolios and use VaR
to quantify how much of each client’s portfolio is genuinely “at risk” and how
far clients are from their stated risk preferences. While part of our
contribution to the literature is a methodology to estimate clients’ risk
preferences, our primary contribution is a methodology to measure the
difference in elicited and revealed risk tolerance.

In Canada, financial advisors and dealers are required by provincial
securities commissions and self-regulatory organizations to collect and
maintain KYC information for investor accounts. With this information,
investors, under their advisor’s guidance, decide on investments that are
presumed to be beneficial to their investment goals and therefore suitable.
Suitability is described by regulators in Canada as a “meaningful dialogue
with the client to obtain a solid understanding of the client’s investment
needs and objectives $\ldots$” [Ontario Securities Commission, 2014], and has
been updated to better serve client needs [Ontario Securities Commission,
2019]. The assumption is that any investment purchases or sales (trading
behaviour) will conform to the KYC attributes and therefore be suitable.

It should be noted that suitability must be informed by both Know Your Product
(KYP) and KYC attributes. Part of the KYP information on securities made
available by the dealership is a risk rating [Ontario Securities Commission,
2009]. These risk ratings are used to place products into risk buckets (for
example, low, medium, or high risk). The KYC attributes give a portfolio
selection across risk buckets, followed by a selection of products within each
risk bucket. Our analysis evaluates a client’s portfolio based on their KYP
attributes–at least in terms of their risk and return characteristics. Our
measures do not delve deeper to complete comprehensive due diligence of the
investment products as would be prescribed by the regulators.

Our analysis shows that advisors and clients use the KYC prescribed risk as a
”guard rail” against which they define–and do not generally exceed–upper and
lower boundaries. However, between the ”guard rails”, they tailor and
customize portfolios in a manner that is consistent with their clients’
elicited KYC risks. Do our findings suggest negative consequences for
investors? We find that advisors are systemically safe and conservative and
thus do not appear to expose clients to undue risk. For clients seeking to
preserve capital, that is good news, but clients seeking to maximize growth
may not be exposed to enough risk to achieve their expressed goals. Under-
risked accounts are just as problematic as over-risked, but easier for
advisors to defend to regulators and lawyers given there is no significant
loss of funds. An example might be older clients who can preserve their
capital but are unable to achieve investment incomes that allow them to
maintain their lifestyles. An accurate answer to the question of client impact
can only be answered through the lens of the advisor/client interactions–which
are by necessity private and confidential.

Every client has a unique story, and we present methods that can digitize
their story to improve personalization, transparency, timeliness and
visualization. Our methodology falls into the interest realms of advisors,
regulators, and robo-advisors; (1) advisors can implement these mathematical
methods into their practice111We have included an example Excel spreadsheet as
part of this article, and this sheet is available upon request from the
corresponding author., (2) regulators will be able to identify inconsistencies
in practitioners’ portfolios, and (3) robo-advisors can use these methods to
help advisors monitor their clients’ portfolios.

The paper from here unfolds as such: Section 2 provides the academic context
for our methodology in the broad area of measuring risk tolerance. Section 3
is an overview of the dataset provided by a private Canadian financial
dealership. Section 4 is a study of financial methodologies, including VaR, to
measure the discrepancy between profile and portfolio risk. Section 5 shows
the results of using the VaR methodology for measuring and comparing risk for
clients, advisors, and dealerships. Lastly, Section 6 summarizes our results
and provides a broader discussion for the impacts of the methodology presented
herein.

##  2 Context

One goal of this paper is to bridge the gap between the ivory tower of
academic computational behavioural finance and the real work of the financial
agent–a client, advisor, dealership, regulator, or otherwise. This section
provides the academic context of how behavioural finance has been applied to
understand risk tolerance from questionnaires and trading behaviour, point to
real applications of behavioural concepts that have directly benefited
practitioners, and elaborate on how our work is helpful to both academics and
practitioners.

###  2.1 Behavioural finance: elicited and revealed risk

In behavioural economics, there is a large body of research dedicated to
measuring and understanding the difference between elicited and revealed
preferences of consumers, originated by Samuelson [1948]. In essence, elicited
preferences are what people say they prefer, whereas revealed preferences are
what they actually prefer, as shown through their behaviours. To apply this
concept in finance, elicited and revealed preferences for financial risk were
developed, with focuses on estimating the utility of wealth of consumers based
on their investing behaviour [Dybvig and Rogers, 1997], applications of
prospect theory [Kahneman and Tversky, 2013], understanding the perceptions of
risk [Diacon and Ennew, 2001], the effect of cognitive ability on risk
preferences [Guillemette et al., 2015], and–the focus of this paper–measuring
risk preferences.

Elicited risk in investment dealerships is typically collected via
questionnaires administered in-person yearly by financial advisors to clients.
In conjunction with the conversations surrounding the questionnaire, the
responses to the questionnaire are used to calculate risk preferences. Finke
and Guillemette [2016] provides a modern review of eliciting risk tolerance
from questionnaires; their summative findings include measuring choices
surrounding income risk and volatility to assess risk tolerance [Guillemette
et al., 2012], financial literacy effects on risk assessment [Linciano and
Soccorso, 2012], and emotional responses to risk and loss aversion
[Loewenstein et al., 2001, Grable and Lytton, 2001, Michael et al., 2015].
Wahl and Kirchler [2020] employed a modern questionnaire methodology that
measures risk propensity, attitude, capacity, and knowledge to elicit risk
tolerance from clients. Our dataset described in Section 3 contains the risk
elicited using a questionnaire under the KYC obligation guidelines. A risk
score is determined from the questionnaire’s responses by calculating a
weighted sum based on their answers. That risk score is then transformed by
the advisor administering the questionnaire into a profile risk allocation
(percentage of assets allocated across risk buckets). Detailed questionnaire
response data is hitherto unavailable, with future plans to exchange and
analyze that information with our industry partner to better evaluate a
client’s elicited risk with guidance from revealed risk.

Revealed risk is typically measured by a client’s recovered utility of wealth
from their trade and transaction behaviours. Utility of wealth is the idea
that income and total wealth affect how a loss is perceived; for example, a
$$10,000$ loss is significant if your total wealth is $$50,000$ and
insignificant if your total wealth is $$50$ million. This concept is not new;
income and total wealth have been shown to affect risk tolerance based on the
utility of wealth[Samuelson, 1975]. A client’s utility of wealth can be
recovered from realizations of how they are investing their wealth [Dybvig and
Polemarchakis, 1981, Dybvig and Rogers, 1997].

We apply the financial measure of VaR to evaluate both elicited and revealed
risk individually. VaR is a quantile of the profit-and-loss distribution–for
example, fixing a time horizon of one year, the 99% VaR is the minimum loss
you should expect on your worst day out of one hundred. This concept is well-
known in financial management, where it is explained in depth in Jorion [2007]
with many different strategies for implementation [Kuester et al., 2006]. In
this paper, VaR is used to accomplish three tasks: (1) quantify elicited risk,
(2) quantify revealed risk, and (3) quantify the discrepancy between elicited
and revealed risk.

Previous research bridging the gap between elicited and revealed has been
conducted to understand risk preferences for crop producers [Sharma and
Schoengold, 2016]. Corter and Chen [2006] used a weighted sum score
methodology to measure revealed risk and compared it with elicited risk
measured with scores from a risk tolerance questionnaire. They found that
higher risk preferences in the questionnaire were positively correlated with
higher risk sum scores. However, our interest is not in reaffirming the
correlation between elicited and revealed risk but in quantifying the
difference between the elicited and revealed preferences. In our approach, VaR
is a risk comparison methodology that allows the user to calculate elicited
and revealed risk using the same criteria for comparison. While part of our
contribution to the literature is using VaR to estimate clients’ risk
preferences, our main contribution is measuring the difference in elicited and
revealed risk tolerance.

##  3 Data description

The dataset used in this paper is drawn from the same database provided by a
private investment dealership discussed in Thompson et al. [2021]. The dataset
in this paper spans from March 29th 2019 to August 12th 2019, and consists of
daily KYC information (age, gender, annual income, risk tolerance) and
financial information (account type, total value of that account, proportion
of total value invested into different securities) for $50,980$ accounts of
$27,644$ clients and 311 advisors. The KYC information is collected via a
questionnaire when the account is opened and is updated at least yearly, and
the financial information is updated each business day. All identifying data
has been anonymized, and individuals will not be identified or referenced in
this paper. Also, any subset of the data cannot be publicly shared. Table 1
shows the variables of the data from Thompson et al. [2021] that are
considered in this paper.

Table 1: Dataset details on clients’ KYC information. Variable |  Summary |  Data type |  Example values
---|---|---|---
Account type |  Canadian account classifications that have different benefits |  Categorical |  Cash, LIRA, RDSP, RESP, RIF, RSP, TFSA, Margin222Account types are cash, locked-in retirement account (LIRA), registered disability savings plan (RDSP), registered education savings plan (RESP), retirement income fund (RIF), retirement savings plan (RSP), tax-free savings account (TFSA), and margin accounts.
Advisory type |  Where the advisor has the discretion to trade for the client |  Categorical |  Discretionary, non-discretionary, or unknown
Age |  Ages range from 18 to 98 years old, with average at 57.4 years |  Continuous |  31 years old
Annual income |  Gross annual income in CAD |  Continuous |  Multiples of $$100$ between $\$0$ and $\$15,000,000$ inclusive
Gender |  $50.5\%$ male and $49.5\%$ female |  Indicator |  M,F
Investment knowledge |  The self-reported investment knowledge of 2.5% sophisticated, 44.7% good, 35.2% fair, or 17.6% poor |  Ordinal |  $1$, $2$, $3$, or $4$
Marital status |  67% married, 18% single, 11% unknown and 4% divorced |  Categorical |  M,D,S, or *
Number of accounts |  Clients can have more than one account |  Ordinal |  $1$,$2$,$3$,$\ldots 8$
Residency |  Province or Country or Region, with approximately $65\%$ from Ontario |  Categorical |  ON, MB, AB, $\ldots$
Retirement indicator |  The client’s retirement status with 73.9% not retired, 18.2% retired, and 7.9% unknown |  Indicator |  Yes, No

The new data in this paper that was previously unavailable is the
discretionary information of the client-advisor relationship. Advisors
associated with the investment dealership have direct control over the
securities owned in each account. Clients cannot make direct changes to the
accounts without going through their financial advisor. There are two types of
advisory relationships: (i) advisors have full discretion on trading for the
client’s accounts, or (ii) advisors must have trades approved by the client.
In our dataset, there are $4,423$ discretionary accounts, $44,712$ non-
discretionary accounts, and $1,845$ unknown. Additionally, there exist
accounts in the dataset that the advisor personally owns, but that information
is currently unavailable due to anonymization. Henceforth, we will treat
advisor accounts as client accounts since we are informed that the dealership
expects advisors to trade on their accounts, similar to how they would trade
for their clients. In fact, internal auditors at the dealership monitor
advisor trades to ensure that they are offering trades to or conducting trades
for clients before trading the same securities in the advisor’s personal
accounts, where regulators mandate this behaviour.

The distribution of account residency is shown in Table 2, with the majority
of accounts owned by clients in the province of Ontario. Figure 1 shows a
graph of the annual incomes where we can see that 25% of all clients earn
$$37,000$ or less, 50% earn $$64,000$ or less, and 75% earn $$100,000$ or
less. There are income spikes at multiples of $50k, and there were also 286
clients that earned $$500,000$ or more. Figure 2 shows the total market values
of client portfolios, where 25% of all clients own assets valued at $$44,041$
or less, 50% earn $$113,147$ or less, and 75% earn $$262,099$ or less.

Table 2: Distribution of residency for client accounts. Locations are Ontario (ON), British Columbia (BC), Alberta (AB), Manitoba (MB) and Nova Scotia (NS). Location | ON | BC | AB | MB | NS | Other
---|---|---|---|---|---|---
Percentage | 65.36 | 13.85 | 12.80 | 4.07 | 2.17 | 1.75
Figure 1: The distribution of annual incomes for clients on August 12th 2019,
with a binwidth of $$10,000$. The three vertical dashed lines represent the
25th, 50th, and 75th percentiles. There are 286 clients not pictured with
annual incomes greater than $$500,000$, with a maximum annual income of
$$15,000,000$. Figure 2: The distribution of the total market value for client
portfolios on August 12th 2019, with a binwidth of $$10,000$. There are 74
clients not pictured with portfolio market values greater than $$3,000,000$,
with a maximum of $$62,684,999$.

Figure 3 shows clients’ ages, where the distribution is unimodal, centred at
57.5 years, has a standard deviation of 14.8 years, and is very slightly left-
skewed. The minimum age is 18 years–the legal age to open an account in
Canada–and the maximum is 104. Table 3 shows the number of accounts for
clients, where 77.3% of clients have two or fewer accounts. Table 4 shows the
account types, where most own RSP and TFSA accounts.

Figure 3: The distribution of client ages, where each bin contains one year. Table 3: The number of accounts owned by clients. Unique accounts | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
---|---|---|---|---|---|---|---|---
Number of clients | 13112 | 8246 | 4380 | 1457 | 330 | 87 | 20 | 12
Table 4: Distribution of account types. Account type | Cash | LIRA | RDSP | RESP | RIF | RSP | TFSA | Margin
---|---|---|---|---|---|---|---|---
Number of accounts | 7843 | 129 | 204 | 2975 | 6169 | $19,979$ | $11,080$ | 944

In addition, we also have data on the elicited and reveal risk distributions.
This data is either collected during the KYC questionnaire (elicited) or
observed from trading behaviours (revealed), and is discussed in the next
section.

###  3.1 Profile risk allocation and portfolio risk selection

The dealer’s method to satisfy the KYC risk tolerance obligation results in
their clients’ risk tolerance represented as a set of five percentages, with
each percentage associated with the following categories: low, low-medium,
medium, medium-high, and high risk. The percentages sum to 100, so all risk
tolerance is allocated. From here on, we will refer to these percentages,
elicited from a client, as the profile risk allocation, or simply profile
risk. Each security available for purchase from the dealer is evaluated to
have a KYP risk rating classified into one of the five risk categories. An
investor’s familiarity with a security’s brand plays a key role in the
optimism of the financial return of an investment in that security [Aspara,
2013], and unfamiliar securities have a higher perception of risk with lower
return [Ganzach, 2000]. Security risk ratings provide numerical guidance to
investors on the realistic risk of an investment. Our dataset’s risk ratings
are based on a weighting of the standard deviation of the security price,
maximum drawdown, downside deviation, and correlations with other securities.
In this paper, the revealed risk distribution on any given day is the
percentage of total market value held in each of the five risk categories.
Henceforth, it will be referred to as the portfolio risk selection, or simply
portfolio risk.

###  3.2 Clusters

Our previous work grouped client behaviours–measured by a recency, frequency,
monetary and profile (RFMP) model–though a $k$-prototypes machine learning
clustering algorithm [Thompson et al., 2021]. In that work, we found the
optimal number of clusters to be five using the silhouette coefficient and the
Davies–Bouldin (DB) score. The cluster attributes yield five personas:

  * •

Cluster 1 – active investors who trade frequently, in large amounts and appear
sensitive to market influences,

  * •

Cluster 2 – younger savers who make regular, smaller deposits using automated
platforms such as preauthorized chequing (PACs) and dollar cost averaging,

  * •

Cluster 3 – “just in time” traders who make infrequent trades at seemingly
random intervals,

  * •

Cluster 4 – older investors who make regular withdrawals and cash out
dividends and interest payments, and

  * •

Cluster 5 – systematic savers who make larger trades but make use of
automation for predictable deposits and re-balancing.

In this paper, we further analyze those clusters to investigate whether the
methodologies presented herein agree with our previous work. We note that risk
tolerance was not an input into the clustering algorithm, and previous
analysis on the profile risk allocation of the clusters showed little
difference between clusters.

##  4 Methodology

Advisors allocate their clients’ wealth to different recommended proportions
of risk categories, and they select particular assets within these categories.
Once set up, advisors continuously aid clients in modifying their portfolio as
markets evolve, based on their stated financial goals. Advisors strive to
construct portfolios that are consistent with stated goals, and thus one
should be able to quantify how close an advisor is coming to the stated goal.
Otherwise, stated goals are inconsequential. Advisors pick assets that are
classified into risk categories–mandatory for all financial institutions. In
our dataset, each asset is classified into one of five risk categories. It is
natural to assume that when an advisor sets up a clients’ proportional risk
allocation, the advisor will ensure that their assets line up with that risk
allocation.

A main issue is quantifying the difference between (i) a client’s prescribed
KYC profile risk and (ii) their actual portfolio risk. A risk allocation or
selection is represented by an ordered collection of five percentages
associated with each risk category (Low, Low-medium, Medium, Medium-high,
High), where the percentages sum to one. An example portfolio risk selection
is (0.2, 0.1, 0.7, 0, 0) with 20% of wealth allocated to the low risk bucket,
10% to low-medium, and 70% to medium. Readers familiar with linear algebra
might (correctly) recognize that we can view an allocation or selection as a
point in five-dimensional Euclidean space. In this case, quantifying
discrepancy is as simple as computing the Euclidean distance between two
points. Readers familiar with probability theory might (correctly) recognize
that we can view an allocation or selection as a probability distribution, in
which case quantifying discrepancy would be possible using well-known measures
such as Kullback-Liebler Divergence. Unfortunately, neither of these
approaches leads to meaningful comparisons to advisors or their clients. See
Appendix A for a detailed investigation of metrics and divergences used for
risk category comparisons. In other words, measuring the discrepancy in a
mathematically valid way is straightforward, but doing so in a manner that is
both mathematically valid and financially meaningful, is surprisingly non-
trivial.

The well-known financial risk measure Value-at-Risk (VaR) allows us to
evaluate and compare risk allocations and selections in a valid and
financially informative way. VaR captures the risk tolerance relative to
actual market indices, while also incorporating the relative and correlated
risk between risk categories. Typically, VaR is quoted in basis points (bps)
so that VaR across clients with different total market values can be compared.
VaR can provide an actual dollar value estimate of the minimum a client should
expect to lose on their worst day out of one hundred, which is an excellent
communication tool between advisors and clients.

To compare profile and portfolio risk, the first step is to estimate the VaR
and the second step is to compute the difference in VaRs to compare them.
Estimating VaR requires an understanding of the assumptions on how the
investor and advisor interpret each risk category, and the statistical
properties of the returns of those risk categories. Our analysis makes
specific choices to calculate VaR, but it is essential to highlight that the
VaR framework is highly customizable by the user.

###  4.1 Value-at-risk

Using the assets in a portfolio is necessary for the exact calculation of
portfolio VaR. However, this approach fails for assigning a profile risk
VaR–perceptions and securities held across advisors can be very different. Our
solution to calculating profile VaR similarly across clients and advisors is
to use an exchange-traded fund (ETF) as a representative investment for each
risk category. This methodology is customizable to any set of ETFs or any
assets of the user’s choosing. The risk representative iShares
ETFs333https://www.blackrock.com/ca/investors/en/products/product-list from a
given class that had the longest publicly-available
data444https://ca.finance.yahoo.com are shown in Figure 4 with their expected
annualized returns, volatility, and correlations555We have included an Excel
spreadsheet [Microsoft Corporation, 2021] that reproduces the VaR calculation,
and Appendix B that provides mathematical details of the calculation..

The expected annualized return of each ETF and the correlations between them
are used to calculate VaR. The VaR calculation gives the minimum expected loss
in basis points (bps) on the worst trading day out of one hundred. Figure 4
shows an example VaR calculation for a single client that is considering two
possible portfolio selections. The client has a profile risk of 100% low-
medium, and their risk is a loss of at least 10.91% of their total market
value. They are considering a portfolio VaR of 100% low, which yields a VaR of
-0.23%–negative means that on your worst day out of one hundred, you expect to
gain 0.23% or less–and a 100% high account, which yields 31.18% VaR. The
discrepancy between the low and low-medium accounts is -11.14%, and between
low and high accounts is 20.27%. A negative VaR discrepancy means the
portfolio is relatively under-risked and a positive VaR discrepancy means it
is relatively over-risked.

Figure 4: VaR calculation using Excel spreadsheet software. The spreadsheet is
available to readers.

##  5 Results

In this section, we apply our risk comparison methodology to a real trade and
transaction dataset. We show the benefits and drawbacks of using VaR to
evaluate and compare profile and portfolio risk. We start by investigating a
single client’s performance over time, followed by an advisor’s portfolio with
a large clientele, and conclude with results at the financial dealership
level.

###  5.1 Client level

First, we look at how the two methodologies can evaluate the difference
between the client’s profile and portfolio risk. This investigation will show
not only the usefulness of the VaR methodology, but also show possible
analyses and visualizations that could appear on a client dashboard. The
example client is a 62-year-old investor with five accounts with five types:
Cash, retirement savings plan (RSP), retirement income fund (RIF), tax-free
savings account (TFSA), registered education savings plan (RESP). The holdings
of each account over time is shown in Figure 5, where the account starts with
approximately $$300,000$ in their RIF accounts, $$75,000$ in each of their
TFSA and RESP accounts, and change in each of their RIF, RSP, and Cash
accounts. A large addition of approximately $$600,000$ is made in the fourth
week of July to their Cash account. Table 5 shows the market value, profile
VaR, portfolio VaR, and VaR discrepancy for each account on on August 12th
2019, where we can see they have the same KYC profile risk (100% medium)
across accounts. The accounts are all under-risked to their stated goal, shown
by the negative differences in VaR.

Figure 5: The portfolio market value of each account of the example client over time. Table 5: The example client market values, profile VaR, portfolio VaR, and VaR discrepancy on August 12th 2019 by account type. The averages in the last row are weighted by the market value of each account’s holdings. Account | Market Value (CAD) | Profile VaR (bps) | Portfolio VaR (bps) | VaR discrepancy (bps)
---|---|---|---|---
A1: Cash | 595152 | 1216 | 971 | -244
A2: RSP | 9883 | 1216 | 1089 | -126
A3: RIF | 288552 | 1216 | 882 | -333
A4: TFSA | 82302 | 1216 | 1089 | -126
A5: RESP | 67028 | 1216 | 1089 | -126
Average | 208583 | 1216 | 965 | -251

Figure 6 shows the client’s profile VaR, individual account portfolio VaRs,
and an overall portfolio VaR (weighted by market value) over time. Similar to
their account holdings in Figure 5, we see there is a re-balancing of asset
risks in their cash and RIF accounts, which affects their overall portfolio
average.

Figure 6: The example client’s account portfolio VaRs over time, with constant
profile VaR of 1216 bps for all accounts. The unseen TFSA account line is same
as the RESP line.

###  5.2 Advisor level

From the individual client view perspective, our methodology allows advisors
to gain insight into their clients’ risk positions and easily compare to their
preferences. In addition, advisors can gain insights on all clients across
their firms by looking at the average metrics across clients and the variation
of each metric. Figure 7 shows the advisor with the fifth-most number of
clients (670) with 1131 accounts over the time period of the dataset. Figure
7(a) shows the mean and median of all the advisor’s client’s profile VaR over
time, where the majority of clients are prescribed less than 1250 bps at risk
on any given day. Figure 7(b) shows that the majority of clients have less
than 1150 bps at risk in the portfolio VaR, and Figure 7(c) shows that most
clients are under-risked with a discrepancy less than 0.

(a) Profile VAR

(b) Portfolio VaR

(c) Discrepancy in VaR

Figure 7: Profile VaR (upper left panel), portfolio VaR (upper right panel),
and the discrepancy between the two VaRs (lower panel) over time for a single
advisor . The advisor starts with 283 clients with $$48,748,731$ CAD in total
asset market value on March 29th 2019 and has 662 clients with $$104,530,928$
in total asset market value on August 12th 2019.

A closer look into VaR distributions for clients is shown using heatmap plots
in Figure 8. The distribution of profile VaR is shown in Figure 8(a) which
shows that over time, most clients are being put into a 100% medium account
shown by the bright yellow strip around $1,110$ bps. The distribution of the
portfolio VaR in Figure 8(b) shows that the advisor appears to place each
client into two swim lanes–one group is at essentially 100% medium risk, and
the other group is slightly below it. This is reflected in the discrepancy in
Figure 8(c) where there is a spread of clients below zero discrepancy.

(a) Profile VaR

(b) Portfolio VaR

(c) Discrepancy in VaR

Figure 8: Heatmaps showing the distributions of profile VaR (upper left
panel), portfolio VaR(upper right panel), and the discrepancy between them
(lower panel) over time for a single advisor.

Figure 9 shows the two-dimensional distribution of portfolio and profile VaR,
where a large proportion of clients are close to the VaR equivalency line.
However, there exist a significant number of clients who have the same profile
risk (100% as mentioned before), which have a portfolio VaR at (approximately
120 clients) and lower (approximately 180 clients) than the profile VaR.

Figure 9: Heatmap of the two-dimensional distribution of portfolio VaR versus
profile VaR on August 12th 2019. The diagonal black line represents the points
at which a client has equal profile and portfolio VaR.

Figure 10 shows the distribution of VaR on August 12th 2019, where we can
specifically see the clear increase in the standard deviation from the profile
VaR in Figure 10(a) to the portfolio VaR in Figure 10(b). The bias to being
under-risked for the advisor’s clientele is shown by the vast majority of the
distribution of discrepancies below zero in Figure 10(c).

(a) Profile VaR

(b) Portfolio VaR

(c) Discrepancy

Figure 10: The single day distributions of profile VaR (upper left panel),
portfolio VaR (upper right panel), and the discrepancy between them (lower
panel) for a single advisor’s clientele on August 12th 2019.

###  5.3 Dealership level

The dealership cross-section of all client account VaRs and discrepancies on
August 12th 2019 are shown in Figure 11. Across all accounts, we can see the
dominant profile VaR in Figure 11(a) is a 100% medium profile (approximately
1200 bps), with lesser spikes at 100% low (approximately 0 bps), 100% medium-
high (approximately 2000 bps), 50% medium and 50% high (approximately 1750
bps) and 100% high (approximately 3200 bps). Figure 11(b) shows that clients
are typically set up around 1000 bps, but spikes exist again at 100% medium,
low, med-high, and high. Figure 11(c) shows that the majority of clients are
under-risked, where 86.7% have a discrepancy at or below zero.

(a) Profile VaR

(b) Portfolio VaR

(c) Discrepancy

Figure 11: The distribution of the profile VaR (upper left panel), portfolio
VaR (upper right panel), and the discrepancy between them (lower panel) for
all clients on August 12th 2019.

The VaR cross-section of a single day is indicative of a global pattern over
the time period of the dataset shown in Figure 12. Figure 12(a) shows that the
median profile VaR is consistently at 1216 bps or, on the worst day out of one
hundred, the minimum loss that is prescribed is 12.16% of the total portfolio
market value. We found that the portfolio risk median in Figure 12(b) is
consistently smaller at 10.90% than the profile risk median. When the VaR are
viewed together in Figure 12(c), we see that profile VaR forms an upper
boundary that the portfolio VaR stays well below. The gap is considerable at
150.2 to 160.2 bps. In our dataset, the median portfolio market value was
$$113,147$, and therefore 10 bps represents $$113.15$ of potential capital
impairment.

(a) Profile VAR

(b) Portfolio VaR

(c) Discrepancy

Figure 12: The 5%, 50%, and 95% quantiles of the profile VaR (upper left
panel), portfolio VaR (upper right panel), and the discrepancy between them
(lower panel) over time.

We further investigated VaR for each of the variables described in Table 1
using the box plots found in Appendix C. We found significant evidence of
advisors tailoring of portfolios to the specific needs or attributes of
individual clients. We found that on average:

  * •

As income increases, profile and portfolio VaR increase.

  * •

As age increases, profile and portfolio VaR decreases.

  * •

Margin accounts have the highest profile and portfolio VaR, while RIF had the
lowest.

  * •

As investment knowledge increases, profile and portfolio VaR decreases.

  * •

People from British Columbia tend to have the lowest profile and portfolio
VaR.

  * •

Across all variables except account type, discrepancies had similar
distributions.

  * •

Retired individuals had lower profile and portfolio VaR.

  * •

Men had slightly higher profile and portfolio VaR than women.

  * •

Single people had the highest profile VaR, and divorced had the lowest.

  * •

40 to 50 year-old’s tended to have higher income, but lower portfolio market
value.

  * •

Investment knowledge is relatively similar across age.

Figures 13(a) and 13(b) show the distributions of profile and portfolio VaR
after a large (50%) injection of funds. Figure 13(c) shows the effect on the
portfolio VaR, where most additions are close to zero and fit into the
portfolio risk selection of the client.

(a) Profile VAR

(b) Portfolio VaR

(c) Change caused by significant investment

Figure 13: The distribution of the profile VaR (upper left panel) and
portfolio VaR (upper right panel) on days when clients added an investment of
at least 50% of their portfolio market value (3109 occurrences). The lower
panel shows the effect that the large investment had on the portfolio VaR.

###  5.4 Cluster value-at-risk

For illustration purposes, we might consider the profile risk as the bumper
rails in a bowling alley. Once the bumper rails have been engaged,
participants are protected from throwing a gutter ball and scoring zero.
However, obtaining a good score, or throwing consistent strikes, still
requires skill, and perhaps, a little luck. The question then becomes, in
setting the risk “guard rails”, do advisors and clients customize the two
risks–prescribed and portfolio–to recognize unique trading behaviour
preferences. We found that advisors and their clients actively manage
differing clients’ needs because the VaR differs for each cluster, as does the
VaR discrepancy (profile VaR minus portfolio VaR).

Cluster personas were determined in previous work, and the results of those
clusters are discussed in Section 3.2. In Figure 14(a), we can see clear
delineations between the profile VaR for each cluster and that the profile VaR
generally behaves as we would expect–the highest profile VaR is with the Early
Savers (longer time horizons) and Systematic Savers (dollar-cost averaging)
while the lowest profile VaR is with the Older Investors. The most significant
increase in profile VaR is with the Active Traders–presumably consistent with
their preferred behaviour.

(a) Profile VaR

(b) Portfolio VaR

(c) Discrepancy in VaR

Figure 14: Mean of the daily VaR for clusters with 95% bootstrapped confidence
intervals ($B=9999$ re-samples). The distribution of the profile VaR is show
in the upper left panel and portfolio VaR in the upper right panel, and the
discrepancy of the VaRs are shown in the lower panel.

Figure 14(b) shows that customized portfolio construction carries through to
the actual portfolio VaR–but within the limits set by the guard rails (the
profile risk). It should also be noted that portfolio VaR remains relatively
flat, indicating active management of the portfolios against background beta.
Figure 14(c) shows the gap between profile and portfolio risk. We observed
that the discrepancy is consistently negative (portfolio risk is safer than
profile risk) and small, but not unique to each cluster. Therefore, clusters
are managed within their guardrails, and portfolios are tailored to each
cluster’s unique, presumably preferred, trading behaviours.

Figure 15 demonstrates each cluster’s account VaRs on the last day of the
dataset. Figure 15(a) shows that the majority of the data has a profile VaR
between 1600 and 850 bps, where systematic savers tend to have the highest
profile risk, and older investors have the lowest profile risk. Figure 15(b)
shows that early savers have a higher overall portfolio risk while older
investors again have the lowest overall portfolio risk. Figure 15(c) shows
that most investors, regardless of cluster, are under-risked.

(a) Profile VaR

(b) Portfolio VaR

(c) Discrepancy in VaR

Figure 15: Profile VaR (top left panel), portfolio VaR (top right panel) and
the discrepancy in VaR (lower panel) on August 12th 2019 by cluster
membership.

###  5.5 Discretionary advising

As further evidence of client customization, we examined the same value-at-
risk metrics against advisor licensing regimes. Within our dataset, there were
two significant advisor licensing regimes–discretionary and non-discretionary.
In Canada, under the IIROC regime, some investment representatives can provide
discretionary portfolio management services666Investment Industry Regulatory
Authority of Canada, Rule 2900, Proficiency and Education
https://www.iiroc.ca/Rulebook/MemberRules/Rule02900_en.pdf and can make
wholesale portfolio decisions on behalf of their clients, without the need for
explicit client permission before placing a trade. In our dataset, 8.7% of
accounts listed investment representatives as licensed to be discretionary
portfolio managers. Non-discretionary advisors, on the other hand, must seek a
client’s permission before every trade. Given these structural differences,
our operating hypothesis was that discretionary managers would have the
capacity to accommodate client behaviour more than non-discretionary advisors
with restricted licenses.

We found that both licensing regimes followed the same patterns noted above
regarding the behaviour of the profile VaR versus the portfolio VaR but that
the discretionary advisors (Figures 16(a), 16(c), 16(e)) appear to be more
effective at maintaining consistent profile VaR and discrepancy gap over time
(except the active traders whom we would expect to look to take advantage of
market outlooks). Non-discretionary advisors, on the other hand, appeared to
allow client profile VaR to systematically drift upwards with the markets, but
portfolios are relatively consistent which results in a growing discrepancy
(Figures 16(b), 16(d), 16(f)).

(a) Discretionary profile VaR

(b) Non-discretionary profile VaR

(c) Discretionary portfolio VaR

(d) Non-discretionary portfolio VaR

(e) Discretionary discrepancy

(f) Non-discretionary discrepancy

Figure 16: The profile VaR, portfolio VaR, and discrepancy between VaRs for
accounts with advisors that have a discretionary licence (top-to-bottom of
left panels) and that have a non-discretionary licence (top-to-bottom of right
panels).

It appears that advisors and their clients are systemically safe and
conservative, which can be in the best interest of many, but not all, clients.
For clients seeking to preserve capital, it is good news–no gutter balls.
However, for clients seeking to maximize growth, it may be inconsistent with
their objectives. An example might be older clients who can preserve their
capital but cannot achieve investment incomes that allow them to maintain
their lifestyles. Maintaining investment income has become a priority in an
era of low interest rates and growing longevity risk.

We also noted that there was little evidence that advisors and clients
actively manage the profile risk once it is set–presumably at the time the
account was opened. We noted that it is relatively rare for advisors to change
the portfolio risk after a change in the KYC (Figure 17(a)) and vice versa, to
change the KYC after a significant change in portfolio risk (Figure 17(b)). It
appears the gap between profile and portfolio VaR is large enough that
subsequent changes provide little incentive to nudge the two portfolios. In
other words, as long as the gutter is protected (”we are fine”) but this
observation would appear to question the inherent utility of the profile risk.
If it is not used to inform or respond to changes in actual portfolio risk,
why bother maintaining it?

(a) Incidence of portfolio changes after a KYC change

(b) Incidence of portfolio changes leading up to a significant KYC change

Figure 17: The left panel is the change in portfolio VaR preceded by a change
in profile VaR two weeks before. The right panel are changes to the portfolio
VaR over a two-week period leading up to a change in the profile VaR.

##  6 Summary of results and benefits to financial agents

This paper found that advisors at the dealership generally similarly under-
risk their clients’ portfolios relative to their stated risk preferences
across the board, regardless of the profile risk. Advisors tend to put clients
into medium or higher profile risks, but consistently medium or lower
portfolios. We found that the clustering methodology from our previous paper
[Thompson et al., 2021] showed that regardless of the cluster, profile VaRs
increased over time, portfolio VaRs remained consistent except for the active
traders, and discrepancies were similarly under-risked and decreasing over
time. We found that VaR aligned well with our cluster personas–older investors
typically had the lowest portfolio VaR while early savers had the highest
portfolio risk, and active traders changed their portfolio risk often.
Discretionary advisors set up their clients in profile and portfolio VaR
consistently over time, while non-discretionary profile risk increased with
time. Discretionary advisors tended to take lower portfolio risks with their
older investors, but higher risks with all other clusters when compared to
non-discretionary advisors.

To compare to similar research, Corter and Chen [2006] took a weighted average
of the allocation to each bucket, with higher weights applied to riskier
buckets. The weights were determined via consultation with a panel of experts.
Though well-founded and entirely reasonable, this approach is designed to
determine if risk questionnaires correlate with hypothetical asset risk
allocations, and is not directly customizable for application to our data. An
alternative option that may meet these goals (design and customization) would
be to consider portfolio volatility. Drawbacks of using portfolio volatility
include the fact that its units are not overly meaningful to clients (what
exactly does a 40% volatility mean, compared to a 20% volatility) and that it
does not reflect the ever-present tension between risk and return.

We assessed the risk of a particular allocation using the well-known VaR
metric to address those issues. Any VaR calculation requires assumptions about
the statistical properties of the underlying asset returns standard deviations
and correlations. In the present context, a VaR calculation requires
assumptions about how investments are made within each risk bucket. We assume
that returns are multivariate normally distributed, an assumption that is very
quickly relaxed to include thicker tails (such as multivariate
$t$-distribution) and asymmetries (such as mixture models). We also assume
that every dollar a client invests in a particular risk bucket is invested in
a representative ETF from that class. The representative asset for a given
class was selected from iShares ETFs. Given more detailed information on an
individual client’s holdings, and an adequate amount of historical data on
those assets to get estimates of expected returns, standard deviations, and
correlations, our approach is even more directly applicable and customizable
for advisors.

We conclude this paper by discussing how we have shown that VaR benefits four
financial agents: clients, advisors, dealers, and regulators. Clients can
monitor their risk using VaR to understand better how their asset portfolios’
changes affect risk at the account level. VaR is a particularly useful
communication tool so that clients can understand risk in dollar amounts
relative to their total market value to conceptualize how well different asset
portfolios match up with stated profile risk preferences.

The advisors in this study appeared to manage risk actively on behalf of their
clients. They did so in the context of their clients’ explicit or derived risk
tolerance or suitability as measured by value at risk. Researchers have
theorized, studied and written about the role of financial advisors [Foerster
et al., 2017, Linnainmaa et al., 2018]. Much of that work has been grounded in
the assumption that advice should generate returns and, in particular, excess
returns or alpha. Our work would appear to suggest that advisors play an
essential role in terms of the perennial balance between return and its
dependant variable risk, and that an advisor’s role is, therefore, more
complex than the simple pursuit of alpha.

By studying this behaviour empirically and using a relatively straightforward
measure, our work can be extended from the chalkboard to the floorboards
through client-facing platforms such as robo-advisors, financial planning
software or client statements. Moreover, it can be done while respecting
existing regulatory frameworks and a client’s best-interest standard of care.
Using the same measure for both profile risk and revealed risk–particularly
the difference between the two–in real-time, we believe VaR could become an
important tool in helping clients, advisors, dealers, and regulators monitor a
careful balance between all the variables and stakeholders.

Advisors can use this methodology to understand each client’s current risk in
terms of a dollar amount or portfolio percentage, and understand how each
client’s stated risk tolerance aligns with any prefabricated investor
portfolios. We believe this will improve communications with the client and
help clients understand better how much of their wealth is at risk and what a
“significant” loss looks like for their worst day out of one hundred.
Additionally, an advisor can quickly understand how a change in the overall
assets across a firm will affect all of their client’s risk positions and
easily detect any clients that may be put into a risky position that they are
not comfortable with stated goals. Advisors can also use VaR to understand all
the assets under their entire firm’s ownership, similar to how VaR is used for
larger financial institutions.

Dealerships and regulators will benefit from VaR by having a methodology for
either advisors to report on how they manage suitability or real-time
monitoring of advisor behaviours. Dealers, and their advisors, can also
demonstrate an additional measure of their value for clients. Moreover,
regulators can use VaR as a tool to monitor suitability or test for intrinsic
stress at the client, dealer and industry level.

## Acknowledgements

The authors would like to thank Nathan Phelps (Wilfrid Laurier University),
Andrew Sarta (Ivey Business School), Poornima Vinoo (Ivey Business School),
Matt Davison (Western University), Lori Weir (Four Eyes Financial), Kendall
McMenamon (Four Eyes Financial), Philip Patterson (Four Eyes Financial), Lucas
Loughead (Four Eyes Financial), and the many members of our data donor team
for their valuable input and insights that improved the content and writing of
this document.

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## Appendix A – Generalized difference between profile and portfolio risk

Before comparing between profiles and portfolios, we first consider a natural
evaluation of an individual risk allocation–the eye-test. Table 6 shows
realistic allocations found in our dataset. Suppose a financial advisor with a
client has prescribed the profile risk allocation in the first row of Table 6,
and the client’s actual portfolio risk selection in the second row. The eye-
test reveals they are clearly different, where the client is under-risked. A
second portfolio selection is shown in the third row, which the eye-test shows
that the portfolio is very under-risked and more under-risked than above
portfolio is over-risked. The eye-test is a natural method of comparing
categories, but an advisor cannot conduct daily eye-tests on, say, over 500
accounts.

Table 6: An example of profile and portfolio risks before and after a re-balancing of assets. The discrepancy is calculated using Equation (1) with $P_{i,i}=i$ and $P_{i,j}=0,$ $i\neq j$. Risk | Low | Low-Medium | Medium | Medium-High | High | Discrepancy
---|---|---|---|---|---|---
Profile | 0 | 0 | 0 | 80 | 20 |
Portfolio (Before) | 0 | 16 | 84 | 0 | 0 | 49280
Portfolio (After) | 0 | 94 | 6 | 0 | 0 | 45380

Next, we shift to mathematical language so that we can implement financial
concepts into our customizable categorical comparisons. Let
$k\in\\{1,2,3,4,5\\}$ be a Likert scale index [Likert, 1932] for the profile
risk categories and $j\in\\{1,2,3,4,5\\}$ for portfolio risk. Consider profile
and portfolio risk selection by percentage $x=(x_{1},x_{2},x_{3},x_{4},x_{5})$
and $y=(y_{1},y_{2},y_{3},y_{4},y_{5})$, which means $x_{k},y_{j}\in[0,1]$ and
$\sum_{k=1}^{5}x_{k}=1,\sum_{j=1}^{5}y_{j}=1$.

Consider a generalized comparison between allocation and selection vectors by
using

$\displaystyle d(x,y)=(x-y)^{T}P(x-y).$ (1)

where $(x-y)$ is a element-by-element comparison of the percent allocations
(same as our first comparison), and $P$ is a $5\times 5$ matrix that controls
customization. Before we choose our perfect customized model, let’s consider
some basic rules on how the function $d(x,y)$ should behave. If the profile
allocation and portfolio selection are the same ($x=y$), then $d(x,y)=0$. The
larger the comparison, the more different the allocation and selection of risk
are.

If we view allocations and selections as points in Euclidean space, it most
natural to compute discrepancy via the now Euclidean metric in 1, where $P$ is
now a positive definite matrix. Some of the issues of this approach are as
follows:

  * •

If $P$ is the identity matrix given as

$\displaystyle P:=\begin{bmatrix}1&0&0&0&0\\\ 0&1&0&0&0\\\ 0&0&1&0&0\\\
0&0&0&1&0\\\ 0&0&0&0&1\end{bmatrix}.$

then $d(x,y)$ is now a sum of the squared deviation between each risk
category. The two portfolio selections of 100% low and 100% low-medium, are
equidistant from the profile allocation in Table 6. This is clearly
unreasonable from a financial perspective, as the difference is much larger
than for the 100% low allocation than the 100% low-medium allocation.

  * •

If $P$ is a diagonal matrix we can alleviate the problem indicated above. For
example, setting $P_{i,i}=i$ gives

$\displaystyle P:=\begin{bmatrix}1&0&0&0&0\\\ 0&2&0&0&0\\\ 0&0&3&0&0\\\
0&0&0&4&0\\\ 0&0&0&0&5\end{bmatrix}.$

where $d(x,y)=\sum_{i=1}^{5}i(x_{i}-y_{i})^{2}$ heavily penalizes
discrepancies in higher risk categories. This ensures that the distance
between 100% high and 100% low-medium is twice as large as that between 100%
low and 100% low-medium. Unfortunately, this approach introduces new
complexities that we discovered when applying it to our data. The client in
Table 6 made a trade on particular date, that clearly moved them further from
their profile allocation. According to the metric, however, the trade moved
them closer, which is clearly unreasonable.

  * •

The previous two matrices have only zeros as off diagonal terms and therefore
the discrepancy that uses the those matrices only compare the same categories.
Consider the penalization matrix that penalizes off diagonal terms given by

$\displaystyle P:=\begin{bmatrix}1&0&0&0&1\\\ 0&1&0&0&0\\\ 0&0&1&0&0\\\
0&0&0&1&0\\\ 0&0&0&0&1\end{bmatrix}.$

which yields
$d(x,y)=\sum_{i=1}^{5}(x_{i}-y_{i})^{2}+(x_{1}-y_{1})*(x_{5}-y_{5})$ The new
term in the sum–a penalization of misallocations in the low category has been
added, but only if there is also a misallocation in the high category (and
vice versa). As it turns out, off-diagonal terms place a heavier penalty on
not just if there is a misallocation, but how they are misallocated. This
comparison will show us that not all misallocations are equal. A penalization
matrix that penalizes how far the misallocations are from the stated goals is

$\displaystyle P:=\begin{bmatrix}0&1&2&3&4\\\ 1&0&1&2&3\\\ 2&1&0&1&2\\\
3&2&1&0&1\\\ 4&3&2&1&0\end{bmatrix}.$

Notice that the diagonal terms are zero, which works in this configuration
since we are considering all misallocations in the off-diagonal terms. We
could also relax the assumption that the penalization matrix is symmetric, and
allow for higher penalties for over-risked profiles. A penalization matrix
where we penalize an over-risked profile more than under-risked could be

$\displaystyle P:=\begin{bmatrix}0&1&2&3&4\\\ 1&0&1&2&3\\\ 1&1&0&1&2\\\
1&1&1&0&1\\\ 1&1&1&1&0\end{bmatrix}.$

Consider another perspective that views profile and portfolio risk allocations
and selections as discrete probability distributions. It is most natural to
compute discrepancy via an information theoretic divergence such as Kullback-
Liebler (K-L) divergence [Kullback and Leibler, 1951]. Unfortunately this
approach also leads to non-useful results. For example, consider a profile of
100% low-medium, and two portfolio selections of 100% low and 100% high give
the same K-L divergence with the same sign, where we lose magnitude of the
difference and whether the portfolio is under- or over-risked. Other
divergences were explored, but these suffered from the same problems as the
K-L divergences.

These metrics and divergences are designed to directly compare risk categories
to evaluate profile and portfolio risk alignment. The advantage of the
proposed metric is that they can be customized to calculate profile and
portfolio risk alignment. We introduced equal and unequal successive
weightings for each category to inject ordering of the categories, and allow
for the natural conceptualization of risk to be included in the calculation.
However, there are ever-present problems with each calculation method that
ruin the overall interpretations of results across a dealership. In
conclusion, we suggest the financial concept of value-at-risk is a much more
feasible method to compare profile risk allocations and portfolio risk
selections.

## Appendix B - Value-at-Risk methodology

Consider $x$ be the profile risk allocation. Let $\mu$ denote the mean return
vector and $\Sigma$ be the covariance matrix of the representative risk
category ETFs. The $\alpha$-level VaR on a KYC risk profile is given by

$\mathrm{VaR}_{\alpha}(x)=x^{T}\mu+\sqrt{x^{T}\Sigma x}\cdot z_{\alpha}\;,$

where $z_{\alpha}$ is an appropriate $\alpha$-quantile from a standard normal
distribution. In this report, we let $\alpha=0.01$. We have chosen five
iShares ETFs with $\mu=[0.52,1.97,2.21,2.93,4.23]$,
$\sigma=[0.13,5.53,6.48,9.68,15.22]$, and

$\displaystyle\rho=\begin{bmatrix}1&-0.22&-0.16&-0.23&0.07\\\
-0.22&1&0.79&0.59&0.12\\\ -0.16&0.79&1&0.78&0.31\\\ -0.23&0.59&0.78&1&0.06\\\
0.07&0.12&0.31&0.06&1\end{bmatrix}$

which yield

$\displaystyle\Sigma=\sigma^{T}\rho\sigma=\begin{bmatrix}0.016900&-0.158158&-0.134784&-0.289432&0.138502\\\
-0.158158&30.580900&28.309176&31.582936&10.099992\\\
-0.134784&28.309176&41.990400&48.926592&30.573936\\\
-0.289432&31.582936&48.926592&93.702400&8.839776\\\
0.138502&10.099992&30.573936&8.839776&231.648400\end{bmatrix}$

## Appendix C - Distribution of VaRs across KYC information

Next, we investigate a series of boxplots for the variables in Table 1. Figure
18 shows boxplots of VaR at for each of the income levels quartiles. We can
see that as income increases, both profile (Figure 18(a)) and portfolio
(Figure 18(b)) show that as annual income increases, VaR increases. The
discrepancy in Figure 18(c) shows that regardless of annual income, most
clients are similarly under-risked.

(a) Profile VaR

(b) Portfolio VaR

(c) Discrepancy in VaR

Figure 18: Boxplots of profile VaR (top left panel), portfolio VaR (top right
panel) and the discrepancy in VaR (lower panel) on August 12th 2019 by annual
income. Each boxplot represents 25% of the data, with the boxplots in each
panel from left to right represent the first, second, third, and fourth
quartiles of annual incomes.

Figure 19 shows boxplots of VaR for each of the quartiles of client ages. We
can see that as income increases, both profile (Figure 19(a)) and portfolio
(Figure 19(b)) show that as client age increases, VaR decreases. The
discrepancy in Figure 19(c) shows that regardless of client age, most clients
are similarly under-risked.

(a) Profile VaR

(b) Portfolio VaR

(c) Discrepancy in VaR

Figure 19: Boxplots of profile VaR (top left panel), portfolio VaR (top right
panel) and the discrepancy in VaR (lower panel) on August 12th 2019 by client
ages. Each boxplot represents 25% of the data, with the boxplots in each panel
from left to right represent the first, second, third, and fourth quartiles of
client ages.

Figure 20 shows boxplots of VaR for each of the account types.

(a) Profile VaR

(b) Portfolio VaR

(c) Discrepancy in VaR

Figure 20: Boxplots of profile VaR (top left panel), portfolio VaR (top right
panel) and the discrepancy in VaR (lower panel) on August 12th 2019 by account
type.

Figure 21 shows boxplots of VaR for each of the investment knowledge types.

(a) Profile VaR

(b) Portfolio VaR

(c) Discrepancy in VaR

Figure 21: Boxplots of profile VaR (top left panel), portfolio VaR (top right
panel) and the discrepancy in VaR (lower panel) on August 12th 2019 by
investment knowledge.

Figure 22 shows boxplots of VaR for each of the residency locations.

(a) Profile VaR

(b) Portfolio VaR

(c) Discrepancy in VaR

Figure 22: Boxplots of profile VaR (top left panel), portfolio VaR (top right
panel) and the discrepancy in VaR (lower panel) on August 12th 2019 by
residency location.

Figure 23 shows boxplots of VaR for retirement status.

(a) Profile VaR

(b) Portfolio VaR

(c) Discrepancy in VaR

Figure 23: Boxplots of profile VaR (top left panel), portfolio VaR (top right
panel) and the discrepancy in VaR (lower panel) on August 12th 2019 by
retirement status.

Figure 23 shows boxplots of VaR for gender.

(a) Profile VaR

(b) Portfolio VaR

(c) Discrepancy in VaR

Figure 24: Boxplots of profile VaR (top left panel), portfolio VaR (top right
panel) and the discrepancy in VaR (lower panel) on August 12th 2019 by gender.

Figure 23 shows boxplots of VaR for marital status.

(a) Profile VaR

(b) Portfolio VaR

(c) Discrepancy in VaR

Figure 25: Boxplots of profile VaR (top left panel), portfolio VaR (top right
panel) and the discrepancy in VaR (lower panel) on August 12th 2019 by marital
status.

Since age is related to the accumulation of experience and wealth, we consider
the distribution of ages against annual income, market value, and investment
knowledge in Figure 26. Figure 26(a) show a comparison of age and income
quartiles. Similarly, Figure 26(b) shows age and total asset market value
quartiles. Figure 26(c) shows age and investment knowledge quartiles.

(a) Annual income

(b) Total market value

(c) Investment knowledge

Figure 26: Boxplots of age against annual income (top left panel), total
portfolio market value (top right panel), and investment knowledge (lower
panel).