E&Y Advisory
Financial Services Risk
Vakpresentatie Caleidoscoop
Contacts
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Diederik Fokkema
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diederik.fokkema@nl.ey.com
(088) 407 08 36
Hans Hellemons
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hans.hellemons@nl.ey.com
(088) 407 21 80
Pagina 2
Agenda
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Introductie
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Terugblik vorige week
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Wie zijn wij?
Wat doen wij?
Wie zoeken wij?
Wat bieden wij?
Theorie portfolio optimalisatie
Opdracht uitwerken
Bank balans optimalisatie
Pagina 3
Wie zijn wij? – Ernst & Young
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Aantal medewerkers in NL: 4.600
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56% man, 44% vrouw
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Gemiddelde leeftijd 34 jaar
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Aantal locaties: 15
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Hoofdkantoor: Rotterdam Boompjes
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Assurance, Advisory, Tax, Transactions
Pagina 4
Wie zijn wij? – Advisory
Geographical Area
Service Lines
EMEIA
Assurance
Risk BeNe
Sub-service lines
FS Risk
Tax
Advisory
Transaction
Advisory
Services
Performance
Improvement
ITRA
Actuarial
Services
Financial Clients
Pagina 5
Wie zijn wij? – FSRisk
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FSRisk bestaat uit ± 55 collega’s
Standplaats kantoor Amsterdam
Lokaal team met een internationale oriëntatie
Jong, ambitieus en snel groeiend team
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Staat bekend om:
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Pagina 6
Kennisgericht
Kwaliteitsgericht
No-nonsense
… en een grote verscheidenheid aan competenties
Wie zijn wij? – FSRisk - quant
Achtergrond
► Econometrie, wiskunde, statistiek, BWI, natuurkunde,
scheikunde en (bedrijfs)economie
► Gezonde affiniteit met de financiële wereld
Uitdagingen
► Je kennis gebruiken op nieuwe terreinen
► Een schakel zijn bij grote projecten
► Commerciële en communicatieve vaardigheden
ontwikkelen
Wat doen wij? - Klanten
Vermogens
beheerders
Retail
Banking
Verzekeraars
Pensioen
fondsen
Wholesale
Banking
(Commodity)
Traders
FSRisk
KLANTEN
Market Makers
Energie
bedrijven
Vastgoed
investeerders
Treasury bij
Corporates en
Non-profit
Pagina 8
Overige
financials
Wie zoeken wij? – ‘de beste mensen’
De FSRisk’er is:
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Pagina 9
Ondernemend
Ambitieus
Innovatief & creatief
Communicatief sterk
Team speler
Vaktechnisch sterk
Programma
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Terugblik naar vorige week
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Terminologie rondom portfolio optimalisatie
Excel demo
Bank balans optimalisatie
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Probleemstelling
Wiskundige problemen
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CVaR optimalisatie
Pagina 10
Terugblik vorige week
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Portfolio optimalisatie
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“Het toewijzen van kapitaal aan een portfolio van activa (e.g.,
aandelen) om zo de winst te maximaliseren en/of het risico te
minimaliseren.”
Terminologie
Mean-variance portfolio optimization
Efficient frontier
Pagina 11
Terugblik vorige week: terminologie
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Consider a portfolio consisting of n assets. The weights to
each asset is given by a decision vector ,
,
with
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being the position in asset i and
The returns on the assets are denoted by a random vector
, such that represents the return on
asset i. Consequently, the expected rates of returns are
defined as
and the expected portfolio
return equals
Pagina 12
Terugblik vorige week: terminologie
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The variance of the return on asset i (i.e., the variance of a
random variable) is given by
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And the covariance between asset i and j is defined as
Pagina 13
Terugblik vorige week: terminologie
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Subsequently, the variance of a portfolio consisting of n
assets is given by
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Or in matrix notation as
, with
representing the covariance matrix
Pagina 14
Terugblik vorige week: terminologie
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An efficient portfolio is defined as an allocation of assets
that maximizes the returns for a certain level of risk
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Here represents the maximum level of risk. Additionally
an extra constraint on the bounds can be added
here
and
bound
Pagina 15
respectively represent the lower and upper
Terugblik vorige week: terminologie
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Alternatively, a combination of assets that minimizes the risk
for a certain level of return is given by
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Here
denotes the minimum level of required return
Pagina 16
Terugblik vorige week: terminologie
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The efficient frontier is defined as a curve that shows all
efficient portfolios in a risk-return framework.
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First calculate the solution to the optimization where the return is
maximized, while ignoring the risk constraint. This gives an upper
bound on the expected return,
Secondly, calculate the solution to the optimization where the risk is
minimized, while ignoring the return constraint. This gives a lower
bound on the expected return,
Finally, the efficient is obtained by solving the last optimization
problem (including the risk constraint) for a certain number of
required returns on the interval
Pagina 17
Terugblik vorige week: opdracht
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Excel demo
Pagina 18
Bank balans optimalisatie
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Portfolio approach for bank balance sheet optimization
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Balance sheet is a summary of the financial balances as of a
specific date (e.g., end of financial year)
Financial balances include
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Assets
Liabilities
Shareholder’s equity
The following should hold:
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Total value assets = total value liabilities + shareholder’s equity
Pagina 19
Bank balans optimalisatie
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A stylized bank’s balance sheet looks as follows
Pagina 20
Bank balans optimalisatie
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Assets on the balance sheet produce a positive return,
whereas liabilities produce a negative return
This is respectively translated to interest income and
interest expenses
Shareholder’s equity does not make any return
Now consider the stylized balance sheet as a portfolio
consisting of 7 asset instruments and 4 liability
instruments (n=11)
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Here the decision vector
denotes the positions
in the optimal portfolio
And each instrument has a certain return, which is denoted by the
return vector
Pagina 21
Bank balans optimalisatie
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Ultimately led to the
following non-linear
optimization problem
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Conditional Value-at-Risk
(CVaR) minimization function
Given a confidence level
, the conditional
expectation of the portfolio
losses above the
-percentile are minimized
(worst-case scenarios)
J is the number of scenarios
Pagina 22
Bank balans optimalisatie
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Ultimately led to the
following non-linear
optimization problem
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Mean retained earnings ( )
should be higher than expected
retained earnings
This constraint is comparable to
Pagina 23
Bank balans optimalisatie
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Ultimately led to the
following non-linear
optimization problem
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Constraints needed for CVaR
optimization objective function,
such that only the worst-case
scenarios are taken into
account
is a complex loss
function (explained later)
Pagina 24
Bank balans optimalisatie
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Ultimately led to the
following non-linear
optimization problem
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The weights to all assets
instruments should add up to 1
Pagina 25
Bank balans optimalisatie
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Ultimately led to the
following non-linear
optimization problem
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The weights to all liability
instruments and equity
instruments should also add up
to 1, such that the balance
sheet is in “balance”
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Total value assets = total value
liabilities + shareholder’s equity
Pagina 26
Bank balans optimalisatie
Ultimately led to the
following non-linear
optimization problem
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Constraints on the maximum
weight shift per portfolio
instrument.
Here
denotes the initial
portfolio weight for asset i
The lower and upper bounds for
the maximum shift are given by
and
respectively
Pagina 27
Bank balans optimalisatie
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Ultimately led to the
following non-linear
optimization problem
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Constraint on the lower and
upper bounds for the portfolio
weights
Pagina 28
Bank balans optimalisatie
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Ultimately led to the
following non-linear
optimization problem
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Lower bound is greater than 0
Pagina 29
Bank balans optimalisatie
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Ultimately led to the
following non-linear
optimization problem
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Banks are subjected to
strict regulation, prescribed
by the BIS (Bank of
International Settlements),
such that a bank resistant
to several types of risk
Recently the BIS released
new regulation, called
Basel III
Constraints to comply to
Basel III
Pagina 30
Bank balans optimalisatie
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Most important risk types for banks
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Credit risk: the risk of loss arising from the default by a creditor or
a counterparty
Market risk: the risk of losses in on and off-balance-sheet
positions arising from movements in market prices
Operational risk: the risk of direct or indirect loss resulting from
inadequate or failed internal processes, people and systems, or
from external events
Liquidity risk: the risk that the bank is unable to meet expected
and unexpected current and future cash flows without affecting
either daily operations or the financial condition of the firm
Pagina 31
Bank balans optimalisatie
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Banks are required by regulation (Basel III) to
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Keep a certain amount of capital available for credit, market and
operational risk
Have a liquidity buffer (consisting of assets that are considered
easily converted into cash) available for liquidity risk
Pagina 32
Bank balans optimalisatie
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Mathematical techniques/difficulties
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Creating the scenario set
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From January 2003 to December 2012
Monthly returns (120 scenarios)
Conditional Value-at-Risk optimization for trading portfolio
Dynamic Conditional Correlation (DCC) GARCH model for trading
portfolio
Non-linear optimization problem was implemented using in R,
software for statistical computing, and optimized using a
sequential quadratic programming (SQP) algorithm from NLOPT
(a free/open-source library for nonlinear optimization)
Pagina 33
Bank balans optimalisatie: CVaR optimization
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The model includes daily returns of 10 equity indices (e.g., AEX,
NASDAQ, S&P 500) and 5 commodity indices (energy, industrial
metals, precious metals, livestock and agriculture)
Each month the trading portfolio had to be optimized
Trading portfolio is subjected to market risk
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The risk of losses in positions arising from movements in market
prices
Regulation requires banks to hold capital against this market risk
Capital consists of the 10-day 99% Value-at-Risk (VaR) of the
trading portfolio
Pagina 34
Bank balans optimalisatie: CVaR optimization
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99%-VaR is minimum amount of capital such that, with probability
99%, the loss will not exceed this amount.
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VaR measure lacks convexity when calculated using scenarios
99%-CVaR is the conditional expectation of the losses exceeding
the 99%-VaR
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CVaR can be optimized by optimization algorithms
By definition: CVaR ≥ VaR
Pagina 35
Bank balans optimalisatie: CVaR optimization
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CVaR optimization problem
Minimize the conditional expectation of
the losses above the threshold (i.e.,
α-VaR level)
For each of the J scenarios, the portfolio
weights ( ) are multiplied by the returns
for a scenario j ( ). The negative of this
value is the loss function:
This leads to J losses. Where the
conditional expectation of the losses
greater than threshold are minimized
Pagina 36
Bank balans optimalisatie: CVaR optimization
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Loss function explained
Pagina 37
Bank balans optimalisatie: CVaR optimization
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The efficient frontier of an optimization for the trading portfolio
looks like:
Pagina 38
Bank balans optimalisatie: CVaR optimization
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The loss function in the final problem is the negative of the
following
Pagina 39
Bank balans optimalisatie: DCC GARCH model
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CVaR optimization was performed for 120 scenarios
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Correlation however is not constant over time
DCC GARCH model is capable of measuring correlations over
time
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First estimate the univariate volatility for each asset
Secondly, construct standardized residuals (i.e., returns divided by
conditional standard residuals)
Finally, estimate the correlations between the standardized residuals
Pagina 40