Managing and Measuring
Financial Risk in Practice
CIMPA School Marrakech
Avril 2007
Nicole El Karoui,
Ecole Polytechnique
1
What is Risk
!Risk exists if there is something you don’t want to
happen – having a chance to happen!!!
! Risk is measured in terms of consequences on your
business and in terms of likelihood.
! Often defined as the standard deviation of the return of
total investment.
! Degree of uncertainty of return on an asset.
! Insurance and financial business are based on risky
positions
2
What Worry About Interest Rate Risk?
Change in Annualized
Return (Percentage
Points)
Monthly Change in U.S. T-Bill
(Annualized) Returns
4
2
0
-2
-4
-6
Month (Jan 1934 through June 2003)
Data per FRED II, St. Louis FRB, for 3-Month T-Bills, Secondary Market
Why Worry About FX Risk?
Time Series of Annual Percentage Changes in Exchange Rates
Japanese Yen / U.S. Dollars
(Data per FRED, St. Louis FRB)
Percentage Change
30.0%
20.0%
10.0%
0.0%
-10.0%
-20.0%
-30.0%
1971
1975
1979
1983
1987
1991
1995
1999
Year
4
New Risks Examples
!Loss of Reputation – business scandals
!Currency swings and the effects of a falling dollar,
!Globalization
!Business failures, and credit risk
!Blow-ups of markets and firms
!Natural and man-made disasters (e.g., industrial
accidents, transport calamities)
5
Reactions to Risks
!
Litigation and law suits
!
Government regulation
!
Businesses become more risk-averse
!
Higher insurance costs
!
Some governments take a more cost-benefit
approach; others put more emphasis on
precautions (“better safe than sorry”)
6
Positive Evolution
!Some risks have become smaller, and more risk is
known today
! New instruments allow to shift some others to
other people. (Derivatives Market)
! Workplaces, the wider environment, and many
diseases have become less hazardous
!Greater focus on corporate governance
7
Five Primary Means of Risk Management
Reduce
1. Reduce the probability that the event will occur
2. Reduce the impact if the event does occur
Transfer
3. Transfer the cost of an undesirable outcome to
someone else
Avoid
4. Completely avoid potential events thus providing a zero
probability that they will occur
Do Nothing
5. Let the risk happen and be ready to bear the
consequences.
8
Risk Management Process
Strategic
Be aware Identify the
risks you faced
Decide on how you will
address the risk:
" reduce, transfer,
" avoid, nothing,
" or some combination
Analyze financial costs
Technological
Evaluate:
" the likelihood that the
risk will occur, and
" how bad the hurt will
be if it does occur
Implement
" scenarios
" criterium
Control the efficiency
9
Prioritizing Which Risks to Address
First
Act if cost
High
effective
Immediate
action
Probability
of Happening
No action
Low required
Small
Action
required
Catastrophic
Potential Impact
Source: Dr. Geoff Benson, North Carolina State University
10
Likelihood
Impact of Financial Risk Management
on Cash Flow Volatility
Cash Flow
11
Financial Risk Management
Financial RisManagement is rocket science :
“Quantitative research group with 20 to 25 Ph.D.’s .” O’Connor
Rocket science builds on
!a theory
!Tests the theory out
!Learns from failures
!Develops new theories
!Continues process until
launches are successful
FRM use
!Complex mathematics
Stochastic Calculus
Partial derivatives
!Learning from failure
Barings Bank,Orange County
Proctor and Gamble
Long Term Capital
!Differences
Limited set of data to test theories
in finance
The rules change
Bank System Risk
Operative
risk
Financial
Risk
Market Risk
Credit Risk
Trade Risk
Strategic
Risk
Liquidity Risk
Operative risk= legal risks+technical risks
Trade risk = changing demand of bank instruments +
competition issues
Strategic risk= danger of a partial failure of financial system
Liquidity risk= re-financing and delay in payment
13
Risk Measurement in Portfolio Management
# The Markowitz’s pioneering work (1959) investigated the
appropriate definition and measurement of risk on portfolio
selection, essentially based on risk =variance and return.
#In 1993-97, the growth of trade activity and instances of
financial instability impose the need to develop reliable risk
measurement techniques.
#Value at Risk is emerging as the industry standard of
integrated risk management by choice or by regulation.
#V@R models aggregate the several components of price risk
into a single quantitative measure.
14
Regulation and V@R
# Recognition of such models by the financial and regulatory
communities are obvious since 1997, where the SEC and the
Basle Committee endorsed the use of V@R models.
#This allows large banks the option to use a Value at Risk
measure to set the capital reserves necessary to cover their
market risk positions.
#Regulators expect social benefit in reducing the likehood of
large-scale financial failures.
#V@R is usually a statistically calculated risk measurement,
offering a degree of practical interpretability often lacking in
other risk measures.
#V@R builts an information report to apprise senior
management of the risk run by trading instrument
15
Controlling traders positions
In order to ensure that all traders act in the best
interests of the firm,
#the risk philosophy is known as « the limits »:
stating what instruments are allowed to be traded,
and by whom and in what amount .
#Decide whether to close the position or not when
the limits are reached
16
II - What is the price of the portfolio ?
17
What is the price of the portfolio ?
Three sources :
! Organized markets for listed products (stocks,
some options on stocks,...)
! Brokers for standard products
! Models for exotic products and more
generally when no market price is available
18
Uncertainty sources
Prices are on uneven quality : eg
! a stock quote given by a stock exchange may be
several weeks old, if there were no exchanges on it,
! two brokers may give two different prices.
Liquidity:
! To
sell (or to buy) a stock will not affect the market,
but to sell in one day several times the average daily
volume will make the stock plummet.
!The liquidation value of a stock is not equal to the
product quantity * quote, but is often less than it.
19
Model Risk
!When no market prices exist, in particular for exotic
products, models need to be used (e.g Black and
Scholes)
!No pricing model is acknowledged by everyone; some
uncertainty exists as soon as one uses a model.
!Given all these points, one cannot give the price of the
portfolio with certainty, ie the quoted market price.
! A « mid-market » result will be computed, from which
a provision will be deducted.
20
IV - TheValue-at-Risk
21
Value-at-Risk
- Definition The VaR of a portfolio is the maximal loss incurred in a
given confidence interval (99 %) and for a given time
horizon (one (ten) day).
10 DAY P&L DISTRIBUTION
Confidence
level
= 99 %
Risk
level
=1%
-25 -20 -15 -10 -5
VaR
0
5 10 15 20 25 30 35
10 days P&L (M$)
22
Value-at-Risk
- Some properties Bad measure of the diversification effect
VaR(A+B) is not less than VaR(A)+VaR(B)
0.4
0.35
0.3
0.25
0.2
P&L 1
P&L 2
0.15
0.1
0.05
0
-8
-6
-4
-2
0
2
4
6
8
23
Value-at-Risk : Back-Testing
Daily comparison between VaR and Profit&Loss (P&L)
gives information on the « quality » of the VaR.
(US D )
3 0 0 0 0 .0
D AILY P /L (M t M ) / V a R 9 9 1 D AY 9 9 %
R e a l / C le a n M t M
R IS K
T h e o re t ic a l P / L
2 0 0 0 0 .0
1 0 0 0 0 .0
0 .0
-1 0 0 0 0 .0
-2 0 0 0 0 .0
-3 0 0 0 0 .0
24
Value-at-Risk : Back-Testing
le 11 Septembre2001
25
Value-at Risk Back-testing
If real losses are often more important than the VaR,
regulators will not accept its use.
! Conversely, if the VaR is much bigger than actual
losses, there will be overestimation of proper funds, and
profitability loss.
!Main official texts :
! Bank
for International Settlements (Basel Comity text
1997) : www.bis.org
! Fed Reserve, Market Authority in France :
www.banque-france.fr
! see also www.gloriamundi.org
26
Value-at-Risk
- back-testing (cont ’ed) Main official texts :
! Bank for International Settlements
(Basel Comity text 1997) : www.bis.org
! Fed Reserve, Market Authority in France :
www.banque-france.fr
! see also www.gloriamundi.org
27
Value-at-Risk Computing methods
There are two ways for computing the VaR :
# Analytic
#Simulation
!
!
- historic
- Monte-Carlo
28
Value-at-Risk
- common first step : risk factors choice
Regulators give minimal constraints.
! But the risk manager has to make choices which may
have important consequences:
#
What kind of interest rates?
# How to distort a curve or a surface ?
! The back-testing constraint can lead to extra risk
factors linked to specific activities (e.g, volatility
arbitrage).
29
- Analytic VaR Assumption: Gaussian risk factors
!The main hypothesis is that the options portfolio is
perfectly represented by its first derivatives.
! In the Gaussian case :
VaR
= 2 . 33
t
δVδ
! where is the sensitivity vector with respect to the
chosen risk factors and V is the variance-covariance
matrix of their variations.
30
Historical Value-at-Risk
Computing steps are :
! make datasets of the risk factors’ variations,
!« apply » them to the risk factors today in order to
get their possible evolutions,
!for each possible evolution, compute the
associated P&L, and then the 1%-quantile : the VaR.
(On 260 data, the 3rd worst P&L)
31
Historical VaR : hypotheses & properties Using historical VaR,
! It allows to measure the risks of non-linear portfolios
(i.e containing options).
! Theoretically, the historical VaR convergence is bad
(estimation of 1% quantile with only 260 data).
! Practically , it is acceptable. One reason might be the
non-independence of daily variations.
! we need to assume something about risk factors, as
stationarity.
32
Monte-Carlo VaR
Similar Methodology to historical VaR, but
! Historical variations are replaced by simulated ones,
! Using statistical estimates of factors distribution
Possibility to simulate the default of an issuer.
33
Monte Carlo VaR : hypotheses & properties
Imprecision sources are :
! Choices of the diffusion (non necessarily
Gaussian) and estimation of its parameter,
!Quantile estimation by simulations.
!This error can be reduced by using a large
number of simulations.
!In the linear Gaussian case, the first one is
also small.
! Precision is quite good.
34
Importance of market datasets !These three methods are « statistical » methods
!They need market datasets that are hard to build
! Depending on the activity, errors in the datasets
may have more or less important consequences.
35
Numerical Example(Historical V@R)
!Statistical scenario based on historical data:1year
!Confidence Level 99%
!Holding period 10 days
!Potential loses : 4,9%
!Currencies : 34%
!Interest rates: 2%
!Volatility: 5%
!Equities:59%
36
Numerical Example(Worst Case)
!Historical maximal value scenario : largest daily
chnages in the risk factors from 1987data
!Holding period 10 days
!Potential loses : 9,4%
!Currencies : 13%
!Interest rates: 3%
!Volatility+Equities: 84%
37
Numerical Example(Sensitivity)
!Stock prices= -10%, Volatility (relative)=20%,
Interest=1% (absolute), Currencies = -10%, Commodity
prices = -10%,
!Holding period 1day
!Potential loses : 8,5%
!Currencies : 40%
!Interest rates: 4%
!Volatility: 7%
!Equities:49%
38
Conclusion
Developing an efficient risk management is a
challenging task of financial institutions.
!Both Academics and Practitioners are working on
the concept of risk measures,
! and on new numerical methods for large portfolios
! Take into account other risks, like operational risk.
39
40
Similarities Between Rocket Science and
Financial Risk Management
# Complex mathematics
!Stochastic Calculus
!Partial derivatives
!Probability
# Learning from failure
!Barings Bank
!Orange County
!Proctor and Gamble
!Long Term Capital
Differences Between Rocket Science and
Financial Risk Management
# FRM researchers can’t hold other variables
constant
# Limited set of data to test theories in finance
# The rules change
“Every time I find the key, they change the lock.”
Kevin Waspi, Finance Professor, University of Illinois
Agenda
The new concept of Market Manager
# controlling traders positions
# validating models used by traders
# helping the “Middle Office” to compute the market
value of the market portfolio
# informing the upper management about financial price
risks
43
II - What is a pricing model ?
44
What is a pricing model ?
A pricing model is the association of :
$ Hypothesis on the diffusion (the law) of the
underlying assets of the option,
%Black & Scholes assumption
(Log Normal Model)
% Hybrid Model (CL)
dS t
= µ .dt + σ .dWt
St
dSt = µ .dt + σ H ( St + mH ).dWt
45
What is a pricing model ? (2)
and of :
& A method which will allow the calculation of a price, the
definition and the calculation of hedging parameters of vanillas
and exotic options :
#
#
#
Closed formulas,
Approximations
Numerical scheme :
' Diffusion Tree
' Monte-Carlo Simulation
' PDE (Partial Differential Equation)
46
What is a pricing model ? (3)
#Closed formulas :
Option Price = f(x1, …, xn) where f is an explicit function
#Approximations :
Option Price = f ( x1 ,..., xn ) ≈ g ( x1 ,..., xn )
where g is an explicit function and f is not.
# Numerical scheme :
' Diffusion Tree
' Monte-Carlo Simulation
' PDE (Partial Differential Equation)
47
What is a pricing model ? (4)
Numerical scheme :
' Diffusion Tree
OptionPrice = EDisc. (PayOff(STn ))
S0
STn
T1 T2
Tn
' Monte-Carlo Simulation
OptionPrice = EDisc. (PayOff (STn ))
' PDE (Partial Differential Equation)
OptionPrice = V /
∂V 1
∂²V
∂V
+ σ ²S ²
+ rS
− rV = 0 ( B & S )
∂t 2
∂S ²
∂S
⇔
Vti , S j = f (Vti+1 , S j+1 , Vti+1 , S j , Vti+1 , S j−1 )
STn
S0
T1 T2
Tn
Sn
S2
S1
T1 T2
Tn
48
What is a pricing model ? (5)
The hypothesis on the underlying diffusion will
let us define hedging parameters that we will be
calculated using either the closed formula or the
numerical scheme.
OptionP rice = P ( S , σ , K , T )
1
⇒ dP ( S , σ , K , T ) = ∆.dS + Γ.(dS ) 2 + V .dσ + Θ.dT
2
Hedging Parameters
49
What is a pricing model ? (5)
# For example, the definition of the volatility, and as a
consequence the calculation of the Vega of a portfolio, is made
through the Black&Scholes assumption.
# The hedging point of view is the main contribution of Black,
Scholes and Merton (1973) to Louis Bachelier’s work (1900).
(The links between pricing and hedging will be developed in the
sequel.)
50
III - What will be the price of
the portfolio tomorrow
51
Price of portfolio evolution
It is not about
! foreseeing trends,
but about
! giving information on the distribution
( of possible gains and losses
( in fews of days.
52
Portfolio Price Evolution
- 1. Risk Factors A first list is quite easy to make :
! interest rates, cross-currency rates,(100)
! stock prices, (6000)
! Implied volatilities,
! counterpart risks,
!...
!Approximately N=10 000 variables
53
Standard Risk Indicators
Widely used indicators are:
! Expected daily variation with associated standard
deviation
! For derivatives, sensitivities, generally Greeks.
! Stress scenarii.
54
Standard Risk Indicators
(cont ’ed)
Their drawbacks:
! they are not using a single figure to summarize the
global risk of a given portfolio,
! they do not allow us to have homogeneous risk
measures for different activities.
They must be adapted to individual management
strategies.
55
Portfolio price evolution
- 3. toward Value-at-Risk Global and homogeneous risk measures are
nevertheless necessary
! to give a unique « meaningful » indicator on the
aggregated portfolio
! to fairly assign to each activity funds to cover induced
risks.
Regulators have thus thoughts after a new method for
estimation of market risks.
56