Risk Management

1
Risk Management

• Andre Horovitz

2
What is Risk?

• Risk (in finance) is interpreted as the
  statistical (mathematical) possibility of
  incurring unforeseen financial losses, above and
  beyond expected, as changes occur in the
  environment (economy, etc.) which affect the
  value of ones assets (also known as investment
  at risk)

3
Risk Management - Definition

• Risk management is the profession (meanwhile an
  entire industry) which deals with the estimation
  (quantification) of risk and employs techniques
  to mitigate or limit excessive risks

4
Types of risks (example)...

• Market Risk - risks attained from adverse changes
  in market parameters such as interest rates,
  foreign exchange rates, equity prices or indexes,
  commodities, etc.
• Credit Risk - risks attained from the failure of
  counterparties to respect their contractual
  obligations (loan losses, settlements, issuer
  defaults, bankruptcies, etc.)

5
Types of risks (contd)

• Business risks - risks attained from unforeseen
  environmental changes (ex. Competitive forces)
  causing an economic entitys drop in earnings
  beyond a forecasted trend
• Operational risks - risks relating to unforeseen
  failures such as fraudulent activities, systems
  failures (IT, technology), natural phenomena
  (catastrophes) or unexpected changes in legal /
  regulatory environments

6
The Risk Management Industry / Profession

• Risk management risk control units in banks,
  insurance/ re-insurance companies, other
  financial or non financial enterprises
• IT departments in charge of providing
  (maintaining) IT systems geared to support risk
  professionals
• Research departments or firms in charge of
  developing/ refining analytical tools to quantify
  risks
• Software providers, business consultants

7
Lessons learned from recent financial disasters

• Losses attributed to Derivatives (1993 through
  1999)

8
Case study 1 Barings

• Nick Leeson (28) lost 1.3 bn in Index Futures
  trading on the Osaka Exchange
• Ran front back office at the same time
• Accumulated 20,000 contracts each worth 200,000
  - approx 20 of the volume
• Big money attracts attention
• YMIS (Young Male Imortality Syndrom)

9
Case Study 2 Metallgesellschaft

• Idea long term oil delivery contracts (180mm
  barrels over 10 years) - equiv. To Kuwaits oil
  production over 85 days
• Rolling hedge with oil futures (3 months
  maturity)
• Basis risk
• Mark to market (margin calls)- lifes a path
  dependent function

10
Case Study 3 Orange County

• 7.5 bn portfolio of public money
• Borrowed 12.5 bn through reverse repos
• Invested in avg. Life 4 years agency bonds /
  notes - refinanced over short term (LIBOR)
• Exposure to yield curve steepness
• Default overmargin calls from short term
  financiers and collateral payments
• No mark to market accounting since held to
  maturity

11
What to expect from this course?

• Risk view from a practitioners perspective
• Familiarity with the market jargon (instruments,
  conventions, methods, systems, players, etc.)
• Merits and pitfalls of applying math in measuring
  risk
• The story of the birth and evolution of the risk
  profession from an eyewitness

12
Fixed Income Fundamentals
Q Prof. Einstein what was in your opinion the
greatest discovery of mankind to date?
A Undoubtedely its been Compound Interest
13
Yield Compounding Conventions

• Yield Internal rate of return
• Typically not a very good measure of return
• Assumes reinvestment prevailing at the same rate
  rarely if ever the case
• finagled in finance by assuming reinvestments
  at the implied forward rates
• Means that indeed, investors expectations of
  future interest rates are reflected in the IFR
  Curve
• Constant refinancing spreads over the lifetime of
  the investment
• Annualized Yield - termed effective annual
  rate (EAR)
• m1 annual
• m2 semiannual
• m12 monthly
• m365 daily

14
Compounding

• n nr of periods
• If n--gt8 - continuous compounding
• Let (1/n) x and applying LHospital Rule
• If n--gt8, F ePTR
• When P1, FeRT
• FP1(m/n)R simple interest compounding

15
Valuation of a Perpetual Bond (Consol)

• Used by the British Empire to finance the Spanish
  Wars
• Used today (albeit in their callable / redeemable
  variants) for Hybrid Tier 1 Funding by Banks in
  Jeopardy of falling under the regulatory Minimum
  Tier 1 Thresholds
• Please Derive the (FINITE) NPV
• Hint CF/y

16
Bond Price Sensitivities first order

• High School Memory (beyond the first kiss!)
• Taylor Expansion around a (Class n) functions
  initial value
• Modified Duration D-(dP/dy)/P0
• P0 represents the Market Price of the Bond at
  time T0 plus accrued interest (dirty price)
• Dollar (effective) Duration DD-(dP/dy)DP0
• DVBP (DV01) DD ?y -(dP/dy) x 0.0001
• MacCauley Duration DMacD(1y)

17
Bond Price Sensitivities second order

• Dollar Convexity C(d2P/dy2)/P0
• Calculate the first and second order
  sensitivities of a Zero Coupon Bond with Maturity
  T years

18
Bond Price Sensitivities finite differences

• Back to Memories from the time of the first kiss
• Delta Duration
• Delta Convexity
• Coupon (Curve) Duration
• Coupon (Curve) Convexity

19
Economic Interpretation of Duration
Duration is the average time of waiting for each
payment, weighted by the present cash flow
wt is the ratio of the PV of each cash flow Ct
relative to the total and summed to unity
Mac Cauley Duration is the average of time to
wait for all cash flows in investment
bankerese average life of a bond
Back to the perpetuity (consol) Find the Mac
Cauley Duration of a Consol and prove that it is
finite and independent of the consols coupon!
20
Economic Interpretation of Convexity
wt a weightet average life of the square of time

As well see later, convexity can be negative for
bonds with uncertain cash flows (MBS or some
callable issues)
21
Duration as a Function of Coupon and Maturity
22
Duration as a Function of Coupon, Maturity,
Coupon Frequency, Yield

• Increasing Maturities correspond to higher
  Durations
• Increased Coupon Values, Yield and Coupon
  Frequencies correspond to lower Durations
• With very high maturities, Durations converge to
  the Consols (finite) Duration
• Duration is a Convex Function of Coupon Size
• Prove!

23
Homework on Duration Curve Shapes

• Obviously Duration of a Zero Bond increases
  linearly with Maturity
• Prove analytically that
• Low coupon bonds (coupon below yield or sub
  par) exhibit a maximum duration at a certain
  maturity
• Determine that maximum duration maturity
• And determine that maximum Duration while proving
  that it is above the consols duration
• then converge to the consols duration (from
  above) while reaching a point of inflexion
• Determine the Maturity corresponding to the
  inflexion point
• Explain why this phenomenon is not true for high
  coupon bonds (coupon higher than yield)
• Prove that the Duration curve is a concave
  function of maturity
• And that it converges at infinity (from below) to
  the consols duration

24
Portfolio Sensitivities

• Let
• PP Portfolio Value
• DPPortfolio Modified Duration
• xi number of units of bond i in the portfolio
• Portfolio Dollar Duration
• If the weights are all the same (for all
  components) the equation also holds for the Mac
  Cauley Duration
• Similar relationship for the portfolio Dollar
  Convexity
• For PP?0, define portfolio weights wixiPi/ PP

25
Portfolio Sensitivities - Homework

• A portfolio of bonds with very short AND very
  long maturities is called a barbell portfolio
• A portfolio of bonds with similar maturities is
  called a bullet portfolio
• Prove that for the same Prortfolio Duration,
  Barbell Portfolios have higher Convexities than
  Bullet Portfolios

26
Shortfalls of Duration and Convexity

• Most Fixed Income Portfolio Managers gauge their
  Portfolio Risk on Duration Convexity
• However, these measures - while simple to use in
  practice are not short of weaknesses
• They represent only first and second order
  sensitivities may be insufficient for portfolios
  with embedded options or highly structured (like
  CDO tranches)
• Do not reflect sensitivities to spreads or credit
  ratings /default probabilities
• Represent changes in portfolio values when very
  small, parallel changes in the interest rate term
  structures occur
• Reality is different most of the time
  especially in those instances when analysts worry
  about high risks

27
Most Common Applications of Parallel Shift
Sensitivities

• Most BondPortfolio Risk is measured and Limited
  using a VAR measures
• Duration is widely used for estimating hedge
  ratios for systematic risk hedging (e.g. via
  futures contracts more on this later)
• Convexity is mainly used to select cheap from
  dear bonds (higher convexity is better, cet.
  par.) in secondary markets
• Convexity (especially negative convexity) plays a
  pivotal role in MBS investments selection /
  pricing and their structured cousins (IOs POs
  more later)

28
A War Story Salomon Brothers Discovered that
Duration can Mislead August 1985
29
Investor holds Bond B and hedges by shorting
bond D

• Not apparently a bad choice
• The two bonds have similar maturities (so low
  yield curve reshaping risk)
• Similar Durations 8,06 vs 8,45
• Investor assumes that closeness of durations
  imply similar price volatilities and choses a
  hedge ratio of 1.0
• Interim Question
• Suppose the investor wants to hedge bond B with
  bond A (a lower duration bond)
• Must she use a higher hedge ratio than 1.0 to
  counter the low volatility of bond A as indicated
  by its duration?
• No because the hedger is interested in Dollar
  Volatility, Not in Percentage Volatility (hedge
  ratio is 0.977)
• Using the PVBP table we can derive the hedge
  ratio to be
• (0.062988 / 0.079345) 0.79 (far from 1.0!)
• Similarly hedging B with A (0.062988/0.064482)0.
  977

30
Hedging as a Function of Absolute Price Volatility
Hedge Ratio depends on Absolute PVBP NOT
NECESSARILLY DURATION!!!
If target security has higher PVBP, then hedge
ratio gt 1, if hedge security, then hedge ratio lt1
31
Break
32
Fundamentals of Probability Theory

• Univariate Distribution Functions
• A Distribution Function F(x)P(Xx) (cumulative
  distribution)
• If R.V. takes DISCRETE values,
• f(x) is the frequency distribution or
  probability density function
• If R.V is CONTINUOUS,
• Whereby the probability density function is

33
An Illustration the Logarithm of Equity Returns
(in theory!)
Cumulative Probability Distribution
Cumulative Probability
34
Moments of a Probability Distribution

• Expected Value (Mean)
  (central tendency)
• Quantile (cutoff)
• There exists a unique probability p1-c / Xx
• The 50 Quantile is called Median of the
  Distribution
• In non symmetrical distribution, Median ? Mean
• The Level of a distribution corresponding with
  the Maximum Density is called Mode
• Distributions can be multi modal
• Some Payoffs occurring in complex structured
  credit products exhibit multi modal behavior

35
A First Encounter of Value at Risk

• VARa the Cutoff / Loss Not To be Exceeded with a
  probability of p a
• If f(u) is the density function of the Loss
  (function),
• Then p being the tail end probability
• VAR can be defined as a quantile, or
• VAR can be defined as the Deviation between the
  Quantile and the Expected Value
  VAR(c)E(x)-Q(x,c)
• Note in market risk we often ignore the expected
  value as we assume that for short holding
  periods, the log distribution of returns is
  typically centered at zero mean not so for
  skewed distributions like in default risk or
  operational risk models

36
VAR A Graphical Representation
VARa
Cumulative Probability
37
Higher Moments of Univariate Distributions

• Variance (Squared dispertion around the Mean)
• Skewness (describes departure from symmetry)
• If ?lt0, the distribution has a long left tail
  (suggesting high probability of negative values)
  if ?gt0, long right tail.
• If we define probable credit losses as positive
  then a typically credit loss distribution would
  be skewed to the right
• Kurtosis (describes the degree of flatness of a
  distribution)
• When dgt3, leptokurtotical distribution (fat
  tails)
• Normal Distribution d3

38
Multivariate Probability Distributions

• If Financial Assets are functions of several
  variables their performance profile is mostly
  described by use of multivariate distributions
• The Joint Bi-Variate Distribution
• The World gets much simpler if the Variables are
  Jointly Independent. This means f(u1,u2)f(u1)
  x f(u2) and F(x1,x2)F(x1) x F(x2)
• It is often useful to characterize the
  distribution of one variable, abstracting from
  the other
• Marginal Density Functions
• Conditional Density Functions (Bayes Rule)
• Note Division by f1( ) resp f2 ( ) keeps
  integration to 1 of f12 ( )

39
A First Encounter of Copulas

• If u1 and u2 are independent, the joint density
  is the product of the marginal densities (we have
  seen that earlier)
• Copulas are functions used to model variable
  dependencies they link marginal distributions to
  joint distributions
• Formally, a Copula is a function of the marginal
  distribution F(x) some parameter ? (specific
  to the copula function)
• In the simple, bi-variate case,
  C12C12F1(x1),F2(x2),?
• Sklars Theorem For Any Joint Density Function,
  there Exists at least One Copula linking it to
  the respective Marginal Distributions
• F12(x1,x2)F (x1) x F(x2) x C12F1(x1),F2(x2),?
• Note If x1, x2 are independent, then C12 is
  identical to 1
• Copulas contain Information on the nature of
  random variable dependencies but NOT on their
  marginal distributions
• Copulas are used extensively in modeling CDOs

40
Covariance as a Copula Measuring Linear Dependency
Define Cov (x1,x2) s12 for x1, x2 being
normally distributed
Further, define the correlation coefficient
In the normal distribution case,?(x1,x2) fully
describes the dependence structure
The two dimentional Gaussian Copula
Note if x1,x2 are independent ( x1 and x2 are
uncorrelated), then
Cov
fS is the conditional Density Function for a
bivariate Normal Distribution with Mean zero and
Covariance Matrix S, or
For ?0, the Gaussian Copula equals the
independence Copula
41
Functions of Random Variables

• Linear Transformations
• E(aXb)aE(X)b
• V(aXb) a2V(X)
• SD(aXb)aSD(X)
• E(X1X2)E(X1)E(X2)
• V(X1X2)V(X1)V(X2)2Cov(X1,X2)
• Portfolio of RandomVariables
• YSwiXiwX

42
Functions of Random Variables (contd)

• E(X1,X2)E(X1)E(X2)Cov(X1,X2)
• If Cov(X1,X2)0 (x1 and x2 are independent), then
• V(X1,X2)E(X1)2V(X2)E (X2)2V(X1)V(X1)V(X2)

43
Distributions of Random Variables

• PYyPg(X) yPX g-1(y)Fx(g-1(y))
• F() is the cumulative distribution function of X
• Lets put some meet around the bone!
• Consider a Zero Coupon Bond
• R6
• T30 Years
• V17.41
• Annual Volatility of Yield Changes 0.80
• Estimate the Probability of the Bonds Value
  falling below 15
• V100 x (1r)-T r(100/V)1/T-1
• Associated Yield Level g-1(y)(100/15)1/30-16.5
  28
• P(V 15)P(r6.528)
• If we assume that yield changes are normally
  distributed, with a volatility of 0.8 - then
  P25.5

44
The Uniform Distribution
f(x)
1/(b-a)
x
a
b
f(x)1/(b-a), axb E(X)(ab)/2 V(X)(b-a)2/2
45
The Normal Distribution
f(x)
68
-2
-1
0
2
1
x
Fully Specified by 2 Parameters µ and s E(X)µ
V(X)s2
For µ0 and s1 Standard Normal
Skewness0 Kurtosis3
46
The Normal Distribution (contd)

• Any Normal Distribution (µ?0s?1) can be
  recovered from the Standard Normal (e)
• By defining X µe s
• E(X) µE(e) sµ0 s µ
• V(X)V(e) s21s2s2
• To find the quantile of X at a given confidence
  level c, replace e by a
• Q(X,c) µ- a s
• VAR aE(X)-Q(X,c) µ- (µ-a s) a s
• Important Property Any Additions or Subtractions
  of Random Variables distributed normally result
  in a normally distributed random variable

47
The Normal Distribution (contd)

• Weve already encountered the normal distribution
  for two correlated random variables (when we
  illustrated the Gaussian Copula)
• For N random variables
• Back to Copulas we can separate into N distinct
  marginal normal distributions a joint Copula
• For the case of only 2 variables X1 and X2
• f12(X1,X 2)f1(X1) x f2(X2) x c12F1(X1),F2(X2),?
  
• f1,f2 are normal marginal distribution and c12 is
  a normal (gaussian) copula (the one we
  encountered earlier, remember?)
• The only parameter is the correlation coefficient
  ?12 determining S
• ?12 defines the strength of the dependency
  between the two variables

48
The Lognormal Distribution

• The (natural or neperian) logarithm of the
  random variable is distributed normally
• Used ad nauseam in modeling equity prices since
  due to the limited liability of the company
  structure, equities cant go negative but can
  raise to technically infinity
• The instantaneous rate of return of equities
  can become negative, but tends to be skewed in
  distribution
• By taking the natural log (of the instantaneous
  returns) finance professionals came up with a
  more sensible description of markets behavior
  (albeit minding higher experienced kurtotic
  effects, predominantly at the negative tail ends)

49
The Lognormal Distribution (contd)
EYElnXµ VYVlnXs2 If we set
EXexp(r), the mean of the associated normal
variable is EYElnXr- s2/2
s1.2
s0.8
s0.5
50
The Student t Distribution

• It is widely used in hypotheses testing as it
  describes the distribution of the ratio of the
  estimated coefficient(s) to its standard error(s)
• Also increasingly used in models where tail end
  probabilities are higher than the ones under
  gaussian distributions has fatter tails than
  the normal distribution (as a marginal
  distribution)
• The Student t Copula allows for stronger
  dependencies in the tails used to model CDO
  mezzanine tranches where default levels once
  exceeding a certain threshold tend to strengthen
  the correlation assumptions (more on it when we
  describe CDOs)

51
The Student t Distribution (contd)
The Density Function
When k converges to infinity, f(x) converges to
the normal density function Mean0 VXk/(k-2)
kgt2 The kurtosis parameter d 36/(k-4)
kgt4 For equity returns, k is btw 4 and 6
52
The ?2 Distribution
EXk VX2k When k is large, the distribution
converges to the normal distribution N(k,2k)
The F
Distribution
F(a,b)?2(a)/a/ ?2(b)/b ratio of independent
chi squared variables divided by their degrees of
freedom Used extensively in hypotheses testing
and regression analyses
53
The Binomial Distribution

• Binomial Variable is the result of independent
  Bernoulli Trials
• Outcome 0 or Outcome1
• EYp VYp(1-p)
• Density Function
• EXpn VXp(1-p)n

54
The Poisson Distribution

• Used to describe the number of events occurring
  over a fixed period of time, assuming events are
  independent of each other
• Used in the CreditRisk Model for quantifying
  credit risk capital (a lot on it later)
• Density Function
• ?gt0 is the average arrival rate during the period
  (success rate)
• EX ? VX ?
• Used for intensity models in credit risk
  (intensity of default)
• Important Properties
• when n is very large, the binomial distribution
  converges to a Poisson distribution
• When the success rate is very small (converges to
  0), the expected value of the binomial
  distribution converges to ? (fixed)
• For very high values of ?, the Poisson
  Distribution converges to a Normal Distribution
  N(?, ?)
• Also known as the central limit theorem used in
  Monte Carlo Simulations

55
Limit Distributions - Averages

• From previous Slide CLT
• The mean of n IID (independent and identical
  distributed) Variables converges to a normal
  distribution, when n increases
• True for Any underlying distribution, as long as
  the realizations are INDEPENDENT
• Same as saying that the Normal Distribution is
  the limiting distribution of the averages
• Since weve seen (previous slide) that binomial
  variableas are made of sums of independent
  Bernoulli trials,
• When n?8, by using CLT we can approximate the
  binomial with a normal by setting

56
Limit Distribution -Tails EVT

• CLT deals with the distributions of means
  (averages)
• Key EVT (extreme value theory) theorem
• Limit distributions for the values of a R.V. x,
  beyond a cutoff point ? belongs to the following
  family (GPD)
• F(y)1-(1-?y) -1/ ? , ??0
• F(y)1-e-y, ?0
• Where y(x- ?)/ß, ßgt0
• Loss defined as positive, such that ygt0
• ß scale parameter
• ? shape parameter determines speed at which
  the tail disappears

57
Limit Distribution -Tails EVT (contd)

• GPT (Generalized Pareto Distribution) because
  it subsumes other distributions as special cases
• ?0, normal distribution tails disappear at an
  exponential speed
• ?gt0, fat tails typical for financial data
• ??0, Gumbel Distribution
• ?gt0, Fréchet Distribution
• ?lt0, Weibull Distribution

58
Case Study Multivariate Distributions

• FX Desk long 70 and 30 reference currency is
  1 Mil.
• / spot 1,2678
• / spot 0,8866
• Joint Density Function of Returns
• Calculate
• The Marginal Density Functions
• The Risk of the Portfolio at a 99 confidence
  level

1,33119
50
1,204411
50
0,97526
45
0,79794
55
59
Case Study The Binomial Distribution

• Your Bank has written a 4th to default CDS over
  one year on a basket containing 50 names
• Assume that all default probabilities (over one
  year) are 0,55 and are uncorrelated with each
  other
• What is the Probability of the Option (assume
  European Feature) being Exercised at the end of
  the year?

60
Solution Case 1
61
Answer Binomial Distribution Case