Transcript Part I:

Session III
Market Risk / Introduction to Insurance Risk
Dr. Peter Kandl
April 23, 2007
Objectives
• To have a basic understanding of the measurement of
market and insurance risk
• To be able to compute Value-at-Risk
• To be able to outline a model verification strategy
2
Financial risk - General remarks
• Modern parlance: danger of loss
• Finance theory: dispersion of unexpected outcomes due to
movements in financial / market variable
– Implies that both, positive and negative deviations,
should be viewed as sources of risk
– Extraordinary performance, both good and bad, should
raise red flags
• Measurement of risk therefore requires definition of the
variable of interest
– Portfolio value, earnings, capital, particular cash flow
3
Variable of interest (risk metric)
• Portfolio Value:
V  A1 ,..., AI    wi v Ai 
i
• Capital(*) (Surplus) : Assets ./. Liabilities: V  A1 ,..., AI   V L1,...,LK 
(*)
needs capital model
• Earnings:
E  R  VC  FC  NC  ID  T
+Revenues
./.Variable costs of operations
./.Fixed cash cost (administrative costs, real estate taxes)
./.Non cash charges (depreciation, deferred taxes)
./.Interest on debt
./.Taxes
• Cash flow: Particular item(s) from earnings statement
4
Market risk
• Usually captures effect on portfolio value
• Four main types of market risk:
– Interest rate risk, exchange rate risk, equity risk,
commodity risk
• Risk is measured by standard deviation of unexpected
outcomes, also called volatility
• Losses can occur through combination of two factors:
– Volatility of underlying financial variable
– Exposure to this source
5
Market jargon
• Measurement of linear exposures to movements in
underlying financial variables:
– Fixed income market: Duration
– Stock market: Systematic risk (Beta)
– Derivatives market: Delta
• Second derivatives (measure quadratic exposure):
– Fixed income market: Convexity
– Derivatives market: Gamma
6
Source of loss: principle
• Can we decompose market loss into its constituent pieces,
exposure and adverse movement of its most influencing
financial variable?
Market Loss = (Exposure) x (Adverse Movement in Financial
Variable)
7
Source of loss: useful approximations
Bond-Position
P  ( D P)  y
*
Change in bond price =
-(Dollar duration) x (Shift in yield)
Equity-Position
Pi  Ri Pi  ( Pi )  RM
Change in price =
(Beta-position) x (Relative change in market)
Option-Position
 dS 
df    S   
 S 
Change in option value =
(Delta-position) x (Relative change in underlying)
8
Measuring asset returns
• Random component: change relative to today‘s price
(level)
• Build random sample from rates of changes in the spot
rate:
Rt  (St  St 1 ) / St 1
• Total return:
RTot
 (St  Dt  St 1 ) / St 1 , where Dt dividend or
t
coupon , reinvested at every end-of-period
9
Measuring asset returns (cont’d)
• Technical reason: construct logarithm of the price ratio:
~
Rt  ln[St / St 1 ]  ln1  Rt 
~
• For small movements: Rt  Rt
– Use Taylor series expansion to show that
for small Rt
ln1  Rt   Rt
• Typical observation: daily data have small movements in
rates of changes
10
Measuring returns: example
Level
108
107
106
105
104
103
102
101
100
99
98
0
5
10
15
20
25
30
35
30
35
Rate
3.0%
2.5%
2.0%
1.5%
1.0%
0.5%
0.0%
-0.5%
-1.0%
-1.5%
-2.0%
0
5
10
15
20
25
11
Sample estimates: mean
• Working assumption: T observations Ri 1iT of (i.i.d.)
random variables (e.g. daily returns in EUR/USD rates)
• Estimate the (unknown!) expected return or mean   ER
by the sample mean:
1 T
m  ˆ   Ri
T i 1
• Principle: Assign to each observation a constant weight 1 T
since all observations have the same probability of
occurrence
12
Sample estimates: variance
• Estimate the (unknown!) variance  2  ER   2 
the sample variance:
by
1 T
2
ˆ
s  ˆ 
(
R


)
 i
T  1 i 1
2
2
1
– Dividing by
ensures unbiasedness of estimate
T 1
• If squared return is of much smaller order than variance
term, than it can be neglected (many financial variables do
have this feature when sampled at daily intervals)
1 T 2
T
ˆ 
R

ˆ 2

i
T  1 i 1
T 1
2
13
Time aggregation
• Parameters can be (under certain conditions) adapted to
other time horizons
• Example: extend daily mean return to monthly mean
return (easy with logarithmic returns!) by using
R02  ln[S2 / S0 ]  ln[(S2 / S1 )(S1 / S0 )]
 ln[(S2 / S1 )]  ln[(S1 / S0 )]  R01  R12
to conclude that
i.i.d. (!)
E( RT )  E( R1 )T
14
Time aggregation (2)
• The same is true for variances:
V ( R02 )  V ( R01 )  V R12   2Cov R01 , R12   V R01   V R12 

 

concluding that
0
V ( RT )  V ( R1 )T
while assuming, that returns have (i.i.d.) properties across time periods
• Expressed in terms of volatilities (“square-root-of-time rule”):
SD( RT )  SD( R1 ) T
(above calculations only hold for logarithmic returns and are only
approximate when using discrete returns)
15
VaR: motivation
• Common task in (market) risk management: Quantify risk
of losses due to movements in financial market variables
• Past concepts
– Notional amounts, sensitivity measures, scenarios
– Well suited for implementation in their defined areas
– Cannot be compared across methods and financial
instruments
• Fundamental question: Is there a method that is more
general for quantifying potential losses in a (arbitrarily
chosen) portfolio?
• Question is too general, must be more specified
16
VaR as downside risk
• VaR (Definition): VaR is the maximum loss over a target
horizon such that there is a low, prespecified probability
that the actual loss will be larger
• Definition requires:
– Specifying loss distribution for a given target horizon
– Choosing an appropriate confidence level
17
VaR (general distributions)
• Define initial setting:
– W0: initial investment, R: rate of return
– Value at end of period: W=W0(1+R)
– R is random variable with (, 2)
• W*=W0(1+R*): lowest value for given confidence level c
• VaR (mean) = E(W)-W* = -W0(R*-) (relative to mean)
• VaR (zero) = W0-W* = - W0R* (absolute loss)
• Relative VaR conceptually more appealing
• For general distributions, cdf must be inverted to find cut-off value R*
(percentile)
18
VaR (parametric distributions)
•
•
•
•
Assume pdf belongs to parametric family (e.g. normal)
Standardize R by  a   R    
Find appropriate value for a confidence level)
Then R*=-a + 
• More generally, parameters  and  are expressed on
annual basis. Then use time aggregation results to show
• VaR(mean) = -W0(R*-) = W0a t
• VaR(zero) = -W0R* = W0(a t  t)
19
Standard Normal Distribution
• Standardization of distribution X~N(, 2) by
Y
X 

• Then, Y~N(0,1): standard normal distribution
• Percentiles of normal distribution
–
–
–
–
–
66% of distribution between [-,  ]
95% of distribution between [-2, 2 ]
95% of distribution below 1.65
99% of distribution below 2.33
99.5% of distribution below 2.58
20
Parametric VaR: examples
• Volatilities of financial variables are usually given on a
yearly basis
• Order of magnitudes
– Stock / market indices:
12% - 50%
– FX (against USD):
5% -15%
– IR (yield changes):
0.6% - 1.2%
– Commodity (base metals, energy):
10% - 100%
• Given a confidence level of a and a fixed position in a
financial variable, what is the “maximum” potential loss
over the next z days? Provide examples!
21
Alternative measures
• Conventional VaR measure is a quantile-estimator of the
loss distribution
• Alternative measures
– Entire distribution: report, e.g., range of VaR numbers
for increasing confidence levels
– Conditional VaR (expected shortfall, Tail VaR):
Expected loss given that loss exceeds VaR
E( X X  VaR)
– Standard deviation
1
N
2




SD X  
x

E
X
 i
N  1 i 1
– Semi-standard deviation
SDL  X  
1
N
2






Min
x
,
0

E
X

i
N L  1 i 1
22
Loss distribution - VaR
Frequency
Prob (L <= VaR)
=a
Losses
VaR
0 
Profits
23
Loss distribution - TVaR
Frequency
E[L | L ≤ VaR]
Losses
VaR
0 
Profits
24
Example
• Given the following 30 ordered, simulated percentage
returns of an asset, calculate the VaR and the expected
shortfall (both expressed in terms of returns) at a 90%
confidence level.
• -16, -14, -10, -7, -7, -5, -4, -4, -4, -3, -1, -1, 0, 0, 0, 1, 2, 2,
4, 6, 7, 8, 9, 11, 12, 12, 14, 18, 21, 23
25
Desirable properties for risk measures
• Monotonicity: If losses in portfolio A are larger than losses in
portfolio B for all possible risk factor return scenarios, then the risk of
portfolio A is higher than the risk in portfolio B
• Translation Invariance: known cash (flows) do not contribute to risk
• Homogeneity: If the size of every position in a portfolio is doubled,
the risk of the portfolio will be twice as large
• Subadditivity: The risk of the sum of the portfolios is smaller or equal
than the sum of their individual risks
• A risk measure satisfying all above criteria is said to be a coherent risk
measure
– VaR fails to satisfy subadditivity property, not a coherent measure
– CVaR is a coherent measure
26
VaR parameters: confidence level
• Confidence Level: Choose appropriate level a
• If a goes up, VaR increases
– But: number of exceptions become very rare
• Choice of a largely depends on use of VaR
• Consistency is what really matters!
– Across business units (trading desks), across time
• Capital adequacy: use high confidence level
• For backtesting purposes: use lower a (e.g. 95%) to ensure
that outcomes “beyond” a actually occur during
investigation period
27
VaR parameters: horizon
• Target horizon: Choose appropriate “holding period” T
• If T increases, VaR increases
– Depends on:
• Behaviour of risk factor(s)
• Portfolio positions
Can we establish basic rule? VaRTdays  VaRNdays T N
– Yes, if returns are i.i.d. and normally distributed
• Choose shorter horizon, if positions change quickly and /
or exposures change as underlying change (e.g. delta)
• But: Target horizon cannot be less than P&L reporting
frequency
•
28
Backtesting
• Involves systematically comparing historical VaR measures with
subsequent returns
• Returns can be based upon
– „Frozen portfolio“
– Actual return („dirty“ P&L) including intra-day trades and other
profit items (actual portfolio changes its composition)
– „Cleaned“ return: actual return minus all non-marked-to-market
items (funding costs, fee income, reserves released,...)
• Model testing is faced with two types of errors:
– Type I: Reject a correct model
– Type II: Accept an incorrect model
• Reduced to classical statistical decision problem: errors of type I and II
must be balanced against each other
29
Model verification: example
• Failure rates: Fix confidence level (p*=1-a) and record VaR figure for
a number of T days. Count number of days where actual loss (profit)
exceeds previous day‘s VaR. This number is N, and the failure rate
N/T
• Outline of a test: Null (H0) is p=p*; Number of exceptions N follows
binomial probability distribution:
T  N
T N
with E(N)=pT, V(N) = p(1-p)T
f ( N )    p 1  p 
N
• If T is large, use CTL and approximate by standard normal
z
N  pT
p(1  p)T
• Under the null, z is standard normally distributed
• Basel framework uses such a type of model verification
30
Introduction to Insurance Risk
• Insurance risk:
– Risk that policyholder claims may prove to be in excess of what
was expected
• Risk metric:
– Volume measure (technical provisions, written premium)
– Requires consistent valuation of Liabilities
• Risk measure:
– “VaR” of volume measure
– Confidence level: 99.5 %
– Holding period: 1 year
31
Reserve risk
•
Reserve risk: non-life underwriting risk that technical provisions, held to cover
incurred claims for coverage already provided, may prove inadequate
•
Principle:
– Use simple factors differentiated by
• Line of Business (different risk characteristics)
• Size of portfolio (Reserve fluctuation of large portfolios tend to be less volatile
because of greater diversification)
– Allow for reinsurance by using net reserves as volume measure
– Factor can be conceptually considered as a multiple of the standard
deviation of the distribution of the technical claim provision. The multiple
is determined such that the required capital plus the MV of claim
provisions combined, are 99.5% sufficient to cover incurred claims for
coverage already provided
32
Reserve risk (2)
• Implementation:
– Required Capital Reserve Risk (LOB) =
MV net claims provisions (LOB) * factor (LOB) * size
factor (LOB)
• Example:
– Reserve risk factor
• Accident and health:
• Motor, third party liability:
28%
12%
– Size factor
• Between 0.7 and 10 (e.g. 2 for net reserves = 25 m CHF)
– CRres (A&H) = 25m CHF * 28% * 2 = 14 m CHF
CRres (M)=
25m CHF * 12% * 2 = 6 m CHF
33
Premium risk
• Defined as risk that volume of ultimate losses for future claims
occurred or still to occur at the valuation date (losses paid during time
horizon of the cover and provisions made at its end) is higher than the
premiums received for the cover period
• Principle:
– Use simple factors differentiated by lines of business and size of
portfolio
– New business and renewals arising over the next year create
additional source of risk
– Business written over upcoming year is uncertain and a proxy for
the written premium of the previous year ca be used
– Use net premiums as volume measure to take Reinsurance into
account
34
Premium risk (2)
• Implementation:
– Required capital premium Risk (LOB) =
(Net written premium (LOB) + Net unearned premiums(LOB)) *
factor (LOB) * size factor (LOB)
• Example:
– Premium risk factor
• Accident & Health
• Motor, third-party liability 12%
18%
– Size factor
• Between 0.7 and 10 (e.g. 2 for net written premium= 25 m CHF)
– CRprem(A&H) = 25m CHF * 18% * 2 = 9m CHF
CRprem(M) =
25m CHF * 12% * 2 = 6m CHF
35
Catastrophe risk
• Principle:
– Include impact of catastrophe events taking into
account reinsurance structure of the company
– Specify certain catastrophe scenarios for the entire
market (e.g. flooding)
– Exposure to catastrophe will be measured by the gross
market share of the insurance undertaking
– Take Reinsurance into account
• Excess of Loss more effective than proportional for catastrophe
events
• Reinsurance limits and retentions should be applied
36
Mortality risk
• Defined as unexpected deviation on the mortality
experience for products providing death coverage
• Risk components are:
– Volatility
– Uncertainty (level risk and trend risk)
– Extreme event (catastrophe)
• Principle:
– Use simple factors
– Volatility risk can be reduced by increasing size of
portfolio
– Uncertainty and catastrophe risk are not diversifiable by
increasing portfolio size
37
Mortality risk (2)
• Implementation:
– Required capital mortality risk = (Factorvol +
Factorterm)* Net sum at risk
– Net sum at risk = sum assured less technical provisions
less reinsurance cover
38
Longevity risk
• Defined as unexpected deviation on the mortality
experience for products providing coverage in case of life
• Implementation:
– CR Longevity Risk = Net Technical Provision *
Longevity factor
39
Lapse risk
• Defined as unexpected deviation on the expected lapse rate
• Implementation:
– CR Lapse Risk = Net Technical Provision * Lapse
factor
40
Appendix
Univariate distribution functions
• Random variable X is entirely characterized by its
probability distribution function (cumulative distribution
function)
F ( x )  P( X  x )
which gives probability that X ends up below x
• If X takes discrete values
F ( x) 
 f (x )
x j x
j
where f(x) is the probability density function (pdf)
41
Appendix
Moments
• Are used to describe characteristics of distributions (in
fact, a distribution is uniquely defined by all of its
moments)
• First order moment: mean (expected value, average)
N
  E ( X )   xi f ( xi )
i 1
• Second order moment: variance

2
 V ( X )  E( x   ) 
2
42
Appendix
Variance and standard deviation
• The variance measures dispersion (uncertainty) of a
random variable
• Standard deviation used as measure in the same units as
random variable X
SD( X )    V ( X )
• In finance, the standard deviation, measured in relative
terms, is often called „volatility“
43
Appendix
Quantiles
• Quantile: given probability c, which cutoff point x is
associated with it?
P( X i  x)  c
• The 50%-quantile is called median
• Example:
VaR for confidence level p=95%:
Loss, which will not be exceeded in (1-p) of all cases
44
Appendix
Linear Transformations
• Let X be a random variable and consider Y=aX+b
• Expectation of Y
E(a  bX )  a  bE( X )
• Variance of Y
V (a  bX )  b V ( X )
2
• Example:
Portfolio with amount a of cash and b units of shares
45
Appendix
Normal Distribution
• The normal distribution is characterized by its first two
parameters:
– mean  (location)
– variance 2 (dispersion)
• The density is given by
f ( x)  ( x) 
1
2 2
exp[
1
2
2
(
x


)
]
2
• There is no analytic expression (involving elementary
functions) for the distribution function (use of tables,
approximations)
46
Appendix
Standard Normal Distribution
• Standardization of distribution X~N(, 2) by
Y
X 

• Then, Y~N(0,1): standard normal distribution
• Percentiles of normal distribution
– 66% of distribution between [-,  ]
– 95% of distribution between [-2, 2 ]
– 95% of distribution below 1.65
– 99% of distribution below 2.33
47
Appendix
0.4
Illustration: Normal Distribution
Frequency
0.3
66%
±1 SD
0.2
0.1
95%
±2 SD
0
-4
-3
-2
-1
0
1
2
3
4
48
Appendix
Stock Prices: Lognormal Distribution
• The normal distribution is often used for modeling because
of its attractive mathematical properties
• The infinite tail on both sides however causes problems in
finance: interest rates, stock prices, ... cannot become
negative (i.e. returns smaller than –1 are not possible)
• X has lognormal distribution if Y=ln(X) is normally
distributed
• The variable X=exp(Y) will always be positive
49
Appendix
Lognormal Distribution (cont´d)
• Y normally distributed with parameters  =E[Y] and 2
=V[Y]
• The density of the lognormal variable X is
f ( x) 
1
x 2 2
exp[
1
2
2
(ln(
x
)


)
], x  0
2
• The mean of the lognormal is
1 2
E ( X )  exp[    ]
2
• The variance of the lognormal is
V [ X ]   2 exp[ 2 1]
50
Appendix
Illustration: Lognormal Density Function
0.8
0.7
Frequency
0.6
0.5
Sigma=1
0.4
Sigma=1.2
Sigma=0.6
0.3
0.2
0.1
0.0
0

2
4
6
8
10
51
Appendix
Multivariate Distribution Functions
• Characterize joint behavior of several random variables
• In the bivariate case, we can write
F12 ( x1 , x2 )  P( X1  x1 , X 2  x2 )
• If the joint density f(u1,u2) is discrete, then
F12 ( x1 , x2 ) 
f
y1  x1 y2  x2
12
( y1 , y2 )
52
Appendix
Independent Random Variables
• If realization of X1 is independent from realization of X2
P( X1  x1 , X 2  x2 )  P( X1  x1 )  P( X 2  x2 )
• Consequently, both variables can be analysed
independently
• Knowledge of value of one variable provides no
information about potential value of other variable
53
Appendix
Conditional Density
• Conditional density: density of X1, given X2
P( X 1  x1 , X 2  x2 )
P( X 1  x1 X 2  x2 ) 
P( X 2  x2 )
P( X1  x1 x2 )  P( X1  x1 )
• If variables are independent:
• The marginal and conditional distributions of multivariate
normal distributions are always normal
54
Appendix
Covariance & Correlation
• Measure of co-movement of 2 random variables
Cov( X1 , X 2 )  E[(x1  1 )(x2  2 )]
• Correlation as unitless measure
 ( X1, X 2 ) 
Cov( X 1 , X 2 )
 1 2
• Correlation lies always in interval [-1,1]
Independence implies correlation of 0, but not
vice versa (exception: normal distribution)
55
Appendix
Sum of Random Variables
• Expectation of 2 random variables
E( X1  X 2 )  E( X1 )  E( X 2 )
• Variance of 2 random variables
V ( X1  X 2 )  V ( X1 )  V ( X 2 )  2Cov( X1 , X 2 )
V ( X1  X 2 )  V ( X1 )  V ( X 2 )  2Cov( X1 , X 2 )
• If the variables are uncorrelated, the variance of the sum is
equal to the sum of variances
• Formulas generalize easily to n random variables
56
Appendix
Portfolios of Random Variables
• Linear combinations with fixed weights wi
N
Y   wi X i
i 1
are a generalization of linear transformations and sums
 N
 N
E  wi X i    wi E ( X i )
 i 1
 i 1
N
 N 2
V   wi X i    wi V ( X i )  2 wi w j Cov( X i , X j )
i j
 i 1
 i 1
• Example: Xi: asset returns, wi: portfolio weights („position
size“)
57