Transcript Topic 1. Introduction to financial derivatives

Topic 3. Measuring Market Risk
3.1 Benefits of measuring market risk
3.2 Mathematical preliminaries
3.3 VaR measure
3.4 RiskMetrics model
3.5 Historical simulation
3.6 Monte Carlo simulation
3.7 Regulatory model
1
3.1 Benefits of measuring market risk

Benefits of market risk measurement (MRM)
 Management information: MRM provides information
on the risk exposure of the trading portfolio of each
FI’s trader or the whole FI to the senior management.
 Setting limit: MRM could provide the information
related to the risk exposure of the trading portfolio.
This facilitate in setting the portfolio position limits.
 Resources allocation: MRM may allow for the
identification of areas with greatest potential return per
unit of risk into which more capital and resources can
be directed.
2
3.1 Benefits of measuring market risk


Performance evaluation: To use return-risk ratio to
assess the performance of traders.
Regulation: FIs can use their internal MRM model to
set their capital requirements when the one set by Bank
of International Settlements (BIS) is too high.
3
3.2 Mathematical preliminaries
Percentile
 If the cumulative distribution function F(x) of a random
variable X is continuous, for 0 < p < 1, the (100p)th
percentile, p, of X is obtained by solving
F 
p
  Pr  X

p

p
(3.1)
4
3.2 Mathematical preliminaries

Example 3.1
Suppose X ~ (0.5, 0.42) where (,) stands for the
normal distribution with mean  and variance .
From Eq. (3.1), 0.9 of X is obtained by solving
Pr  X   0 .9   0 . 9
 0 .9  0 . 5 

Pr  Z 
  0 .9
0 .4


 0 .9  0 . 5
where
Z ~  0 ,1 
 1 . 282
0 .4
 0 .9  1 . 013
5
3.2 Mathematical preliminaries

If F(x) is not continuous (may be piecewise
continuous), the (100p)th percentile, p, of X is
obtained

p
 min l : Pr  X  l   p 
(3.2)
6
3.2 Mathematical preliminaries

Example 3.2
x
2
4
7
12
15
Pr(X = x)
0.2
0.02
0.48 0.22
0.08
Pr(X x)
0.2
0.22
0.70 0.92
1
From Eq. (3.2),
0.7 = 7
0.9 = 12
7
3.2 Mathematical preliminaries



In realistic, the probability distribution of X is
unknown. So, we need to base on the observations of
X to find its (100p)th percentile p.
There is no universal agreement upon the definition
of the percentiles of sample of data.
We define {x1, x2, …, xn} be a sample of size n of a
random variable X. We assume the observations in the
sample are already arranged in ascending order i.e.
x1< x2< …< xn .
8
3.2 Mathematical preliminaries
Definition S1
 The probability distribution of X is approximated by
Pr  X  x i  
1
for i  1, 2 ,  , n .
(3.3)
n
The p is defined as


p
 min x i : Pr  X  x i   p 
(3.4)
If np is an integer, then p = xnp.
9
3.2 Mathematical preliminaries
Definition S2 (Usually given in the statistics textbook)
 The probability distribution of X is approximated by
 i
for x  x i , i  1, 2 ,  , n
n 1

Pr  X  x    j  x  x j  1
(3.5)
for x j  x  x j  1 ,
 n  1 x j 1  x j n  1

j  1, 2 ,  , n  1

The second line in Eq. (3.5) is obtained by taking the
linear interpolation between (xj, Pr(X  xj)) and
(xj+1, Pr(X  xj+1)).
10
3.2 Mathematical preliminaries
F(x)
F(xj+1)
F(x)
F(xj)
Note: F(x) = Pr(X  x)
xj
x
xj+1
x
11
3.2 Mathematical preliminaries
Define r = p  (n +1). Set r = k + d where k is the integer
part of r and d is its decimal part. Then
p = xk + d  (xk+1 – xk)
(3.6)
Exercise: Using Eq. (3.5), verify Pr(X  p ) = p.
In this definition, the range of p is
 100 % 100 %  n 
 n  1 , n  1 .


One possible way to handle the case for p outside the
above range, the minimum or maximum values in the
observations of X are assigned to percentiles for p
outside that range.
12
3.2 Mathematical preliminaries
Definition S3 (Excel)
 The probability distribution of X is approximated by
 i 1
for x  x i , i  1, 2 ,  , n
n 1

Pr  X  x    j  1  x  x j  1
for x j  x  x j  1
 n  1 x j 1  x j n  1

j  1, 2 ,  , n  1

Define l = p  (n – 1) + 1.
Set l = k + d where k is the integer part of l and d is its
decimal part. Then
p = xk + d  (xk+1 – xk)
(3.8)
(3.7)
13
3.2 Mathematical preliminaries

Example 3.3
By considering the following 10 observations
1000, 800, 600, 400, 0, 50, 300, 400, 900, 1000
Find 0.1.
Definition S1:
0.110 = 1  0.1= x1 = 1000.
Definition S2:
r = 0.1  (11) = 1.1
k = 1 and d = 0.1
0.1 = x1 + 0.1 (x2 – x1)
= – 1000 + 0.1  (– 800 – (– 1000)) = –980
14
3.2 Mathematical preliminaries
Definition S3:
l = 0.1  (9) + 1 = 1.9
k = 1 and d = 0.9
0.1 = x1 + 0.9 (x2 – x1)
= – 1000 + 0.9  (– 800 – (– 1000)) = –820
15
3.2 Mathematical preliminaries

If the sample size n is large, p in Definition S1, S2
and S3 will be similar to each other.
16
3.2 Mathematical preliminaries
Covariance and correlation
 Let X1 and X2 be 2 random variables.
The covariance of X1 and X2 is denoted as cov(X1, X2) and
defined as
cov  X 1 , X 2   E  X 1  E  X 1  X 2  E  X 2 
 E  X 1 X 2   E  X 1 E  X 2 
(3.9)
where E(X) denotes expectation (expected value) of a
random variable X.
 If cov(X1, X2) is positive (negative), then X1 and X2 are
said to be positively (negatively) correlated.
17
3.2 Mathematical preliminaries

If the two random variables are positively (negatively)
correlated, then they tend to move in the same (opposite)
direction.
18
3.2 Mathematical preliminaries

Example 3.4
Suppose var(X1) > 0. (Fact: var(X1)  0.)
If X2 = 2 X1, then
2
2
cov  X 1 , X 2   E 2 X 1   2  E  X 1 
  
 2 E X 1   E  X 1 
2
2

 2 var  X 1   0
So, X1 and X2 are positively correlated.
From the definition of X2, we see that X2 will increase
(decrease) if X1 increase (decrease).
19
3.2 Mathematical preliminaries

If cov(X1, X2) = 0, then X1 and X2 are said to be
uncorrelated.
If X1 and X2 are independent, then X1 and X2 are
uncorrelated. This can be seen by replacing E(X1X2 ) with
E(X1)E(X2) in Eq. (3.9).
 Uncorrelated does not imply independent.
 Example 3.5
Define X1 as

Pr  X 1   1   Pr  X 1  0   Pr  X 1  1  
X2  X
2
1
1
3
20
3.2 Mathematical preliminaries
E  X 1   1
1
 0
3
1
 1
3
 
E X 2   E X 1  1
2
 
2
0
3
 0
3
E  X 1 X 2   E X 1  1
3
1
1
1

2
3
3
 0
1
3
 1
3
1
0
3
From Eq. (3.9),
cov  X 1 , X 2   0  0 
2
 0.
3
So, X1 and X2 are uncorrelated. It is obvious that X1 and
X2 are not independent.
21
3.2 Mathematical preliminaries

The correlation, corr(X1, X2), between X1 and X2 is
defined as
cov  X 1 , X 2 
cov  X 1 , X 2 
corr  X 1 , X 2    X , X 

XX
var  X 1  var  X 2 
1
2
1
(3.10)
2
where X is the standard deviation of a random variable X.
22
3.2 Mathematical preliminaries

Formulae for variance, covariance and correlation:
2
2
i. var  aX   a  X
 N

ii. var   X i  
 i

N

i 1
N
2
Xi
 2
i , j 1 j  i
iii. cov  aX , bY   ab cov  X , Y
m
 n


iv. cov  a i X i ,  b j Y j  


j 1
 i 1


n
X i ,X
X
i
X
j

(3.11)
m
ab
i
i 1
j
j
cov  X i , Y j 
j 1
v.  aX , bY  sgn( ab )  X ,Y
where
1 if x  0

sgn( x )   0 if x  0
  1 if x  0

X, Y, Xi, Yj are random variables and a, b, ai, bj are constants.
23
3.2 Mathematical preliminaries

Denote the observations of X1 and X2 as follow:
X 1 : x11 , x12 ,  , x1 N 
X 2 : x 21 , x 22 ,  , x 2 N 
The unbiased estimate of cov(X1, X2) and Xj are
cov  X 1 , X 2  

X
j

where
1
N 1
1
N
x

1
1i
 x1  x 2 i  x 2 
(3.12)
i 1
N
 x
N
 xj
2
ji
for j  1, 2
(3.13)
i 1
 N

x1    x1 i  / N ;
 i 1

 N

x2    x2i  / N
 i 1

Based on Eqs. (3.12) and (3.13), the estimate of
also be obtained from Eq. (3.10).
X
1,X 2
can
24
3.2 Mathematical preliminaries
Monte Carlo simulation
 A random variable X is said to be uniform distributed
over the interval (a, b), a < b, if its probability density
function is given by
 1

f x    b  a
 0
if a  x  b
(3.14)
otherwise
We use X ~ U(a, b) to denote that X follows a uniform
distribution over (a, b).
25
3.2 Mathematical preliminaries
f(x)
1/(b – a)
a
b
x
 In
other words, X is uniformly distributed over (a, b) if all
its possible values are restricted on that interval and it is
equally likely to pick any sub-interval with equal length
on that interval.
26
3.2 Mathematical preliminaries

Example 3.6
Suppose X ~ U(4, 10).
Using f(x) in Eq. (3.14),
Pr 0  X  1  
Pr 3  X  5  
Pr 5  X  7  

1
f ( x ) dx 
0
0 dx  0
0

5

7
f ( x ) dx 
3
Pr 8  X  10  

1

4
0 dx 
3
1
10  4
5

10
8
dx 
1
10  4
5
1
4
10  4

dx 
1
6
1
3
dx 
1
3
 Pr 5  X  7 
27
3.2 Mathematical preliminaries

Theorem (Inverse transform theorem):
Let U ~ U(0,1) (Standard Uniform distribution). For
any continuous distribution function F the random
variable X defined by
–1
X = F (U)
(3.15)
has distribution F. [F –1(u) is defined to be that value
of x such that F(x) = u.]
28
3.2 Mathematical preliminaries

The procedures to generate a sample of N
observations from (, 2) are as follows:
1.
2.
3.
4.


Generate a random number, u, from U(0,1).
Solve z from Pr(Z  z) = u where Z ~ (0, 1).
Set x =  +  z. (X =  +  Z ~ (, 2).)
Repeat steps 1 to 3 N times.
In Excel, the steps 1 to 3 can be done by just using a
single command:
=NORMINV(RAND(),,)
Example 3.7 (To be discussed in the tutorial.)
simulation-normal.xls
29
3.2 Mathematical preliminaries

X1, X2, …, Xn are said to follow a multivariate normal
distribution if

X i~   i ,  i

2

and corr  X i, X
j
 
ij
The covariance matrix, , of X1, X2, …, Xn is defined
as
   ij 
n n
and
 ij  cov  X i , X
j
 
ij
 i
j
30
3.2 Mathematical preliminaries

In case of n = 2 and i > 0 (for i = 1, 2), we can
generate a sample with sample size of N for X1 and X2
with following procedures:
1. Generate two random numbers, u1 and u2, from U(0,1).
2. Solve zi (i = 1, 2) from Pr(Z  zi) = ui where Z ~ (0, 1).
3. Set x1 = 1 + 1 z1;
x 2   2   2  12 z1   2 1   12 z 2 .
2
4. Repeat steps 1 to 3 N times.

The general case will not be discussed in this course.
Those interested may refer to the supplementary
material “Generating Multivariate Normals” in course
webpage.
31
3.3 VaR measure
Let P be a portfolio of financial assets.
Statement S:
“We are X % certain that the portfolio P will not loss
more than $V in the next N days.”
V is defined as the VaR (value at risk) of the portfolio P.
V depends on the time horizon (N days) and the
confidence level (X %). To avoid confusion in some cases,
we will state VaR as “N-day X % VaR”.
 VaR is a single number which attempts to summarize the
total risk in a portfolio of financial assets.

32
3.3 VaR measure
Define LN as a random variable which stands for the
portfolio loss over the next N days.
 Positive LN = loss; Negative LN = gain.
 When the distribution of LN is continuous, the statement S
can be expressed mathematically as

Pr  L N  V   X %.
(3.16)
33
3.3 VaR measure


From Eq. (3.1), the VaR in Eq. (3.16) is the Xth percentile
of the distribution of LN.
With Eq. (3.2), or Definition S1, S2 or S3, we can also
define the VaR when the distribution of LN is not
continuous or unknown.
34
Source: Alexander J. McNeil et. al. “Quantitative Risk Management”,
Princeton University Press 2005.
35
3.3 VaR measure
The 1-day VaR is termed as daily earnings at risk (DEAR).
 If the portfolio loss on successive days have independent
identical normal distributions with mean zero, then

N  day VaR 
N  DEAR
(3.17)
Exercise:
Generalize Eq. (3.17) for the case of the portfolio loss on
successive days which have independent normal
distributions but with unequal non-zero means.
36
3.3 VaR measure

X % DEAR of a portfolio with current portfolio value of
Q (>0) can be expressed as
DEAR  Q  DEAR_U
(3.18)
where DEAR_U is the X % DEAR of the same portfolio
with the current value being scaled to $1.
DEAR_U is called the price volatility.
37
3.3 VaR measure
Proof (Eq. (3.18)):
Let L1 be the random variable for the portfolio loss over 1 day.
By Eq. (3.16),
Pr  L1  DEAR   X %
L

Pr  1  DEAR
 X%
Q
Q


~
Pr L1  DEAR_U  X %

( Q  0 )

where L~1 is the portfolio loss of the original portfolio with the
current value being scaled to 1 over 1 day.
By comparing the second and third line of the above derivation,
we prove (3.18).
38
3.3 VaR measure

Example 3.8
X = 95 and N = 5.
Assume L5 ~(0,102).
Pr  L 5  V   0 . 95
V 
 L5
Pr 

  0 . 95
 10 10 
V
 1 . 645
10
V  16 . 45
The 5-day 95% VaR is $16.45.
39
3.4 RiskMetrics model



The RiskMetrics model is developed by J.P. Morgan in 1994.
Let Q(y1, y2, …, yn) be the value of a FI’s portfolio which
depends on the market risk factors y1, y2, …, yn. The market
risk factors can be interest rate, FX rate, equity price and
others.
Let
Q = Q(y1+  y1, y2+  y2, …, yn+ yn) – Q(y1, y2, …, yn)
40
3.4 RiskMetrics model

By using the Taylor expansion with first order approximation,
we have
Q 
Q
 y1
 y1 
Q
y 2
y2   
  1Q   2 Q     n Q
Q
y n
yn
(3.19)
where iQ stands for the changes of the portfolio value due
to the changes of yi.
41
3.4 RiskMetrics model


From Eq. (3.19), Q is linear with respect to each  yi. So,
Eq. (3.19) is termed as the linear model for Q.
In terms of the portfolio loss (L = –Q), Eq. (3.19) can be
expressed as
L  L  L   L
1
2
n
(3.20)
where Li stands for the portfolio loss due to the changes of yi
i
and L

Q
yi
yi .
42
3.4 RiskMetrics model

Assume

 y i ~  0,  yi
2

for i  1, 2 ,  , n
corr  y i ,  y j    ij

(3.21)
From Eq. (3.11) and Eq. (3.21), we deduce that
L ~ (0, 2)
(3.22)
where
n

2


i 1
 Q

  y  yi
 i
2
n

 Q
  2    ij 

 y
i
,
j

1
j

i

 i
  Q

  y
j


 
 yi y j

43
3.4 RiskMetrics model

From (3.22), the X % DEAR of the portfolio,
DEARportfolio, under the changes of all market risk factors
y1, y2, …, yn is given by
DEAR
portfolio
 z X % 
(3.23)
where z X % is given by
Pr( Z  z X % )  X % where Z ~  0,1 .
44
3.4 RiskMetrics model

Example 3.9
Suppose a FI has a portfolio which consists of
• a zero coupon bond of 7 years to maturity with the face
value of $1 million. (current annual bond yield = 7%).
• a foreign cash deposit of €1.6 million.
• 1 million shares of Stock A.
Denote
y1: interest rate;
y2: exchange rate ($/€);
y3: price of one share of Stock A.
45
3.4 RiskMetrics model
Suppose

 y i ~  0,  yi
2

for i  1, 2 ,3
corr  y i ,  y j    ij
where   y
1
 0 . 001 ;   y 2  $ 0 . 00565 / pound;   y 3  $ 0 . 5 .
46
3.4 RiskMetrics model
ij are given in the following table,
i
ij
1
2
3
1
1
0.2
0.4
j 2
 0.2
1
0.1
3
0.4
0.1
1
47
3.4 RiskMetrics model
The value of the portfolio is given by
Q ( y1 , y 2 , y 3 ) 
Q
 y1
Q
y 2
Q
y3

D
1  7%
$ 1M
1 
y1 

1M
7
 €1 . 6 M  y 2  1M  y 3 .
1  7% 
7

7
1  7%

1M
1  7% 
7
  $ 4 , 074 , 064 ;
 €1 . 6 M;
 1M.
48
3.4 RiskMetrics model
From Eq. (3.22),  is given by
 
  4 , 074 , 064  0.001 2  1 . 6 M  0.00565 2  1M  0 . 5 2 
2 (  0 . 2 )   4 , 074 , 064  0.001 1 . 6 M  0.00565  
2 ( 0 . 4 )   4 , 074 , 064  0.001 1M  0 . 5  
2 0 . 1 1 . 6 M  0.00565 1M  0 . 5 
 $ 499 , 387 . 05
From Eq. (3.23), the 95% DEAR of the portfolio is given
by
DEARportfolio = 1.645$499,387.05 = $821,491.7.
49
3.4 RiskMetrics model
Weakness
 The portfolio loss is assumed to follow a normal
distribution which is a symmetric distribution. In
reality, the distribution of the portfolio loss is not
symmetric. Under the recent financial crisis, its
distribution tends to skew to the right (fatter right tail
and thinner left tail).
50
3.5 Historical simulation




Historical simulation involves using past data as a
guide to predict what will happen in the future.
There is no need to specify the probability
distribution for the changes in the risk factors or the
portfolio loss.
Contrast to RiskMetrics model, more complicated
form of Q(y1, y2, …, yn) can be handled.
Steps in calculating X % DEAR:
1. To identify the market variables affecting the portfolio.
2. We then collect data on the movement in these market
variables over the most recent N+1days (usually 501
days). Today – Day N, …, First data available day – Day
0.
51
3.5 Historical simulation

Steps in calculating X % DEAR (cont.):
3. Assume the % changes of the market variables between today
and tomorrow are the same as they were between Day i  1
and Day i for 1  i  N.
4. Define vi as the value of a market variable on Day i and
suppose today is Day N. The generated value of the market
variable for tomorrow in scenario i will be
vi
vN
for i  1,2,  , N
v i 1
Then, we generate N scenarios for the value of market
variables for tomorrow.
52
3.5 Historical simulation

Steps in calculating X % DEAR (cont.):
5. Based on the values of market variables for each scenario in
step 4, to calculate the portfolio value for tomorrow.
6. By comparing the value of the portfolio today and tomorrow,
to calculate the change of the portfolio value between today
and tomorrow.
7. X % DEAR = – [(1 – X%)N]th worst number in step 6.
53
3.5 Historical simulation

Example 3.10
(John Hull, “Options, futures and other derivatives”, 7th ed., Prentice Hall)
Today
Q = $23.5 m
54
3.5 Historical simulation
=25.85(20.78/20.33)
55
3.5 Historical simulation
Weakness
 Backward-looking.
 If N is small, the confidence interval around the
estimated DEAR will be wide. Increasing N, past
observations may become decreasingly relevant in
predicting future DEAR.
 Unpleasant window effect. When N + 1 days have
passed since the certain financial crisis, the crisis
observation drops out of our window for historical
data, and the reported VaR suddenly drops from one
day to the next.
56
3.6 Monte Carlo simulation



To overcome problem of limited number of past
observations.
Let Q(y1, y2, …, yn) be the value of a FI’s portfolio
which depends on the market risk factors y1, y2, …, yn.
Procedure:
1. Value the portfolio today in the usual way using current
values of market variables, y1, y2, …, yn .
2. Sample y1, y2, …, yn once from their joint
distribution (eg. multivariate normal).
3. Use the sampled values of y1, y2, …, yn in step 2 to
determine the value of y1, y2, …, yn at the end of one day.
57
3.6 Monte Carlo simulation

Procedure (cont.):
4. Revalue the portfolio at the end of the day by using the
value of yi in step 3.
5. Subtract the value calculated in step 1 from the value in
step 4 to determine a sample of the portfolio loss L.
6. Repeat steps 2 to 5 many times to build up a probability
distribution of L.
7. The X % DEAR can be calculated as the X percentile of
the probability distribution of L.
58
3.6 Monte Carlo simulation
Weakness
 It tends to be computationally slow for the portfolio
involving a large number of different types of
financial assets since it involves to revalue the
portfolio for each sampled value of yi.
59
3.7 Regulatory model



A standardized approach for the market risk which is
proposed by BIS for the FIs to measure their market
risk.
Subject to regulatory permission, large banks may be
allowed to use their internal models (such as
RiskMetrics, historical simulation or Monte Carlo
simulation) as the basis for determining their capital
requirements.
For the standardized approach in Hong Kong, may
refer to
http://www.hkma.gov.hk/eng/key-functions/bankingstability/basel-3/banking_capital_rules_gazette_b.shtml
(Section 279 to 322)
60
61
3.7 Regulatory model
(continued)
62
3.7 Regulatory model
Derived from the residual in the Section “Horizontal Offset
within Same Time Zones” and “Between Time Zones”.
(continued)
*Residual amount carried forward for additional offsetting as appropriate.
Note: Qual Corp is an investment-grade debt issue (e.g., rated BBB and above). Non Qual is a belowinvestment-grade debt issue (e.g., rated BB and below), that is, a junk bond.
63
3.7 Regulatory model
Fixed income
 Specific risk charge:
A charge reflecting the risk of a decline in the
liquidity or credit risk quality of the trading portfolio.
Eg. The weight of Treasuries is 0% while the weight
of 10-15 years nonqualifying (Non Qual) bond is 8%.
64
3.7 Regulatory model

General market risk charge:
Charges reflecting the modified duration and interest
rate shocks expected for each maturity.
Weight =MDE(ΔR)
where E(ΔR) is the expected interest rate shock.
Eg. For 10 – 15 years Treasuries, MD = 8.75 years
and E(ΔR) = 0.6%, the weight in “general market
risk” is 8.75 0.6% = 5.25%.
65
3.7 Regulatory model

Vertical offsets (disallowances):
• Additional capital charges assigned because long and short
positions in the same maturity bucket but in different
instruments cannot perfectly offset each other.
• Charge (time band i)= disallowance (time band i) × offset
(time band i)
where
offset = the smallest absolute value of the general market
risk charge of long and short positions of time band i.
• Eg. For 3-4 years time band, offset is 45. Additional 10%
charge (disallowance) on the offset is 10%45 = 4.5.
66
3.7 Regulatory model

Horizontal offset (disallowances):
• Additional capital charges required because long and short
positions of different maturities do not perfectly hedge each
other.
• Within time zones:
The imperfect correlation of interest rates on debts of
different maturities within the time zone.
• Between time zones:
The interest rates on short maturity debt and long maturity
debt do not fluctuate exactly together.
• Charges (within or between time zones) = Disallowance ×
offset (within or between time zones)
67
3.7 Regulatory model
Foreign exchange
 Convert the total long and short FX positions to reporting
currency.
 Capital requirement
= 8%  max(|Aggregate long FX position (reporting
currency)|, |Aggregate short FX position (reporting
currency)|)
where |x| = absolute value of x.
68
3.7 Regulatory model

Example 3.11
The figures in the table are in millions of dollars.
Capital requirement
= 8%  max(300 million, 200 million) = 24 million.
69
3.7 Regulatory model
Equities
 Capital requirement
= 4%  Gross position in the stock (unsystematic
risk) +
8%  Net position in the stock (systematic risk).
70
3.7 Regulatory model

Example 3.12
71