Topic 1. Introduction to financial derivatives
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Transcript Topic 1. Introduction to financial derivatives
Topic 4. Measuring Market Risk
4.1 Benefits of measuring market risk
4.2 Mathematical preliminaries
4.3 VaR measure
4.4 RiskMetrics model
4.5 Historical simulation
4.6 Monte Carlo simulation
4.7 Regulatory model
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4.1 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.
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4.1 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.
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4.2 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 PrX p p
(3.1)
4
4.2 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 whereZ ~ 0,1
0.4
0.9 0.5
1.282
0.4
0.9 1.013
5
4.2 Mathematical preliminaries
If F(x) is not continuous (may be piecewise
continuous), the (100p)th percentile, p, of X is
obtained
p minl : Pr X l p
(3.2)
6
4.2 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
4.2 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
4.2 Mathematical preliminaries
Definition S1
The probability distribution of X is approximated by
Pr X xi
1
n
for i 1,2, , n.
(3.3)
The p is defined as
p minxi : Pr X xi p
(3.4)
If np is an integer, then p = xnp.
9
4.2 Mathematical preliminaries
Definition S2
The probability distribution of X is approximated by
i
n 1 for x xi , i 1,2,, n
j
x xj
1
Pr X x
(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
4.2 Mathematical preliminaries
F(x)
F(xj+1)
F(x)
F(xj)
Note: F(x) = Pr(X x)
xj
x
xj+1
x
11
4.2 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.
100% 100% n
,
.
In this definition, the range of p is
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.
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4.2 Mathematical preliminaries
Definition S3
The probability distribution of X is approximated by
i 1
n 1 for x xi , i 1,2,, n
j 1
x xj
1
Pr X x
(3.7)
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)
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4.2 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:
0.110 = 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
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4.2 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
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4.2 Mathematical preliminaries
If the sample size n is large, p in Definition S1, S2
and S3 will be similar to each other.
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4.2 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 EX 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.
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4.2 Mathematical preliminaries
If the two random variables are positively (negatively)
correlated, then they tend to move in the same (opposite)
direction.
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4.2 Mathematical preliminaries
Example 3.4
Suppose var(X1) > 0. (Fact: var(X1) 0.)
If X2 = 2 X1, then
2E X E X
cov X 1 , X 2 E 2 X 12 2E X 1
2
2
1
2 var X 1 0
2
1
So, X1 and X2 are positively correlated.
From the definition of X2, we see that X2 will increase
(decrease) if X1 increase (decrease).
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4.2 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
1
Pr X 1 1 Pr X 1 0 Pr X 1 1
X2 X
3
2
1
20
4.2 Mathematical preliminaries
1
1
1
E X 1 1 0 1 0
3
3
3
2
1 2
E X 2 E X 12 1 0
3
3 3
1
1
1
E X 1 X 2 E X 13 1 0 1 0
3
3
3
From Eq. (3.9),
2
cov X 1 , X 2 0 0 0.
3
So, X1 and X2 are uncorrelated. It is obvious that X1 and
X2 are not independent.
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4.2 Mathematical preliminaries
The correlation, corr(X1, X2), between X1 and X2 is
defined as
corr X 1 , X 2 X1 , X 2
cov X 1 , X 2
cov X 1 , X 2
X 1 X 2
var X 1 var X 2
(3.10)
where X is the standard deviation of a random variable X.
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4.2 Mathematical preliminaries
Formulae for variance, covariance and correlation:
i. varaX a 2 X2
N
N
N 2
ii. var X i X i 2 X i , X j X i X j
i , j 1 j i
i
i 1
iii. covaX , bY ab cov X , Y
m
n
n m
iv. cov a i X i , b jY j ai b j covX i , Y j
j 1
i 1
i 1 j 1
v. aX ,bY sgn(ab) X ,Y
(3.11)
1 if x 0
where sgn( x) 0 if x 0
1 if x 0
X, Y, Xi, Yj are random variables and a, b, ai, bj are constants.
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4.2 Mathematical preliminaries
Denote the observations of X1 and X2 as follow:
X 1 : x11 , x12 ,, x1N
X 2 : x21 , x22 ,, x2 N
The unbiased estimate of cov(X1, X2) and Xj are
1 N
x1i x1 x2i x2
cov X 1 , X 2
N 1 i 1
X
j
1 N
2
x
x
ji
j
N 1 i 1
for j 1,2
(3.12)
(3.13)
N
N
where x1 x1i / N ; x2 x2i / N
i 1
i 1
Based on Eqs. (3.12) and (3.13), the estimate of X ,X can
also be obtained from Eq. (3.10).
1
2
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4.2 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).
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4.2 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
4.2 Mathematical preliminaries
Example 3.6
Suppose X ~ U(4, 10).
Using f(x) in Eq. (3.14),
Pr0 X 1 f ( x)dx 0dx 0
1
1
0
0
1
1
Pr3 X 5 f ( x)dx 0dx
dx
3
3
4 10 4
6
7
1
1
Pr5 X 7
dx
5 10 4
3
10
1
1
Pr8 X 10
dx Pr5 X 7
8 10 4
3
5
4
5
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4.2 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
4.2 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(),,)
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4.2 Mathematical preliminaries
X1, X2, …, Xn are said to follow a multivariate normal
distribution if
X i~ i , i2 and corrX i, X j ij
The covariance matrix, , of X1, X2, …, Xn is defined
as
ij nn and ij covX i , X j ij i j
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4.2 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;
x2 2 2 12 z1 2 1 122 z 2 .
4. Repeat steps 1 to 3 N times.
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4.3 VaR measure
Let P be a portfolio of financial assets.
Statement S:
“We are X % certain that the portfolio P will not lose
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
4.3 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
PrLN V X %.
(3.16)
33
4.3 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
4.3 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
4.3 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
4.3 VaR measure
Proof (Eq. (3.18)):
Let L1 be the random variable for the portfolio loss over 1 day.
By Eq. (3.16),
PrL1 DEAR X %
L
Pr 1 DEAR X %
( Q 0)
Q
Q
~
Pr L1 DEAR_U X %
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
4.3 VaR measure
Example 3.8
X = 95 and N = 5.
Assume L5 ~(0,102).
P rL5 V 0.95
L5 V
P r 0.95
10 10
V
1.645
10
V 16.45
The 5-day 95% VaR is $16.45.
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4.4 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
4.4 RiskMetrics model
By using the Taylor expansion with first order approximation,
we have
Q
Q
Q
Q
y1
y2
yn
y1
y2
yn
1Q 2Q nQ
(3.19)
where iQ stands for the changes of the portfolio value due
to the changes of yi.
41
4.4 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 L1 L2 Ln
(3.20)
where Li stands for the portfolio loss due to the changes of yi
Q
and L yi .
yi
i
42
4.4 RiskMetrics model
Assume
yi ~ 0, 2yi
corryi , y j ij
for i 1,2, , n
(3.21)
From Eq. (3.11) and Eq. (3.21), we deduce that
L ~ (0, 2)
(3.22)
where
n
Q
Q Q
y y
yi 2 ij
i j
y
y
i 1 yi
i
,
j
1
j
i
i
j
n
2
2
43
4.4 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
DEARportfolio z X %
(3.23)
where z X % is given by
Pr(Z z X % ) X % where Z ~ 0,1.
44
4.4 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
4.4 RiskMetrics model
Suppose
yi ~ 0, 2yi
corryi , y j ij
for i 1,2,3
where y 0.001; y $0.00565/ pound; y $0.5.
1
2
3
46
4.4 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
4.4 RiskMetrics model
The value of the portfolio is given by
$1M
Q( y1 , y 2 , y3 )
€1.6M y2 1M y3 .
7
1 y1
Q
D
1M
7
1M
$4,074,064;
7
7
y1
1 7% 1 7%
1 7% 1 7%
Q
€1.6M;
y2
Q
1M.
y3
48
4.4 RiskMetrics model
From Eq. (3.22), is given by
4,074,064 0.0012 1.6M 0.005652 1M 0.52
2(0.2) 4,074,064 0.0011.6M 0.00565
2(0.4) 4,074,064 0.0011M 0.5
20.11.6M 0.005651M 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
4.4 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
4.5 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
4.5 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
vi 1
Then, we generate N scenarios for the value of market
variables for tomorrow.
52
4.5 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
4.5 Historical simulation
Example 3.10
(John Hull, “Options, futures and other derivatives”, 7th ed., Prentice Hall)
Today
Q = $23.5 m
54
4.5 Historical simulation
=25.85(20.78/20.33)
55
4.5 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
4.6 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
4.6 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
4.6 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
4.7 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
4.7 Regulatory model
BIS (including Federal Reserve) approach: Market risk
may be calculated using standard BIS model:
-- Specific risk charge
-- General market risk charge
-- Offsets
61
4.7 Web resources
For information on the BIS framework, visit:
Bank for International Settlement
www.bis.org
Federal Reserve Bank
www.federalreserve.gov
62
4.7 Regulatory model
--Specific risk charge:
Risk weights × absolute dollar values of long and
short positions
--General market risk charge:
reflect modified durations expected interest rate
shocks for each maturity
--Vertical offsets:
Adjust for basis risk
--Horizontal offsets within/between time zones
63
64
4.7 Regulatory model
(continued)
65
4.7 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.
66
4.7 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%.
67
4.7 Regulatory model
General market risk charge:
Charges reflecting the modified duration and interest
rate shocks expected for each maturity.
Weight =MDE(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%.
68
4.7 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.
69
4.7 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)
70
4.7 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.
71
4.7 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.
72
4.7 Regulatory model
Equities
Capital requirement
= 4% Gross position in the stock (unsystematic
risk) +
8% Net position in the stock (systematic risk).
73
4.7 Regulatory model
Example 3.12
74