11 Topic 4. Measuring Market Risk 4.1 Benefits of measuring market risk 4.2 Mathematical...

transcript

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

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.

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.

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

(3.1) Pr pXF pp

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

282.14.0

1,0~ where9.04.0

4.2 Mathematical preliminaries If F(x) is not continuous (may be piecewise

continuous), the (100p)th percentile, p, of X is obtained

(3.2) Pr:min plXlp

From Eq. (3.2),

0.7 = 7

0.9 = 12

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

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 .

Definition S1 The probability distribution of X is approximated by

The p is defined as

If np is an integer, then p = xnp.

(3.3) .,,2,1for 1

Pr nin

(3.4) Pr:min pxXx iip

Definition S2

The probability distribution of X is approximated by

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)).

1,,2,1

,for 1

,,2,1 ,for 1

xxxnxx

xj xj+1x

F(xj+1)

Note: F(x) = Pr(X x)

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

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.

Definition S3

The probability distribution of X is approximated by

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)

1,,2,1

,,2,1 ,for 1

xxxnxx

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

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

If the sample size n is large, p in Definition S1, S2 and S3 will be similar to each other.

4.2 Mathematical preliminariesCovariance 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

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.

221121

XEXEXXE

XEXXEXEXX

4.2 Mathematical preliminaries If the two random variables are positively (negatively) correlated,

then they tend to move in the same (opposite) direction.

Suppose var(X1) > 0. (Fact: var(X1) 0.)

If X2 = 2 X1, then

So, X1 and X2 are positively correlated.

From the definition of X2, we see that X2 will increase (decrease) if X1 increase (decrease).

22,cov

XEXEXX

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

11Pr0Pr1Pr 111 XXX

212 XX

From Eq. (3.9),

So, X1 and X2 are uncorrelated. It is obvious that X1 and X2 are not independent.

200,cov 21 XX

4.2 Mathematical preliminaries The correlation, corr(X1, X2), between X1 and X2 is

defined as

where X is the standard deviation of a random variable X.

(3.10)

varvar

,cov,corr

4.2 Mathematical preliminaries Formulae for variance, covariance and correlation:

X, Y, Xi, Yj are random variables and a, b, ai, bj are constants.

(3.11)

)sgn( v.

,cov,cov iv.

,cov,cov iii.

2var ii.

var i.

YXbYaX

ji ijXXXX

YXbaYbXa

YXabbYaX

0 if 10 if 0

0 if 1)sgn( where

4.2 Mathematical preliminaries Denote the observations of X1 and X2 as follow:

The unbiased estimate of cov(X1, X2) and Xj are

Based on Eqs. (3.12) and (3.13), the estimate of can also be obtained from Eq. (3.10).

222212

112111

NxxNxx

/;/ where

(3.13) 2,1for 1

(3.12) 1

21 ,XX

4.2 Mathematical preliminariesMonte 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

We use X ~ U(a, b) to denote that X follows a uniform distribution over (a, b).

(3.14) otherwise 0

bxaabxf

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.

1/(b – a)

Suppose X ~ U(4, 10).

Using f(x) in Eq. (3.14),

1108Pr

10)(53Pr

00)(10Pr

dxdxdxxfX

dxdxxfX

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

X = F –1(U) (3.15)

has distribution F. [F –1(u) is defined to be that value of x such that F(x) = u.]

4.2 Mathematical preliminaries The procedures to generate a sample of N

observations from (, 2) are as follows:1. Generate a random number, u, from U(0,1).

2. Solve z from Pr(Z z) = u where Z ~ (0, 1).

3. Set x = + z. (X = + Z ~ (, 2).)

4. 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(),,)

ijjiiii XXX ,corr and ,~ 2

X1, X2, …, Xn are said to follow a multivariate normal distribution if

The covariance matrix, , of X1, X2, …, Xn is defined as jiijjiijnnij XX

,cov and

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;

4. Repeat steps 1 to 3 N times.

.1 22122112222 zzx

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.

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

(3.16) %.Pr XVLN

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.

Source: Alexander J. McNeil et. al. “Quantitative Risk Management”, Princeton University Press 2005.

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

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.

(3.17) DEARVaRday NN

4.3 VaR measure X % DEAR of a portfolio with current portfolio value of Q (>0) can be

expressed as

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.

(3.18) DEAR_UDEAR Q

4.3 VaR measureProof (Eq. (3.18)):

Let L1 be the random variable for the portfolio loss over 1 day.

By Eq. (3.16),

where 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).

%DEAR_U~

)0( %DEARPr

%DEARPr

4.3 VaR measure Example 3.8

X = 95 and N = 5.

Assume L5 ~(0,102).

The 5-day 95% VaR is $16.45.

645.110

95.01010

95.0Pr

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.

Q = Q(y1+ y1, y2+ y2, …, yn+ yn) – Q(y1, y2, …, yn)

4.4 RiskMetrics model By using the Taylor expansion with first order approximation, we have

where iQ stands for the changes of the portfolio value due to the changes of yi.

(3.19) 21

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

where Li stands for the portfolio loss due to the changes of yi

(3.20) 21 nLLLL

4.4 RiskMetrics model Assume

From Eq. (3.11) and Eq. (3.21), we deduce that

L ~ (0,2) (3.22)

(3.21) ,corr

,,2,1for ,0~ 2

ji ijyy

ijii y

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

.0,1~ where%)Pr(

by given is where

(3.23) DEAR

%portfolio

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.

4.4 RiskMetrics modelSuppose

3,2,1for ,0~ 2

.5.0$ pound;/00565.0$ ;001.0321

4.4 RiskMetrics modelij are given in the following table,

ij 1 2 3

1 1 0.2 0.4

j 2 0.2 1 0.1

3 0.4 0.1 1

4.4 RiskMetrics modelThe value of the portfolio is given by

.M1M6.1

M1$),,( 327

321 yyy

;064,074,4$7%1

4.4 RiskMetrics modelFrom Eq. (3.22), is given by

From Eq. (3.23), the 95% DEAR of the portfolio is given by

DEARportfolio = 1.645$499,387.05 = $821,491.7.

05.387,499$

5.0M10.00565M6.11.02

5.0M10.001064,074,4)4.0(2

0.00565M6.10.001064,074,4)2.0(2

5.0M10.00565M6.10.001064,074,4 222

4.4 RiskMetrics modelWeakness 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).

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.

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

Then, we generate N scenarios for the value of market variables for tomorrow.

iN ,1,2,for

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.

4.5 Historical simulation Example 3.10

(John Hull, “Options, futures and other derivatives”, 7th ed., Prentice Hall)

Today Q = $23.5 m

4.5 Historical simulation

=25.85(20.78/20.33)

4.5 Historical simulationWeakness 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.

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.

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.

4.6 Monte Carlo simulationWeakness 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.

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/banking-stability/basel-3/banking_capital_rules_gazette_b.shtml

(Section 279 to 322)

BIS (including Federal Reserve) approach: Market risk may be calculated using standard BIS model:

-- Specific risk charge

-- General market risk charge

-- Offsets

4.7 Web resourcesFor information on the BIS framework, visit:

Bank for International Settlement www.bis.org

Federal Reserve Bank www.federalreserve.gov

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

4.7 Regulatory model (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 below-investment-grade debt issue (e.g., rated BB and below), that is, a junk bond.

Derived from the residual in the Section “Horizontal Offset within Same Time Zones” and “Between Time Zones”.(continued)

4.7 Regulatory modelFixed 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%.

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%.

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)

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.

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)

4.7 Regulatory modelForeign 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.

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.

4.7 Regulatory modelEquities Capital requirement

= 4% Gross position in the stock (unsystematic risk) +

8% Net position in the stock (systematic risk).

4.7 Regulatory model Example 3.12

11 Topic 4. Measuring Market Risk 4.1 Benefits of measuring market risk 4.2 Mathematical...

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