11 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

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.

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.

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

5 4.2 Mathematical preliminaries  Example 3.1 Suppose X ~  (0.5, ) where  ( , ) stands for the normal distribution with mean  and variance. From Eq. (3.1),  0.9 of X is obtained by solving

6 4.2 Mathematical preliminaries  If F(x) is not continuous (may be piecewise continuous), the (100p)th percentile,  p, of X is obtained

7 4.2 Mathematical preliminaries  Example 3.2 From Eq. (3.2),  0.7 = 7  0.9 = 12 x Pr(X = x) Pr(X  x)

8 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 {x 1, x 2, …, x n } 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. x 1 < x 2 < …< x n.

9 Definition S1  The probability distribution of X is approximated by The  p is defined as  If np is an integer, then  p = x np. 4.2 Mathematical preliminaries

10 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 (x j, Pr(X  x j )) and (x j+1, Pr(X  x j+1 )). 4.2 Mathematical preliminaries

Mathematical preliminaries x F(x)F(x) xjxj x j+1 x F(xj)F(xj) F(x j+1 ) F(x)F(x) Note: F(x) = Pr(X  x)

12 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 = x k + d  (x k+1 – x k ) (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. 4.2 Mathematical preliminaries

13 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 = x k + d  (x k+1 – x k ) (3.8) 4.2 Mathematical preliminaries

14  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 = x 1 =  Definition S2: r = 0.1  (11) = 1.1 k = 1 and d = 0.1  0.1 = x  (x 2 – x 1 ) = –  (– 800 – (– 1000)) = – Mathematical preliminaries

15 Definition S3: l = 0.1  (9) + 1 = 1.9 k = 1 and d = 0.9  0.1 = x  (x 2 – x 1 ) = –  (– 800 – (– 1000)) = – Mathematical preliminaries

16  If the sample size n is large,  p in Definition S1, S2 and S3 will be similar to each other. 4.2 Mathematical preliminaries

Mathematical preliminaries Covariance and correlation  Let X 1 and X 2 be 2 random variables. The covariance of X 1 and X 2 is denoted as cov(X 1, X 2 ) and defined as where E(X) denotes expectation (expected value) of a random variable X.  If cov(X 1, X 2 ) is positive (negative), then X 1 and X 2 are said to be positively (negatively) correlated.

Mathematical preliminaries  If the two random variables are positively (negatively) correlated, then they tend to move in the same (opposite) direction.

Mathematical preliminaries  Example 3.4 Suppose var(X 1 ) > 0. (Fact: var(X 1 )  0.) If X 2 = 2 X 1, then So, X 1 and X 2 are positively correlated. From the definition of X 2, we see that X 2 will increase (decrease) if X 1 increase (decrease).

Mathematical preliminaries  If cov(X 1, X 2 ) = 0, then X 1 and X 2 are said to be uncorrelated.  If X 1 and X 2 are independent, then X 1 and X 2 are uncorrelated. This can be seen by replacing E(X 1 X 2 ) with E(X 1 )E(X 2 ) in Eq. (3.9).  Uncorrelated does not imply independent.  Example 3.5 Define X 1 as

Mathematical preliminaries From Eq. (3.9), So, X 1 and X 2 are uncorrelated. It is obvious that X 1 and X 2 are not independent.

Mathematical preliminaries  The correlation, corr(X 1, X 2 ), between X 1 and X 2 is defined as where  X is the standard deviation of a random variable X.

Mathematical preliminaries  Formulae for variance, covariance and correlation: X, Y, X i, Y j are random variables and a, b, a i, b j are constants.

Mathematical preliminaries  Denote the observations of X 1 and X 2 as follow: The unbiased estimate of cov(X 1, X 2 ) and  X j are Based on Eqs. (3.12) and (3.13), the estimate of can also be obtained from Eq. (3.10).

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

Mathematical preliminaries  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. x f(x)f(x) a b 1/(b – a)

Mathematical preliminaries  Example 3.6 Suppose X ~ U(4, 10). Using f(x) in Eq. (3.14),

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.]

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(), ,  )

Mathematical preliminaries  X 1, X 2, …, X n are said to follow a multivariate normal distribution if  The covariance matrix, , of X 1, X 2, …, X n is defined as

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 X 1 and X 2 with following procedures: 1.Generate two random numbers, u 1 and u 2, from U(0,1). 2.Solve z i (i = 1, 2) from Pr(Z  z i ) = u i where Z ~  (0, 1). 3.Set x 1 =  1 +  1 z 1 ; 4.Repeat steps 1 to 3 N times.

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.

VaR measure  Define L N as a random variable which stands for the portfolio loss over the next N days.  Positive L N = loss; Negative L N = gain.  When the distribution of L N is continuous, the statement S can be expressed mathematically as

VaR measure  From Eq. (3.1), the VaR in Eq. (3.16) is the Xth percentile of the distribution of L N.  With Eq. (3.2), or Definition S1, S2 or S3, we can also define the VaR when the distribution of L N is not continuous or unknown.

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

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.

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.

VaR measure Proof (Eq. (3.18)): Let L 1 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).

VaR measure  Example 3.8 X = 95 and N = 5. Assume L 5 ~  (0,10 2 ). The 5-day 95% VaR is $16.45.

RiskMetrics model  The RiskMetrics model is developed by J.P. Morgan in  Let Q(y 1, y 2, …, y n ) be the value of a FI’s portfolio which depends on the market risk factors y 1, y 2, …, y n. The market risk factors can be interest rate, FX rate, equity price and others.  Let  Q = Q(y 1 +  y 1, y 2 +  y 2, …, y n +  y n ) – Q(y 1, y 2, …, y n )

RiskMetrics model  By using the Taylor expansion with first order approximation, we have where  i Q stands for the changes of the portfolio value due to the changes of y i.

RiskMetrics model  From Eq. (3.19),  Q is linear with respect to each  y i. 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 L i stands for the portfolio loss due to the changes of y i and.

RiskMetrics model  Assume  From Eq. (3.11) and Eq. (3.21), we deduce that L ~  (0,  2 ) (3.22) where

RiskMetrics model  From (3.22), the X % DEAR of the portfolio, DEAR portfolio, under the changes of all market risk factors y 1, y 2, …, y n is given by

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 $1 million. (current annual bond yield = 7%). a foreign cash deposit of €1.6 million. 1 million shares of Stock A. Denote y 1 : interest rate; y 2 : exchange rate ($/€); y 3 : price of one share of Stock A.

RiskMetrics model Suppose where

RiskMetrics model  ij are given in the following table, i  ij  j2 

RiskMetrics model The value of the portfolio is given by € €

RiskMetrics model From Eq. (3.22),  is given by From Eq. (3.23), the 95% DEAR of the portfolio is given by DEAR portfolio =  $499, = $821,491.7.

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).

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(y 1, y 2, …, y n ) 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.

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 v i 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.

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.

Historical simulation  Example 3.10 (John Hull, “Options, futures and other derivatives”, 7 th ed., Prentice Hall) Today Q = $23.5 m

Historical simulation =25.85  (20.78/20.33)

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.

Monte Carlo simulation  To overcome problem of limited number of past observations.  Let Q(y 1, y 2, …, y n ) be the value of a FI’s portfolio which depends on the market risk factors y 1, y 2, …, y n.  Procedure: 1.Value the portfolio today in the usual way using current values of market variables, y 1, y 2, …, y n. 2.Sample  y 1,  y 2, …,  y n once from their joint distribution (eg. multivariate normal). 3.Use the sampled values of  y 1,  y 2, …,  y n in step 2 to determine the value of y 1, y 2, …, y n at the end of one day.

Monte Carlo simulation  Procedure (cont.): 4.Revalue the portfolio at the end of the day by using the value of y i 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.

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 y i.

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 stability/basel-3/banking_capital_rules_gazette_b.shtml (Section 279 to 322)

Regulatory model BIS (including Federal Reserve) approach: Market risk may be calculated using standard BIS model: -- Specific risk charge -- General market risk charge -- Offsets

Web resources For information on the BIS framework, visit: Bank for International Settlement Federal Reserve Bank

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

64

Regulatory model (continued)

Regulatory model * 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)

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 years nonqualifying (Non Qual) bond is 8%.

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%.

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.

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)

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.

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.

Regulatory model Equities  Capital requirement = 4%  Gross position in the stock (unsystematic risk) + 8%  Net position in the stock (systematic risk).

Regulatory model  Example 3.12