An Introduction to Risk Management Financial Risks: definition - - PowerPoint PPT Presentation

1

An Introduction to Risk Management

Nattawut Jenwittayaroje, Ph.D., CFA NIDA Business School National Institute of Development Administration Financial Risk Management

2

Topics covered

 The concept and practice of risk management  Types of risks  Financial Risks: definition  Basic stat review  Value at Risk (VaR): Basics  Value at Risk (VaR): Motivation & Definition  Value at Risk (VaR): Three methods of VaR calculation  Benefits and extensions of VaR

3

 Definition of risk management: the practice of defining the

risk level a firm desires, identifying the risk level it currently has, and using derivatives or other financial instruments to adjust the actual risk level to the desired risk level.

4

Why Practice Risk Management?



The Motivation for Risk Management



Firms practice risk management for several reasons:



Interest rates, exchange rates and stock prices are more volatile today than in the past. These factors create risks over which most businesses have little

• expertise. Therefore, it makes sense for a business to manage and largely

eliminate these risks.



Significant losses incurred by firms that did not practice risk management



Improvements in information technology – without enormous developments in computing power, it would not have been possible to do the complex calculations necessary for pricing derivatives and for keeping track of positions taken.



Favorable regulatory environment – the growth of derivatives (i.e., main tools for managing risks) was fueled by the favorable regulatory environment.

5

Risks of Financial Intermediation



Interest rate risk resulting from intermediation:



Mismatch in maturities of assets and liabilities.



Interest rate sensitivity difference exposes equity to changes in interest rates



Balance sheet hedge via matching maturities of assets and liabilities is problematic for FIs.



Inconsistent with asset transformation role



Refinancing risk



The risk that the cost of rolling over or reborrowing funds (e.g., deposits) will rise above the returns being earned on asset investments (e.g., loans)



Reinvestment risk.



The risk that the returns on funds to be reinvested (e.g., making loans) will fall below the cost of funds (e.g., deposits)

6

Risks of Financial Institutions

 The risks associated with financial intermediation:

 Interest rate risk  Market risk  Credit risk  Operational risk  Liquidity risk  Legal and regulatory risk

Financial Risk

7

Market Risk



Incurred in trading of assets and liabilities (and derivatives).

 Examples: Barings & decline in ruble.



1995 Barings Bank, forced into insolvency due to losses on its trading in Japanese stock index futures



1996 Sumitomo Corp. lost $2.6 billion in commodity futures trading



1997 market volatility in Eastern Europe and Asia



1998 losses on Russian bonds and Ruble currency: Big US banks have to write off hundreds of millions of dollars in losses on their holdings of Russian government securities



DJIA dropped 12.5 percent in two-week period July, 2002.

 Heavier focus on trading income over traditional activities increases

market exposure.

8

Credit Risk

 Risk that promised cash flows are not paid in full.

 Firm specific credit risk  Systematic credit risk

 Credit risk is the oldest form of risk in the financial

institutions.

9

Operational Risk

 BIS definition: Risk of losses resulting from inadequate or

failed internal processes, people, and systems or from external events.

 Operational risk is the risk that existing technology or support

systems may malfunction or break down.

 Power failures, computer problems such as viruses, the failure

• f staff personnel to monitor and record transactions properly,

etc..

 However, operational risk losses do not occur often (but

can lead to tremendous losses), so it is difficult to do the analysis.

10

Liquidity Risk

 Liquidity risk is the risk that a sudden surge in liability

withdrawals may leave a bank in a position of having to liquidate assets in a very short period of time and at low prices.

 May generate runs.

 Runs may turn liquidity problem into solvency problem.  Risk of systematic bank panics.

11

Legal and Regulatory Risks

 Legal risk is the risk that the legal system will fail to enforce

a contract.

 For example, suppose a dealer enters into a swap with a

counterparty that, upon incurring a loss, then refuses to pay the dealer, arguing that the dealer misled it or that the counterparty had no legal authority to enter the swap.

 Regulatory risk is the risk that regulations will change.

 Regulatory risk means that certain existing or contemplated

transactions can become illegal or regulated.

12

Financial Risks



Market risk is the uncertainty of a firm’s value or cash flow that is associated with movements in an underlying source of risk (e.g., interest rates, foreign exchange rates, stock prices, or commodity prices).

 E.g., interest rate risk faced by financial intermediary, FX risks faced

by FX dealers, oil price risks faced by oil distributors, and stock price risk faced by equity fund managers.

 However, because many sources of risk are partially correlated, the

combined effects of all sources of risk must be considered.



Credit risk is the uncertainty and potential for loss due to a failure to pay on the part of a counterparty. It is the risk that promised cash flows are not paid in full.

 Credit risk (or default risk) of loans faced by any lender, and credit

risk of derivatives, i.e., the risk of default faced by any party that may receive obligation payments from another party.

13

Basic Stat Review

 Want to know the height of Thai population

 Randomly select 100 people and measure each person’s height  What is the average height of Thai people?   What is the standard deviation of height of Thai people?   How many people have height below 161?  What is the percentage of people having height below 161?

14

Basic Stat Review (Con’t)

 Want to know the height of Thai population (con’t)

 Sort from the lowest to the highest  How many people have height below 161?

 15 people

 What is the percentage of people having height below 161?

 15 from 100 people  15%

15

Basic Stat Review (Con’t)

 Want to know the height of Thai population (con’t)

 How many people have height above 185?  What is the percentage of people having height above 185?  How many people have height between 161 and 185?  What is the percentage of people having height between 161

and 185?

 About 5% of people have height below………………?  About 95% of people have height at least………………?

16

Value at Risk (VaR) Basics

 Want to know the risk of my stock portfolio….

 For example, I now have 100,000 shares of AA stock @$10

each.

 So my portfolio is now worth $1,000,000.  What is the probability that tomorrow (i.e, one day later) my

portfolio will be worth below $930,000?

 There is 5% chance that tomorrow (i.e, one day later) my

portfolio will be worth below $xxxxxx?

17

Value at Risk (VaR) Basics (Con’t)

 Want to know the risk of my stock portfolio….

 Collect data of AA share price over the recent past 100 days as

follows..

 Sort the data from the smallest price to the highest price..  What is the probability that tomorrow (i.e, one day later) my

portfolio will be worth below $930,000?

 There is 5% chance that tomorrow (i.e, one day later) my portfolio

will be worth below $xxxxxx?

18

Value at Risk (VaR) Basics (Con’t)

 Want to know the risk of my stock portfolio…A Better Way

 Collect data of AA share price over the recent past 100 days;  Compute daily return of AA share price over the recent 100 days..  Sort the 100 daily returns from the smallest to the highest..

(price at 4Jun – price at 3Jun)/price at 3 Jun = (9.57-9.33)/9.33

19

Value at Risk (VaR) Basics (Con’t)

 Want to know the risk of my stock portfolio….

 There is 5% chance that tomorrow (i.e, one day later) my

portfolio will be worth below $xxxxxx?  $993,000

 Equivalently, there is 5% chance that tomorrow (i.e, one day

later) my portfolio will lose more than $7,000

Value at Risk - VaR

20

Value at Risk (VaR): Motivation

 Starting Point: “At close of business each day, tell me

what the market risk are across all business and locations”

 In a nutshell, A chairman wants a single dollar amount

at 4.15pm that tells him JPM’s market risk exposure

• ver the next day – especially if that day turns out to be

a “bad” day.  If tomorrow turns out to be a bad day, how much will we make loss?

21

Value at Risk (VaR) defined

 A dollar measure of the minimum loss that would be

expected over a given time with a given probability.

 Example:

 VAR of $30,000 for one day at 5% means that the firm

could expect to lose at least $30,000 over a one day period 5% of the time.

22

Value at Risk (VaR) defined (Con’t)

 Equivalently, Value at Risk can be defined as the

maximum loss that might be expected from holding a security or portfolio over a given period of time (say, a single day), given a specified confidence level.

 VAR of $30,000 for one day at 95% confidence level

means that the firm could expect to lose no more than $30,000 over a one day 95% of the time.

23

Calculating VaR

 Three methods of estimating VAR

(I): Historical Method (or Back Simulation approach) (II): Analytical Method (or Variance/Covariance approach) (III): Monte Carlo Simulation

24

(I) Historical method

 Historical method: Uses actual data from a recent historical

period to determine the VaR.

 Specifically, the historical method estimates the distribution

• f the portfolio’s performance by collecting data on the past

performance of the portfolio and using it to estimate the future probability distribution.

25

Historical method (Con’t)

 Collect a sample of actual daily prices (e.g., $US/Baht,

SET50 index, $US three-month interest rate, oil prices, etc..)

• ver a given period of time, say 254 days.

 Revalue the portfolio based on those daily prices every day

for the previous 254 days, and then compute daily portfolio returns

 Construct the histogram of portfolio returns and identify the

VaR that isolates the 5th percentiles of the distribution in the left-hand tail (e.g., about 13th lowest value of 254 days), if VAR is derived at the 95% confidence level.

26

Historical method: Example

 Today (19 Sep 2012), our portfolio consists of 10,000 barrels

• f crude oil, and 20,000 shares of IBM stock.

 On 19 Sep 2012, crude oil is now worth $100 per barrel, and

IBM $203 a share.

 Therefore, our portfolio is currently worth

 = (10,000 x $100) + (20,000 x $203)  = $5,060,000

 What is the 5% daily VaR of our portfolio??

27

Historical method (Con’t)



For example, our portfolio consists of 10,000 barrels of crude oil, and 20,000 shares of IBM stock. Sort from the lowest to the highest returns

• 254
• 253
• 252

28

Historical method (Con’t)

 Sort from the lowest daily return to the highest daily returns.  Then locate the 13th observation (as a fifth percentile). That value

is the value below which 5% of the data lie.

 Suppose such 13th observation is about 10%.  For portfolio of $5.06 million, VaR at 5% is approximately a loss

• f 10% or $5,060,000(0.10) = -$506,000.

 The portfolio would be expected to lose at least $506,000 in one

day about 5 percent of the time (which is about once a month).

29

Historical method – Advantages and Disadvantages

 Advantages:

 Simplicity  Does not require normal distribution of returns (which is a critical

assumption for Analytical Method)

 Does not need correlations or standard deviations of individual

asset returns.

 Disadvantage:

 254 observations is not very many from statistical standpoint.  However, increasing number of observations by going back

further in time is not desirable. That is, the greater the sample, the

• lder are some of the data and the less reliable they become.

 It also is limited by the results of the chosen time period, which

might not be representative of the future.

30

Historical method – More examples

 Example1: A Thai bank has a long position in Japanese Yen of

30,000,000 and US dollar of 700,000. The current rates are yen30/1Baht, and 35Baht/1USD. What is 1-day VAR at 5%?

 Example2: A Thai investor currently holds 5,000 shares of PTT,

40,000 shares of BBL, and 10,000 shares of ADVAN. The current PTT share price is 280 baht, BBL 160 baht, and ADVAN 300 baht. What is 1-day VAR at 1%?

31

Basic Stat Review revisited



Histogram

 The height of 100 randomly selected Chula students



Normal distribution.

 The height of all people in Thailand (i.e., population) 32

Basic Stat Review revisited (Con’t)

Normal Distribution (x) vs Standard Normal Distribution (z)

x

• 3 -2 -1 0 +1 +2 +3

33

Basic Stat Review revisited (Con’t)

 Standard normal distribution

 Table Z scores

 Assume the Thai people height is normally distributed with mean of

165 and standard deviation of 3.5. (where mean and SD are estimated from the sample 100 people).

 1) What proportion of Thai people have height of less than 155?  2) What proportion of Thai people have height between 160 and

170?

 3) About 1% of Thai people have height of less than…???..cm.  4) About 5% of Thai people have height of less than…???..cm.

34

Basic Stat Review revisited (Con’t)



1) What proportion of Thai people have height of less than 155?

 z = (155-165)/3.5 = -2.86  P(z < -2.86) = 0.0021 

2) What proportion of Thai people have height between 163 and 170?

 z = (163-165)/3.5 = -0.57 and z = (170-165)/3.5 = +1.43  P(-0.57 < z < +1.43) = 0.6393 

3) About 1% of Thai people have height of less than…???..cm

 P(z < Z) = 0.01  z = -2.33  -2.33 = (x-165)/3.5  x = 156.85 

4) About 5% of Thai people have height of less than…???..cm

 P(z < Z) = 0.05  z = -1.65  -1.65 = (x-165)/3.5  x = 159.23 35

VaR: Calculation idea of VaR

 Idea behind VAR is to determine the probability distribution

(e.g., normal distribution) of the underlying source of risk (e.g., gold price, stock prices, oil price) and isolate the worst given percentage (e.g., 1%, 5%) of outcomes.

 Using 5% as the critical percentage, VAR will determine the

5 percent of outcomes that are the worst. The performance at the 5 % mark is the VAR.

36

VaR: Calculation idea of VaR (Con’t)

For discrete probability: VAR at 5% is $3 million loss  there is a 5% probability that over the given time period, the portfolio will lose at least $3million For continuous distribution. In a normal distribution, a 5% VAR

• ccurs 1.65 standard deviations

(z=1.65) from the expected value (i.e., mean). A 1% VAR occurs 2.33 standard deviations (z=2.33) from the mean.

37

(II) Analytical Method

 Example: suppose that a portfolio manager holds 20

million baht of PTT stock.

 PTT stock has an expected return (i.e., mean) of 0.0476%

per day and volatility (i.e., SD) of 0.945% per day.

 What is daily 5% VaR for our portfolio???

 With a normal distribution, we have a daily 5% VAR = 0.0476 -

1.65(0.945) = -1.51%

 So a daily 5% VAR is 20,000,000(-1.51%) = 302,300 baht.  In other words, the portfolio would be expected to lose more than

302,300 baht in one day 5 percent of the time or one out of twenty days.

38

Analytical Method (Con’t)

 For a weekly 5% VAR, convert the daily figures to weekly figures.

 Expected return = 0.0476% x 5 = 0.238%  Volatility = 0.945% x = 2.113%.

 With a normal distribution, we have a weekly 5% VAR = 0.238 -

1.65(2.113) = -3.248%

 So the weekly 5% VAR is 20,000,000(-3.248%) = -649,690 baht.  In other words, the portfolio would be expected to lose at least

649,690 baht in one week 5 percent of the time or one out of twenty weeks

39

Analytical Method (Con’t)

 1) What about monthly VAR at 5% ?  2) What about yearly VAR at 5% ?  3) What about daily VAR at 1% ?  4) What about weekly VAR at 1% ?  5) What about monthly VAR at 1% ?  6) What about yearly VAR at 1% ?

40

Analytical Method (Con’t)

1) 0.0476%*22 – 1.65*0.945%*sqrt(22) 2) 0.0476%*252 – 1.65*0.945%*sqrt(252) 3) 0.0476% – 2.33*0.945% 4) 0.0476%*5 – 2.33*0.945%*sqrt(5) 5) 0.0476%*22 – 2.33*0.945%*sqrt(22) 6) 0.0476%*252 – 2.33*0.945%*sqrt(252)

41

Analytical Method: Two Assets

Consider the following two r isky assets. T her e is a 1/ 3 c hanc e of eac h state of the ec onomy and ther e ar e two assets available - a stoc k fund and a bond fund.

42

Analytical Method: Two Assets

Note that stoc ks have a higher expec ted r etur n than bonds and higher r

• isk. L

et us tur n now to the r isk and r etur n a por tfolio that is 50% invested in bonds and 50% invested in stoc ks.

Expected Return and SD of each asset

43

Analytical Method: Two Assets

The variance of the rate of return on the two risky assets portfolio is where BS is the correlation coefficient between the returns on the stock and bond funds.

Variance of a Portfolio – 50% in each

44

Analytical Method: Two Assets



Analytical method: Uses knowledge of the parameters (expected return and standard deviation) of the probability distribution and assumes a normal distribution.



VAR calculations require use of formulas for expected return and standard deviation of a portfolio:

 where



E(R1), E(R2) = expected returns of assets 1 and 2



1, 2 = standard deviations of assets 1 and 2



 = correlation between assets 1 and 2



w1, w2 = % of one’s wealth invested in asset 1 or 2

The method requires the input values (i.e., mean, SD, covariance) and any necessary pricing models along with an assumption of a normal distribution.

Page 45

Formula of Standard Deviation

• f a portfolio with n assets.

Analytical Method: n assets

46

Analytical Method: Example of a two-asset portfolio

 Example: suppose that a portfolio manager holds two

distinct classes of stocks.

 $20 million of S&P 500 with expected return of 12% per

year and (annualized) volatility of 15% and

 $12 million of Nikkei 300 with expected return of 10.5%

per year and (annualized) volatility of 18%.

 Correlation between the Nikkei 300 and the S&P 500 is

0.55.

47

Analytical Method: Example of a two-asset portfolio (Con’t)

 Using the above formulas, the overall portfolio expected return is

0.1144 and volatility is 0.1425.

 With a normal distribution, we have a yearly VAR =

0.1144 - 1.65(0.1425) = -0.12073

 So a yearly VAR is $32,000,000(-0.12073) = -$3,863,200.  In other words, the portfolio would be expected to lose no more

than $3,863,200 in one year 95 percent of the time or nineteen out

• f twenty years

48

Analytical Method: Example of a two-asset portfolio (Con’t)

 For a weekly VAR, convert the yearly figures to weekly

figures.

 Expected return = 0.1144/52 = 0.0022 or 0.22% per week.  Volatility = 0.1425/ = 0.0198 or 1.98% per week.

 With a normal distribution, we have a weekly VAR = 0.0022

• 1.65(0.0198) = -0.0305 or -3.05% per week.

 So the weekly VAR is $32,000,000(-0.0305) = -$976,000.  In other words, the portfolio would be expected to lose at

least $976,000 in one week 5 percent of the time or one out of twenty weeks

49

Analytical Method: Example of a two-asset portfolio (Con’t)

 What about monthly VAR with 99% confidence level?

0.1144/22 – 2.33*0.1425/sqrt(22)

 What about daily VAR with 99% confidence level?

0.1144/252 – 2.33*0.1425/sqrt(252)

50

Analytical Method: calculation of parameters in practice

51

Analytical Method – VaR for Fixed Income Securities

 For example, a bank wants to know its market risk in terms

• f 1-day VAR.

 1-day VAR = dollar market value of position × price sensitivity

× potential adverse changes in yield

 where price sensitivity = (-Modified Duration)  If we assume that yield changes are normally distributed, we

can construct confidence intervals around the expected value.

 Assuming normality, 90% of the time the yield changes will

be within 1.65 standard deviations of the mean.

52

Analytical Method – VaR for Fixed Income Securities

 Suppose that a bank has a $1 million market value position in

7-year zero-coupon bonds with a market value of $1,000,000.

 Today’s yield on these bonds is 7.243% per year. And we define

“bad” yield changes such that there is only 5% chance of the yield change being exceeded in either direction.

 Assuming normality, 90% of the time yield changes will be within

1.65 standard deviations of the mean.

 If the yield change has a standard deviation is 10 basis points (and

mean is zero), this corresponds to 16.5 basis points.

53

Analytical Method – VaR for Fixed Income Securities



Assume that the modified duration of the bond is 6.527. (Since bond is zero-coupon, D = 7 years, MD = D/(1+R) = 7/(1+0.07243) = 6.527, where Modified duration = Maculay duration/(1+R)



1-day VAR = (Market value of position)  (-MD)  (Potential adverse change in yield) = ($1,000,000)  (-6.527)  (0.00165) = ($1,000,000)  -1.077% = -$10,770



To calculate the potential loss for more than one day:  (VARN) = 1-day VAR × N



Example:

 For a five-day period, VAR5 = -$10,770 × 5 = -$24,082 54

Analytical Method – VaR for Equity Portfolios



VaR for equity portfolios



For equities, if the portfolio is well diversified then VaR = dollar value of position × stock market return volatility where the market return volatility is taken as 1.65 M at 5% where M the volatility of the returns of the market portfolio (e.g., S&P500, SET index)



Suppose the bank holds a 1 million baht trading position in 100 stocks listed on SET. Empirically, the 100-stock portfolio has a β of 1.1. What is 1-day VaR and 5-day VaR?



VaRP = β .VaRM since P = β . M



5-day VaR = 5 x 1-day VaR

55

Analytical Method – VaR for Equity Portfolios



VaR for equity portfolios



Suppose the bank holds a 1 million baht trading position in 100 stocks listed on SET. Empirically, the 100-stock portfolio has a β of 1.1. What is 1-day VaR at 5% and 5-day VaR at 5%?



Suppose that M is empirically estimated to be 1.58% per day.



VaRP = 1.1 x VaRM since P = 1.1 x M



P = 1.1 x 1.58% = 1.738%



1-day VaR at 5% = 1.65 x 1.738% x 1,000,000 baht = 28,677 baht



5-day VaR at 5% = 5 x 1-day VaR at 5% = 5 x 28,677 = 64,123 baht

56

(III) Monte Carlo Simulation Method

 Monte Carlo Simulation method



Inputs on the expected returns, standard deviations, and correlations for each security are required.



Make sure that the portfolio’s returns properly account for the correlations among the securities in the portfolio (i.e., one set of returns can be generated but the other set of returns must reflect any correlation between the two sets of returns.



To obtain the VAR, Monte Carlo simulation generates random

• utcomes based on an assumed probability distribution.



Benefits: flexible  allow the user to assume any known probability distribution and can handle relatively complex portfolios.



Disadvantages: computer-intensive.

57

A Comparison of Analytical vs Historical

 Suppose we have a portfolio of 1,000 troy ounce of 99.5% gold.  The current price (as of 20 Sep 2013) of gold is 1,349.25. So our

gold position is now $1,349,250.

 A sample of daily prices of gold for the past five years is

provided.

 Compute VaR using Analytical vs Historical methods… see

Excel sheet attached..

58

A Comparison of Analytical vs Historical (con’t)

 VaR 1% is (always)

higher than VaR 5%.

 For 1% VaR,

Historical method gives higher VaR numbers than Analytical method does. VaR – 5% Analytical Historical 1 year 29,136 25,312 3 years 26,830 25,607 5 years 29,637 27,873 VaR – 1% Analytical Historical 1 year 40,590 44,805 3 years 37,966 44,686 5 years 42,090 51,693

59

A Comparison of Analytical vs Historical: Two assets



As of today (i.e., 20 Sep 2013), we have a portfolio consisting of the following assets;

 A group of stocks, which have a beta of 1, using DJIA as a market

• portfolio. The current market value (as of the end of 20 Sep 2013) is

$1,500,000.

 1,000 ounces of gold, in which its current value (as of 20 Sep 2013)

$1,349,250 (=1,000x1,349.25).



A sample of daily prices of DJIA index and gold for the past three years is provided.



Compute VaR using Analytical vs Historical methods… see Excel sheet attached..

60

A Comparison of Analytical vs Historical: Two assets(Con’t)

Analytical Historical VaR – 5% 36,084 36,683 VaR – 1% 51,265 58,410

 Again, VaR 1% is (always) higher than VaR 5%.  For 1% VaR, Historical method gives higher VaR numbers

than Analytical method does.

61

A Comparison of the Two Methods

Key considerations:

 VaR is a number that is quite sensitive to how it is calculated.

 Wide ranges such as this are therefore common.  Even more variation if our data is collected over different time

periods, or over different frequency (weekly or monthly instead of daily).

 A wide range of VaR numbers does not mean VaR is useless or

unreliable.

 Instead, knowing the potential range of VaR is itself very useful. 62

A Comparison of the Two Methods

Key considerations: (con’t)

 Always follow up the calculation with an ex post evaluation,

called “back testing”.

 For example, if we settle on a VaR of 540,000, this figure should be

exceeded NO more than 5% of the time.

 Over a long period of time, the risk manager can determine whether

such VaR figure is a reasonable reflection of the true risk. That is, if the $540,000 VaR is exceeded far more or less than 5% of the time, the figure was not a good estimate, and the methodology used to compute VaR should be re-evaluated.

63

Benefits of VaR



Benefits of VaR



Widely used by nearly every major (derivatives) dealer and an increasing number of end users.



Facilitates communication with senior management. A VaR number conveys a lot of useful information that the CEO can easily grasp.



Acceptable in banking regulation. Most banking regulators use VaR as a measure of the risk of a bank.



Used to allocate capital within firms.



Banks engaged in significant trading activities commonly use VaR as a measure to allocate capital, that is, they set aside a certain amount of capital to protect against losses.



Used in performance evaluation.



The modern approach to performance evaluation is to adjust the return performance for a measure of the risk taken in achieving that performance. VaR is often used as a measure of risk in this context.

64

Extensions of VaR



Stress testing



Stress testing involves estimating how the portfolio would have performed under some of the most extreme market moves.



In other words, a stress test determines how badly the portfolio will perform under some of the worst and most unusual circumstances.



To test the impact of an extreme movement, let’s presume that in a given day markets perform terribly.



Consider the portfolio of $20 million in the S&P500 and $12 million in the Nikkei300.



Let us presume that in a given day both markets perform terribly, for example, with S&P500 losing 6% (roughly 6 SD) and Nikkei300 losing 5% (also roughly 6 SD).



Then a total loss for the portfolio will be 5.625% or $1.8 million.



If the performance is tolerable under such extremely unlikely situations, then the portfolio risk is assumed to be acceptable.

65

Extensions of VaR



Stress testing can be considered as a way of taking into account extreme events that do occur form time to time but that virtually impossible according to the probability distribution typically assumed for market variables.

 A 5-standard-deviation daily move in a market variable is one such

extreme event.

 Under the assumption of a normal distribution, it happens about

• nce every 7,000 years! But, in practice, it is not uncommon to see a

5-standard-deviation daily move once or twice every 10 years.



Stress testing can be quite valuable as a supplement to VaR.



Therefore, one major advantage of the stress testing is that it could cover situations commonly absent from the historical data.