Lecture 8

Lecture 5Market Risk

Todays TopicsThis Chapter discussess the nature of market and appropriate measures:Market riskValue at riskVaR parametersMethods of Calculating VAR:RiskMetricsHistoric or back simulationMonte Carlo simulationStress testingMethods of calculating VaRLinks between market risk and capital requirements

Need for banks capitalTo ensure the soundness of banks, regulators specify minimum capital ratio that banks need to maintainIf banks are not regulated, what will be the tendency of banks towards capital-to-assets ratio?

Will deposit insurance solve the problem?Deposit insurance will create moral hazard for insurance companies,Again for the solution of moral hazard problem, banks should be required by an apex authority to main enough capital

The pre-1988 period of banks regulationsPrior to 1988, regulators in different countries tended to regulate bank capital by setting minimum level for the ratio of capital to total assets

However, definition of capital and ratios fluctuated widely from country to country

Banks were competing globally and a bank in a country with tight capital regulations was considered to have competitive disadvantage as compared to a bank in country with slack capital requirements

At the same time, many banks were heavily involved in derivative contracts, that do not appear on balance sheet but increase the risk

These derivative contracts were not covered by the capital-to-assets ratio

These problems led supervisory authorities in 12 countries to form a Basel committee on banking's supervision

They met regularly in Basel, Switzerland, under the patronage of the bank for International Settlements.

Assets to Capital must be less than 20. Assets includes off-balance sheet items that are direct credit substitutes such as letters of credit and guarantees

Cooke Ratio: Capital must be 8% of risk weighted amount. At least 50% of capital must be Tier 1.

Tier 1 Capital: common equity, non-cumulative perpetual preferred shares, minority interests in consolidated subsidiaries less goodwill

Tier 2 Capital: cumulative preferred stock, certain types of 99-year debentures, subordinated debt with an original life of more than 5 years

Risk-Weighted CapitalA risk weight is applied to each on-balance- sheet asset according to its risk (e.g. 0% to cash and govt bonds; 20% to claims on OECD banks; 50% to residential mortgages; 100% to corporate loans, corporate bonds, etc.)

For each off-balance-sheet item we first calculate a credit equivalent amount and then apply a risk weight

Risk weighted amount (RWA) consists of:sum of risk weight times asset amount for on-balance sheet itemsSum of risk weight times credit equivalent amount for off-balance sheet items

1996 Amendment Implemented in 1998

Requires banks to measure and hold capital for market risk for all instruments in the trading book including those off balance sheet (This is in addition to the BIS Accord credit risk capital)

The Market Risk CapitalThe capital requirement is

Where k is a multiplicative factor chosen by regulators (at least 3), VaR is the 99% 10-day value at risk, and SRC is the specific risk charge (primarily for debt securities held in trading book)

Basel IIIn 1999 the Basel committee proposed new rules that have become known as Basel IIExpected to be implemented in 2007

Three pillars1-New minimum capital requirements for credit and operational risk

2-Supervisory review: more thorough and uniform

3-Market discipline: more disclosure

Soundness of FIs and Capital AdequacyOverview:An important objective of government is to provide a stable economic environment for businesses and individuals

One of way of accomplishing this objective is to provide stable and reliable banking system where bank failures are rare and depositors are protected

Soundness of a bank mainly depends upon the banks ability to absorb unexpected losses

This ability is critically dependent upon the banks capital and other components of equity that the bank holds

Soundness of contIt is widely accepted that the capital a financial institution requires should cover the difference between the expected losses and worst-case losses over some time horizonThe worst-case loss is the loss that is not expected to be exceeded with some high degree of confidenceThe high degree of confidence might be 99%Why only worst-cases losses?

Market riskOne source of risk for financial institution comes from market variables

Market risk is the risk that the value of on and off-balance sheet positions of a financial institution will be adversely affected by movements in market rates or prices such as interest rates, foreign exchange rates, equity prices, credit spreads and/or commodity prices resulting in a loss to earnings and capital

Convergence of Economies Easy and faster flow of information Skill Enhancement Increasing Market activity Why the focus on Market Risk Management ?Leading to Increased VolatilityNeed for measuring and managing Market RisksRegulatory focusProfiting from Risk

Value at RiskTo ensure the adequate capital is there to cover losses above expected levels, financial institutions need to know a single number that summarizes the total risk in portfolio of financial assets

Value at Risk (VaR) is such measure which was pioneered by J.P. Morgan

VaR Historical perspectiveThe chairman Dennis Weatherstone was dissatisfied with the long risk reports

He asked for something simpler that focused on the bank total exposure over the next 24 hours

This measure helped management to know the risk being taken by the bank and were better able to allocate capital within the bank

Definition of VaRWhat loss level is such that we are X% confident it will not be exceeded in N business days?

Value at risk is a function of two variables:- The time horizon - and the confidence level

VaR in Basel II

The VAR measure used by regulators for market risk is the loss on the trading book that can be expected to occur over a 10-day period 1% of the time

The value at risk is $1 million means that the bank is 99% confident that there will not be a loss greater than $1 million over the next 10 days.

The capital that it requires is a k times of the VaR measure

Capital requirement for VaRK is chosen on a bank-by-bank basis by the regulators and must be at least 3

It historical data shows that VaR measure did not perform well in the last 250 trading days, k may be set as high as 4

Example of Var (in %)A percentage VaR is very simpleValue at risk is the worst expected loss over some selected time period with some selected confidenceAccording to the above, Var is the product of VaRa = zaZa is the chosen level of probabilityIf we set z value at 95% confidence, it will return the value 1.65 of critical z which means that we are 95% confident that the worst negative change in our given variable will be 1.65 times of the standard deviation value Similarly, 99% confidence returns the critical value 2.33 of z

The critical value of z at a given confidence level can be calculated by the =normsinv function of ExcelFor 95% confidence, we shall input values like:=normsinv(.05)Example in excel sheet given (Var Parametric)

VaR ParametersTime HorizonVaR should be calculated over a horizon of how many days?An appropriate choice for the time horizon depends on the applicationFor example, trading desks of banks calculate the profit and loss dailyTheir positions are usually fairly liquid and actively managedThis is why daily VaR will be the right choice for internal useIf VaR turns out to be beyond the acceptable limit, composition of portfolio can be changed quickly

Choice of Parameters

The confidence intervalBasel committee has chosen a time horizon of 10-days and a confidence level of 99% for market risks in the trading booksThe confidence interval is primarily dependent upon the accuracy of VaR measures over timeIf VaR threshold is breached several times in the past, it can be suspected that the VaR is underestimatedShould we increase or decrease confidence level in such a scenario?

Choice of Parameters cont

Dilema?The higher the confidence level , the greater the VAR measure

It is not clear, however, whether one should stop at 99%, 99.9%, 99.99% and so on. Each of these values will create an increasingly larger loss, but less likely.

Another problem is that, as confidence level increases, the number of occurrences below VAR shrinks, leading to poor measures of large but unlikely losses.

The choice of the confidence level depends on the use of VAR. For most applications, VAR is simply a benchmark measure of downside risk. If so, what really matters is consistency of the VAR confidence level across trading desks or time.

ImplicationsEmphasizes importance of:Measurement of exposureControl mechanisms for direct market risk and employee created risksHedging mechanismsOf interest to regulators Ch 10-26

Market RiskMarket risk is the uncertainty resulting from changes in market prices Affected by other risks such as interest rate risk and FX riskCan be measured over periods as short as one dayUsually measured in terms of dollar exposure amount or as a relative amount against some benchmarkCh 10-27

Market Risk MeasurementImportant in terms of:Management informationSetting limitsResource allocation (risk/return tradeoff)Performance evaluationRegulationBIS and Fed regulate market risk via capital requirements leading to potential for overpricing of risksAllowances for use of internal models to calculate capital requirementsCh 10-28

Calculating Market Risk ExposureGenerally concerned with estimated potential loss under adverse circumstancesThree major approaches of measurement:JPM RiskMetrics (or variance/covariance approach)Historic or Back SimulationMonte Carlo SimulationCh 10-29

RiskMetrics ModelIdea is to determine the daily earnings at risk = dollar value of position price sensitivity potential adverse move in yield or, DEAR = dollar market value of position price volatility.Where, price volatility = price sensitivity of position potential adverse move in yield

Ch 10-30

RiskMetricsDEAR can be stated as: DEAR = (MD) (potential adverse daily yield move)

where,MD = D/(1+R).

MD = Modified duration D = Macaulay duration

Ch 10-31

Confidence IntervalsIf we assume that changes in the yield are normally distributed, we can construct confidence intervals around the projected DEAR (other distributions can be accommodated but normal is generally sufficient)Assuming normality, 90% of the time the disturbance will be within 1.65 standard deviations of the mean(5% of the extreme values remain in each tail of the distribution)Ch 10-32

Adverse 7-Year Rate MoveCh 10-33

Confidence Intervals: ExampleSuppose that we are long in 7-year zero-coupon bonds and we define bad yield changes such that there is only a 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 standard deviation is 10 basis points, this corresponds to 16.5 basis points. Concern is that yields will rise. Probability of yield increases greater than 16.5 basis points is 5%.Ch 10-34

Confidence Intervals: ExampleYield on the bonds = 7.243%, so MD = 6.527 yearsPrice volatility = (MD) (Potential adverse change in yield)= (6.527) (0.00165) = 1.077%DEAR = Market value of position (Price volatility)= ($1,000,000) (.01077) = $10,770Ch 10-35

Confidence Intervals: ExampleTo calculate the potential loss for more than one day:Market value at risk (VARN) = DEAR Example: For a five-day period, VAR5 = $10,770 = $24,082

Foreign ExchangeIn the case of foreign exchange, DEAR is computed in the same fashion we employed for interest rate riskDEAR = dollar value of position FX rate volatility, where the FX rate volatility is taken as 1.65 sFXCh 10-37

EquitiesFor equities, total risk = systematic risk + unsystematic risk If the portfolio is well diversified, then DEAR = dollar value of position stock market return volatility, where market volatility taken as 1.65 sm If not well diversified, a degree of error will be built into the DEAR calculation

Ch 10-38

Aggregating DEAR EstimatesCannot simply sum up individual DEARsIn order to aggregate the DEARs from individual exposures we require the correlation matrix.Three-asset case:DEAR portfolio = [DEARa2 + DEARb2 + DEARc2 + 2rab DEARa DEARb + 2rac DEARa DEARc + 2rbc DEARb DEARc]1/2 Ch 10-39

DEAR: Large US Banks 2005 & 2008Ch 10-40

Estimation of VAR: ExampleConvert todays FX positions into dollar equivalents at todays FX ratesMeasure sensitivity of each positionCalculate its deltaMeasure risk Actual percentage changes in FX rates for each of past 500 daysRank days by risk from worst to bestCh 10-41

Other methods of finding VaRThere are other two approaches to calculate VaRHistorical simulationMonte Carlo simulation

Historical SimulationHS involves using past data as a guide to what will happen in the future

Historic or Back SimulationBasic idea: Revalue portfolio based on actual prices (returns) on the assets that existed yesterday, the day before that, etc. (usually previous 500 days)Then calculate 5% worst-case (25th lowest value of 500 days) outcomesOnly 5% of the outcomes were lowerCh 10-43

Estimation of VAR: ExampleConvert todays FX positions into dollar equivalents at todays FX ratesMeasure sensitivity of each positionCalculate its deltaMeasure risk Actual percentage changes in FX rates for each of past 500 daysRank days by risk from worst to bestCh 10-44

44

Historic or Back SimulationAdvantages:SimplicityDoes not need correlations or standard deviations of individual asset returnsDoes not require normal distribution of returns (which is a critical assumption for RiskMetrics)Directly provides a worst case value

Ch 10-45

HS - MethodologySuppose we wan to calculate VaR for a portfolio using 1-day horizon, a 99% confidence level, and 500 days of dataCollect data on the daily movements in the given market variables Conduct 500 trials assuming as if todays prices will change at a past rate of change in each of the 500 daysThis way forecasted value for tomorrow will be: Where Vn is todays value of the variableVi is the variable value in past daysVi-1 is the variable value one-day before the vi value

HS - MethodologyAfter that, calcualte the value of portfolio based on the trial values of each variableFind the difference between the forecasted values and todays value of the portfolio in all 500 trialsFind the given percentile of these differences, and that will be the VaR estimate

Percentile The 1st percentile in 500 observations mean the 5th worst loss in all 500 observationsIn Excel, we do this by:=percentile(range, .01) if it is for 1st percentile

WeaknessesDisadvantage: 500 observations is not very many from a statistical standpointIncreasing number of observations by going back further in time is not desirableCould weight recent observations more heavily and go further backCh 10-49

Monte Carlo SimulationTo overcome problem of limited number of observations, synthesize additional observationsPerhaps 10,000 real and synthetic observationsEmploy historic covariance matrix and random number generator to synthesize observationsObjective is to replicate the distribution of observed outcomes with synthetic data

Ch 10-50

AccuracyPast may not be accurate estimate of the future, Kendall and Stuart describ how to calculate a confidence for the percentile of the probability distribution when it is esitmated from a sample dataSuppose that x is the qth percentileof the loss distribution when it is estimated from n observations. The standard error of x is

where f(x) is an estimate of the probability density of the loss at the qth percentile calculated by assuming a probability distribution for the loss

Regulatory ModelsBIS (including Federal Reserve) approach:Market risk may be calculated using standard BIS modelSpecific risk chargeGeneral market risk chargeOffsetsSubject to regulatory permission, large banks may be allowed to use their internal models as the basis for determining capital requirementsCh 10-52

BIS ModelSpecific risk charge: Risk weights absolute dollar values of long and short positionsGeneral market risk charge:reflect modified durations expected interest rate shocks for each maturityVertical offsets:Adjust for basis riskHorizontal offsets within/between time zones Ch 10-53

q= NORMINV(0.01, mean, Stdev)

f(x) = NORMDIST(q, mean, stdev, false)

Error is the +/- in the value of VaR