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Analytics of Risk Management III: Motivating Risk Measures Risk Management Lecturer: Mr. Frank Lee Session 5
Analytics of Risk Management III:
Motivating Risk Measures
Risk Management
Lecturer:
Mr. Frank Lee
Session 5
Overview • Risk Measurement Applications,
Scenario building and Simulations– JPM RiskMetrics– Historic or Back Simulations– Monte Carlo Simulations– Hedging and risk limits
Risk Management Application
• Uncertainty or changes in value resulting from changes in the underlying parameter. Can be measured over periods as short as one day.
• Usually measured in terms of ‘dollar’ exposure amount or as a relative amount against some benchmark
• Find value at risk, e.g. market risk, interest rate risk, foreign exchange risk etc.
Application: Market Risk Measurement
• Important in terms of: – Management information– Setting limits– Resource allocation– Performance evaluation– Regulation
Calculating Market Risk Exposure
• Generally concerned with estimating potential loss under adverse circumstances.
• Three major approaches of measurement– JP Morgan RiskMetrics (variance/covariance
approach)– Historic or Back Simulation– Monte Carlo Simulation
JP Morgan RiskMetrics Model
– Idea 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.
– Can be stated as (-MD) × adverse daily yield move where,
MD = D/(1+R)Modified duration = MacAulay duration/(1+R)
Confidence Intervals
– If 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.
– Also, 98% of the time the disturbance will be within 2.33 standard deviations of the mean
Confidence Intervals: Example
– Suppose that we are long in 7-year zero-coupon bonds 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 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%.
*(suppose YTM=7.25%)
Confidence Intervals: Example
• Price volatility = (-MD) (Potential adverse change in yield)= (-6.527) (0.00165) = -1.077%DEAR = Market value of position (Price volatility)
= ($1,000,000) (.01077) = $10,770
Confidence Intervals: Example
• To calculate the potential loss for more than one day:Market value at risk (VAR) = DEAR × N
• Example: For a five-day period, VAR = $10,770 × 5 = $24,082
Foreign Exchange & Equities
• In the case of Foreign Exchange, DEAR is computed in the same fashion we employed for interest rate risk.
• For equities, if the portfolio is well diversified then DEAR = dollar value of position × stock market return volatility where the market return volatility is taken as 1.65 M.
Aggregating DEAR Estimates
• Cannot simply sum up individual DEARs.• In order to aggregate the DEARs from individual
exposures we require the correlation matrix. • Three-asset case:
DEAR portfolio = [DEARa2 + DEARb
2 + DEARc2 +
2ab × DEARa × DEARb + 2ac × DEARa × DEARc + 2bc × DEARb × DEARc]1/2
Historic or Back Simulation• Advantages:
– Simplicity– Does not require normal distribution of returns
(which is a critical assumption for RiskMetrics)– Does not need correlations or standard deviations
of individual asset returns.
Historic or Back Simulation
• Basic idea: Revalue portfolio based on actual prices (returns) on the assets that existed yesterday, the day before, etc. (usually previous 500 days).
• Then calculate 5% worst-case (25th lowest value of 500 days) outcomes.
• Only 5% of the outcomes were lower.
Estimation of VAR: Example
• Convert today’s FX positions into dollar equivalents at today’s FX rates.
• Measure sensitivity of each position– Calculate its delta.
• Measure risk – Actual percentage changes in FX rates for each of
past 500 days.
• Rank days by risk from worst to best.
Weaknesses
• Disadvantage: 500 observations is not very many from statistical standpoint.
• Increasing number of observations by going back further in time is not desirable.
• Could weight recent observations more heavily and go further back.
Monte Carlo Simulation
• To overcome problem of limited number of observations, synthesize additional observations.– Perhaps 10,000 real and synthetic observations.
• Employ historic covariance matrix and random number generator to synthesize observations.– Objective is to replicate the distribution of
observed outcomes with synthetic data.
Monte Carlo Simulation
• Step 1: Modeling the Project• Step 2: Specifying Probabilities• Step 3: Simulate the Results (e.g. cash flows, values
etc.)
• Monte Carlo simulation is conceptually simple, but is generally computationally more intensive than other methods.
Modeling Process
• The generic MC VaR calculation goes as follows:– Decide on N, the number of iterations to perform. – For each iteration:
• Generate a random scenario of market moves using some market model. • Revalue the portfolio under the simulated market scenario. • Compute the portfolio profit or loss (PnL) under the simulated scenario. (i.
e. subtract the current market value of the portfolio from the market value of the portfolio computed in the previous step).
– Sort the resulting PnLs to give us the simulated PnL distribution for the portfolio.
– VaR at a particular confidence level is calculated using the percentile function. For example, if we computed 5000 simulations, our estimate of the 95% percentile would correspond to the 250th largest loss, i.e. (1 - 0.95) * 5000.
• Note that we can compute an error term associated with our estimate of VaR and this error will decrease as the number of iterations increases.
Monte Carlo Simulation
Monte Carlo Simulation
• Monte Carlo simulation is generally used to compute VaR for portfolios containing securities with non-linear returns (e.g. options) since the computational effort required is non-trivial.
• For portfolios without these complicated securities, such as a portfolio of stocks, the variance-covariance method is perfectly suitable and should probably be used instead.
• MC VaR is subject to model risk if our market model is not correct.
Regulatory Models
• BIS (including Federal Reserve) approach:– Market risk may be calculated using standard BIS
model.• Specific risk charge.• General market risk charge.• Offsets.
– Subject to regulatory permission, large banks may be allowed to use their internal models as the basis for determining capital requirements.
BIS 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
Large Banks: BIS versus RiskMetrics
– In calculating DEAR, adverse change in rates defined as 99th percentile (rather than 95th under RiskMetrics)
– Minimum holding period is 10 days (means that RiskMetrics’ daily DEAR multiplied by 10.
– Capital charge will be higher of:• Previous day’s VAR (or DEAR 10)• Average Daily VAR over previous 60 days times a
multiplication factor 3.
Websites
Bank for International Settlements www.bis.orgFederal Reserve www.federalreserve.govCitigroup www.citigroup.comJ.P.Morgan/Chase www.jpmorganchase.comMerrill Lynch www.merrilllynch.comRiskMetrics www.riskmetrics.com
Hedging and Derivatives
General idea of hedgingNeed to look for hedge that has opposite characteristic to underlying price risk
Change in price
Change in valueUnderlying risk
Hedge position
Money Market Hedges
• Locking in a Rate of InterestLoan in 3 MonthsBorrow Now Deposit for 3 Months
• Locking in Exchange RateExchange £ for $ and Invest in US Money Market Now
Forwards and futures
• Forward is agreement today to buy at future time but at price agreed today---OTC and counter-party risk
• Futures contract is similar but in standard bundles on an organized exchange so risk is different and margining means that futures are like a string of daily forward contracts.
cash
FX or commoditycontract
Hedging with Futures and Forwards - Difficulties
• Asset Hedged may not be the same as that underlying the Futures Contract
• Hedger may be uncertain as to when asset will be Bought or Sold
• Hedge may have to be closed out with Futures contract well before Expiry Date
• These problems give rise to Basis Risk
Basis Risk and Hedging
Basis = Spot price of an - Futures price of asset to be hedged Contract Used
Price Obtained with Short Hedge = S2 + F1 - F2 = F1 + b1
Price Paid for with Long Hedge = S2 + F1 - F2 = F1 + b1
Where Hedge Contract Different from Underlying Asset
S2 + F1 - F2 = F1 + (S*2 - F2) + (S2 - S*2)
Optimal Hedge Ratios
Varianceof Position
Hedge Ratio hh*
OHR -The ration of the size of the position taken in futures contract to the size of the exposure.
SWAPs
• Interest Rate - Fixed for Variable• Currency - Principle (Paid and Repaid) and
Interest Payments
OC = Ps[N(d1)] - S[N(d2)]e-rt
OC- Call Option Price
Ps - Stock Price
N(d1) - Cumulative normal density function of (d1)
S - Strike or Exercise price
N(d2) - Cumulative normal density function of (d2)
r - discount rate (90 day comm paper rate or risk free rate)
t - time to maturity of option (as % of year)
v - volatility - annualized standard deviation of daily returns
Black-Scholes Option Pricing ModelBlack-Scholes Option Pricing Model
Options - Application
Protective Put - Long stock and long put
Share Price
Pos
itio
n V
alue
Long Stock
Options - Application
Protective Put - Long stock and long put
Share Price
Pos
itio
n V
alue
Long Put
Options - Application
Protective Put - Long stock and long put
Share Price
Pos
itio
n V
alue Protective Put
Long Put
Long Stock
Options - ApplicationStraddle - Long call and long put - Strategy for profiting from high volatility
Share Price
Pos
itio
n V
alue Long call
Options - ApplicationStraddle - Long call and long put - Strategy for profiting from high volatility
Share Price
Pos
itio
n V
alue
Long put
Options - ApplicationStraddle - Long call and long put - Strategy for profiting from high volatility
Share Price
Pos
itio
n V
alue
Straddle
Trading Strategies with Options1. Vertical: same maturity, different exercise price2. Horizontal: same ex. price, different maturity3. Diagonal: different ex. price and different
maturities
• Bull Spread• Bear Spread• Butterfly Spreads• Calendar Spreads• Straddles• Strangles
Trading Strategies Involving Options
Trading Strategies involving Options
• Single Option and a Stock strategies• Spreads• Combinations• Other Payoffs
Single option and a Stock• Writing a Covered Call (long stock and short call)• the long stock protects a trader from the payoff of the short
call if there is a sharp rise in the stock price
Share Price
Pos
itio
n V
alue
Short Call
Long Stock
Single option and a Stock• Writing a Covered Call (long stock and short call)• the long stock protects a trader from the payoff of the short
call if there is a sharp rise in the stock price
Share Price
Pos
itio
n V
alue
Covered Call
Short Call
Long Stock
Single option and a Stock• Short stock and long call• Reverse of writing a covered call
Share Price
Pos
itio
n V
alue
Long Call
Short Stock
Share Price
Pos
itio
n V
alue
Long Call
Short Stock
Single option and a Stock• Short stock and long call• Reverse of writing a covered call
Single option and a Stock• Writing a Protective Put (buying a put and the stock itself)
Share Price
Pos
itio
n V
alue
Long Put
Long Stock
Share Price
Pos
itio
n V
alue Protective Put
Long Put
Long Stock
Single option and a Stock• Writing a Protective Put (buying a put and the stock itself)
Single option and a Stock• Short stock and short put• reverse of protective put
Share Price
Pos
itio
n V
alue
Short Put
Short Stock
Share Price
Pos
itio
n V
alue
Short Put
Short Stock
Single option and a Stock• Short stock and short put• reverse of protective put
Spreads
• A spread trading strategy involves taking a position in two or more options of the same type (i.e. two or more calls or two or more puts)
Bull Spread
Profit
St
Buy a call and sell a call with a higher strike price (on the same stock ) or buy a put with a low strike price and sell a put with a high strike price
Bear Spread
Profit
St
Buy a call with a higher strike price and sell a call (on the same stock). Hope that the stock price will decline.
Butterfly Spread
Profit
St
Three different strike prices (on the same stock). Buy a call with a relatively low strike price x1, buy a call with a relatively high strike price x3 and sell two calls with a strike price half way x2. Can use put options too.
x2x1 x3
Calendar Spread
Profit
St
Same strike price, different expiration dates. Sell a call and buying a call with the same striking price but longer maturity.
StraddleProfits
St
Strangle
Profit
St
Buy a call and a put with the same expiration date and different strike price
Option Hedging Strategy
• With option, we can engineer a portfolio with the underlying asset and the option
• The nature of the new portfolio can be either:– Riskier (to pursue higher return)– Or risk free
Standard option strategies in investment – a summary
• Protective put– Long stock – Long put
• Covered call – Long stock– Short Call
• Straddle– Same X– Long call + long put
• Spreads– Combination of two or more
options of same type and on same asset
– Different X or T– Vertical-money spread
• Same T• Different X
– Horizontal-time spread• Different T• Same X
– Diagonal spreads
Exercises
• Draw Diagrams for Sell Call and Sell Put • Draw Diagram for Long in Call and Short in Call
where the Long Call has the lower strike Price • Can the Payoff of the Previous question be
replicated with Put Options ?
Exercises
Additional Revision Themes
• Merits and demerits of sensitivity, statistical and downside measures of risk.
• Use of derivatives for risk management.• VaR application and calculation.• Duration, bond prices.• Brush up on mean, standard deviation,
normal distribution; calculation in the ‘portfolio’ or ‘weighted’ average/risk context.
Where to find out more on Risk Management?
• Risk Management and Derivatives by R.M.Stulz; Thomson 2003
• Beyond Value at Risk: The New Science of Risk Managemnt J Wiley 2003
• Measuring Market Risk, 2nd edition, by K. Dowd; John Wiley & Sons, 2005.
• Value-at-Risk: Theory and Practice by G.A. Holton, Academic Press, 2003.
• Financial Institutions Management: A Risk Management Approach, 5th edition, by A.Saunders and M. Cornett; McGraw-Hill 2006
Where to find out more on Risk Management?
• www.riskglossary.com – main concepts• www.contingencyanalysis.com - Glyn Holton• www.theirm.org – The Institute of Risk
Management• www.risk.net – Risk Magazine
