Transcript No Slide Title

Risk
Management
Session 5
Analytics of Risk
Management III:
Motivating Risk Measures
Lecturer:
Mr. Frank Lee
Overview
• Risk Measurement Applications,
Scenario building and Simulations
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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 sM.
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 + DEARb2 + DEARc2 +
2rab × DEARa × DEARb + 2rac × DEARa × DEARc +
2rbc × 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
Modeling Process
• 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.
Monte Carlo Simulation
• 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 is generally used to compute
VaR for portfolios containing securities with nonlinear 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.org
Federal Reserve www.federalreserve.gov
Citigroup www.citigroup.com
J.P.Morgan/Chase www.jpmorganchase.com
Merrill Lynch www.merrilllynch.com
RiskMetrics www.riskmetrics.com
Hedging and Derivatives
General idea of hedging
Need to look for hedge that has opposite characteristic to underlying price risk
Change in value
Underlying risk
Change in price
Hedge position
Money Market Hedges
• Locking in a Rate of Interest
Loan in 3 Months
Borrow Now Deposit for 3 Months
• Locking in Exchange Rate
Exchange £ 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.
contract
FX or commodity
cash
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
asset to be hedged
Futures price of
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
OHR -The ration of the size of the position taken
in futures contract to the size of the exposure.
Variance
of
Position
h*
Hedge Ratio h
SWAPs
• Interest Rate - Fixed for Variable
• Currency - Principle (Paid and Repaid) and
Interest Payments
Black-Scholes Option Pricing Model
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
Options - Application
Protective Put - Long stock and long put
Position Value
Long Stock
Share Price
Options - Application
Position Value
Protective Put - Long stock and long put
Long Put
Share Price
Options - Application
Protective Put - Long stock and long put
Position Value
Long Stock
Protective Put
Long Put
Share Price
Options - Application
Position Value
Straddle - Long call and long put
- Strategy for profiting from high volatility
Long call
Share Price
Options - Application
Straddle - Long call and long put
- Strategy for profiting from high volatility
Position Value
Long put
Share Price
Options - Application
Position Value
Straddle - Long call and long put
- Strategy for profiting from high volatility
Straddle
Share Price
Trading Strategies with Options
1. Vertical: same maturity, different exercise price
2. Horizontal: same ex. price, different maturity
3. Diagonal: different ex. price and different
maturities
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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
Position Value
Long Stock
Share Price
Short Call
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
Position Value
Long Stock
Covered Call
Share Price
Short Call
Single option and a Stock
• Short stock and long call
• Reverse of writing a covered call
Position Value
Short Stock
Long Call
Share Price
Single option and a Stock
• Short stock and long call
• Reverse of writing a covered call
Position Value
Short Stock
Long Call
Share Price
Single option and a Stock
• Writing a Protective Put (buying a put and the stock itself)
Position Value
Long Stock
Long Put
Share Price
Single option and a Stock
• Writing a Protective Put (buying a put and the stock itself)
Position Value
Long Stock
Protective Put
Long Put
Share Price
Single option and a Stock
• Short stock and short put
• reverse of protective put
Position Value
Short Stock
Short Put
Share Price
Single option and a Stock
• Short stock and short put
• reverse of protective put
Position Value
Short Stock
Short Put
Share Price
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
 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
Profit
St
Bear Spread
 Buy a call with a higher strike price and sell a call (on the
same stock). Hope that the stock price will decline.
Profit
St
Butterfly Spread
 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.
Profit
x1
x2
x3
St
Calendar Spread
 Same strike price, different expiration dates. Sell a call and
buying a call with the same striking price but longer maturity.
Profit
St
Straddle
Profits
St
Strangle
 Buy a call and a put with the same expiration date and
different strike price
Profit
St
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