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)  ( ) = % 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 = [DEAR a 2 + DEAR b 2 + DEAR c  ab × DEAR a × DEAR b + 2  ac × DEAR a × DEAR c + 2  bc × DEAR b × DEAR c ] 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 (25 th 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. ( ) * 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 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 Federal Reserve Citigroup J.P.Morgan/Chase Merrill Lynch RiskMetrics

Hedging and Derivatives

General idea of hedging Need to look for hedge that has opposite characteristic to underlying price risk Change in price Change in value Underlying risk 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. cash FX or commodity contract

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 hedgedContract 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 Variance of Position Hedge Ratio h h*h* 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

O C = P s [N(d 1 )] - S[N(d 2 )]e -rt O C - Call Option Price P s - Stock Price N(d 1 ) - Cumulative normal density function of (d 1 ) S - Strike or Exercise price N(d 2 ) - Cumulative normal density function of (d 2 ) 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 Model

Options - Application Protective Put - Long stock and long put Share Price Position Value Long Stock

Options - Application Protective Put - Long stock and long put Share Price Position Value Long Put

Options - Application Protective Put - Long stock and long put Share Price Position Value Protective Put Long Put Long Stock

Options - Application Protective Put - Long stock and long put Share Price Position Value Protective Put

Options - Application Straddle - Long call and long put - Strategy for profiting from high volatility Share Price Position Value Long call

Options - Application Straddle - Long call and long put - Strategy for profiting from high volatility Share Price Position Value Long put

Options - Application Straddle - Long call and long put - Strategy for profiting from high volatility Share Price Position Value Straddle

Options - Application Straddle - Long call and long put - Strategy for profiting from high volatility Share Price Position Value Straddle

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 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 Position Value 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 Position Value Covered Call Short Call Long Stock

Single option and a Stock Short stock and long call Reverse of writing a covered call Share Price Position Value Long Call Short Stock

Share Price Position Value 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 Position Value Long Put Long Stock

Share Price Position Value 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 Position Value Short Put Short Stock

Share Price Position Value 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. x2 x1x3

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.

Straddle Profits 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, 2 nd edition, by K. Dowd; John Wiley & Sons, Value-at-Risk: Theory and Practice by G.A. Holton, Academic Press, Financial Institutions Management: A Risk Management Approach, 5 th edition, by A.Saunders and M. Cornett; McGraw-Hill 2006

Where to find out more on Risk Management? – main concepts - Glyn Holton – The Institute of Risk Management – Risk Magazine