Applied Derivatives Risk Management Value at Risk Risk Management, ok but what s risk? risk is the pain of being wrong Market Risk: Risk of loss due to a change in market price Counterparty Risk: Risk of loss due to credit deterioration or failure to perform of a counterpart, and/or due to the actions of a counterpart against you Liquidity Risk: Risk of loss due to inadequate funding Model Risk: Risk that the model used to trades is inadequate Operational Risk: Risk of failed processes/systems/people s actions Legal / Compliance Risk: Risk of loss due to a failure in legal structures, client or firm compliance issues, etc. Regulatory Risk: Risk of sanction due to a failure to comply with regulatory requirements Reputational / Headline Risk: many examples out there
Market Risk The Normal Distribution Market Risk management math is easier assuming normal returns: Standard deviation is a good measure of risk as there are only two moments If individual security returns are symmetric, then also portfolio returns will be normally distributed Assuming normality, future scenarios can be estimated using just mean and standard deviation The Normal Distribution
Normality and Risk Measures What if returns are not normally distributed? Standard deviation is no longer a complete measure of risk Need to consider higher moments: Skew: measures asymmetry Kurtosis: measures thickness of tails Skew and Kurtosis skew kurtosis 3 R R Expect. ˆ this is zero for symmetric distributions 3 4 R R Expect. 3 4 ˆ this equals 3 for a Normal distribution
Normal and Skewed Distributions Normal and Fat-Tailed Distribution (mean =.1, SD =.2)
Value-at-Risk (VaR) What is VaR? Estimate of the potential loss in value due to adverse market movements Based on historical analysis of changes in asset or portfolio values Specified by a time horizon and a frequency (or probability) of occurrence Banks use 1-day, 95% VaR as an estimate of the potential loss in value of trades Meaning: there is a 5% (or 1 in 2) chance that daily revenues will fall below expected daily revenues by an amount > VaR Drawbacks Why is VaR useful? Trend analysis and product comparisons over short time horizons Objective measure Can aggregate across portfolios and products (can t do that with greeks) Can back-test against P&L Embraced by regulators and industry Assumes that historical data accurately describe current market conditions Does not account for relative liquidity of assets Assumes normality - Does not capture tail events 2.5 Normal Distribution and VaR 2 Percentile 1.5 1.5 VaR 1..8.6.4.2..2.4.6.8 1.
Expected Shortfall (ES) a.k.a. Conditional Tail Expectation (CTE), a.k.a. C-VaR A more conservative measure of downside risk than VaR: VaR takes the highest return from the worst cases Real life distributions are asymmetric (skew) and have fat tails (kurtosis) ES (or CTE) takes an average return of the worst cases Let s see it with a chart 2.5 Normal Distribution, VaR, and Expected Shortfall 2 The area is the percentile 1.5 1.5 Expected Shortfall VaR 1..8.6.4.2..2.4.6.8 1.
A game with a coin Let s play a game: flip a (fair) coin, and receive $1 if heads Assume Pr[Heads]= p (if fair, then p=5%) Q. What is the game s expected outcome? Q. What is the Variance? Q. What is the St.Dev? A game with two coins Let s play a game: flip 2 fair coins, and receive $1 for each head Q. What is the portfolio expected return? Q. What is the portfolio Variance? Q. What is the portfolio St.Dev?
A lot more coins Let s play a game: flip 3 fair coins, and receive $1 for each head Q. What is the portfolio expected return? Q. What is the portfolio Variance? Q. What is the portfolio St.Dev? A Portfolio of 2 stocks Portfolio =.5 * A +.5 * B A: r A =.8 StDev A =.1 B: r B =.1 StDev B =.1 Q. What is the portfolio expected return? Q. What is the portfolio Variance? Q. What is the portfolio Standard Deviation?
A Portfolio of 3 stocks Portfolio = w A * A + w B * B + w C * C Q. What is the portfolio expected return? Q. What is the portfolio Variance? Q. What is the portfolio Standard Deviation? Q. What is if you have N stocks? Arrange P&L data into buckets 6 Daily P+L of $1. Billion $/Yen Short Position 6 5 4 4 3 2 2 Millions of Dollars -2 Jan-86 Jan-88 Jan-9 Feb-92 Feb-94 1 Feb-96-1 -2 P+L Bucket ``` Feb-98 Feb- Mar-2 Best Result = $58.8 Worst Result = -$32.2 Mar-4 Mar-6 Mar-8 Mar-1-3 -4-4 -5-6 -6 9.% 8.% 7.% 6.% 5.% 4.% 3.% Percent of Total Observations 2.% 1.%.%
Arrange P&L data into buckets 9.% Distribution of 25 years of Daily P+Ls ($1. Billion Short $/Yen) 8.% Percent of Total Observations 7.% 6.% 5.% 4.% 3.% 2.% Daily P+L Frequency Best Result = $58.8 Worst Result = -$32.2 1.%.% -6-55 -5-45 -4-35 -3-25 -2-15 -1-5 5 1 15 2 25 3 35 4 45 5 55 6 P+L Bucket... and VaR can be estimated. 9.% 8.% Distribution of 25 years of Daily P+Ls ($1. Billion Short $/Yen) 5.% of Results 95.% of Results 7.% Daily P+L Frequency Percent of Total Observations 6.% 5.% 4.% 3.% 2.% VaR is expressed in terms of a confidence interval. In this case we note the 95.% confidence point, which is $1.3 million. That means that 5.% of the time the result was $1.3 million or worse. 5% corresponds to one out of every 2 business days, or approximately once per month. 95% VaR ($1.3mm) Best Result = $58.8 Worst Result = -$32.2 1.%.% -6-55 -5-45 -4-35 -3-25 -2-15 -1-5 5 1 15 2 25 3 35 4 45 5 55 6 P+L Bucket
... and VaR can be estimated. 9.% Distribution of 25 years of Daily P+Ls ($1. Billion Short $/Yen) Percent of Total Observations 8.% 7.% 6.% 5.% 4.% 3.% 2.% 1.% 5.% of Results 95.% of Results Most VaR calculations are based on a probability distribution inferred from the data. A distribution allows p+l data to be condensed into a single number, the standard deviation (or volatility). VaR can be computed as a multiple of the standard deviation. 1 Day Prob. Distribution Daily P+L Frequency 95% VaR ($11.6mm) Best Result = $58.8 Worst Result = -$32.2 The inferred probability distribution may provide a different estimate of VaR than the price data itself. In this case the VaR estimate changes to $11.6 million..% -6-55 -5-45 -4-35 -3-25 -2-15 -1-5 5 1 15 2 25 3 35 4 45 5 55 6 P+L Bucket... and VaR can be estimated. 9.% 8.% Distribution of 25 years of Daily P+Ls ($1. Billion Short $/Yen) 5.% of Results 95.% of Results Percent of Total Observations 7.% 6.% 5.% 4.% 3.% 2.% 1.% Most VaR calculations are based on a probability distribution inferred from the data. A distribution allows p+l data to be condensed into a single number, the standard deviation (or volatility). VaR can be computed as a multiple of the standard deviation. 1 Day Prob. Distribution 95% VaR ($11.6mm) Best Result = $58.8 Worst Result = -$32.2 The inferred probability distribution may provide a different estimate of VaR than the price data itself. In this case the VaR estimate changes to $11.6 million..% -6-55 -5-45 -4-35 -3-25 -2-15 -1-5 5 1 15 2 25 3 35 4 45 5 55 6 P+L Bucket
Assume: Short $1bn $/Yen, Long $65mm S&P, and Long 8mm barrels of crude oil 8 Daily P+L of $1. Billion $/Yen Short Position 8 Daily P+L of $65 Million S&P Long Position 8 Daily P+L of 8 Million Barrels of Crude Oil Long Position 6 6 4 3 4 2 2 Millions of Dollars Jan-86-2 -4 Jan-9 Feb-94 Feb-98 `` Mar-2 Mar-6 Mar-1 Millions of Dollars Jan-86-2 -7 Jan-9 Feb-94 Feb-98 `` Mar-2 Mar-6 Mar-1 Millions of Dollars Jan-86-2 -4 Jan-9 Feb-94 Feb-98 `` Mar-2 Mar-6 Mar-1-6 -6-8 -12-8 -1-12 VaR = $12 mm -17 VaR = $13 mm -1-12 VaR = $14 mm 15 Daily P+L of $/Yen, S&P, & Crude Oil Portfolio 1 Millions of Dollars 5-5 -1-15 Jan-86 Jan-88 Jan-9 Feb-92 Feb-94 Feb-96 Feb-98 Feb- Mar-2 What is the VaR of this Portfolio? 12+13+14=39mm? Mar-4 Mar-6 Mar-8 Mar-1 We can estimate VaR for this portfolio just as we did for a single position. Distribution of 25 years of Daily P+Ls ($/Yen, S&P, & Crude Oil Portfolio) 12.% 1.% Daily P+L Frequency 95% VaR ($21.2mm) 1 Day Prob. Distribution Percent of Total Observations 8.% 6.% 4.% 5% of the daily P+L results for this combined portfolio of short $1 billion of $/Yen, long $5 million of the S+P 5 Index, and long $25 million of crude oil are losses in excess of $17 million. 2.%.% -14-128 -117-15 -93-82 -7-58 -47-35 -23-12 12 23 35 47 58 7 82 93 15 117 128 14 P+L Bucket
Overall VaR depends on the correlations of price changes among themselves 12.% Distribution of 25 years of Daily P+Ls ($/Yen, S&P, & Crude Oil Portfolio) Q.What would the overall VAR be if all assets were Gaussian and uncorrelated? Percent of Total Observations 1.% 8.% 6.% 4.% 2.%.% $/Yen VaR = $12 mm S&P VaR = $13 mm Crude Oil VaR = $14 mm Sum of VaRs = $38 mm Portfolio VaR = $21 mm The difference is called the diversification effect. It arises because asset prices do not always move in concert. The degree to which prices move together is called correlation. -14-128 -117-15 -93-82 -7-58 -47-35 -23-12 12 23 35 47 58 7 82 93 15 117 128 14 A. Sqrt(12 2 + 13 2 + 14 2 ) = Sqrt( 59 ) = 22.56 mm P+L Bucket The P&L distribution for this portfolio reflects both the volatility of individual assets and their correlations. For perfectly correlated assets (1) the P&L distribution would look like the sum of the individual risks (blue area) and the VaR would equal the sum of the individual VaRs = ~39mm Old info counts less 15 Daily P+L of $/Yen, S&P, & Crude Oil Portfolio Assign greater weight to recent events 1 5 Millions of Dollars Jan-86 Jan-88 Jan-9 Feb-92 Feb-94 Feb-96 Feb-98 Feb- Mar-2 Mar-4 Mar-6 Mar-8 Mar-1-5 -1-15 P+L 95% VaR (Last = $29.9mm) Avg 95% VaR ($21.2mm)
Market Risk Management Scenario Analysis and Measures Scenario Analysis (Stress Tests) What is it? Why is it useful? Estimate of potential loss Highlights catastrophic risks to from large unexpected or businesses rare events Additional dimension for Stresses are applied to allocation of risk capital positions based either on Loss measure that can be historical analysis or on compared to expected revenues projected stressful In conjunction with position scenarios limits, it promotes hedges aimed at mitigating catastrophic risks Drawbacks Estimates losses without specifying likelihood of occurrence Risk aggregation across multiple products is difficult
Scenario Analysis: Examples Fall 98 Credit Spreads Widening Equity Derivatives Market Crash Analysis Emerging Markets Stress-Test Emerging Markets Country Default Analysis Macro-Economic Scenarios Analysis Assesses the potential P&L impact of a credit spread-widening event to global credit-sensitive products, as observed in the Fall 1998. Estimates potential P&L impact using full portfolio revaluation by incrementally changing market level and volatility, both at the portfolio and underlier levels. Estimates the potential P&L impact of an economic crisis at the country, regional or global level, based on current Emerging Market inventory. Estimates the potential P&L impact of a sovereign default, based on current Emerging Market inventory. Estimates the potential P&L impact of macroeconomic scenarios involving various assumptions of recession, inflation or deflation. Draw stress tests from history? How to combine individual stress tests? 15 Daily P+L of $/Yen, S&P, & Crude Oil Portfolio 1 5 Millions of Dollars -5 Jan-86 Jan-88 Jan-9 Feb-92 Feb-94 Feb-96 Feb-98 Feb- Mar-2 Mar-4 Mar-6 Mar-8 Mar-1-1 -15 Sum of largest individual moves =-$28 million But they did not occur simultaneously The largest 1-day move was -$134 million Worst $/Yen Day -$32 Worst S&P Day -$133 Worst Crude Oil Day -$114