Financial Engineering 6

Lecture 24

Introduction to Risk Management

References: Villalobos

Lecture Topics • Definition of Financial Risk • Definition of Value at Risk (VaR) • Historical Method • Variance-Covariance Method • Monte Carlo Simulation Method • Example of VaR • Back Testing (Portfolio) • Conclusions

Definition • Financial risk management is the process of identification,

analysis and either acceptance or mitigation of uncertainty in investment decision making.

• Risk management is a three step process: – Identifying the risks that exist in an investment – Assessing those risks – Mitigating those risks in a way best suited to your

investment objectives

Value at Risk (VaR) • A technique used to estimate the probability of portfolio losses

based on the statistical analysis of historical price trends and volatilities.

• VaR is commonly used by banks, security firms, and companies that are involved in trading energy and other commodities.

• VaR is able to measure risk while it happens and is an important consideration when firms make trading or hedging decisions.

Value at Risk (VaR) • The most popular and traditional measure of risk is volatility.

• Volatility is static and does not take into account the direction of an investment's movement.

• VaR answers the questions: – What is my worst-case scenario? – How much could I lose in a really bad month? – What is the maximum percentage I can expect to lose over

the next year?

Value at Risk (VaR) • VaR analysis has three elements:

– A relatively high level of confidence, typically either 95% or 99%. – A time period such as a day, a month, or a year. – An estimate of investment loss expressed either in dollars or

percentage terms.

VaR = rc x V

f(r) = distribution of returns V = portfolio value rc = critical return

Returnsrc

Example • Consider a trading portfolio:

– The investment bank holding that portfolio might report that its portfolio has a 1-day VaR of $4 million at the 95% confidence level.

– This implies that under normal trading conditions the bank can be 95% confident that a change in the value of its portfolio would not result in a decrease of more than $4 million during 1 day.

• This is equivalent to saying that there is a 5% confidence level that the value of its portfolio will decrease by $4 million or more during 1 day.

– A 95% confidence interval does not imply a 95% chance of the event happening.

– We are not predicting the probability of the event happening.

• The key point is that the target confidence level, 95% in the above example, is the given parameter here.

– The output from the calculation, $4 million in the above example, is the maximum loss, or the value at risk, at that specified confidence level.

VaR Importance • It is a requirement for any trading organization or a major

user of derivatives such as: – Banks – Insurance companies – Investment funds

• Internal factors include: – A summary of risks for senior management and the

Board of Directors – A tool for setting trading limits and for performance

measurement

• External factors include: – Regulatory agencies – Credit rating agencies – Creditors.

Three Methods for Calculating VaR • Historical Method:

– The historical method simply re-organizes actual historical returns, putting them in order from worst to best.

– It then assumes that history will repeat itself from a risk perspective.

• The Variance-Covariance Method: – This method assumes that stock returns are normally distributed. – It requires that we estimate only two factors, an expected (or average)

return, and a standard deviation. – Allows us to plot a normal distribution curve of the actual return data.

• The Monte Carlo Simulation Method: – Future asset returns are randomly simulated. – This method requires the generation of random scenarios of market

moves using a model of the market. – The value of the portfolio is estimated under the simulated market

scenario. – This method is used for securities with nonlinear returns such as

options.

Historical Method So, what data do we need?

• Current prices, mark to market, of the portfolio.

• Historical data for the required period such as day, month, etc.

• Calculate the change (Rate of Return) in one period for all the securities and stocks in the portfolio.

• Sort the change in value from lowest value to highest and determine VaR based on the desired confidence level.

• Assuming that sometime ago an original portfolio consisting of stocks of YHOO, GOOG, JNJ, KO, UNP, F worth $100,000 was built using the minimum variance portfolio.

– The current value of the portfolio is $145,145.01 • The rates of returns observed in the last 100 days are:

• What are the 95% and 99% VaR?

Historical Method Example

-0.030

-0.020

-0.010

0.000

0.010

0.020

0.030

30-Aug-07 30-Oct-07 30-Dec-07

Daily RoR

0

5

10

15

20

25

Portfolio Return RoR Order 150090.39 3233.40 0.021543 1 137554.16 2774.33 0.020169 2 153019.46 3044.08 0.019893 3

152469.91 -1976.24 -0.012962 94 156293.06 -2062.62 -0.013197 95 154237.11 -2166.74 -0.014048 96 157740.63 -2235.42 -0.014171 97 156434.95 -2313.97 -0.014792 98 150493.67 -2369.36 -0.015744 99 148124.31 -3280.84 -0.022149 100

Historical Method Example 100 Days • Current price of the portfolio is $145,145.01. • Historical data for the required period, such as day or month, are

shown in the Excel file. • Calculate the change (RoR) in one period for all the securities and

stocks in the portfolio; see Excel file. • Sort the change in value from highest to lowest value and determine

VaR based on the desired confidence level. – Note, this is nonparametric.

VAR (95%) = (-0.013197)(145,145.01) = -$1,915.50

VAR (99%) = (-0.015744)(145,145.01) = -$2,285.15

Historical Method Example • Do you have any concerns with the 100 day estimates? • Is it enough to get the last 100 days? • How this might affect your estimation of risk exposure? • If we consider a bigger sample of 863 data points (from August

2004 to January 2008), then we get the following results:

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

19-Aug-04 19-Aug-05 19-Aug-06 19-Aug-07

Daily RoR

0

50

100

150

200

250

300

Historical Method Example 863 Days • Same basic process as before. • Current price of the portfolio is $145,145.01. • Convert the order column from 1– 863 to approximately 0–100 by

dividing by 863. • Is this really better than 100 data ponts?

Portfolio Return RoR Order Order/863 126378.52 3417.50 0.027042 1 0.1 113528.26 2580.01 0.022726 2 0.2 150090.39 3233.40 0.021543 3 0.3

134256.03 -1392.54 -0.010372 818 94.8 139019.18 -1465.02 -0.010538 819 94.9 135981.39 -1433.28 -0.010540 820 95.0 113533.93 -1198.72 -0.010558 821 95.1 129366.39 -1376.58 -0.010641 822 95.2

150493.67 -2369.36 -0.015744 852 98.7 138666.97 -2254.60 -0.016259 853 98.8 107439.72 -1771.91 -0.016492 854 99.0 101002.17 -1698.45 -0.016816 855 99.1

VAR (95%) = (-0.010540)(145,145.01) = -$1,529.87

VAR (99%) = (-0.016492)(145,145.01) = -$2,393.75

Historical Method Pros

• No assumed distributions • Volatility/correlations as they

actually occurred • Outliers included in the price

list • Aggregations across markets

is straightforward • Spreadsheet programmable

Cons • Trends reflected in historical

data can distort risk measurement

• VaR measurements sensitive to outliers in data

• Pricing models required for all instruments

• Difficult to conduct sensitivity analyses and stress tests

Variance-Covariance Method 1. Specify distribution and payoff profiles:

– Risk manager makes assumptions of the shape of distribution (normal).

– With normality assumption, all the information is summarized by the mean, standard deviation and covariance of market factors.

– Assumes that value changes in the transactions are linear.

2. Decompose the transaction in the portfolio into simpler transactions. – The analytic variance-covariance method assumes that the financial

instruments in the portfolio can be decomposed into a set of simple instruments that are exposed to only one market factor.

3. Estimate variance and covariance of the factors.

4. Calculate VaR. – Because we assumed that the individual distributions are normal

and the returns from the factors are linear, then the distribution of the returns of the portfolio is also normal.

Variance-Covariance Method Variance-Covariance method formulas include:

σ µ−

= x

z

2 2 2 2

.05

.01

2 1.65 2.33

P X Y XY X Y

P aX bY

a b ab Z Z

σ σ σ ρ σ σ

= +

= + +

= −

= −

Variance-Covariance Example

• Let’s use the same portfolio that we used to calculate the VaR with the historical method.

YXXYYXP abba σσρσσσ 2 2222 ++=• Using the formula

• We get the following $1, 031.52Pσ =

• The VaR(95%)

• The VaR(99%)

92.83 1.65 1, 031.52 $1, 603.86− × = −

92.83 2.33 1, 031.52 $2, 306.84− × = −

Results Summary

Method 95% 99% Historical -$ 1,529.87 -$ 2,393.75 Variance-Covariance -$ 1,603.86 -$ 2,306.84

Variance-Covariance Method Pros

• No price models are required • Data sets are readily available • Off-the-shelf software is

available

Cons • Decomposing cash flows can

be complex • Difficult to conduct sensitivity

analysis • Does not provide a reliable

risk measure for options • We assume univariate normal

distributions when we really have multivariate distributions.

Monte Carlo Simulation Method 1. Specify a probability distribution for each market factor:

– The distributions can be estimated based on the historical information.

2. Obtain estimates of the parameters for each of the distributions specified in step 1, and the correlations between pairs of markets. – These parameters might be obtained from the historical data or

selected by the managers for testing and sensitivity analysis.

3. Use Monte Carlo simulation techniques to randomly sample from the distributions identified in steps 1 and 2. – With adequate software, multivariate, and if needed, distributions

other than normal.

4. Obtain the VaR in the same manner as the historical method. – Instead of using observed historical data, with the Monte Carlo

simulation method we simulate them!

Monte Carlo Simulation Method • Use multivariate normal to simulate 1,000 data points for each

one of the components of the portfolio. • Use the vector of means and the covariance matrix to sample

the values using Matlab.

mu = [-0.00005017 0.00225163 0.00016253 0.00039124 0.00101418 -0.00072043]; sigma= [0.000489 0.000191 0.000023 0.000054 0.000064 0.000101;

0.000191 0.000460 0.000018 0.000036 0.000053 0.000055; 0.000023 0.000018 0.000066 0.000024 0.000024 0.000025; 0.000054 0.000036 0.000024 0.000069 0.000038 0.000047; 0.000064 0.000053 0.000024 0.000038 0.000229 0.000100; 0.000101 0.000055 0.000025 0.000047 0.000100 0.000423];

M = mvnrnd(mu,sigma,1000);

• The VaR at the 95% confidence level is the 50th worst value of the combined portfolio for a loss of $1,537.

Monte Carlo Simulation Method Pros

• Specifying distributions builds in more theory

• Aggregation across markets is simple and consistent

• Ability to do sensitivity analysis and stress test

• Must specify a distribution for each market factor

Cons • Pricing models are required

for all instruments • Outliers are not included in

price lists • Simulated prices are one level

removed from the “real-world” prices

• Need to validate results • More complex programming is

required

Comparing Methods • For transactions with reduced nonlinearities and one-day VaR

portfolios, the variance-covariance may be the best choice.

• If transactions are nonlinear, and the time horizon is longer, then the Monte Carlo simulation may be more appropriate.

• If sensitivity analysis and stress tests are required, then the Monte Carlo method is the most attractive.

VaR Related Concepts • Stress Testing: Moving beyond the ad hoc scenario analysis,

we will want to stress-test the VaR number by changing the parameters of the model.

– We would like to see what happens if values move beyond confidence intervals.

• Back Testing: Test how the VaR performed relative to realized losses.

– By comparing actual losses to the extreme losses generated by the VaR, we can get some insight whether the VaR model is appropriate.

• Delta: The change in the value of the asset associated with changes in the underlying financial price; ΔV/ΔP.

• Gamma: The change in the delta associated with changes in the underlying financial price; Δdelta/ΔP.

• Vega: The change in the value of the asset associated with changes in the volatility of the underlying financial price; ΔV/ΔσP.

Back Testing the Portfolio • Using the portfolio with the minimum variance, as our portfolio

we estimate the daily loss in the previous months to determine if the results are robust

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97

Daily RoR

5% VAR

99% VAR

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97

last 100 days

95 VAR

99% VAR

863 Days100 Days

• As we can see for the current environment, the 95% and 99% for the estimate of 100 days behaves better than the 863 days

Another VaR Example • Inputs:

– Positions for each commodity (gamma – delta approach). – Forward price curve for each commodity (provided by traders). – Volatility curve for each commodity to be used in simulations. – Correlations across commodities as well as across forward

contracts of different maturity. • Outputs:

– Distribution of returns for each portfolio in the portfolio hierarchy.

Company

Gas Power Liquids

Taylor Series Expansion

( ) ( ) ( ) ( ) ( ) 1! 2! 3!

f x a f x a f x a f x f a

′ ′′ ′′′− − − = + + + +

VaR Example • Portfolio holdings (positions) are represented through Deltas ∆i

and Gammas Γi, i=1,2, … N.

• Obtain a new price, Fi, sim, from Monte Carlo simulations.

• Change in position value for the contract i is:

• A number of simulations of the Portfolio Value Change are calculated.

• The 5th percentile is used to obtain VaR.

( ) ( )21, , 2i i i i

F sim F F sim F∆× − + Γ× −

VaR Example • Calibrate the model: Choose appropriate price processes,

parameter estimation, and volatility specification.

• Capture patterns for energy prices: Seasonality, gaps, and behaviors.

• Capture extreme events such as the tails of distributions and non-normal returns.

• Move from “risk measurement” to “risk management” by making the risk modeling system an active tool in the portfolio management process.

Conclusions • VaR is related to the volatility of the positions in an investment

portfolio.

• However VaR only measures the downside effects of the variability on those positions.

• VaR can be used as an estimate of risk, and can be used to determine the amount of losses that can be incurred in a defined period of time.

• Banks use VaR or similar methods to determine the cash reserves needed for covering their accounts.

• VaR is very dependent on the historical information used. – For better results it is recommended to use stress testing

and scenario analysis.

Assignments • Review the VaR processes used by J.P. Morgan in the Risk

Management Technical Document.

• Briefly read The J.P. Morgan Guide to Credit Derivatives.