Bloomberg VaR Methodology

Value at Risk Methodology April 2007

Table of Contents Preface ...................................................................................... 1 A Note about Risk Factors.............................................................. 1 What is a Scenario?...................................................................... 1 Value at Risk (VaR)...................................................................... 2 Time Horizons............................................................................ 2 Monte Carlo Scenarios .................................................................. 3 The Variance Covariance Matrix ....................................................... 3 Monte Carlo Scenario Generation ...................................................... 4 Geometric Brownian Motion Model ................................................... 4 Standard Random Sampling............................................................ 5 Summary of Parameter Settings ...................................................... 6 Historical Scenario Generation........................................................ 6 Parametric VaR ............................................................................ 7 Examples of Parametric VaR Computation .......................................... 9 Interest Rate Risk ......................................................................... 9 Equity Risk ................................................................................ 10 FX VaR ..................................................................................... 11 Comparison of Risk Methodologies .................................................. 12

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Preface The Value at Risk Methodology document outlines the various “Risk” methodologies that are incorporated into the Bloomberg Value at Risk (VAR) application.

A N o t e a b o u t R i s k Fa c t o r s A large Part of this document is dedicated to scenarios and scenario generation. Algorithmics uses a “Full Valuation Methodology” to Theoretically Value every security we process. This means that security is broken down into its building blocks of risk, or Risk Factors. There are three broad Risk Factor classifications:

• • •

Interest Rates Base Bank and Treasury Rates, Credit Spreads, Sector Spreads. Market Indexes Major equity indexes, Multi-Factor Model Indexes. Exchange Rates Most major global FX rates.

The specific risk factors are tracked daily and the CLOSING price or rate (for that particular day) history kept in a time series database. The database is then used to generate scenarios. The scenarios move the risk factors, which change the Theoretical Valuation and thus create a Theoretical Profit or Loss. The time series database contains data for three years of history. This is a rolling period, in that there are always three years of price/rate history in the time series database.

Wh a t i s a S c e n a r i o ? A scenario is a factor that is applied to the current risk factor level resulting in a change in the risk factor value. Subsequently, the security “Theoretical Value” will be affected. Scenarios can be described as a series of factors applied to current risk factor levels. Changing the risk factor levels results in a new Value for a particular security and thus a “Theoretical Profit or Loss” (Theoretical Value Scenario 1 – Current Value = Theoretical Profit or Loss). The Theoretical Profit/Loss for an individual security is then sorted for VaR, or simply reported for sensitivity or stress test purposes.


2

V a l u e a t Ri s k ( V a R) There are three different VaR measures within the Algorithmics for Everyone (A4E) suite:

• • •

Monte Carlo Historical Parametric.

The first two VaR methodologies are identical, other than they use different scenario sets. Both methodologies are Non-Parametric, which means that the profits/losses for each scenario (and each time horizon) are sorted from largest loss to largest gain. Based on the confidence interval, a particular scenario profit/loss (in most cases a loss) is reported as the VaR. For example, a 99% confidence interval with 1000 scenarios will select the 990 th worst scenario and report it as the VaR. Parametric VaR uses an approximation methodology based on the volatilities and correlations of the risk factors to compute a map of the instrument’s value to a particular risk factor associated with it. A simple equation then aggregates the VaR map into a VaR measure. This will be outlined further later in the document.

T i m e H o r i zo n s Value at Risk must be associated with a time horizon. A series of profits or losses does not occur instantaneously, and typically for risk measurement is measured over a minimum of one day. Depending on the selected scenario set, there is a one-day horizon available for Historical Simulation and an additional two- week horizon for Monte Carlo. It is worth noting that the A4E product suite revalues securities across time accurately, meaning forward interest and FX rates are computed, and growth rates applied to market indexes. In addition, using full valuation will capture cash flows from such events as coupon payments and incorporate them in the valuation at that time point and affect the risk accordingly. A time horizon for either the Historical or Monte Carlo scenarios impacts the span over which historical returns for Risk Factors are computed. Therefore, longer time horizons would be expected to be more volatile than shorter time horizons. Consider each scenario to be a factor applied to current risk factor levels. Different scenario methodologies result in different factors that are applied, but each scenario generation methodology also has it own way of adjusting the factor for a time horizon. Where Monte Carlo uses a simulation model that is described below, Historical scenarios on the other hand, simply aggregate daily For 24 hour global customer support call +1 212 318 2000, +44 20 7330 7500, +65 6212 1000, +81 3 3201 8900 or press the HELP key twice on the BLOOMBERG PROFESSIONAL ® service.

3 observed returns into a return over a time period. Parametric approaches simply scale a daily VaR number (since the historical returns are daily) into a longer horizon. Algorithmics uses full forward evolution for the security valuation in the future. This implies several key points about looking at future time horizons:

•

Forward Curve/FX rates used. This means that at any forward time horizon all rates are evolved, encompassing “rolling down the yield curve” (interest rate curves) and “interest rate parity” (FX rates) properties.

•

All security cash payments incorporated as fixed income securities will move towards par.

M o n t e Ca r l o S c e n a r i o s There are two stages to Monte Carlo scenario generation: 1. Compute risk factor volatilities and correlations (variance/covariance matrix). 2. Use a variance-covariance (VcV) matrix to generate the actual scenarios, or factor shifts of current risk factor levels. Currently in A4E, 1000 Monte Carlo scenarios are used across two time steps: one day and two weeks.

T h e V a r i a n c e Co v a r i a n c e M a t r i x A variance-covariance (VcV) matrix is a set of numbers that describes the relationships within a set of risk factors. Fundamentally, risk factors are treated as random variables. The time series data for the risk factors and one of three statistical methods are used to create the covariance matrix. A covariance matrix has one row and one column for each risk factor. The offdiagonal elements represent the covariances between risk factors while the diagonal elements represent the variance of individual risk factors. A covariance or correlation is a measure of the relation between the values of two risk factors. A variance or volatility is a measure of the variability of the value of a risk factor. This standardization ensures compatibility with scenario generation and parametric VaR computation. Algorithmics uses a RiskMetrics model to compute the VcV matrix. The RiskMetrics model is based on the recommendations published by the RiskMetrics Group and uses an exponentially weighted moving average model.


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The parameters used to compute the VcV Matrix include:

•

decay factor = 0.94 (determines how quickly the weighting of past return data declines)

•

k value = 74 (or the number of business days in the past that are used for estimate). The decay factor and k value used in the estimation depend on the length of time into the future to be forecast and the desired accuracy of the predictions. Both the decay factor and the k value serve similar purposes, to be used to control sensitivity to more recent events.

M o n t e Ca r l o Sc e n a r i o Ge n e r a t i o n Monte Carlo simulation allows you to find solutions to problems that require a probability analysis. Monte Carlo methods solve highly complex financial problems, and are frequently used in pricing derivatives, or estimating a portfolio’s VaR. The process to generate Monte Carlo Scenarios typically involves three steps:

• • •

Determine the risk factors Assign a simulation model for the risk factors Produce draws (sample) from the model (representing the actual scenarios).

Ge o m e t r i c B r o w n i a n M o t i o n M o d el The simulation model is a Geometric Brownian Motion model that uses the VcV matrix outlines above. This model is intended to outline how the risk factors will move across time. Also known as the Black Scholes Model with Zero Drift, this can be pictured as a random walk in continuous time. The movements in prices from one day to another are represented as a series of returns. The distribution of these continuously compounded returns at the end of any finite time interval is a Log Normal distribution. Another feature of this model is that the dispersion of the scenarios increase without bound as simulation time increases. Graphically, this is shown in Figure 1.


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Figure 1: Sample Paths of a Brownian Motion

Discrete approximation to Brownian motion is used to produce scenarios. This approximation is based on an important property of Brownian Motion: the movement of the risk factor between two points in time is normally distributed with variance proportional to the square root of the elapsed time. Beginning at time zero, a jump to time 9 would have a variance of 3(sigma)^2, where sigma is the one-step variance.

St a n d a r d Ra n d o m Sa m p l i n g Standard Random Sampling is the simplest sampling technique and can be used with Monte Carlo scenario generation. With Monte Carlo, the Standard Random Sampling method is the classic way to use a pseudorandom number generator to generate a sequence of random variables. Standard Random Sampling draws values randomly from a uniform distribution until a desired number of scenarios with a desired number of trigger times have been created. A transformation is then applied to convert the uniform random variates into random variates that follow a normal distribution.


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Su m m a r y o f Pa r a m e t e r Se t t i n g s Parameter settings are summarized in Table 1. Table 1: Summary of Parameter Settings

Parameter

Setting

Comment

Number of Scenarios

1000

A standard number of scenarios for Market Risk simulation.

Delta Type

Ratio

The shift is defined by the ratio of one value versus the other.

Covariance Matrix

Internally generated

All historical risk factor data used to generate a VcV matrix.

Multi Step Trigger Time Definition

One day, one week, two weeks, one month, one quarter.

Trigger times correspond to the various investment horizons desired.

Simulation Model

Geometric Brownian Motion (see note).

Model of evolution of risk factors. Detailed explanation beyond scope of this document.

Simulation Schema

Standard

Sampling technique, with no limits on dimensionality, no prerequisite for Principal Component Analysis, and is based on “pseudo-random” numbers.

Decay Factor

0.94

Latest Risk Factor Data weighted heavier than less recent dates.

Risk Factor Dates

Note: Simulation models of this type are also known as stochastic processes because they specify the (marginal) evolution of a risk factor through time. The equation of the process implies a distribution for risk factor value at any particular point in time.

H i s t o r i c a l Sc e n a r i o G e n e r a t i o n This scenario uses a replication sampling method to draw observations directly from historical market data. This data consists of one full year of values for all the risk factors. Actual variable shifts and “shocks” are applied to the historical data and a relative one day change is calculated for each risk factor. The investment horizon for the scenario is set to one day (making it a one- day holding period). The return is calculated by comparing the current day values to previous day values found in the historical sample and applying changes one day at a time to For 24 hour global customer support call +1 212 318 2000, +44 20 7330 7500, +65 6212 1000, +81 3 3201 8900 or press the HELP key twice on the BLOOMBERG PROFESSIONAL ® service.

7 re-value the portfolio holdings. This allows a one- day VaR to be calculated (e.g., picking the worst 5% of days for 95% VaR). Historical scenarios are drawn directly from historical market data. When using historical scenarios as the basis of statistical analyses, the distribution will be similar to the historical distribution. That is, the future will resemble the past. When compared to Monte Carlo scenarios, historical scenarios are based on less restrictive distributional assumptions and are often easier to explain. In fact, the typical process for generating historical scenarios is simple: 1. Select the risk factors. 2. Track these risk factors over time. 3. Determine a time window that serves as the basis for the scenarios. 4. Calculate returns for the selected risk factors over this time window. 5. Use the returns to create scenarios.

Pa r a m e t r i c V a R The third type of VaR analysis available within A4E is the Parametric VaR. This type of VaR is not scenario based, rather an approximation made from volatilities and correlations. The Parametric VaR measure is a statistical risk measure that calculates the market risk of a portfolio for a specific confidence level. The VaR gives an indication of a loss with a given probability. The analytic engine uses a tweaking method to calculate a risk factor (VaR) map from the risk factors specified in the covariance data files. The tweak is used to approximate a partial derivative of the instrument, with respect to risk factor node. The tweak controls the amount the risk factor is shifted when instrument sensitivity is measured. The Parametric methodology then uses the VaR map to calculate VaR. This methodology involves analytical measures and is independent of simulation. For each security a “VaR Map” is computed, using the Theoretical Value and the data in the VcV files for the risk factors applicable to that particular security. RiskWatch (the A4E simulation engine) computes the map values by calculating the delta of an instrument for each risk factor. A risk factor can be an exchange rate, an equity index, or a bucket on a zero or swap curve in a specific currency, which is used to shock the RiskWatch curve.


8 RiskWatch performs VaR mapping in two main steps: 1. Determine all the risk factors that could affect the value of an instrument. 2. Compute the map value for each factor found in Step 1. Conceptually, each map value is calculated in the following manner: Map Value = (Change in Present Value)/(Relative Change in Risk Factor) This change is a result of the tweaking or moving the relevant curves up one basis point (0.01 movement for equity indexes). The general notion of VaR is simply a percentile of the (probabilistic) distribution of financial losses. More precisely, let the value of a financial instrument, or portfolio of such, be denoted by V , and the difference of today’s value from the (random) value at the end of a fixed time horizon be denoted by ∆V . Then VaR, at the two-sided confidence level α , is defined implicitly through the relation: P (− ∆V ≤ VaR ) =

1 + α 2

⎛ 1 + α ⎞ ⎟ -percentile of the losses, − ∆V . Note that a negative ⎝ 2 ⎠

so that VaR is the ⎜

loss is a positive gain. In general, exact analytical valuation of VaR is usually intractable. Parametric approaches involve a first-order numerical approximation to ∆V and a parametric hypothesis on the joint distribution of the risk factors underlying V . Specifically, the Parametric model assumes that: 1. V = f ( P1 , P2 ,...., Pn ) where the risk factors, P1 , usually represent prices. 2. ∆V is approximated by the differential, dV = df .

3. The joint distribution of the returns, dPi / Pi , is a multivariate normal, usually with mean zero. (An exception is Equity VaR, in which there is a growth effect which implies a non-zero mean.) Assuming for simplicity that the multivariate mean in the third model assumption is zero, it follows that dV is a mean-zero normal random variable with variance of 2 σ dV ; and the approximate equation:


9 P (− dV ≤ VaR) ≈

1 + α 2

implies that: VaR ≈ z[(1+α ) / 2] • σ dV • dt

Thus the approximate evaluation reduces to the calculation of the variance in terms of the variances and covariances of the returns dPi / Pi of the risk

σ

2

dV

factors Pi . These latter statistical parameters are typically read by RiskWatch from data files supplied by the Algorithmics Scenario Engine. In RiskWatch, the partial derivatives,

∂ f ∆ f , are evaluated numerically as , a process that was ∂Pi ∆Pi

referred to above as tweaking . The resulting numerical gradient vector is called the VaR map. In practice, the approximation works reasonably well for linear financial instruments when the volatilities of the risk factors are of low to moderate size. For nonlinear instruments such as options, the approximation can be very poor.

Ex a m p l e s o f Pa r a m e t r i c V a R Co m p u t a t i o n There are different ways in which Interest Rate, Equity, or FX VaR is computed parametrically. For the purpose of illustration in this document, an outline of how Parametric Interest Rate VaR is computed will be described below.

I n t e r e s t R a t e Ri s k The Parametric VaR Calculation uses three inputs for Interest Rates: 1. The VaR Map (as described in Parametric VaR). M i =

( pv1 − pv0) ( df 1 − df 0)

• df 0

Where “pv0” = Security Present Value “untweaked”, and 1 is “tweaked” “df0” = Discount Factor at that particular time point


10 2. Yield and Price Volatility, from the Variance Covariance Matrix.

⎛ ⎛ dy ⎞ 2 ⎞ Yield Volatility = σ 2 yi • dt = E ⎜ ⎜⎜ i ⎟⎟ ⎟ ⎜ ⎝ y i ⎠ ⎟ ⎝ ⎠ Price Volatility = σ pi = ( D A )i • yi • σ yi 3. Risk Factor correlation, from the Variance Covariance Matrix. Let the two risk factors in question be “i” and “j”:

ρ ij

⎛ dy dy j ⎞ ⎟ / (σ • σ yj ) = E ⎜ i • ⎜ y i y j ⎟ yi ⎝ ⎠

With these inputs we can represent the Parametric VaR, for an specific unwind period: (

η =

SDF • UnwindPeriod , where SDF=Standard Deviation Factor = Z [(1+α ) / 2 ]

and α represents the confidence interval) the Parametric VaR can be represented as:

VaR = η •

n

n

∑∑ M σ i

pi ρ ij

M j σ pj

i =1 j =1

Eq u i t y Ri s k With typical equity models, the change in the Equity Value, V, is represented through a change in a Market Index Price, P m. In this case the equity VaR map is represented by: M =

∂V Pm ∂Pm


11 Subsequently, the VaR is computed: VaR = η • M • σ m

In the case of a Multi-Factor equity model, the VaR Map has multiple sensitivities to multiple market indexes. The VaR calculation is then changed by adding a term to represent the correlation of the various indexes, similar to the equation above for interest rates. Note that there is no “specific” risk included in the parametric equity VaR Calculation.

FX VaR The main difference between the FX VaR and Interest Rate VaR is in the VaR Map procedure. To generate the VaR map for FX sensitive securities, the FX rates are tweaked: M i =

∂ f I pv1 = pv 0 • spot 0 r = I i spot 1 − spot 0 ∂r i

Where spot0 is the current exchange rate from the foreign currency to the currency of the security, that is spot0= r i I Once the VaR map is created, the VaR looks very similar to the Interest Rate VaR:

VaR = η •

n

n

∑∑ M σ i

pi ρ ij

M j σ pj

i =1 j =1


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Co m p a r i s o n o f R i s k M e t h o d o l o g i e s A comparison of risk methodologies is shown in Table 2. Table 2: Comparison of Risk Methodologies

Topic

Monte Carlo

Historical

Parametric

Time Series

Estimation period = 74 days (in the VcV matrix).

One year of scenarios.

Estimation period = 74 days (in the VcV matrix).

Normality Assumption

Lognormal Distribution.

None

Time Series Weights

Exponentially weighed, meaning recent results have larger impact on VcV matrix.

Equally weighted returns determine the scenarios.

Exponentially weighed, meaning recent results have larger impact on VcV matrix.

Time Horizon Methodology

Geometric Brownian Motion outlines risk factor movement through time.

Observed daily returns sampled and concatenated to form time horizon factor.

Daily results scaled by square root of the unwind period.

Security Forward Evolution across time horizon.

Forward Curves, FX rates used, all cash payments incorporated.

Forward Curves, FX rates used, all cash payments incorporated.

Scaling is does not encompass forward curve evolution or cash payments.

Equity Specific Risk Simulated?

Yes, residual volatility from multi factor model used as basis for specific risk (normal assumption of specific risk).

Yes, residual volatility from multi factor model used as basis for specific risk (normal assumption of specific risk).

No


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