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PORTFOLIO & RISK ANALYTICS
A Bloomberg Professional Service Offering
PORTFOLIO
VALUE AT RISK
Author(s): David Frank
Date: January 2019
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PORTFOLIO & RISK ANALYTICS
A Bloomberg Professional Service Offering
Contents
Chapter 1: Introduction .......................................................................................... 3
Chapter 2: Bloomberg Fundamental Factor Models ............................................. 4
Structure of Factor Models ................................................................................ 4
Estimating Portfolio Volatility ............................................................................. 5
Chapter 3: VaR Methodologies............................................................................. 6
Definition of VaR ............................................................................................... 6
VaR Methodologies in PORT ............................................................................ 6
Chapter 4: Parametric VaR .................................................................................. 7
Non-Linear Instruments in Parametric VaR ....................................................... 7
Chapter 5: Scenario-Based VaR - Monte-Carlo and Historical ............................. 8
Monte-Carlo VaR .............................................................................................. 8
Historical VaR ................................................................................................... 9
Stress Matrix Pricing ......................................................................................... 9
Computing Portfolio VaR ................................................................................... 9
Chapter 6: Comparison of VaR Methodologies .................................................. 11
Chapter 7: VaR Attribution .................................................................................. 12
Partial VaR ...................................................................................................... 12
Component VaR and Marginal VaR ................................................................ 12
Chapter 8: Conditional VaR (Expected Shortfall)................................................ 13
References ......................................................................................................... 15
Contact Us .......................................................................................................... 15
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PORTFOLIO & RISK ANALYTICS
A Bloomberg Professional Service Offering
Chapter 1: Introduction
Managing portfolio risk is an integral component of any disciplined investment process.
Volatility, given by the standard deviation of portfolio returns, is perhaps the most widely
used risk measure. Roughly speaking, volatility describes the range of “typical” portfolio
outcomes. Tail risk, by contrast, focuses on the frequency and severity of extreme
outcomes.
Bloomberg’s Portfolio and Risk Analytics solution (PORT), available via the Bloomberg
Professional® service, offers a comprehensive suite of advanced tools for portfolio
management. These include sophisticated analytics for performance attribution, risk
forecasting, risk attribution, scenario analysis, and portfolio construction.
This document describes the methodology for portfolio Value at Risk (VaR) computation
used by Bloomberg Portfolio and Risk Analytics. Three types of VaR are provided:

Parametric VaR

Historical VaR

Monte Carlo VaR
In this document, we describe the different components of the Bloomberg VaR
methodology, including the Bloomberg factor models, security valuation methods, and
details of Parametric, Historical and Monte Carlo VaR. We also discuss attribution of VaR
to individual securities in the portfolio. Finally, we discuss a complementary measure of
tail risk known as Conditional VaR, or Expected Shortfall.
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PORTFOLIO & RISK ANALYTICS
A Bloomberg Professional Service Offering
Chapter 2: Bloomberg Fundamental Factor Models
Accurately estimating portfolio volatility is the key first step towards computing reliable
VaR estimates. Bloomberg uses linear factor models to estimate portfolio volatility. Factor
models have become an indispensable tool for modern portfolio management. In addition
to forecasting portfolio volatility, factor models are widely used for performance attribution,
risk attribution, and portfolio construction.
Structure of Factor Models
Factor models are based on the premise that security returns are driven by a parsimonious
set of common factors. Therefore, portfolio risk depends on the volatility and correlation
of these factors and on the portfolio exposure to these factors.
Factors represent the systematic drivers of asset returns, but do not explain the totality of
asset returns. The component of return not explained by the factors is known as the
specific return of the asset, or the non-factor return. This return component is considered
idiosyncratic and unique to the particular asset. The volatility of the specific returns is
known as the specific risk of the asset.
Bloomberg’s approach to constructing risk factor models uses a combination of explicit
and implicit factors. Implicit factor models “observe” factor exposures and estimate factor
returns by cross-sectional regression. Examples of implicit factors include industry,
country, and style factors in the Bloomberg equity factor models.
By contrast, in an explicit factor model, the exposures are computed analytically (or
estimated) and the factor returns are directly observed. For instance, changes in the yield
curve along certain points (known as key rates) are used as explicit factors in fixed income
factor models. In this case, factor exposures (i.e., key-rate durations) are computed
analytically.
The single-period return of security n is modeled by Bloomberg factor models as
K
rn   X nk f k   n ,
(2.1)
k 1
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PORTFOLIO & RISK ANALYTICS
A Bloomberg Professional Service Offering
where X nk is the exposure of security n to factor k , f k is the return of the factor, K is
the total number of factors, and  n is the non-factor return of the security.
The model assumes that the specific returns of different assets are uncorrelated1 and that
the specific returns are uncorrelated with the factor returns. The number of factors and
their definition depend on the particular factor model under consideration. Bloomberg
offers a wide range of factor models categorized by asset class and region. Please see
the model white papers for an in-depth description of individual factor models.
The factor model shown in Equation (2.1) for security returns may be written in matrix
notation as
r  Xf  ε ,
(2.2)
where r is an N  1 vector of security excess returns, X is the N  K factor exposure
matrix (known at start of period), f is a K 1 vector of factor returns over the period, and
ε is an N 1 vector of specific returns over the period.
Estimating Portfolio Volatility
The factor returns are used to estimate the factor covariance matrix F , whose entry in
row i , column j , is the covariance of returns on factors i and j . See Menchero and Ji
(2016) for details of the covariance matrix construction methodology.
We must also consider the N  N covariance matrix Δ of the specific returns ε . As
discussed previously, all off-diagonal elements are zero, except in the case of linked
securities.
Given a K 1 vector x P of portfolio factor exposures, and an N  1 vector w P of portfolio
weights, the predicted variance of the portfolio is given by
 P 2  xPFxP wPΔw P .
(2.3)
Exceptions occur for “linked” securities, such as ADRs or different share classes of the same
company.
1
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One shortcoming of portfolio volatility is that it is a symmetric risk measure. In other words,
it does not distinguish between positive outcomes (upside potential) and negative
outcomes (i.e., downside risk). Tail-risk measures are designed to address this issue.
Chapter 3: VaR Methodologies
This section provides the definition of VaR, followed by a description of the three
approaches used to compute VaR.
Definition of VaR
The VaR of a portfolio, with a confidence level  over a time horizon T , is defined as the
smallest portfolio loss in the worst 1   of outcomes over the specified horizon. In other
words, it represents “the best outcome in a bad period.” For instance, a 95% one-day VaR
of $1M means that with probability 0.95, losses on the portfolio over the next day will not
exceed $1M.
To compute VaR, we must make some assumptions about the distribution of portfolio
returns. Different distributional assumptions lead naturally to different VaR methodologies.
VaR Methodologies in PORT
As noted above, there are three distinct methodologies used in PORT for computing VaR.
The oldest and simplest method is Parametric VaR, which assumes that portfolio returns
are normally distributed. In this case, the VaR is simply a multiple of the portfolio volatility.
Hence, parametric VaR provides no additional information about portfolio risk beyond that
which can be obtained from the volatility measure.
The other two VaR methods involve creating scenarios and evaluating the portfolio returns
under the various scenarios. For Monte-Carlo VaR, we create many scenarios by
simulating factor returns under an assumed distribution. For Historical VaR, we chose a
historical period (e.g., the most recent 1 year, 2 years, or 3 years) and for each day within
the period, the full set of factor returns realized on that day becomes a scenario. In both
of these scenario-based methods, we evaluate the portfolio under the scenarios and look
at the level demarcating the worst 1   percent of the scenarios.
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PORTFOLIO & RISK ANALYTICS
A Bloomberg Professional Service Offering
In the next two sections, we illustrate in greater detail at how VaR is computed within
PORT under each of the three methodologies. In each case, we consider one-day VaR
with confidence level  . PORT computes VaR for horizons longer than one day by
multiplying one-day VaR by the square root of the number of days in the horizon period.
Note that the “square root of time” scaling assumption may not be valid for some types of
portfolios with embedded options.
Chapter 4: Parametric VaR
Parametric VaR assumes that portfolio returns are normally distributed. We compute
portfolio variance using Equation (2.3). The portfolio VaR is then found by taking the lower
 tail of a normal distribution with the appropriate variance.
There is a simple relationship between Tracking Error (volatility) and Parametric VaR.
Tracking Error is defined as the standard deviation of portfolio returns over the specified
horizon. Since the probability of being more than one standard deviation below the mean
is 0.1587, Tracking Error is conceptually equivalent to Parametric VaR at an 84.13%
confidence level. In practice, however, Tracking Error and 84.13% Parametric VaR will
differ in PORT because Tracking Error uses volatilities derived from weekly returns
whereas Parametric VaR uses volatilities derived from daily returns.
The Parametric VaR method offers some advantages, particularly the simplicity and
uniformity of its approach. However, these advantages come at a cost. In reality, we know
that portfolio returns tend to exhibit fat tails. Therefore, extreme moves occur with a higher
frequency than that predicted by the normal distribution. Moreover, Parametric VaR
assumes that all security returns are linear functions of factor and specific returns. This
may be a poor assumption for derivatives, or even for securities with embedded
derivatives, such as callable bonds.
Non-Linear Instruments in Parametric VaR
Non-linear instruments (e.g., options, callable bonds, and mortgages) are covered in
Parametric VaR using the Delta-Gamma-Vega approximation. Delta and Gamma
represent, respectively, the first and second partial derivatives of option price with respect
to the underlying price. Vega is the first partial derivative of option price with respect to
volatility.
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Options will have exposures to the same factors as the underlying security of the option,
but multiplied by the option's Delta. This reduces the exposure to the factors so that, for
instance, a deep out-of-the-money option has much lower exposure to the stock's factors
than would an equal size position in the stock itself. Similarly, the option has exposure to
a Market-Return-Squared factor proportional to the option's Gamma; and exposure to a
Volatility factor proportional to the option's Vega. This will not in general capture the
option's full risk profile due to omission of higher-order partial derivatives. For this reason,
Parametric VaR should be used with caution for portfolios with derivatives or other highly
non-linear instruments.
Chapter 5: Scenario-Based VaR - Monte-Carlo and Historical
This section describes how scenarios are generated using the Monte-Carlo and Historical
VaR methods. The distribution of portfolio losses under the scenarios is used to determine
the VaR level.
Monte-Carlo VaR
Monte-Carlo VaR is computed in PORT by simulating 10,000 scenarios of factor returns
and specific returns drawn from the appropriate distribution, as described below. The
returns of each security are computed under every scenario to determine the portfolio
return within each scenario. The resulting distribution of portfolio returns is then used to
compute the portfolio VaR.
Simulating factor returns requires an estimate of the factor covariance matrix. For
purposes of computing Monte-Carlo VaR, PORT estimates the factor covariance matrix
using daily observations, allowing users to select among three sets of half-life parameters
(11d, 23d, and 26w). To ensure that the factor covariance matrix is full rank, the underlying
correlation matrix is shrunk by 0.01 toward the identity.
To specify the multivariate distribution required for creating these scenarios, we separate
the modeling of the distribution of factor returns (i.e., the marginal distribution) from that
of the co-movement of factor returns. We use fat-tailed t-distributions with six degrees of
freedom (DOF) to model the distribution of individual factor returns. We use a fat-tailed tcopula (with six DOF) to model the co-movement of factors. Finally, specific returns are
simulated using a t-distribution, also with six DOF.
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Historical VaR
The scenarios in historical VaR are built from the realized daily factor returns over the
historical window. Note, however, that idiosyncratic returns are not taken from the
historical specific returns of the assets, since some the assets may not have even existed
historically (e.g., IPO’s). Instead, specific returns are simulated using a t-distribution with
six DOF. The standard deviation of the t-distribution is selected to match the specific-risk
forecast of the asset.
Stress Matrix Pricing
Once a scenario is created, we know the return of each factor under the scenario, as well
as the specific return of each asset. The standard evaluation for most securities is to sum
up the product of the securities factor exposure times the factor returns. We then add the
specific return to this sum to get the security return under the scenario, as in Equation
(2.1). However, securities such as callable bonds and derivatives tend to have returns that
are highly non-linear in the factors. For such securities, this standard evaluation may not
work well. Instead, we adopt the method of Stress Matrix Pricing (SMP) for valuation of
these securities, as described by Frank (2015). SMP is used for options and bonds with
embedded options.
Here, we illustrate the idea behind SMP with a simple example: a call option on a stock.
The best approach for pricing the call option under a scenario is to run a derivatives pricing
model (e.g., Black Scholes) for each scenario, where the factor returns determine the
stock price level, interest rate levels, and levels of other model inputs. This approach
(known as full valuation) is computationally expensive. Instead, we price the call option on
a parsimonious grid in the main return drivers of the call option (stock price and volatility),
and interpolate the grid values for pricing under the individual scenarios. For most
derivatives, SMP is highly accurate for purposes of computing VaR. That is, the error in
SMP pricing versus full valuation is generally negligible within a VaR context.
Computing Portfolio VaR
Having computed the return of all securities in the portfolio under all scenarios, we now
have a distribution of portfolio returns, and are ready to compute VaR. One approach to
computing VaR is to select the “borderline” scenario’s loss level (e.g., the 500th worst
scenario of 10,000 for 95% VaR). While in principle this provides a good estimate of VaR,
it does not produce stable estimates for VaR attribution. The reason is that individual
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assets may have very different returns in adjacent scenarios. What is required for VaR
attribution is to average over a number of scenarios centered about the VaR estimate.
In PORT, we use a Gaussian kernel to smooth results of several scenarios surrounding
the VaR estimate. Let
L j denote the portfolio loss under scenario j . Let V denote the
portfolio VaR, computed as a weighted average of portfolio losses in scenarios
surrounding the VaR estimate,
V   v j Lj .
(5.1)
j
where
v j denotes the weight assigned to scenario j by the Gaussian kernel.
As an example, consider the Gaussian kernel as applied to scenarios when computing
one-year Historical VaR at the 95% confidence level. Given 262 trading days, the 95%
VaR is centered about scenario 249. In Figure 5.1, we show the scenario weights applied
in the computation of one-year Historical VaR. The weights are given by a Gaussian
distribution centered about the 249th scenario.
Figure 5.1: Plot of the weights of scenarios used in Computation of 1 Year Historical VaR using the
Gaussian kernel.
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PORTFOLIO & RISK ANALYTICS
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Chapter 6: Comparison of VaR Methodologies
The different VaR methodologies all attempt to answer the same question, but make
different assumptions about the distribution of portfolio returns to derive the answer. The
Parametric VaR methodology follows the traditional approach of assuming a jointly normal
distribution among all assets in the portfolio to compute a VaR estimate analytically. The
advantages of the parametric approach to computing VaR are very high speed of
computation and compatibility with traditional risk measures such as volatility.
However, as is increasingly recognized by risk practitioners, realized distributions of
portfolio returns may be significantly non-normal: they exhibit fat-tailed behavior, which
means that extreme moves in portfolio return may occur with much higher frequency than
that predicted by a normal distribution. Therefore, Parametric VaR tends to underestimate
VaR at very high confidence levels (typically, this becomes visible at the 97.5% confidence
level and becomes more visible as we go further into the tail). Parametric VaR also
imposes the restriction of linear pricing, which is not suitable for highly non-linear
securities. Historical and Monte Carlo VaR estimates aim to overcome these drawbacks
of Parametric VaR, but come at the expense of higher computational cost.
Historical VaR models fat-tailed behavior of returns by using the distribution of realized
(historical) factor returns, instead of assuming that factor returns are normally distributed.
The main advantage of Historical VaR over Monte Carlo VaR is the fact that it makes no
assumptions about the joint return distribution other than that the future return distribution
is the same as the historical distribution. This often makes Historical VaR easier to
interpret and explain. On the other hand, one may question the validity of using the
historical distribution for the distribution of future returns, since future market conditions
may be quite different from those experienced in the past. While we may think it
advantageous to use a limited amount of recent historical data based on the belief that
the future will be more like the recent than the distant past, using a small number of
historical scenarios will lead to a lower statistical confidence in the VaR estimate.
By contrast, Monte Carlo VaR simulates many scenarios (10,000) by making distributional
assumptions on factor returns and specific returns. Monte-Carlo VaR also increases the
statistical precision of VaR estimation compared with Historical VaR, due to the use of a
very large number of scenarios.
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Chapter 7: VaR Attribution
It is useful to decompose VaR into contributions from individual securities, factors, or
groups. Below, we present several methods for gaining insight into how VaR depends on
individual securities.
Partial VaR
Partial VaR represents the change in VaR that would result if one were to completely
eliminate the position in a single security or group, and re-compute VaR for the new
portfolio. Note that the sum of Partial VaR does not equal the portfolio VaR. Hence, Partial
VaR does not constitute a VaR decomposition.
Component VaR and Marginal VaR
Component VaR represents the contribution to VaR from an individual security or group
of securities. The sum of Component VaR gives the portfolio VaR. Hence, Component
VaR represents a risk attribution methodology. In this section, we derive the formula for
VaR attribution. We also discuss and interpret the concept of Marginal VaR.
Let wn be the portfolio weight in security
under scenario
n , and let lnj
denote the loss of the security
j . The portfolio loss under scenario j is therefore
L j   wnlnj .
(7.1)
n
Substituting Equation (7.1) into Equation (5.1), we obtain
V   v j  wnlnj ,
j
(7.2)
n
which can be rewritten as
V   wn  MVn ,
(7.3)
n
where MVn is known as the Marginal VaR of security
n , given by
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PORTFOLIO & RISK ANALYTICS
A Bloomberg Professional Service Offering
MVn   v jlnj .
(7.4)
j
Taking the partial derivative of Equation (7.3) with respect to wn , we see that Marginal
VaR can also be interpreted as the change in portfolio VaR given an infinitesimal change
in the weight of security
n , i.e.,
MVn 
V
.
wn
(7.5)
Component VaR is given by wn  MVn , and represents the contribution to VaR from
security
n . Component VaR adds up to the total portfolio VaR.
Chapter 8: Conditional VaR (Expected Shortfall)
VaR addresses the likelihood of observing a tail loss greater than or equal to a given
threshold (i.e., the VaR) over a specified horizon. A basic limitation of VaR is that it says
nothing about the expected portfolio loss conditional on the VaR threshold being
breached.
Conditional VaR, also known as Expected Shortfall, is designed to answer this question.
As such, Expected Shortfall represents an important complementary risk measure. While
VaR represents the best outcome in a bad period, Expected Shortfall represents the
expected outcome in a bad period.
To compute Expected Shortfall, first order scenarios
j from smallest loss to largest loss.
Let scenario j denote the first scenario that breaches the VaR threshold. Expected
Shortfall
S is computed as the average loss over all scenarios in which the VaR threshold
has been breached,
S
1
 Lj ,
J j j
(8.1)
where J is the total number of violations.
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PORTFOLIO & RISK ANALYTICS
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Substituting Equation (7.1) into Equation (8.1), we obtain a decomposition for Expected
Shortfall,
1

S   wn   lnj  .
n
 J j j 
(8.2)
Analogous to Marginal VaR, we can define the Marginal Shortfall, given by
MS n 
1
 lnj .
J j j
(8.3)
The Marginal Shortfall can also be interpreted as the change in Expected Shortfall given
an infinitesimal change in the weight of security
MS n 
The Shortfall Contribution of security
n
n . That is,
S
.
wn
(8.4)
is given by wn  MS n , which adds up to the total
Expected Shortfall.
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PORTFOLIO & RISK ANALYTICS
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References
Frank, David, "Stress Matrix Pricing in PORT", 2015, Bloomberg whitepaper.
Menchero, Jose, and Lei Ji, "Multi-Asset Class Risk Model (MAC2)", 2016, Bloomberg
whitepaper.
Note: Bloomberg whitepapers may be found by typing “PORT HELP <GO>” to access the
PORT help page.
Contact Us
To learn more about Bloomberg’s Portfolio & Risk Analytics, contact your
Bloomberg account representative or press the <HELP> key twice on the
Bloomberg Professional® service.
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