CHAPTER 7
Value-at-Risk Contribution
INTRODUCTION
• The output from a VaR calculation includes the
following reports that can be used to identify the
magnitude and source of each risk:
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–
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Total VaR for the trading operation
Stand-alone VaR for each subportfolio
Stand-alone VaR for each risk factor
Sensitivity (or duration) for each risk factor
Duration matrix for each subportfolio
Duration matrix for Total portfolio
VaR Contribution for each subportfolio
VaR Contribution for each risk factor
INTRODUCTION
• The first six of these reports are generated
easily from the analyses we discussed in
the previous chapter
• The total VaR is calculated by including all
of the bank's instruments and risk factors.
• The stand-alone VaR for a subportfolio is
the VaR that the portfolio would have if we
ignored the rest of the bank.
INTRODUCTION
• Similarly, the stand-alone VaR for each
risk factor is calculated by setting the
standard deviation on all the other risk
factors equal to zero.
• The sensitivity of the value of the portfolio
to changes in risk factors is given by the
derivative vector that is used in Parametric
VaR.
INTRODUCTION
• The main problem with the stand-alone
VaR is that the sum of the stand-alone
VaRs does not, in general, equal the total
VaR.
• Also, the stand-alone VaR ignores the
correlation with the rest of the portfolio.
INTRODUCTION
• For clarity, consider the following Parametric
VaR example for a portfolio with two
subportfolios, A and B.
• Here, SVaR represents the stand-alone VaR:
(1) [1,-1]
(2) If the value =1, then VARP=SVARA+SVARB
INTRODUCTION
• The VaR Contribution (VaRC) technique is
useful because it gives us a measure of risk for
each individual subportfolio that includes the
interportfolio correlation effects.
• Furthermore, VaRC is constructed so that the
sum of VaRC for all the subportfolios equals the
total VaR for the portfolio.
• This allows us to make straightforward
statements such as, "The VaR for the bank is $8
million, caused by contributions of $2 million
from the equities desk, $3 million from bonds, $2
million from FX, and $1 million from derivatives."
INTRODUCTION
• As explored in the following chapters,
VaRC is also useful for allocating the
bank's capital to those units causing the
risk and for setting limits on the amount of
risk that individual traders may take
• The process used to define VaRC is the
same as the process that is used later in
the credit-risk chapters to define ULC, the
Unexpected Loss Contribution
INTRODUCTION
• This chapter will show the derivation of
VaRC for individual risk factors and for
individual subportfolios.
• We will show how VaRC can be
calculated in algebraic, summation, and
matrix forms
• In each case, we will start with a portfolio
of just two risks and then generalize to a
portfolio of many risks
DERIVATION OF VARC IN
ALGEBRAIC NOTATION
• Consider a portfolio exposed to two sources
of risk, A and B
the average correlation between the given risk
and the rest of the portfolio
DERIVATION OF VARC IN
ALGEBRAIC NOTATION
DERIVATION OF VARC IN
ALGEBRAIC NOTATION
DERIVATION OF VARC IN
ALGEBRAIC NOTATION
DERIVATION OF VARC IN
ALGEBRAIC NOTATION
DERIVATION OF VARC IN
SUMMATION NOTATION
Summation notation can
be useful if there are
many risk factors
because it can express
long equations in a
compact form.
DERIVATION OF VARC IN
MATRIX NOTATION
• Matrix notation also gives a good
shorthand way of writing equations.
• It also allows us to easily show how the
VaRC can be calculated either for single
risk factors affecting many positions or for
single positions affected by many risk
factors.
DERIVATION OF VARC IN
MATRIX NOTATION
DERIVATION OF VARC IN
MATRIX NOTATION
DERIVATION OF VARC IN
MATRIX NOTATION
DERIVATION OF VARC IN
MATRIX NOTATION
Example of Calculating VaRC
Using Matrix Notation
Example of Calculating VaRC
Using Matrix Notation
VARC CALCULATED FOR
SUBPORTFOLIOS
• In the derivation above, we showed the
VaR Contribution for different risk factors.
• VaRC can also be calculated for different
business units, or subportfolios, each of
which may share some risk factors with
the other desks.
• This is most easily shown in matrix
notation
VARC CALCULATED FOR
SUBPORTFOLIOS
• In this case, we break down the D vector
into the sensitivity vector for each
subportfolio
• Consider the following bank consisting of a
number of portfolios, a to z
• The sensitivity vector for the bank as a
whole has an element for each of the N
risk factors: 1 to N
VARC CALCULATED FOR
SUBPORTFOLIOS
VARC CALCULATED FOR
SUBPORTFOLIOS
VARC CALCULATED FOR
SUBPORTFOLIOS
VARC CALCULATED FOR
SUBPORTFOLIOS
VARC CALCULATED FOR
SUBPORTFOLIOS
VARC CALCULATED FOR
SUBPORTFOLIOS
CALCULATING VARC WHEN USING
MONTE CARLO ORHISTORICAL
SIMULATION
• With the Monte Carlo and Historical
simulation methods, we can calculate a
VaRC by going through all the simulation
results and examining all the scenarios in
which the VaR is exceeded by the
experienced losses
• For example, if we run 5000 scenarios, the
99% VaR would be defined by the 50thworst result
CALCULATING VARC WHEN USING
MONTE CARLO ORHISTORICAL
SIMULATION
• VaRC can be calculated using these 50
cases when the losses equal or exceed
the VaR
• On each occasion when the VaR is
exceeded, we record the losses from the
individual position
• This gives the percentage contribution of
each position to the portfolio's loss in
those particularly bad scenarios
CALCULATING VARC WHEN USING
MONTE CARLO ORHISTORICAL
SIMULATION
Homework
• Consider a case of a foreign bond+foreign
cash, same with the example in the
textbook.
• Calculate VaRC via the three different VaR
approaches