1. science
2. mathematics
3. statistics

risk t u t orial
How to Estimate and Calibrate
Analytical VaR for Interest Rate Risk
These absolute one-week interest rate changes are specified
for each maturity bucket and for each principal component
or scenario — the first scenario being a proportional shift of
the yield curve, the second scenario a rotation of the term
structure and the third scenario a change of the curvature.
Combinations of the scenarios are applied to the interest rate
sensitivities.
In the remainder of this article, we will apply our interest rate scenarios to the bp sensitivities illustrated in Figure 1
(below).
What is principal component analysis (PCA)? How can a firm use
PCA to calculate analytical VaR for interest rate risk? And how does
analytical VaR compare with Monte Carlo VaR?
By Marco Folpmers and Niels Eweg
Interest Rate Scenarios Derived by Principal
Component Analysis
Principal Component Analysis (PCA) is applied to highly
correlated data series in order to reduce dimensionality. The
underlying idea is that highly correlated time-series data, such
as weekly observations of the term structure, can be summarized into a small number of principal components.
As is explained in the appendix “PCA 101” (see pgs. 4950), term structure fluctuations can often be captured in three
meaningful components; hence, the “effective dimensionality”
is three rather than the 18 maturity buckets of the original data
set (up to a given confidence level, which in our case is 96.3%).
The results of the PCA decomposition, as described in the
appendix, are three sets of independent 99% interest rate
shocks ∆Rij (i = 1, …, 3, j = 1, …, 18 for all 18 maturities).
The three sets of shocks capture three generic movements in
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the term structure: a parallel change, a rotation and a change
of the curvature.
In Table 1 (below), we present the outcomes in basis points
(bp) per year, stressed at a 99% level. We conclude that the
parallel shift contained in the first principal component is
roughly consistent with the 200 bp shock scenario proposed
by CEPS.1 However, the combination of the three principal
components allows a more realistic scenario that includes a
rotation and a change of the curvature of the term structure.
0.5
0.0
Euro
-0.5
PC 1
PC 2 PC 3
Maturity
PC 1
PC 2 PC 3
3M
26
-152
48
7Y
234
7 -36
6M
81
-138
77
8Y
228
23 -26
9M
120
-124
44
9Y
224
36 -17
1Y
151
-118
14
10Y
220
48
-9
2Y
221
-95 -42
12Y
212
63
7
3Y
241
-67 -64
15Y
201
81
31
4Y
247
-49 -62
20Y
189
103
61
5Y
243
-30 -55
25Y
179
112
76
6Y
239
-11 -46
30Y
173
120
87
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-1.0
-1.5
-2.0
-2.5
Table 1 Eigenvector Values for Principal Components 1 through 3 in Basis Points per Year (Stressed
to the 99% Confidence Level)
Maturity
x 104 Sensitivities: change in EUR per 1 bp increase in interest rate
1.0
-3.0
-3.5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Maturity bucket
Our stylized example shows a typical pattern in which the
bank is funded with the help of short-term liquidity, whereas
its assets have longer maturities (an increase in interest rate
implies a decrease in present value of the assets). We denote
these key rate durations (KRDs) as KRDj , j =1,...,18.
The sum of the KRDs is equal to € - 220,000. This means
that a 1 bp parallel increase in the term structure leads to a
€220,000 decline in market value. The one-week 99% VaR
(with bp equal to 10-4) 2 equals:
3
Charts by Desiree Nunez
ince analytical value-at-risk (VaR) is easier to
understand and to apply than Monte Carlo
VaR, the hunt for analytical VaR methodologies is worthwhile, particularly if they can
compete with the complexities of Monte Carlo
modeling.
This article demonstrates how analytical interest rate risk
VaR can be estimated using principal component analysis
(PCA). As we will show below, the analytical VaR can be backtested with the help of historical simulation. This means that
relevant term-structure scenarios can be developed quickly
and conservatively. These scenarios can be used for risk measurement, such as VaR estimation, but also for the immunization/hedging of fixed income portfolios.
VaR 1week (0.99)=
√
{
18
∑ ∑
i=1
j=1
2
{
S
Figure 1 Basis Point Sensitivities
(∆Rij • KRDj/bp)
Note that a Euclidean distance in cubic space is calculated
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(square root of the sum of three squares), which is justified since
the principal components are independent (see the appendix).
For our example, the one-week 99% VaR equals €4.3 million.
Calibration and Back-Testing of the PCA-based
Interest Rate Risk VaR
Whereas the normal approximation illustrated earlier is
often used, the principal components for the weekly growth
rates exhibit fat tails. A more conservative approach would
be to use a t-distribution. The parameters for standard deviation and degrees of freedom can be derived by fitting t-distributions to the principal component scores using maximum
likelihood estimation (MLE).
Instead of multiplying by a z-score when calculating the
stressed principal components, we multiply with a score from
the inverse t-distribution.3 We derive the degrees of freedom
parameter needed from a MLE estimation of a fitted t-distribution applied to the principal component scores. The degrees of freedom for the fitted-t distribution of the principal
component scores are relatively low ( [ 7.1, 2.4, 2.3 ] for the
first three principal components), signaling a clear departure
from normality because of fat tails.
In Figure 2 (below), we compare the VaR calculated with
the help of the normal and t-approximation. We conclude
that, in our example, fat tails are relevant for confidence levels
above 97%.
Figure 2 VaR, according to Normal and
t-approximation
8
PCA normal approx
x 106
PCA t-approx
7
I Week Value at Risk (Euro)
risk t u t orial
6
5
4
3
2
90
91
92
93
94 95 96
Confidence level
97
98
99
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risk t u t orial
Below the 97% confidence level, the normal approximation
has higher VaR outcomes since the standard deviations of the
MLE-fitted t-distributions are slightly lower than the standard
deviations from the PCA procedure (which are equal to the
square roots of the eigenvalues, as shown in the appendix).
The 99.9% 1-week VaR that we estimate with the help of the
t-distribution equals €7.6 million.
This result can be compared to the VaR that we calculated
with the help of a back-test. This back-test is performed using
a historical simulation in which we select the 99.9th historical
decline in market value due to a weekly change in the term
structure.
The historical 99.9% weekly VaR that we estimate equals
€7.7 million, which is very close to the €7.6 million that we estimate with the help of the PCA approach using t-distribution
scaling. At the same time, this back-test illustrates the necessity of using the t-distribution scaling instead of the normal
approximation for an adequate VaR level.
{
18
∑ ∑
2
{
3
Immune Portfolios
(∆R • KRDj/bp)
outcomes of the PCA are notijonly useful
for VaR esVaR 1weekThe
(0.99)=
timation, but also for the establishment of portfolios that are
j=1
i=1
immune to √
interest rate shock. An immune portfolio is defined as a non-trivial mix of KRDs, determined in such a way
that the VaR equals zero.4
We consider a portfolio immunized if VaR = 0. For this constraint to hold, the sum of the products ∆Rij • KRDj needs to
be zero for each principal component; hence, the portfolio is
immune if and only if
This is a set of three linear equations in four unknowns. A
possible solution approach is to set Yj=4=1. An example with
maturities {1,6,11,16} results in the following equations:
[
1.98 18.73 17.79 14.75
-11.86 -5.24 1.78 8.02
3.73 -4.99 -2.05 4.78
Figure 3 Example of a Basis Point Sensitivity
Profile for an Immunized Portfolio
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x 104
Immunized Sensitivity Profile
1.5
1.0
j=1
48
=a•
-0.44
1.67
-2.53
1
where a is an arbitrary multiplicative factor. This method
of finding the KRDs can be applied to all groups of four
maturities. The complete portfolio now is a linear combination of the solutions of the complete set of groups of
maturities. An example of an immunized portfolio is given
in Figure 3 (below).
(∆Rij • KRDj) = 0
Q•y=0
[ ][]
[ ][]
KRD1
KRD6
KRD11
KRD16
2.0
0.5
Euro
for all principal components i = {1,2,3}. Since the system
consists of three equations with 18 variables (KRDj), it cannot
be solved directly. We therefore suggest a piecewise solution in
which the equation above is solved for groups of four maturities ( ˆj= {1,2,3,4}) at a time.
We construct a matrix Q with elements ∆ Rij (dimension [3
* 4]) and a column vector y with elements KRDj (dimension [4
* 1]). The problem can now be stated as follows:
]
KRD1
0
• 10 * KRD6 = 0
0
KRD11
1
-4
Note that, although the VaR of the aforementioned portfolio equals zero, the sum of the KRDs equals € - 4,095. This
is the change in market value if all interest rates increase simultaneously with one basis point. Although this portfolio is
not equal to a portfolio with duration = 0, it is more effective
in reducing the interest rate risk. In contrast to the durationmatching portfolio, it is immune for variations described by
the first three principal components (96.3% of the total variance, as shown in the appendix).
This gives the following KRDs:
18
∑
risk t u t orial
0.0
-0.5
-1.0
-1.5
-2.0
-2.5
Closing Thoughts
This article demonstrates how an analytical VaR can be derived for interest rate risk. With the help of a standard procedure from multivariate statistics, (principal component analysis) coherent interest rate scenarios can be computed that take
into account the co-movement of interest rate changes across
maturities.
This VaR procedure has existed for approximately 12 years
(since Golub & Tilman coined the term “principal component duration” in 1997) and is integrated in some commercial software applications for the management of interest rate
risk. It is much superior to single duration values that assume
a parallel shift of the term structure (e.g., the standard 200
basis point shock proposed by CEPS).
We demonstrated the usefulness of the PCA method for
constructing immunized portfolios, and we have also shown
that a multiplier from a normal distribution may underestimate the VaR for the higher confidence levels. (This is easily
remedied by using parameters from a MLE-fitted t-distribution.) With the help of this feature, we think that analytical
VaR methods for interest rate risk VaR remain interesting for
a straightforward and quick computation, as an alternative to
Monte Carlo simulation.
Appendix: Principal Component Analysis 101
We start with a dataset, R, containing for each week t the
euro zero-coupon interest rate for maturity bucket j (j=1, ...,18).
In our example, R contains the term structure data of a sixyear history ending on December 31, 2008.
For the PCA, the input data consists of the weekly growth
rates of the interest rates (so X1t = RR –1) for the 18 maturity
buckets after normalization to z-scores (zero mean, unit standard
deviation). This data-matrix is referred to as XT = [X1,..., X18].
The first principal component is found with the help of
a linear transformation using coefficients contained in a
1t
1t-1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Maturity bucket
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[18 * 1 ] vector a1, as follows:
Y1 = aT1X
The vector a1 is chosen so that Y1 has the largest possible
variance, subject to the constraint that aT1 a1 =1. The second
principal component is found by choosing a2 so that Y2 has
the largest possible variance, subject to the constraints that
aT2a2 =1 and a2Ta1 =0, or Y2YT1=0.(The principal components
are orthogonal: i.e., they are independent and, hence, uncorrelated). Similarly, Y3, ..., Y18 are derived, so as to be orthogonal and to have decreasing variance.
We will not go into the constrained optimization procedure
to derive the a- vectors and the principal components (this
procedure is covered in standard textbooks on multivariate
statistics5), but we will only point out that this procedure is
based on a (singular value) decomposition of the [ 18 * 18 ]
covariance matrix6 and that it delivers the a vectors that are
called eigenvectors, along with the variances of the principal
components 1, ..., 18, which are called eigenvalues.
The sum of these lambdas are equal to the sum of the variances of the original X vectors. Since we used normalized X
vectors, the sum of the variances of the original X vectors
and, hence, the sum of the lambdas, equals 18.
The optimization procedure outlined above warrants that
the eigenvalues 1, ..., 18 are decreasing in value. Since it
is convenient to claim that the first m principal components
account for n percent of the variance, it is useful to calculate
the proportion ∑mi=1 i / ∑18
 . Since we use normalized raw
i=1 i
data, the formula can be simplified to ∑mi=1 i/18.
When applying the PCA to yield curve data, often the following interpretation (which is an empirical finding rather
than a mathematical truth) can be given to the first three principal components:7 (1) the first principal component is highly
correlated with the rates of all maturities and the correlations
are of the same sign. So the first principal component is a factor that shifts the whole term structure (“level”); (2) the second
principal component is negatively correlated with short-maturity rates and positively correlated with long-maturity series.
This allows for an interpretation of the second principal component as a factor that tilts the term structure (“steepness”);
and (3) the third principal component is positively correlated
with short- and long-maturity rates and negatively correlated
with intermediate-maturity rates, and is therefore a factor
that effects the curvature.
The reduction of dimensionality consists of the modeling
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of the entire 18-dimensional term structure with the help of
the first three principal components. For the computation of
the interest rate risk VaR, a “shocked” term structure will be
derived from stressed values of the first three principal components.
The [18 * 18] matrix of eigenvectors is denoted by A, where
A = [a1, ..., a18]. The principal components are denoted by Y,
where Y = ATX.
Since we want to apply a shock to the three main principal
components and convert these shocked values to a stressed
term structure across 18 maturities, we need to incorporate
the inverse transformation (from principal components to the
normalized weekly growth rates8), using the following formula: X = AY.
Figure 4 Eigenvectors 1 through 3
Eigenvectors 1, 2 and 3
0.5
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
Table 2 Symbols Used for PCA
Symbol Interpretation
-0.5
Dimension
ROriginal dataset, weekly term structure for 18 maturity
buckets; 6 year history ending
on December 2008
[ Nr of weeks * 18 ]
XWeekly growth rates for
18 maturity buckets
[ 18 * Nr of weeks ]
A
Eigenvectors
[ 18 * 18 ]
Y
Principal components of X
[ 18 * Nr of weeks ]
i,i=1,...,18Eigenvalues, or variances
of the principal components
[ 18 * 1 ]
For our example, we use weekly growth rates of interest rate
across 18 buckets (3M, 6M, 9M, 1Y, 2Y, 3Y, 4Y, 5Y, 6Y, 7Y,
8Y, 9Y, 10Y, 12Y, 15Y, 20Y, 25Y, 30Y), from January 1, 2003,
until December 31, 2008 (six years). The X matrix comprises the normalized growth rates across these 18 buckets and
313 weeks. The first three principal components account for
96.3% of the total variance.
This proves that the dimensionality can be reduced to three
while losing only a small fraction of the original total variance.
The 96.3% variance is sufficient for many risk applications.
The first three eigenvectors are shown in Figure 4 (above).
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Maturity bucket
Our decomposition of the term structure into three principal components shows the typical patterns outlined above.
The blue line illustrates the first eigenvector and shows a fairly
stable value across maturity buckets. Hence, the first principal
component captures proportional increases and decreases of
the term structure. The green line illustrates the second eigenvector and depicts the typical increasing trend of the term
structure. The red line (third eigenvector) accounts for the
curvature of the yield curve.
A 99% VaR is calculated with the help of stressed values
of the principal components. The stressed values of the three
principal components are equal to
strPC = a • √ • G (0.99), i = 1,...,3, j = 1,..., 18
ij
i
ij
for a one-week 99% VaR with G(.) the inverse standard normal distribution. The result is one vector for each principal
component (i) containing stressed values for the normalized
one-week growth rates for the 18 maturities (j). Note that a
normal distribution is assumed since we multiply with a zscore. This inherent normality assumption is discussed the
“calibration and back testing” section of the article.
The vectors have to be multiplied by the standard deviations of the growth rates for all maturities, and, subsequently,
by the last interest rates in the dataset (term structure of December 31, 2008 = T) in order to convert the stressed growth
rates to stressed absolute interest rate changes, as follows:
∆R =strPC •  ( ∆R ) • R , i = 1,...,3, j = 1,..., 18
ij
ij
j
T,j
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risk t u t orial
FOOTNOTES
REFERENCES
1. CEPS, 2006, 11.
Agca, S. “The performance of alternative interest
rate risk measures and immunization strategies under
a Heath-Jarrow-Morton framework.” Virginia Polytechnic Institute and State University (published thesis, March 2002).
2. Note that we implement a delta-normal approach.
More precise alternatives are: delta-gamma (correcting for convexity) and full revaluation. The delta-normal approach follows Golub & Tilman, 2000.
3. See Fiori & Iannotti (see references), 2006, p.12,
about the t-distribution as a valid fat-tailed distribution for stressing the eigenvectors.
CEPS, CP11, March 2006. Consultation paper on
technical aspects of the management of interest rate
risk arising from non-trading activities and concentration risk under the supervisory review process.
4. “Non-trivial” excludes the solution of zero KRDs
for all maturity buckets.
Chatfield, C. and A. Collins. “Introduction to Multivariate Analysis,” Chapman & Hall,1981.
5. See, e.g., Chatfield & Collins, 1981, Ch. 4.
Fiori, R. and S. Iannotti. “Scenario based principal
component value-at-risk: an application to Italian
banks’ interest rate profile,” Banca d’Italia, September 2006.
6. Since we use normalized X vectors, the decomposition is applied to the correlation matrix of the data
before normalization. Some statistical software packages use an option when calling the PCA procedure to
distinguish between a PCA based on the covariance
matrix or a PCA based on the correlation matrix. A
PCA based on the correlation matrix is equivalent to
a PCA based on the covariance matrix if the raw data
have been normalized first. For the decomposition of
interest rate data, it is generally recommended to use
the correlation matrix (or use normalized raw data)
rather than the covariance matrix (see Loretan, 1997,
p. 26). The singular value decomposition of the variance matrix ∑ delivers the decomposition ∑ = U U1,
with the matrix with the eigenvalues  on the main
diagonal and zeros elsewhere. The matrix U contains
the eigenvectors.
V
V
7. See Loretan (1997, pg. 28), Golub & Tilman (2000,
pg. 94), Rachev (e.a., 2007, pg. 453) and Fiori & Iannotti.
8. Note that A is an orthogonal matrix whose transpose is its inverse.
Golub, W. and L. Tilman. “Risk Management: Approaches for Fixed Income Markets,” John Wiley &
Sons, 2000.
Loretan, M. “Generating market risk scenarios using
principal component analysis: methodological and
practical considerations,” Federal Reserve Board,
March 1997.
McNeil, A.J., R. Frey and P. Embrechts. « Quantitative Risk Management: Concepts, Techniques, Tools,
» Princeton, 2005.
Rachev, S.T., S. Mittnik, F.J. Fabozzi, S.M. Focardi
and T. Jasic. “Financial Econometrics,” John Wiley
& Sons, 2007.
M. Weiskopf. “Immunization of fixed income portfolios.” (published thesis, 2009).
Marco Folpmers (PhD, FRM) is the leader of the financial risk management service
line at Capgemini Consulting in the Netherlands. He can be reached at [email protected]
Niels Eweg (M.Sc., FRM) works as a structured finance and asset and liability management consultant at Capgemini. He can be reached at [email protected].
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