Market risk modelling for counterparty risk measurement B&H RESEARCH

ENTERPRISE RISK SOLUTIONS
MAY 2013
B&H RESEARCH
Market risk modelling for counterparty risk
measurement
Steven Morrison PhD
Laura Tadrowski PhD
Overview
MAY 2013
ESG
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Moody's Analytics Research
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Monte Carlo simulation of market risk factors is a key step in the models used by banks to
measure counterparty exposure, and requires the bank to make subjective assumptions around
‘real world’ distributions of risk factors. The sensitivity of results to such highly subjective
assumptions is a source of model risk for firms, and naturally an area of particular scrutiny by
regulators who require firms to be able to justify assumptions and quantify the sensitivity to
them.
There are a number of technical challenges associated with building and calibrating market risk
models for this type of application – Monte Carlo scenarios need to be generated under realworld probabilities, over relatively long time horizons, and over multiple multi risk factors. In this
paper we illustrate the impact of key real-world scenario generation modeling and calibration
assumptions on standard measures of counterparty exposure using an example interest rate
scenario generation model.
Firstly, we consider the sensitivity of volatility assumptions to the weighting put on current
market conditions, comparing a Point-in-Time volatility assumption (heavily weighted on current
market conditions as implied by market option prices) to a Through-the-Cycle volatility
assumption (based on long-term historical data and therefore a relatively low weighting on
current market conditions). Such assumptions can have a material impact on the entire
distribution of future exposure at all future horizons, including Expected Exposure over one year.
Secondly, we consider the impact of risk premia assumptions. Though such assumptions have a
relatively low impact on the distribution of exposure in the short term, they can have a material
effect over the life of a netting set, and hence in the calculation of PFE 95 for the purpose of
assessing exposure against credit limits.
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Contents
2
1. Introduction
3
2. The challenges of scenario generation for counterparty risk
4
3. An example scenario generator and calibration choices
Model description
Volatility calibration: Point-In-Time or Through-The-Cycle?
Term premium calibration: zero, constant or time varying?
5
5
6
7
4.The impact of calibration choices on counterparty exposure measures
10
5. Summary
12
References
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1. Introduction
The Global Financial Crisis highlighted counterparty risk as one of the major risks facing banks and other financial institutions, and
has led to a growing interest and investment in models to understand and measure future counterparty exposure. In general,
future exposure is a highly complex function of a large number of market risk factors, due to complexities such as netting and
collateralization. This has naturally led to Monte Carlo simulation being adopted as the standard tool in estimating the distribution
of future exposure, with the general approach consisting of the following steps:
1.
2.
3.
4.
Generate Monte Carlo scenarios for all relevant market risk factors.
Revalue instruments in each scenario.
Calculate exposures in each scenario, accounting for netting and collateralization.
Estimate statistics such as Expected Exposure (EE) and Potential Future Exposure (PFE).
7
7
6
6
5
5
4
Calibration 1
3
Calibration 2
2
PFE 95 (£m)
Expected Exposure (£m)
In this note, we discuss the market risk simulation models used in the first of these steps, sometimes referred to as ‘scenario
generators’. Standard measures of counterparty exposure are likely to be highly sensitive to subjective modeling and calibration
assumptions embedded within these scenario generators. For example, the following charts compare EE and PFE 95 profiles for an
example swaps portfolio under three different calibration assumptions for the underlying interest rate model:
4
Calibration 1
3
Calibration 2
2
Calibration 3
1
Calibration 3
1
0
0
0
1
2
3
4
5
6
7
0
1
Projection horizon (years)
2
3
4
5
6
7
Projection horizon (years)
6%
6%
5%
5%
4%
Calibration 1
3%
Calibration 2
2%
Calibration 3
Forward rate
1%
0%
Expected ten year rate
Expected ten year rate
In particular, the PFE 95 peaks at between £2.92m and £4.54m depending on the choice of calibration, which in this case reflects
different assumptions for the expected path of future interest rates relative to market forward rates:
4%
Calibration 1
3%
Calibration 2
2%
Calibration 3
Forward rate
1%
0%
0
1
2
3
4
5
Projection horizon (years)
6
7
0
1
2
3
4
5
6
7
Projection horizon (years)
However, the expected path of future rates cannot be unambiguously inferred from market prices and/or historical data. This, and
other, calibration assumptions rely on the subjective judgement of the modeller. The sensitivity of results to such highly subjective
assumptions is a source of model risk for firms, and naturally an area of particular scrutiny by regulators who require firms to be
able to justify assumptions and quantify the sensitivity to them.
In this note we aim to share some of our experiences in building and calibrating scenario generators for use in similar long-term,
real-world, applications and demonstrate the sensitivity of counterparty exposure measures to key calibration assumptions.
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The paper is structured as follows:
»
Section 2 outlines some of the challenges in scenario generation for assessment of counterparty exposures.
»
Section 3 discusses some of the key considerations and choices in setting calibration assumptions, using an example interest
rate model.
»
Section 4 demonstrates the impact of key calibration choices on standard counterparty risk measures.
2. The challenges of scenario generation for counterparty risk
Scenario generators are stochastic models for the market risk factors that drive exposures, implemented via Monte Carlo
simulation. Stochastic models and Monte Carlo implementation techniques are familiar to banks’ risk managers and quantitative
analysts, both in the pricing of derivatives and in the assessment of Value-at-Risk over short horizons (e.g. one day or 10 days).
However, assessment of counterparty risk gives rise to a number of new technical challenges, in particular:
»
The requirement to model under the real world measure 1, over relatively long time horizons. Time horizons of interest here
vary from one year up to the maturity of the portfolio, which could be 30 years or more.
Calibration of such models requires us to make statements about the real-world probabilities of different scenarios, including
the effects of risk premia. Unlike in pricing applications, these probabilities cannot be directly inferred from market prices. We
may believe that market prices provide useful information about such probabilities, but they cannot objectively tell us what
they are.
To the extent that the model reflects past history, analysis of historical data can provide useful calibration information in
addition to market prices. Given a number of historical data points, one can easily fit a stochastic model using standard
statistical techniques such as maximum likelihood. However, the length of projection horizon of interest here, and
correspondingly low volume of data points, means that any such estimators are likely to be subject to a large amount of
estimation error.
Furthermore, statistical estimation requires us to make subjective assumptions about data period, frequency and weighting
scheme (e.g. fixed vs. exponentially weighted) and estimates can be highly sensitive to these assumptions. These choices are
also important in determining how sensitive calibration assumptions are to new information 2.
History is of course only a guide to the future and economic judgement may provide useful additional forward-looking
information particularly over ultra long horizons. This requires subjectivity not only in setting forward looking assumptions
themselves but also in deciding how much weight to put on these relative to other source of information.
In summary, setting of long-term real-world assumptions involves a huge amount of subjectivity, and shouldn’t be thought
of simply as an exercise in statistical handle-turning.
»
Netting agreements mean that exposure cannot be considered on a deal-by-deal basis but needs to be aggregated at the level
of portfolios (netting sets). This results in the requirement to generate joint scenarios for a number of risk factors, capturing
the dependency between them. There is a huge amount of literature on models describing the behavior of individual asset
classes, but relatively little on the interactions between them, particularly over the long projection horizons of relevance here.
»
The complexity and path dependency of exposure, including the effects of collateral. This results in a requirement for models
that generate paths for risk factors and not just their values at a single future time horizon.
»
The computational cost associated with having to revalue large, complex, portfolios under a large number of scenarios.
It is worth noting that the insurance industry has also been grappling with similar technical challenges for a number of years.
Insurance liabilities are typically long-term, path dependent and depend on multiple risk factors, while actuaries have traditionally
thought of reserves and capital in terms of the real-world run off of these liabilities. The requirement to project the value of
complex derivative portfolios is another area of overlap, with techniques such as Least Squares Monte Carlo being successfully
adopted by both banks (Cesari, Aquilina, Charpillon, Filipovic, Lee, & Manda, 2009; Davidson, 2012) and insurers (Morrison,
Turnbull, & Vysniauskas, 2013).
1
Of course, there is a separate requirement to generate scenarios under the risk-neutral measure, for pricing purposes (e.g. calculation of CVA). We will not
consider such applications in this note.
2
They also determine the sensitivity of calibration assumptions are to old information. Fixed weighting schemes are sensitive to old data suddenly dropping
out of the estimation window which can have large, spurious, impact on estimates.
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In this note we consider the measurement of counterparty exposure using a scenario generator that is widely used by insurers in
their assessment of risk and capital: the B&H Economic Scenario Generator (ESG), using calibrations that have been designed for
use in long-term real-world applications.
We focus our attention here on interest-rate scenario generation, illustrated using a netting set consisting of vanilla interest-rate
swaps in a single currency, though the key messages are applicable to other types of risk factor and the dependencies between
them.
3. An example scenario generator and calibration choices
Model description
The analysis in this note uses the B&H Economic Scenario Generator (ESG), with interest rate scenarios being generated using a 2factor version of the Black-Karasinski model (Morrison, 2007). Here we provide a brief technical description of the model and its
main features, before going on to discuss its calibration.
The B&H 2-factor Black-Karasinski (2FBK) model describes the joint behaviour of the short-rate 𝑟𝑡 and ‘mean-reversion level’ 𝑚𝑡
according to the following system of stochastic differential equations:
�𝑡1
𝑑𝑙𝑛(𝑟𝑡 ) = 𝛼1 �𝑙𝑛(𝑚𝑡 ) − 𝑙𝑛(𝑟𝑡 )�𝑑𝑡 + 𝜎1 𝑑𝑊
�𝑡2
𝑑𝑙𝑛(𝑚𝑡 ) = 𝛼2 �𝜇(𝑡) − 𝑙𝑛(𝑚𝑡 )�𝑑𝑡 + 𝜎2 𝑑𝑊
�𝑡2 are independent Brownian motions under the risk-neutral probability measure. Written in this form, the 2FBK
�𝑡1 , 𝑊
where 𝑊
model can be viewed as a lognormal version of the standard 2-factor Hull-White model (Hull & White, 1994), or a 2-factor
extension of the standard Black-Karasinski model (Black & Karasinski, 1991). Like these, and other, ‘short-rate’ models the 2FBK
model describes the risk-neutral stochastic behavior of the short-rate, with all other interest-rates being calculated using riskneutral pricing. In particular, we can calculate the initial term structure of interest-rates, and choose the deterministic function
𝜇(𝑡) to match the term structure observed in the market.
Similar short-rate models are sometimes used in valuation of interest-rate derivatives, and in this context it is sufficient to describe
the model under the risk-neutral measure only. However, here we also require a description of the model under the real-world
measure. Theoretically, this change of probability measure corresponds to a change of drift to the two Brownian motions:
𝑑𝑙𝑛(𝑟𝑡 ) = 𝛼1 �𝑙𝑛(𝑚𝑡 ) − 𝑙𝑛(𝑟𝑡 )�𝑑𝑡 + 𝜎1 (𝑑𝑊𝑡1 + 𝛾1 (𝑡)𝑑𝑡)
𝑑𝑙𝑛(𝑚𝑡 ) = 𝛼2 �𝜇(𝑡) − 𝑙𝑛(𝑚𝑡 )�𝑑𝑡 + 𝜎2 (𝑑𝑊𝑡2 + 𝛾2 (𝑡)𝑑𝑡)
where 𝑊𝑡1 , 𝑊𝑡2 are independent Brownian motions under the real-world probability measure.
𝛾1 (𝑡), 𝛾2 (𝑡) are commonly referred to as ‘market prices of risk’ (MPR) and in this case control the risk premium on bonds. In the
special case where both market prices of risk are zero, the real-world measure is identical to the risk-neutral measure, and so (by
definition) the expected returns on bonds, of all maturities, are equal to the return on cash. More generally, non-zero market prices
of risk will pull the expected path of interest rates away from the risk-neutral expected path, and in turn give risk to average bond
returns that are different from the cash return i.e. bond risk premiums (sometimes referred to as ‘term premiums’). For example,
negative market prices of risk will pull rates down relative to the risk-neutral expected path and therefore give rise to bond returns
that exceed cash returns on average i.e. positive risk premiums.
This model has a number of features that make it attractive for long-term real-world scenario generation. The model explicitly
assumes that the short-rate mean-reverts, which is consistent with historical short-rates observed over long time periods. The
short-rate is at any future point in time has a lognormal distribution, and hence is positive. This in turn gives rise to positive
interest rates of all maturities. While small negative short-term interest-rates have been observed historically in a number of
economies, the frequency of such observations has been relatively small.
The model is also arbitrage-free by construction, in the sense that we cannot construct risk-free portfolios that are guaranteed to
return more than the risk-free rate. Given a particular choice of stochastic model, we can construct immunisation strategies i.e.
trade in bonds of different maturities in such a way that the resulting portfolio is riskless. In order to avoid arbitrage the resulting
portfolio cannot earn a risk-premium, which in turn constrains the expected movements of different points on the yield curve
(Heath, Jarrow, & Morton, 1992). While individual bonds may earn a risk-premium, these must exactly cancel out when we
construct a risk-free portfolio. Our model specification ensures that this is the case. This property shouldn’t be taken for granted –
not all stochastic interest-rate models are arbitrage-free in this way.
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Finally, the 2FBK model provides a relatively intuitive and parsimonious description of yield curve dynamics, with just 4 parameters
(𝛼1 , 𝛼2 , 𝜎1 𝜎2 ) describing the volatility and correlation structure, and two functions ( 𝛾1 (𝑡), 𝛾2 (𝑡)) describing term premia. This
allows us to test sensitivity of model results to a relatively small number of key assumptions 3. Having introduced the model, we
outline some of the key considerations in choosing these parameters, and in testing sensitivity to them.
Volatility calibration: Point-In-Time or Through-The-Cycle?
In the 2FBK model, interest rate volatility is controlled via the parameters 𝛼1 , 𝛼2 , 𝜎1 𝜎2 . If we were using the model for pricing
purposes, these parameters to be consistent with market implied volatilities of certain instruments (typically caps and/or
swaptions). This type of ‘market consistent’ calibration exercise is standard practice for pricing quants 4.
However, for the applications considered in this note, we are interested in the real-world distribution of rates. While we might
expect market prices to tell us something about this distribution, there are additional factors that might contribute to the market’s
pricing. In general, the Moody’s Analytics approach to real world calibration considers historical data in addition to market prices
and the ‘forward looking’ views of experts. Ultimately, the weight that is put on each of these pieces of information is subjective,
and depends on the intended use of the model, in particular the time horizon that it is to be used to project over. Moody’s
Analytics currently produce regular updates of two different types of volatility calibration of the 2FBK model:
»
Point-in-Time calibration
The Moody’s Analytics Point-in-Time (PIT) calibration of the 2FBK model (Hibbert & Skrk, 2008) is intended to describe our
best estimate of the distribution of rates at a one year horizon, in particular the extreme tails of the distribution. At short
horizons, we take the view that option prices provide a good estimate of forward looking volatility, and so market swaption
implied volatilities (where available) with a maturity of one year are used to infer the proportional volatility of par yields of a
number of selected tenors. As such, the PIT calibration is highly responsive to changes in market observables, and calibrations
can exhibit relatively large changes in the values of model parameters at different calibration dates.
Note that the Moody’s Analytics Point-in-Time calibration is intended to be used over a one year horizon only, and is not
intended for use over longer horizons.
»
Through-the-Cycle calibration
In our PIT calibration, the parameters 𝛼1 , 𝛼2 , 𝜎1 𝜎2 change at each calibration date to reflect current market conditions.
However, if we consider using the model to project beyond one year, the model assumes that these parameters are held
constant. The model doesn’t capture any variability of these parameters due to changing market conditions in future. As a
result a PIT calibration, while providing a sensible description of volatility over the next year, may overestimate or
underestimate future volatility beyond the one year horizon.
In particular, note that the 2FBK model assumes that the absolute volatility of rates is (approximately) proportional to the
level of rates i.e. proportional volatility is (approximately) constant. In reality the volatility structure observed in the market is
somewhat different from this – there is some evidence to suggest that proportional volatility is, on average, greater when
rates are low than when rates are high. As a result, a calibration which reflects the market’s view of volatility when rates are
initially at relatively low levels is likely to overstate volatility when rates revert to normal (or higher than normal) levels, while
a calibration which reflects the market’s view of volatility when rates are initially at relatively high levels is likely to
understate volatility when rates revert to normal (or lower than normal) levels.
Rather than setting the parameters 𝛼1 , 𝛼2 , 𝜎1 𝜎2 to reflect current market conditions, Moody’s Analytics Through-the-Cycle
(TTC) calibration sets parameters that are reflective of ‘average’ market conditions 5. As such, the calibration is designed to be
used for projection over very long time horizons covering changing market conditions.
Figure 1 compares the one year ahead distributions of GBP one and ten year spot interest rates, using end-December 2012 PIT and
TTC calibrations. This illustrates just how different these distributions can be using the different calibration approaches. The
current relatively low interest rate environment, and correspondingly high swaption implied volatilities, gives rise to a far wider
distribution of rates using a PIT calibration to a TTC calibration.
6
3
The ability to specify term premia as a function of time increases the number of parameters significantly, though in practice we will usually parameterise
these functions in terms of a small number of assumptions e.g. unconditional expected short rate and rate of mean reversion towards this.
4
Market consistent calibration is also used by insurance firms who use such models to put a value on options embedded within their liabilities to
policyholders.
5
More specifically, the calibration uses volatility and dispersion estimated using Exponentially Weighted Moving Average estimators with a mean age of 25
years (Liu & Jessop, Real-world interest rate calibration: Long term distributional targets for inflation, nominal and real rates, 2012; Tadrowski & Jessop,
2012).
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The PIT and TTC volatility assumptions considered here represent two extremes, with the PIT volatility assumption being heavily
weighted on current market conditions, while the TTC volatility assumption puts a relatively low weighting on current market
conditions. Alternative weighting assumptions are of course possible.
Figure 1: Interest rate distributions at one year horizon using end-December 2012 calibrations –
One year rate (LHS) and ten year rate (RHS)
5%
2.0%
1.5%
Percentiles 95% to 99%
1.0%
Percentiles 75% to 95%
Percentiles 50% to 75%
0.5%
Percentiles 25% to 50%
Ten year rate at year 1
One year rate at year 1
2.5%
Percentiles 5% to 25%
0.0%
PIT
TTC
TTC
TTC
(zero term (positive (negative
premium)
term
term
premium) premium)
4%
3%
Percentiles 95% to 99%
2%
Percentiles 75% to 95%
Percentiles 50% to 75%
1%
Percentiles 25% to 50%
Percentiles 5% to 25%
0%
Percentiles 1% to 5%
PIT
TTC
(zero term
premium)
Average
TTC
(positive
term
premium)
TTC
(negative
term
premium)
Percentiles 1% to 5%
Average
Calibration Type
Calibration Type
While these differences observed in Figure 1 are primarily due to different volatility assumptions, though there is a second order
difference as a result of different term premium assumptions. We will discuss term premia in more detail below, but for now note
that our PIT calibration assumes that there is no term premium which, over a one year horizon, corresponds loosely to expected
rates being equal to forward rates. In contrast our TTC calibrations make different term premia assumptions depending on the
choice of the user. For completeness, Figure 1 shows TTC calibration results using three different choices of term premium – zero
(where forward rates are approximately equal to expected rates), positive (where forward rates exceed expected rates) and
negative (where forward rates are lower than expected rates).
Term premium calibration: zero, constant or time varying?
In addition to the volatility of rates, we need to specify their expectations. As described in Section 2, the market price of risk
functions 𝛾1 (𝑡), 𝛾2 (𝑡) determine how the real-world expected path of interest rates differs from the risk-neutral expected path,
and hence the resulting risk premiums on bonds.
Here we consider and compare three different assumptions for the market price of risk: zero, constant or time-varying. In all cases,
we have fixed the parameters 𝛼1 , 𝛼2 , 𝜎1 𝜎2 according to our end-December 2012 Through-the-Cycle calibration and fitted the
deterministic function 𝜇(𝑡) to match the UK Government bond term structure observed in the market at end-December 2012.
Figure 2 shows the corresponding expected path of one and yen year spot rates and compares these with market forward rates at
end-December 2012.
Figure 2: Expected interest rate paths – one year rate (LHS) and ten year rate (RHS)
6%
6%
Expected rate
(zero term premium)
4%
Forward rate
3%
2%
Expected rate
(time varying term
premium)
1%
0%
0
5
10
15
20
25
Projection horizon (years)
30
Expected rate
(constant term
premium)
Expected rate
(zero term premium)
5%
Ten year rate
One year rate
5%
4%
Forward rate
3%
2%
Expected rate
(time varying term
premium)
1%
0%
0
5
10
15
20
25
30
Expected rate
(constant term
premium)
Projection horizon (years)
The three different term premium assumptions used in Figure 2 are described below.
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»
Zero term premium
If 𝛾1 (𝑡) = 𝛾2 (𝑡) = 0 there is no risk premium on bonds i.e. bond returns are risk-neutral. The resulting expected path of
interest rates approximately follows initial forward rates. Actually, due to the technical definition of the risk-neutral
probability measure, expected paths always lie somewhat above corresponding forward rates, with the difference depending
on the assumed volatility of rates 6 and on the projection horizon.
»
Constant term premium
In the zero term premium calibration, the expected path of rates is entirely determined by initial forward rates plus our
assumed level of volatility. This may be viewed as a strong assumption, and more generally we might want to assume that
rates follow a different expected path. The inclusion of non-zero market prices of risk allows us to incorporate such views.
For comparison, we show the expected paths produced by an alternative calibration in which 𝛾1 (𝑡) = 𝛾2 (𝑡) = −0.14 at all
times. The assumption of negative market prices of risk results in rates being pulled downwards relative to the risk-neutral
path (hence a positive bond risk premium), and in this case the market prices of risk are sufficiently large that the expected
path of rates is pulled below initial forward rates. In this case we have chosen the market price of risk according to Moody’s
Analytic’s standard Constant Term Premium calibration of the GBP yield curve at end-December 2012. This reflects our views
on the ‘unconditional’ expected short-rate, relative to the corresponding unconditional forward rate. Of course other views
are possible, though the assumption of a constant market price of risk will always give rise to an expected path which follows
the broad shape implied by market forward rates – the constant MPR allows us to change the level of the expected path but
not its general shape. As a particular consequence of this, the expected paths produced can vary significantly due to changes
in market forward rates, even at relatively long projection horizons.
»
Time varying term premium
To give further control over the shape of the expected path, while still fitting the current yield curve, we need a more general
assumption for the market price of risk. For example, rather than assuming that the market prices of risk are constant, we can
assume that they vary deterministically with time, and choose this deterministic path so as to control not just the expected
rate at a particular point in time but at all points in time. Figure 2 also shows the expected rate paths produced using
Moody’s Analytics ‘Time Varying Term Premium’ calibration of the GBP yield curve at end-December 2012. These expected
rate paths are based on an assumption of a relatively smooth path of expectations, along with targets for the unconditional
short-rate plus the rate of mean-reversion towards this (Liu & Sorensen, 2009).
In contrast to the Constant Term Premium (CTP) calibration, the Time Varying Term Premium (TVTP) calibration is designed
to give relatively stable expected rate paths, but with time varying market prices of risk. The CTP calibration corresponds to a
view that risk premia are relatively stable and movements in the yield curve arise primarily due to changing expectations,
while the TVTP calibration approach takes the alternative view that these yield curve movements are largely driven by
changes in term premia rather than expectations.
These differences in term premium assumption don’t just affect the expected path of rates, but also impact on their entire
distribution. Figures 3 to 5 show the distributions of one and ten year rates over a 30 year horizon, comparing Zero Term Premium
(Figure 3), Constant Term Premium (Figure 4) and Time Varying Term Premium (Figure 5).
Due to the proportional volatility structure of the 2FBK model, the absolute standard deviation of the rate distribution increases
with the expected level of rates. As a result, changes in assumed expectations can have a large effect on the rest of the distribution
in absolute terms. For example, at 20 years, the average short rate is 5.3% (zero term premium) compared to 3.4% (constant term
premium), while the 95th percentile is 12.6% (zero term premium) compared to 7.9% (constant term premium).
Another feature of the proportional volatility structure is that any variation in expectations over time feeds through to a
corresponding time variation in percentiles. For example, both Figure 3 (Zero Term Premium) and Figure 4 (Constant Term
Premium) show distributions that begin to narrow after around 20 years, as a result of the humped shape of forward rates, and
hence expectations. By assuming an expected path that is more stable over time, the Time Varying Term Premium model gives
rise to a more stable time variation in the entire distribution of rates.
6
8
The difference occurs due to the non-linear relationship between prices and forward rates, and is sometimes referred to as a ‘convexity term’.
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Figure 3: Interest rate distributions (Zero Term Premium) – one year rate (LHS) and ten year rate (RHS)
25%
25%
20%
Percentiles 95% to 99%
15%
Percentiles 75% to 95%
Percentiles 50% to 75%
10%
Percentiles 25% to 50%
Ten year rate
One year rate
20%
Percentiles 5% to 25%
5%
Average
0
5
10
15
20
25
Percentiles 75% to 95%
Percentiles 50% to 75%
10%
Percentiles 25% to 50%
Percentiles 5% to 25%
5%
Percentiles 1% to 5%
0%
Percentiles 95% to 99%
15%
Percentiles 1% to 5%
Average
0%
30
0
5
10
15
20
25
30
Time (years)
Time (years)
Figure 4: Interest rate distributions (Constant Term Premium) – one year rate (LHS) and ten year rate (RHS)
25%
25%
20%
Percentiles 95% to 99%
15%
Percentiles 75% to 95%
Percentiles 50% to 75%
10%
Percentiles 25% to 50%
Ten year rate
One year rate
20%
Percentiles 5% to 25%
5%
Average
0
5
10
15
20
25
Percentiles 75% to 95%
Percentiles 50% to 75%
10%
Percentiles 25% to 50%
Percentiles 5% to 25%
5%
Percentiles 1% to 5%
0%
Percentiles 95% to 99%
15%
Percentiles 1% to 5%
Average
0%
30
0
5
10
Time (years)
15
20
25
30
Time (years)
Figure 5: Interest rate distributions (Time Varying Term Premium) – one year rate (LHS) and ten year rate (RHS)
25%
25%
20%
Percentiles 95% to 99%
15%
Percentiles 75% to 95%
Percentiles 50% to 75%
10%
Percentiles 25% to 50%
Percentiles 5% to 25%
5%
Percentiles 1% to 5%
Average
0%
0
5
10
15
Time (years)
20
25
30
Ten year rate
One year rate
20%
Percentiles 95% to 99%
15%
Percentiles 75% to 95%
Percentiles 50% to 75%
10%
Percentiles 25% to 50%
Percentiles 5% to 25%
5%
Percentiles 1% to 5%
Average
0%
0
5
10
15
20
25
30
Time (years)
Figures 3to 5 illustrate the sensitivity of interest rate distributions to subjective assumptions about risk premia. While such
assumptions are largely irrelevant for pricing applications, in general they can have a large impact on risk measures that depend on
the real-world distribution of rates, as we will see in Section 5 below.
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4. The impact of calibration choices on counterparty exposure measures
We consider the measurement of counterparty exposure for a single netting set, consisting of vanilla GBP swap contracts maturing
between 5 and 7 years of the start date. This netting set has an initial exposure of £0.39m at end-December 2012, and is generally
exposed to rises in interest rates.
Firstly we compare the effect of different volatility assumptions on the distribution of exposure over a one year horizon. Figures 6
and 7 show the distributions of projected mark-to-market values and exposures 7 at monthly time steps over one year. Figure 6
shows the distributions under a Point-in-Time calibration, while Figure 7 shows the distributions under a Through-the-Cycle
calibration. Both calibrations assume zero term premium, with the yield curve approximately expected to follow the initial forward
curve on average. However, the calibrations differ significantly in their volatility assumptions, with the PIT calibration being far
more volatile than the corresponding TTC calibration at this date (recall Figure 1).
10
9
8
7
6
5
4
3
2
1
0
-1
-2
-3
Percentiles 95% to 99%
Percentiles 75% to 95%
Percentiles 50% to 75%
Percentiles 25% to 50%
Exposure (£m)
MtM value (£m)
Figure 6: Counterparty exposure distributions (Point-in-Time volatility) – Mark-to-Market value (LHS) and exposure (RHS)
Percentiles 5% to 25%
Percentiles 1% to 5%
Average
0
1
2
3
4
5
6
7
8
10
9
8
7
6
5
4
3
2
1
0
-1
-2
-3
9 10 11 12
Percentiles 95% to 99%
Percentiles 75% to 95%
Percentiles 50% to 75%
Percentiles 25% to 50%
Percentiles 5% to 25%
Percentiles 1% to 5%
Expected Exposure
PFE 95
0
1
2
3
Time (months)
4
5
6
7
8
9 10 11 12
Time (months)
10
9
8
7
6
5
4
3
2
1
0
-1
-2
-3
Percentiles 95% to 99%
Percentiles 75% to 95%
Percentiles 50% to 75%
Percentiles 25% to 50%
Percentiles 5% to 25%
Percentiles 1% to 5%
Average
0
1
2
3
4
5
6
7
8
Time (months)
9 10 11 12
Exposure (£m)
MtM value (£m)
Figure 7: Counterparty exposure distributions (Through-the-Cycle volatility) – Mark-to-Market value (LHS) and exposure (RHS)
10
9
8
7
6
5
4
3
2
1
0
-1
-2
-3
Percentiles 95% to 99%
Percentiles 75% to 95%
Percentiles 50% to 75%
Percentiles 25% to 50%
Percentiles 5% to 25%
Percentiles 1% to 5%
Expected Exposure
PFE 95
0
1
2
3
4
5
6
7
8
9 10 11 12
Time (months)
We note that expected path of the mark-to-market value of the portfolio is approximately the same for both calibrations, being
£1.01m after one year in the PIT calibration, compared to £0.93m in the TTC calibration. This reflects the fact that the term
premium assumption is the same in both cases, while expected value is relatively insensitive to the assumed volatility of rates.
Since exposure is a non-linear function of value, the expected path of the exposure of the portfolio is more sensitive to the
volatility assumption, being £1.37m after one year in the PIT calibration, compared to £0.96m in the TTC calibration.
As expected, upper percentiles are even more sensitive to the volatility assumption, with the 95th percentile of exposure (PFE 95)
being £5.41m after one year in the PIT calibration, compared to £2.31 in the TTC calibration.
7
10
We define the exposure as max(MtM value,0) and assume that there are no collateral agreements for simplicity.
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Secondly, we compare the effect of different term premium assumptions. Figures 8-10 show the distributions of projected markto-market values and exposures at annual time steps over seven years (the maturity of the longest swap in the portfolio).
Figure 8: Counterparty exposure distributions (Zero Term Premium) – Mark-to-Market value (LHS) and exposure (RHS)
7
7
6
6
5
4
Percentiles 95% to 99%
3
Percentiles 75% to 95%
2
Percentiles 50% to 75%
1
Percentiles 25% to 50%
Exposure (£m)
MtM value (£m)
5
Percentiles 5% to 25%
0
Percentiles 1% to 5%
-1
Average
-2
0
1
2
3
4
5
6
4
Percentiles 95% to 99%
3
Percentiles 75% to 95%
2
Percentiles 50% to 75%
1
Percentiles 25% to 50%
0
Percentiles 5% to 25%
-1
Percentiles 1% to 5%
-2
Expected Exposure
-3
PFE 95
0
7
1
2
3
4
5
6
7
Time (years)
Time (years)
Figure 9: Counterparty exposure distributions (Constant Term Premium) – Mark-to-Market value (LHS) and exposure (RHS)
7
7
6
6
5
4
Percentiles 95% to 99%
3
Percentiles 75% to 95%
2
Percentiles 50% to 75%
1
Percentiles 25% to 50%
Exposure (£m)
MtM value (£m)
5
Percentiles 5% to 25%
0
Percentiles 1% to 5%
-1
0
1
2
3
4
5
6
Percentiles 75% to 95%
3
Percentiles 50% to 75%
2
Percentiles 25% to 50%
1
Percentiles 5% to 25%
0
Percentiles 1% to 5%
-1
Average
-2
Percentiles 95% to 99%
4
Expected Exposure
-2
PFE 95
0
7
1
2
Time (years)
3
4
5
6
7
Time (years)
Figure 10: Counterparty exposure distributions (Time Varying Term Premium) – Mark-to-Market value (LHS) and exposure (RHS)
7
7
6
6
5
4
Percentiles 95% to 99%
3
Percentiles 75% to 95%
2
Percentiles 50% to 75%
1
Percentiles 25% to 50%
Percentiles 5% to 25%
0
Percentiles 1% to 5%
-1
Average
-2
0
1
2
3
4
Time (years)
5
6
7
Exposure (£m)
MtM value (£m)
5
Percentiles 95% to 99%
4
Percentiles 75% to 95%
3
Percentiles 50% to 75%
2
Percentiles 25% to 50%
1
Percentiles 5% to 25%
0
Percentiles 1% to 5%
-1
Expected Exposure
-2
PFE 95
0
1
2
3
4
5
6
7
Time (years)
While differences in term premia are expected to have a relatively small effect on exposure over short horizons (where the
volatility assumption is likely to have a more significant impact), we observe that differences in term premia assumed here have a
significant impact over the life of the portfolio.
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To illustrate further, Figure 11 compares expected exposure and PFE 95 over the life of the portfolio, for the three different term
premium assumptions. In all cases the expected exposure and PFE 95 peak at around 3 years, 8 months. At this point the expected
exposures are £1.02m (constant term premium), £1.34m (zero term premium) and £1.91m (time varying term premium), while the
PFE 95 is £2.92m (constant term premium), £3.53m (zero term premium) and £4.54m (time varying term premium). These results
are consistent with the order of expected paths produced by the three different calibrations, as shown in Figure 2.
7
7
6
6
5
5
4
Time Varying Term
Premium
3
Zero term Premium
2
PFE 95 (£m)
Expected Exposure (£m)
Figure 11: Profiles of expected exposure (LHS) and PFE 95 (RHS)
4
Time Varying Term
Premium
3
Zero term Premium
2
Constant Term
Premium
1
0
Constant Term
Premium
1
0
0
1
2
3
4
5
Projection horizon (years)
6
7
0
1
2
3
4
5
6
7
Projection horizon (years)
Of course these results, and the relative differences between them under different calibration assumptions, are dependent on the
specific characteristics of the netting set being analysed. For example, if we changed the sign of the notionals in this particular
netting set, the EE and PFE are then greatest under the constant term premium calibration and smallest under time varying term
premium calibration.
5. Summary
Monte Carlo simulation of market risk factors is a key step in the models used by banks to measure counterparty exposure, and
requires the bank to make subjective assumptions around ‘real world’ distributions of risk factors. The sensitivity of results to such
highly subjective assumptions is a source of model risk for firms, and naturally an area of particular scrutiny by regulators who
require firms to be able to justify assumptions and quantify the sensitivity to them.
There are a number of technical challenges associated with building and calibrating market risk models for this type of application
– Monte Carlo scenarios need to be generated under real-world probabilities, over relatively long time horizons, and over multiple
multi risk factors. In this paper we have illustrated the impact of key real world scenario generation modeling and calibration
assumptions on standard measures of counterparty exposure using the Barrie & Hibbert Economic Scenario Generator (ESG), a
scenario generator widely used in the insurance industry for similar long-term, real-world, multi-asset applications.
Firstly, we have considered the sensitivity of volatility assumptions to the weighting put on current market conditions, comparing
a Point-in-Time volatility assumption (heavily weighted on current market conditions as implied by market option prices) to a
Through-the-Cycle volatility assumption (based on long term historical data and therefore a relatively low weighting on current
market conditions). Such assumptions can have a material impact on the entire distribution of future exposure at all future
horizons, including Expected Exposure over one year.
Secondly, we have considered the impact of risk premia assumptions. Though such assumptions have a relatively low impact on
the distribution of exposure in the short-term, they can have a material effect over the life of a netting set, and hence in the
calculation of PFE 95 for the purpose of assessing exposure against credit limits.
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