Download 01 Risk aggregation: generalising dependency Relationships in the Barrie & Hibbert ESG

Global insurance risk management report
Risk aggregation: generalising dependency
Relationships in the Barrie & Hibbert ESG
Steven Morrison PhD
September 2010
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Contents
Introduction ...................................................................................................... 3
Dependency in the Barrie & Hibbert ESG ................................................................... 4
Alternative shock distribution choices ....................................................................... 6
Quantifying tail dependency .................................................................................. 7
1-year VaR case study .......................................................................................... 9
Single-factor models for interest rates and equity returns ........................................ 9
Multi-factor models for interest rates and equity returns ........................................ 12
Summary ......................................................................................................... 16
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Introduction
The use of 1-year Value-at-Risk (VaR) in insurance groups’ principle-based capital assessment has become
increasingly common in recent years, both for group-wide Economic Capital purposes and in regulatory
capital assessments such as Solvency II’s Solvency Capital Requirement (SCR). 1-year VaR capital
assessments depend critically on the assumed joint distribution of all risk factors affecting the balance sheet at
a 1-year time horizon, or, in other words, how the different sources of risk are aggregated.
When thinking about the joint distribution of a number of risk factors, it can be convenient to separate out the
distribution into two distinct components:
1.
The marginal distributions of each individual risk factor.
2.
The copula, which is a description of the dependency between risk factors.
Thinking of the distribution in this way, we can consider the effect of changing one of the individual
components while leaving the others unchanged. In particular, we can consider changing the dependency
structure through different choices of copula and examine the effect on model results (for example the SCR).
In recent years this separation of marginal distributions and copula has provided a useful tool in educating
modellers about dependency, in particular that there is more to dependency than ‘just’ correlation. This has
highlighted the concept of ‘tail dependency’, which refers to the tendency for different risk factors to
simultaneously take extreme values. Furthermore, we usually have more confidence in our view of individual
marginal distributions than on the dependency structure, so may be interested in stressing these dependency
assumptions while keeping marginal distributions fixed.
In this note, we describe how dependency arises in the ESG and how we can change dependency through
changing the distribution of the random shocks used to drive the ESG models. We then demonstrate the
effect of changing dependency in this way using a simple 1-year VaR case study.
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Dependency in the Barrie & Hibbert ESG
First, let’s clarify how dependency between risk factors arises in the Barrie & Hibbert ESG.
Barrie & Hibbert’s ESG is a multi-period, structural model. It is multi-period in the sense that it describes the
path of risk factors at multiple points in time and not just their values at a single point in time. In fact, most of
the models adopted by Barrie & Hibbert are fundamentally continuous-time models, i.e. they describe the
entire continuous path of risk factors, though in practice the models need to be implemented in such a way
that the path is sampled at discrete (e.g. monthly or annual) time steps.
It is also a structural model, in the sense that different risk factors are connected via specified structural
relationships. To give a simple example, equity total returns are constructed by adding an excess return onto
the risk-free short-term interest rate, so that when risk-free interest rates are relatively large (small) we
expect, on average, that the total nominal returns of risky assets’ will also be relatively large (small). The use
of a multi-period model also builds in certain structural relationships over time, such as mean-reversion of
interest-rates, i.e. when interest rates are relatively large (small) relative to their average levels we expect, on
average, that they will decrease (increase). So dependency between risk factors at any given time horizon
comes about partly because of the assumed structural relationships between them and the assumed
structural relationships over time.
Exhibit 1 indicates the main structural relationships between risk factors in Barrie & Hibbert’s ESG.
Exhibit 1
Overview of Barrie & Hibbert’s ESG structure
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The green boxes in Exhibit 1 indicate stochastic models for individual risk factors. Within each of these
models, stochastic changes in risk factors (over a model time step, such as one month or one year) arise due
to random ‘shocks’, sampled using a random number generator. Furthermore, the shocks to different risk
factors are, in general, dependent. They are sampled from some joint distribution, with a dependency
structure. So dependency between risk factors also arises due to the assumed dependency between shocks.
In summary, dependency between risk factors at any given time horizon arises through a combination of:
1.
The assumed model structure.
2.
The assumed joint distribution of the shocks.
If we want to change the dependency between risk factors, we can do so either by changing model structure,
or the joint distribution of the shocks, or a combination of the two.
Changing the dependency structure through choice of model structure is attractive, since it helps us to
explain dependency in fundamental terms. As noted above, we usually have little confidence in dependency
relationships based on analysis of historical data alone, largely because there are so few historical
observations to base such an analysis on. Imposing structure on the model naturally ‘shapes’ the dependency
structure in a way that reflects our fundamental economic views.
Furthermore, model structure allows us to capture fundamental relationships between dependency and
marginal distributions. Though it can be convenient to think about dependency separately from marginal
distributions, this doesn’t reflect the fact that both are driven by the same fundamental economic drivers. For
example, it seems natural that correlation between different equity markets increases at times of high
volatility in all individual markets. The fundamental economic driver here is the volatility of the global
influences of equity market returns, and this affects both the marginal distribution of individual markets and
their dependency at the same time. It turns out that this feature can be built into the model structure relatively
easily, and Barrie & Hibbert provide a model that does this – the SVJD model. So, in this case the user can
change dependency via changing model structure, though the choice of structure also means that the
marginal distributions change (naturally) at the same time.
We believe that coherent economic structure has a crucial role to play in rigorous multi-asset real-world
modelling. However, making fundamental changes to model structure can be a significant task. It may also be
useful to have some flexibility to make quicker changes to dependency relationships in the model, for
example, for the purposes of testing capital assessment results’ sensitivities to dependency relationships
where we may not have strong views on model structure to help guide these assumptions. Interest in such
functionality could arise from a firm’s consideration of the model risk in their Internal Model, or from specific
requests arising from their regulator’s review of their Internal Model implementation. The alternative method
of changing dependency by changing the joint distribution of shocks provides such an approach, and can be
implemented easily using the Barrie & Hibbert ESG’s Shock Import functionality1.
1
See ESG help for details on how to do this.
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Alternative shock distribution choices
Using the Shock Import functionality, the user can import any shocks of their choosing. In this note we will
focus on shocks generated from a particular class of distributions – the joint distribution having standard
normal marginal distributions and a t copula.
As described in the introduction, a general multivariate distribution can be described in terms of its individual
marginal distributions along with its copula, which separately describes its dependency. Such a separation
allows one to construct ‘meta-distributions’ by combining different copulas and marginal distributions. For
example, one could construct a bivariate distribution where one variable has a uniform distribution, the other
has a normal distribution and the copula is that of a t distribution with 5 degrees of freedom.
Here we will consider shocks generated from the class of meta-distributions having standard normal marginal
distributions but with different t copulas. This means that the marginal distributions of shocks are unchanged
from those assumed in the ESG’s internal random number generator, but the dependency structure is more
general. By retaining the same marginal distributions the intention is to have minimal impact on the resulting
marginal distributions of the risk factors. It should be noted that in practice, depending on the ESG model
structure chosen, changing the shock copula may have some effect on the marginal distributions of risk
factors (even if the marginal distributions of the shocks are unchanged). This is discussed further in the case
study below.
The choice of the t copula is by no means the most general assumption we can make. However, it is a natural
generalisation of the Gauss copula (the copula assumed in the ESG’s internal random number generator).
Like the Gauss copula, it is parameterised by a correlation matrix specifying the ‘overall level’ of correlation
between risk factors. However, we now have a single additional parameter – the ‘number of degrees of
freedom’ of the corresponding t distribution, with the Gauss copula being the limiting case as the number of
degrees of freedom tends to ∞. This additional degree of freedom allows us to control the level of ‘tail
dependency’ - the tendency for different risk factors to simultaneously take extreme values.
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Quantifying tail dependency
Dependency can be described in many ways. There are a number of correlation coefficients describing the
overall level of dependency, and a number of measures of tail dependency, which measure the level of
dependency at the extremes of the distribution. Though such statistics do not tell us everything about the
dependency structure, they summarise some of the key features of dependency and allow us to compare
different joint distributions in relatively simple terms. Here we will quantify tail dependency as follows.
Firstly, we define the Probability of Joint Quantile Exceedance, at various probability levels. For a given
probability level, this measures the probability that both risk factors simultaneously exceed that probability
level. Mathematically:
where
is the probability level and
distributions of the two risk factors
.
are cumulative probability distributions for the marginal
We then define the Conditional Probability of Joint Quantile Exceedance as the conditional probability of one
risk factor exceeding each probability level, given that the other risk factor exceeds that level.
Mathematically:
Note that we have defined these statistics in terms of the left hand tail of each of the marginal distributions,
but similar statistics can be defined for the other ‘quadrants’ of the joint distribution.
The calculation of these statistics is indicated in Exhibit 3, which shows 10,000 samples from three different
copulas – the Gauss, t5 and t2 copulas2. In all cases the marginal distributions are standard normal and the
linear correlation is approximately +0.53. The red lines indicate quantiles at the p=1% level. The JPQE
measures the proportion of samples which lie in the area below the dashed red line and to the left of the solid
red line, while the CPJQE measures this number as a proportion of the samples to the left of the solid red line.
Note that the CPJQE at the 1% level is 12.9% for the Gauss, 25.9% for the t5 copula and 39.5% for the t2
copula (despite the fact that the overall level of correlation is the same in both cases), indicating a stronger
level of tail dependency as we decrease the number of degrees of freedom for the t copula.
Exhibit 3
10,000 pairs of shocks with standard normal marginal distributions, correlation = 0.5 but different copulas
2
Here we refer to the copula of the t distribution with 5 degrees of freedom as the t5 copula, and the copula of the t
distribution with 2 degrees of freedom as the t2 copula.
3
Strictly speaking, the rank correlation is exactly the same in all cases, but there are small differences in the linear
correlation.
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Exhibit 4 further illustrates the difference in tail dependency for different copulas, showing the CPJQE at
different probability levels from 1% to 10%. This shows not just the differences at different fixed probability
levels, but also indicates how the CPJQE changes with a decrease in the probability level i.e. as we go further
into the tail. In fact, for the Gauss copula, the CPJQE tends to zero in the limit as the probability level tends to
zero, while for the t copulas this limiting CPJQE is non-zero (20.7% for the t5 copula for this level of
correlation)4.
Exhibit 4
Conditional Probability of Joint Quantile
Exceedence
Tail dependency for three different choices of copula (correlation = 0.5)
40%
35%
30%
25%
t5 Copula
20%
15%
Gauss Copula
10%
5%
0%
0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%
Probability level
4
This limiting statistic is a popular measure of tail dependency, sometimes known as the ‘coefficient of (lower) tail
dependence’.
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1-year VaR case study
To illustrate the effect of changing the shock copula, we will calculate the 1-year VaR for a straightforward
asset-liability position: we will define the liability as a vanilla 10-year put option on an equity total return
index, struck at the forward price; and assume that the assets backing this liability are invested in cash.
In this case, the value of the asset-liability position in one year depends on the end-year values of three key
risk factors: the equity total return index, the 9-year risk-free rate, and the 9-year equity implied volatility. For
simplicity we will assume a fixed implied volatility5 of 20% and generate scenarios for the other two risk
factors using the Barrie & Hibbert ESG under real-world calibration assumptions.
Barrie & Hibbert’s ESG provides a flexible modelling framework with a number of different model choices
and calibrations for each risk factor. For the specific purpose of 1-year real-world projection, Barrie &
Hibbert’s recommended ESG configuration adopts ‘multi-factor’ models for both interest rates and equity
returns. By ‘multi-factor’ here, we mean that changes in each individual risk factor are generated using a
number of shocks. Our recommended real-world interest rate model, the 2-factor Black-Karasinksi model, is a
model in which interest rate changes are driven by two shocks, while our recommended equity model is a
model in which equity returns on each individual market are generated by at least 7 shocks (6 ‘systematic’
shocks, which are common to all equity markets, and a single ‘idiosyncratic’ shock per market) 6. The use of a
multi-factor interest rate model means that different points on the yield curve are less than perfectly
correlated, while the use of a multi-factor equity model provides a convenient way of describing correlation
across numerous equity markets, as well as providing a natural mechanism for generating elevated tail
dependency between equity markets (if the SJVJD model is used). So this model structure, in which both
interest rates and equity returns are generated using numerous underlying shocks, has been chosen
deliberately to give certain realistic market features that can be important for 1-year real-world projection.
However, a potential drawback of such a multi-factor approach is that there is a relatively complicated
relationship between the joint distribution of risk factors and that of the shocks. In such a model, changing
the distribution of the shocks (either by changing their marginal distributions or their copula), has a complex
effect on the distribution of the risk factors that are derived from them. For example, changing the shock
copula changes the marginal distribution of the risk factors even if the marginal distribution of the shocks is
unchanged. In order to clarify this, and other related effects, below we will also consider a simpler Barrie &
Hibbert ESG model structure in which each individual risk factor is generated using a single shock (hence a
‘single-factor’ model). The price that one pays for such a simpler model is that (depending on the exact
requirements of the model) it is potentially less realistic than the multi-factor model. However, it can provide
some useful insights before moving on to the more complex, and realistic, multi-factor model.
Of course, in considering different choices of copula between equity returns and interest rates, this naturally
leads us to the question of which is the ‘right’ one. Empirical data analysis will only be able to take us so far
here – there is unlikely to be sufficient historical annual data to allow many meaningful statements about the
precise statistical nature of this tail dependency. However, this highlights the advantage of this flexible, fast
approach to stress testing the copula assumption.
Single-factor models for interest rates and equity returns
Firstly, we consider the following single-factor models:
Interest rates: 1-factor Libor Market Model.
Equities: 1-factor constant volatility model7.
Barrie & Hibbert don’t provide a standard real-world calibration of this model configuration, but we have
prepared a calibration to be broadly consistent with our standard real-world calibrations of our recommended
(multi-factor) models at end-December 2009.
Note that for the purpose of illustration in this particular case study, since capital requirements increase when
both equities and interest rates fall, we have chosen a (small) positive correlation between equity returns and
5
Note that B&H ESG 7 does provide a stochastic equity option-implied volatility model but we have chosen to keep this
risk factor fixed in this analysis as a two-dimensional case study is simpler than a three-dimensional one!
6
In fact, further additional shocks are required if a Stochastic Volatility Jump Diffusion model is used.
7
In the ESG, this is set up using the ‘ParentEquityAssetCorrelationModel’, which is our recommended method for
modelling equities for market-consistent applications.
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interest rate changes. Our standard real-world calibrations actually assume a small negative correlation here,
which is consistent with long-term ‘through the cycle’ (TTC) correlation behaviour, though recent historical
data indicates that it would be reasonable to assume a positive correlation on a ‘point in time’ (PIT) basis.
For our case study, we have generated two sets of 100,000 scenarios under two different copula
assumptions: the Gauss copula (generated using the ESG’s internal random number generator) and the t5
copula (generated externally and imported using the shock import functionality). Apart from the different
choices of copulas for the shocks, all other model and calibration assumptions are the same in both cases. In
particular both the equity shock and the interest rate shock have standard normal marginal distributions and
we assume the same rank correlation between them.
Firstly we check the effect of changing the shock copula on the marginal distributions of our risk factors. Due
to the simple one-to-one relationship between risk factors and shocks, the marginal distributions of each of
the risk factors should be unchanged if the marginal distributions of the shocks are unchanged.
To check this, we have measured the marginal distributions of our two risk factors – the equity return over
the year, and the 9-year spot rate at the end of the year – using 100,000 scenarios. Exhibits 5 and 6 show the
left-hand tail of the distributions of the equity log-return and 9-year spot rate respectively. Both probability
distributions are shown in cumulative terms. As expected, we have been able to change the copula
relationship between the random shocks driving the ESG without changing the marginal distributions of the
risk factors.
Exhibit 5
Comparison of marginal distributions of equity return under different shock copulas (‘single-factor’ models)
10%
8%
7%
Gauss Shock Copula
6%
5%
4%
3%
2%
Cumulative Probability
9%
t5 Shock Copula
1%
0%
-45%
-35%
-25%
-15%
Equity return
Exhibit 6
Comparison of marginal distributions of 9-year spot rate under different shock copulas (‘single-factor’ models)
10%
8%
7%
Gauss Shock Copula
6%
5%
4%
3%
2%
Cumulative Probability
9%
t5 Shock Copula
1%
0%
2.5%
3.0%
3.5%
4.0%
9-year spot rate
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Changing the shock copula doesn’t change the marginal distributions of risk factors, but should change the
level of tail dependency between risk factors. To demonstrate this, Exhibit 7 compares tail dependency
statistics for the two choices of shock copula. Tail dependency is quantified using the Conditional Probability
of Joint Quantile Exceedance. We estimate these statistics empirically using the 100,000 scenarios generated
under our two different shock copula assumptions.
As expected, moving from a Gauss to a t5 copula increases the tail dependency between the shocks, and this
gives rise to increased tail dependency between the risk factors derived from these shocks.
Furthermore in this case, due to the simple one-to-one relationship between risk factors and shocks, we
expect the copula between the risk factors to be exactly the same as the copula between the shocks. As a
simple check of this, we have also calculated our tail dependency statistics under ‘equivalent’ Gauss and t5
copulas. By ‘equivalent’ we mean that these copulas have the same overall level of correlation8. Though these
tail dependency statistics don’t tell us everything about dependency, these results indicate that the
dependency between risk factors looks like a Gauss copula if the underlying shocks are generated using a
Gauss copula, and looks like a t5 copula if the shocks are generated using a t5 copula. Thus we have direct
control over the copula between risk factors via our choice of shock copula, and the tail dependencies can be
changed without impacting on the marginal distributions.
Exhibit 7
Conditional Probability of Joint Quantile
Exceedence
Tail dependency between risk factors under two different choices of shock copula (‘single-factor’ models)
Gauss Shock Copula
20%
18%
16%
14%
B&H ESG
12%
10%
Equivalent t5
Copula
8%
6%
Equivalent Gauss
Copula
4%
2%
0%
0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%
Conditional Probability of Joint Quantile
Exceedence
Probability level
t5 Shock Copula
20%
18%
16%
14%
B&H ESG
12%
10%
Equivalent t5
Copula
8%
6%
Equivalent Gauss
Copula
4%
2%
0%
0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%
Probability level
8
Here we measure ‘overall’ correlation using Kendall’s tau rank correlation measure.
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Finally, we examine the effect of the copula assumptions on the 1-year VaR for our case study –a liability of a
10-year put option on an equity fund of £1m, struck at the forward price and with backing assets invested
wholly in cash. We assume that the equity implied volatility is fixed at 20%. At end-December 2009 (using the
Government yield curve) this implies an initial liability value of £0.248m.
Exhibit 8 shows the tail of the distribution of the capital requirement under the two different choices of shock
copula. In particular, the 99.5% VaR (in excess of the time-0 liability value) is £0.261m and £0.278m under
Gauss and t5 shock copulas respectively. Unsurprisingly, the VaR capital requirement increases as we move
from the Gauss to t5 copula – the capital requirement is driven by exposures to falls in equity values and falls
in interest rates; assuming that these jointly occur with greater joint ‘severity’ fattens the tail of the distribution
of possible capital requirements, and hence increases the capital assessment. Put another way, the move
from the Gauss to t5 copula has reduced the diversification credit produced in the capital assessment. The
stand-alone equity and interest rate VaR requirements are £0.211m and £0.083m respectively9. So moving
from a Gauss to t5 copula has roughly halved the diversification benefit, from £0.034m to £0.016m.
Exhibit 8
Distribution of capital requirement under different shock copulas (‘single-factor’ models)
100.0%
99.9%
99.8%
Probability Level
99.7%
99.6%
99.5%
99.4%
99.3%
Gauss Shock Copula
99.2%
99.1%
t5 Shock Copula
99.0%
98.9%
0.20
0.25
0.30
0.35
0.40
Capital Requirement (£m)
Multi-factor models for interest rates and equity returns
In the models considered so far there is a simple one-to-one relationship between risk factors and shocks. As
we have seen, this allows us to change the copula between risk factors without changing their marginal
distributions and allows us direct control over the type of copula between risk factors.
Now we will consider a more complex multi-factor model, consisting of the following components:
Interest rates: 2-factor Black-Karasinski model
Equities: 6-factor equity model with Stochastic Volatility Jump Diffusion (SVJD)
This is Barrie & Hibbert’s standard recommended ESG configuration for 1-year real-world modelling. Here we
have used Barrie & Hibbert’s standard multi-year real-world calibrations of these models at end-2009 but, as
above, flipped the sign of the equity vs rate correlation for the purpose of illustration in this particular case
study.
As before, we have generated two sets of 100,000 scenarios under two different copula assumptions: the
Gauss copula (generated using the ESG’s internal random number generator) and the t5 copula (generated
externally and imported using the shock import functionality). Apart from the different choices of copulas for
the shocks, all other model and calibration assumptions are the same in both cases. In particular all shocks
have standard normal marginal distributions and the same rank correlation parameters are assumed.
9
These ‘stand-alone’ capital requirements are calculated by setting one risk factor to its 99.5% level and the other to its
forward level.
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Firstly we check the effect of changing the shock copula on the marginal distributions of our risk factors.
Now, since each risk factor is dependent on a number of shocks, we might expect the marginal distribution of
the risk factors to depend not only on the marginal distribution of the shocks, but also their dependency. So
even though we have assumed standard normal marginal distributions for the shocks, changing their copula,
or indeed simply changing their correlation, is expected to have some effect on the marginal distributions of
risk factors derived from these (multiple) shocks.
Exhibits 9 and 10 show the left-hand tail of the distributions of the equity log-return and 9-year spot rate
respectively, under our two different assumptions for the copula between shocks. For both risk factors
considered here, the changes in the shock copula have had a small impact on the marginal distributions of the
risk factors: the 99.5th percentile of the equity return is estimated as -45.4% and -46.3% for the Gauss and t5
copulas respectively, whilst the 99.5th percentile of the 9-year spot rate (continuously compounded) is 2.91%,
and 2.83% for Gauss and t5 copulas respectively10.
Exhibit 9
Comparison of marginal distributions of equity return under different shock copulas (‘multi-factor’ models)
10%
8%
7%
Gauss Shock Copula
6%
5%
4%
3%
2%
Cumulative Probability
9%
t5 Shock Copula
1%
0%
-45%
-35%
-25%
-15%
Equity return
Exhibit 10
Comparison of marginal distributions of 9-year spot rate under different shock copulas (‘multi-factor’ models)
10%
8%
7%
Gauss Shock Copula
6%
5%
4%
3%
2%
Cumulative Probability
9%
t5 Shock Copula
1%
0%
2.5%
3.0%
3.5%
4.0%
9-year spot rate
Exhibit 11 compares tail dependency statistics for the two choices of shock copula. As before, changing the
underlying shock copula changes the tail dependency between risk factors derived from these shocks, and
10
Of course, all percentiles are estimated using 100,000 scenarios and so are subject to sampling error.
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increasing the tail dependency of the shocks (by decreasing the number of degrees of freedom) also
increases the tail dependency between risk factors11.
However, unlike in the single-factor model case, here the copula between risk factors is, in general, different
from the copula used to generate the shocks. Exhibit 11 illustrates the difference between the (derived)
copula between risk factors and the assumed copula between underlying model shocks. In the special case
where the shocks are generated using a Gauss copula, the level of tail dependency between the risk factors is
similar to that of a Gauss copula with the same rank correlation. However, for the t5 shock copula, the copula
between risk factors is quite different from the copula between underlying shocks. If a t5 copula is used for
the shocks, the resulting tail dependency between risk factors is stronger than an equivalent Gauss copula,
but not quite as strong as an equivalent t5 copula. There is some ‘dilution’ of dependency as we move from
the model shocks to the risk factors derived from these. This comes about because of the assumed multifactor structure of the models.
Exhibit 11
Conditional Probability of Joint Quantile
Exceedence
Tail dependency between risk factors, compared to ‘equivalent’ Gauss and t5 copulas (‘multi-factor’ models)
Gauss Shock Copula
12%
10%
B&H ESG
8%
6%
Equivalent t5
Copula
4%
Equivalent Gauss
Copula
2%
0%
0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%
Conditional Probability of Joint Quantile
Exceedence
Probability level
t5 Shock Copula
12%
10%
B&H ESG
8%
6%
Equivalent t5
Copula
4%
Equivalent Gauss
Copula
2%
0%
0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%
Probability level
It is worth noting that this dilution effect, and the effect on marginal distributions, will be present to some
degree in any model in which risk factors are built up using multiple shocks. In particular, most yield curve
models are likely to be built using such a structure, unless interest rates of different maturities are assumed to
be perfectly correlated. More realistic yield curve models are likely to be driven by 2 or 3 shocks (commonly
referred to as Principal Components). So, any multi-factor interest rate model, whether an arbitrage-free
short-rate model or a non-arbitrage free PCA model, will have this same issue: changing equity / interest rate
11
The ‘lumpiness’ of these curves simply indicates sampling error in the estimation of CPJQEs, particularly as we go
further into the tail.
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tail dependency by changing the copula relationships between the multiple factor shocks will not result in the
same copula relationships between equities and rates.
Nevertheless, the ability to change the shock copula gives us control over the level of tail dependency
between risk factors, and for this particular choice of copula family, the ‘degrees of freedom’ parameter
provides a simple dial allowing us to change the overall level of tail dependency between risk factors.
Finally, we examine the effect of the copula assumptions on the 1-year VaR for our case study. Exhibit 12
shows the tail of the distribution of the capital requirement under the two different choices of shock copula.
In particular, the 99.5% VaR (in excess of the time-0 liability value) is £0.296m and £0.309m under Gauss and
t5 shock copulas respectively. Unsurprisingly, the VaR capital requirement increases as we move from the
Gauss to t5 copula – as in the single-factor case, joint large falls in equity prices and interest rates (which
increase capital requirements) are more likely when we move to the t5 copula, and hence increases the
capital assessment. In addition here the marginal distributions for both equity returns and interest rates are
fattened as we move to the t5 copula, and this also contributes to an increased capital requirement.
Due to these two contributing components to the increase in capital requirements – increased tail
dependency and increased marginal tails – it is hard to quantify the effect attributable to each component
individually. Now the stand-alone equity and interest rate VaR requirements are £0.245m and £0.091m using
the Gauss copula but £0.251m and £0.098m using the t5 copula, so the sum of these marginal contributions
to capital changes by roughly the same amount as the total capital requirement when we change copulas. In
other words, the diversification benefit is similar for both choices of copula, being £0.040m under the Gauss
copula and £0.041m under the t5 copula. This simply reflects the fact that this measure of diversification
benefit doesn’t adequately separate out the effect of dependency alone.
Exhibit 12
Distribution of capital requirement under different shock copulas (‘multi-factor’ models)
100.0%
99.9%
99.8%
Probability Level
99.7%
99.6%
99.5%
99.4%
99.3%
Gauss Shock Copula
99.2%
99.1%
t5 Shock Copula
99.0%
98.9%
0.20
0.25
0.30
0.35
0.40
Capital Requirement (£m)
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Summary
In this note, we have described a relatively straightforward way of changing dependency in the Barrie &
Hibbert ESG, through changing the distribution of the random shocks used to drive the ESG models. This
makes use of the ‘shock import’ functionality of the ESG, allowing the user to import shocks with a general
dependency structures of their choosing. Whilst Barrie & Hibbert strongly believe in the need for
economically coherent model structure for the purposes of rigorous multi-asset real-world modelling, we also
recognise the benefits of being able to easily test the sensitivity of model results to changes in tail
dependency relationships.
In this note we have developed a case study that considers the impact of changing the copula driving the
model’s shocks from a Gauss to a t copula. We have measured the change in tail dependency between risk
factors produced by this simple approach, and illustrated the effect on marginal distributions of risk factors.
We have shown the effects for different model choices – ‘single-factor’ models, where we assume a one-toone relationship between risk factor and shock, and ‘multi-factor’ models, where risk factors are built up from
a number of shocks.
In the single-factor case (e.g. a single-factor interest rate model and single-factor equity model), changes in
the copula relationship between the shocks driving the risk factors are directly transformed into changes in
the copula relationship between the risk factors, e.g. moving from a Gauss to a t5 copula between shocks
results in the equity / interest rate copula moving from Gauss to t5. Furthermore, these changes in tail
dependency can be made without having any impact on the marginal distributions of the risk factors. Thus
direct control of the risk factors’ tail dependencies can be exerted using this ESG functionality.
However, the multi-factor case (e.g. a multi-factor interest rate model and multi-factor equity model) is more
complex. Here, the one-to-one relationship between the shocks and the risk factors does not hold, e.g.
changing the copula relationship for the underlying shocks from Gaussian to t5 does not result in a t5 copula
relationship between equities and interest rates. It also results in changes in the marginal distributions of the
risk factors. An important point to note here is that this multi-factor modelling limitation does not apply only
to structural or arbitrage-free models but to any multi-factor modelling such as a purely statistical (e.g. PCA)
approach to modelling the yield curve. This is not a limitation caused by economic structure, but by the
general need for multiple factors to be used in the efficient modelling of closely related variables such as
different points on a yield curve.
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Disclaimer
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prior written permission of Barrie & Hibbert Limited (SC157210) registered in Scotland at 7 Exchange Crescent,
Conference Square, Edinburgh EH3 8RD.
The information in this document is believed to be correct but cannot be guaranteed. All opinions and estimates included
in this document constitute our judgment as of the date indicated and are subject to change without notice. Any opinions
expressed do not constitute any form of advice (including legal, tax and/or investment advice).
This document is intended for information purposes only and is not intended as an offer or recommendation to buy or sell
securities. The Barrie & Hibbert group excludes all liability howsoever arising (other than liability which may not be
limited or excluded at law) to any party for any loss resulting from any action taken as a result of the information provided
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of the securities mentioned.
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of Barrie & Hibbert Limited.
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