Yield curve stress Insights

Insights
Yield curve stress
Some thoughts on the challenges of building yield
curve stress scenarios for solvency capital assessment
John Hibbert
[email protected]
The purpose of this note is to provide a summary of the high-level approach taken by
Barrie & Hibbert to modelling extreme movements in yield curves for the purposes
of setting solvency capital within a VaR framework. I would like to give an overview
of the resources clients can use to understand the choices available to them and
associated challenges.
Our general approach
For the purposes of projecting financial market risk factors (including yield curves) Barrie &
Hibbert provides its clients with access to:
▪▪
A range of alternative models
▪▪
Target joint distributional assumptions for risk factors and supporting analysis
▪▪
A set of ‘standard’ calibration assumptions designed to match model outputs to target
distributional characteristics
▪▪
Tools and advice to allow clients to calibrate model outputs to be consistent with their own
views.
It is important to highlight that the target distributional assumptions may be developed
independently of models. Given a forward-looking target distribution, different models will
offer different numbers of degrees of freedom and so users should consider how this freedom
is used. Since all models provide only imperfect representations of real-world dynamics, model
users should understand the material sensitivities of any risk management problem and aim to use
model calibration freedom in the most efficient way. As an example, consider a situation where
risk capital is driven by a certain quantile of a risk factor distribution. If we were to set targets for
the first 4 moments of the distribution (mean, standard deviation, skew and kurtosis) but the
model used for projection generates a simple Gaussian (Normal) distribution, it would usually
only be possible to match 2 moments exactly. In practice the modeller would be expected to use
this freedom to exactly match the key quantile together with some other quantile point or moment
of the distribution. The normal model is not capable of matching the full target distribution in this
case.
Conditional forecast?
A fundamental question in specifying stresses to a risk factor is whether they should be constructed
conditional on current financial market conditions (‘point-in-time’ / PIT) or, alternatively, whether
stresses should be unconditional and designed to be consistent with long-run behaviour
(‘through-the-cycle’ / TTC). The views of market practitioners, empirical observation and
the typical elevated cost of option protection would all suggest that the PIT measures should
probably be more severe during periods of stress. This might be viewed as procyclical since these
conditional stresses will tend to raise up-front capital requirements at times when new capital is
costly and difficult to raise. As a result, regulators have been reluctant to provide clear guidance
on this choice which can have a major impact on the magnitude of stress tests. Solvency II
regulations are especially perverse on this point. For the standard formula, stresses appear to
be either TTC or dampened post-stress1. Firms operating internal models are required (by the
various tests specified for model approval) to apply objective stresses which would be expected
to look quite different to standard formula stresses. There is certainly a considerable challenge
for regulators but this (procyclicality) can be viewed as an inevitable consequence of using a
market-consistent valuation approach in conjunction with a short-term VaR-style capital measure.
Indeed, it is possible to argue that use of TTC stresses could have procyclical effects in that, whilst
it might hide the up-front capital requirement, firms will still be required to source additional
capital when they mark to market at year end following extreme movement in risk factors.
Given this uncertainty in the regulators’ approach, Barrie & Hibbert provides both unconditional and
conditional calibrations for short-term stress testing purposes. Note that, in practice, some features of
the conditional VaR calibration are matched to unconditional views (e.g. distribution type, correlation
targets, term premium) and some features of our ‘multi-year’ calibrations are conditional (initial
yield curve, term premium and absolute yield curve stress – since the proportional volatility is fixed).
Finally, it is worth pointing out that a more sophisticated and complete model – in the sense of
describing all real-world dynamics – might remove the need for this dual model and parameter
selection. We actively research more powerful models but it should be stressed that this power
nearly always comes at the cost of additional model complexity which places a burden on users
to understand and communicate. In practice, the choice of model will depend on how users
choose to manage this trade-off.
Barrie & Hibbert calibrations at end-2010
Given these considerations, at year-end 2010 Barrie & Hibbert produced a number of alternative
‘standard’ model calibrations for clients. In addition, some users will have developed their own
assumptions and used Barrie & Hibbert software tools to calibrate. In addition, firm-specific
calibrations have been tailored to client specifications on an individual basis. Let us look at the
‘standard’, off-the-shelf choices:
a. A ‘multi-year’ calibration for real-world projection (as opposed to valuation).
The main consideration in this calibration is the long-term properties of the model where
we avoid making strong statements about differences between economies and currencies
on an ‘unconditional’ basis. We believe this is suitable for long-horizon projection work
and, given that the interest rate model typically used for this purpose projects proportional
(lognormal) changes, it will tend to reduce extreme changes when initialised low (and
increase them when initialised high). The model offers limited degrees of freedom and
so care should be taken to understand which properties of the projection have a material
impact on model application.
Note that this type of calibration is prepared with two alterative treatments for the behaviour
of interest rate term premia. Firstly, where the term premium is fixed over the projection.
Whilst this is a simple assumption, it can result in paths for expected (i.e. mean) short rates
that may be viewed as unreasonable. Secondly, where term premia are allowed to vary over
the projection horizon and we manage the expected path for future short-term interest
rates so it is consistent with expectations. Calibrations are prepared across 29 different
economies. Each note provides a high-level description of the methods used for estimation.
b. A conditional 1-year ahead VaR calibration of government yield curves (24 currencies).
Here, the targets for key quantiles are estimated using market at-the-money swaption
implied volatilities (IV). We recognise this approach is fairly crude for number reasons.
First, we believe swaption IVs provide an upward-biased estimate for ‘true’ forwardlooking volatility – an argument for scaling them down for use in deriving a forward-looking
estimate. Second, at-the-money IVs will tend to be lower than low-strike options because of
fat-tails (relative to lognormal) in the ‘true’ distribution of interest rate changes and because
1 See also CEIOPS-DOC-03/06, Answers to the European Commission on the third wave of Calls for Advice in the
framework of the Solvency II project, May 2006 which provides a useful discussion of the two approaches and appears
to favour a PIT approach to the SCR (see paragraph 22.25).
https://eiopa.europa.eu/en/fixed-width/consultations/consultation-papers/2006-2004-consultation-papers/
consultation-papers-20-1/consultation-paper-no09/index.html
https://eiopa.europa.eu/fileadmin/tx_dam/files/publications/submissionstotheec/CEIOPS-DOC-0306Answerstothirdwave.pdf
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low-strike options will contain other hedging and capital costs which will tend to exaggerate
this bias. We implicitly assume these effects offset each other, albeit without any strong
evidence. Finally, we impose a lognormal distributional assumption.
There are other possibilities. The use of a Normal assumption for swap rate changes (rather
than lognormal) was discussed as an alternative at end-20072.
c. Conditional 1-year ahead calibration of swap curves (24 currencies)
As for government curves the forward-looking distribution is estimated using interest rate
options. Each document (across the 24 currencies) sets out targets for the tail of interest
rate distributions and compares model performance in matching them. The table below is
an extract for the year-ahead GBP swap rates at end-September 2011.
Exhibit 1.1.3: Targeted GBP swap rates: Various maturities & quantiles
Summary of standard model calibrations at end-September 2011
Primary application
Interest rate model
Multi-year projection
2F B-K+
CTP
Multi-year projection
2F B-K+
TVTP
1-year VaR
2F B-K+
Multi-year projection
LMM
Note:
CTP
# Economies
Swaps
Govt
4
29
Equity model Credit model
Constant vol/
SVJD
Constant vol/
SVJD
JLT+
-
29
24
24
SVJD
JLT+
4
11
Constant vol/
SVJD
JLT+
JLT+
CTP = Constant term premium
TVTP = Time-varying term premium
Some additional comments
Although the main emphasis of the discussion above has been on quantiles of a forward-looking
distribution for interest rates, there are other characteristics of the joint distribution that may
deserve analysis. Firstly, the correlation structure of possible changes across a yield curve may
be important. These dependencies might be specified in terms of a full correlation structure or,
commonly, by using the statistical shorthand of ‘principal components analysis’. Secondly, some
model users may focus on the expected (i.e. mean) short-term rate at a 1-year horizon. Our
models and approach generally ‘respects’ the forward rate so that the mean projected rate is
fixed in line with the forward rate. This is based on the view that shorter maturity nominal forward
rates tend to do a better forecasting job than longer maturity forward rates i.e. they have more
informational content about future interest rates than longer dated forward rates”3. Someone
who believes that there are other strong influences on short rates – such as short-term risk premia
and convexity effects which undoubtedly shape the behaviour of long rates – might then choose
to shift the mean away from the forward rate.
For multi-year projections a global, uniform assumption is adopted for interest rate volatility. We
have used this neutral assumption because we do not believe it is credible to claim that we can
differentiate between the behaviour of different economies on a 30 to 50 year horizon. Looking
back over the past century, it is not at all obvious that information on economic and inflation
performance in, say, 1950 is useful in predicting outcomes today. Of course, CEIOPS did adopt a
different approach to calibrating unconditional forward interest rates for yield curve extrapolation
purposes and we acknowledge that other (reasonable) views exist.
Real-World Interest Rate Distribution: End 2007 - 1-year ahead interest rate tails: initial analysis and discussion. http://
www.barrhibb.com/knowledge_base/article/1_year_ahead_interest_rate_tails/
3
Real-world interest rate calibration: How to set a target path for interest rates, August 2009 http://www.barrhibb.com/
knowledge_base/article/real-world_interest_rate_calibration_how_to_set_a_target_path_for_interest_/
2
www.barrhibb.com
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