Universal regimes for rates and inflation: the effect of local elasticity on market and counterparty risk

Abstract

The dependence of interest rate volatility on the level of rates has both general macroeconomic significance and direct consequences on computing market risk metrics such as VAR, SVAR or ES, and counterparty credit risk modelling. Such dependence is investigated and viewed in terms of local elasticity. A new regime at low and negative rates with volatility independent of the level of the rates is found, and three other regimes reported by Deguillaume et al. (The nature of the dependence of the magnitude of rate moves on the rates levels: A universal relationship. Quant. Financ., 2013, 13(3), 351–367] are confirmed with more recent data and a larger pool of currencies. A preliminary study into the existence of regimes for break-even inflation is also conducted and indications of regimes are found. One of these regimes has no equivalence in interest rates; it exhibits negative elasticity slope which may imply a similar regime if rate levels also reach sufficiently negative values. The overall shape of inflation elasticity resembles a strangle payoff, and we hypothesise that this directly reflects markets’ response to macroeconomic policy of inflation targeting and also indirectly links such policy to the nominal rate regimes. We demonstrate that the incorporation of such regimes in market risk modelling improves its predictive capacity, and for counterparty risk modelling has significant impact on risk and regulatory calculations.

Keywords:

• Interest rates
• Inflation
• Elasticity
• Volatility
• Market risk
• VAR
• Counterparty risk
• PFE

Subject Classification Codes:

• C50
• E40
• G17

Acknowledgements

We would like to thank and acknowledge our colleague Fares Triki for major contribution to programming enabling efficient aggregation of results, assistance with calculations, helpful discussions and the review of this manuscript; Mario Bolognesi for helpful discussions; Guillaume Cheron for helpful discussions and assistance with the investigation of evolution of volatility vs. rate level via chronological plots; our former colleague Mark Lichtner, who was employed by BNP Paribas in the early part of the project, for helpful discussions and contribution to the research into least-squares minimisation scaling method; Vera Minina, our co-investigator in the separate study of stochastic volatility for the kind permission to refer to the study’s results.

Disclosure statement

The authors report no conflicts of interest. The views expressed in this paper are those of the authors and do not necessarily reflect the views and policies of BNP Paribas.

Notes

† Although neither our nor the DRP approach is designed to capture regime changes specific for higher moments, knowing the general behaviour of the second moment as a function of rates already delivers a strong improvement to the quality of market and counterparty risk modelling. This will be demonstrated later in the article.

‡ As introduced in the context of Derman and Kani (1994) and Dupire (1994).

§ Locality implies the dependence on $r$, so an elasticity function is by definition local. On the other hand, the elasticity coefficient in the studies reviewed in this section is treated as a constant, i.e. a global coefficient, so making it dependent on $r$ will make it local elasticity coefficient.

† In the period from October 1979 to September 1982 Federal Reserve Board targeted monetary aggregates.

‡ He investigated tenors from 3 months to five years. We have also observed that elasticity is dependent on a tenor, as reported in the next section.

§ We investigated elasticity for 1-day and 10-day returns, the latter in the context of FRTB, and also found common patterns. The results are discussed in the next section. DRP find the three regimes persist between 1- and 100-day returns, with absolute regime being restricted to a progressively narrower range for longer returns.

¶ We will return to results of Ball and Torous (1999) later in this section in the context of the discussion on stochastic volatility.

‖ In our study, we also investigate the impact of high kurtosis by exploring elasticity of ‘filtered’ data, with jumps removed. This indeed leads to some reduction of elasticity at high rates.

** In our investigation, we do find fundamental differences between tenors with longer tenors generally having lower relative elasticity (a concept to be introduced later).

† An approach based on ‘relative elasticity’ will be introduced in the next section, but for reader’s convenience we also give a flavour here. Recall the assumption that local volatility can be presented in two parts: $G\left(r,t\right)=E\left(r\right)\sigma \left(t\right)$. As written, neither the elasticity nor volatility parts are uniquely defined. Our aim is to define and effectively normalise $E$. By introducing relative elasticity $E^{\mathrm{\prime }}\left(r\right)$ as the ratio of $E\left(r\right)$ to the elasticity at some anchor rate, for example zero: $E^{\mathrm{\prime }}\left(r\right)=E\left(r\right)/E\left(0\right)$, the whole value of $G$ could be transferred into the volatility part: ${\sigma }^{\mathrm{\prime }}\left(t\right)=E\left(0\right)\sigma \left(t\right)$, resulting in the representation of local volatility as $G\left(r,t\right)=E^{\mathrm{\prime }}\left(r\right){\sigma }^{\mathrm{\prime }}\left(t\right)$, where relative elasticity is 1 for all currencies at zero rate. Then rate regimes corresponds to the slope or relative changes of $E^{\mathrm{\prime }}$ (which is also the relative change of $E$) and this slope can be aggregated across currencies, whilst the resulting function $E^{\mathrm{\prime }}$ can be used to improve market and counterparty risk model.

‡ Negative rates of Swiss franc in 1972–1978 are a notable exception.

§ Exact time of the start of the rise depends on the index type; see Garriga (2016).

¶ Appendix also provides the rate level coverage per currency.

‖ For historical completeness the earlier results of super calibration of generalised Hull-White model presented in Hull and White (2001), already indicated the existence of three regimes: close to lognormal at low rates, close normal at mid-rate and close to lognormal again at high rates.

** This matches findings by Koutmos (1998) for realised volatility which we discussed earlier.

† Both Andersen and Lund (1997) and Ball and Torous (1999) introduce into CKLS a stochastic volatility via mean-reverting process for logarithm of rate’s variance, with its innovations uncorrelated with the rates’ innovations. However the latter article offers additional observations on the dynamics of stochastic volatility pertinent to our later discussion. Ball and Torous (1999) compare the dynamic of rates’ volatility with equity and find that the correlation between volatility and rates returns is negligible, unlike the negative correlation in the case of equity and its volatility; whilst mean reversion of rate’s volatility is faster than that for equity. In a separate investigation we also found negligible correlation between (daily) implied volatility returns and that of rates returns for a more recent period (2008–2016) for EUR, GBP, JPY and USD (results not presented).

‡ FRTB: fundamental review of the trading book.

† There are a few additional comments with regards to the use of filtering. We need to mention the possibility that some of this filtering may remove not ‘true’ extreme returns, but possible data errors in the input time series (obtained from Bloomberg) as the errors often manifest themselves as discontinuity of level and thus jumps in the returns. It is not possible to cross check manually every return for 40 currencies up to 30 years, but visual examination reveals overall good quality of data. Also it is not clear a priory whether removal of bad returns should impact elasticity in any direction. Finally, in case of fully automated calibration of elasticity for risk management systems, the use of filtered data might prove more stable and reliable, as it constitutes a primitive filter against poor data which may be especially relevant for emerging currencies.

‡ See Chorniy and Greenberg (2015) for further discussion of jump detection in financial time series.

§ Function ksmooth in R was used. We are grateful to Mark Lichtner for suggestions and help with Nadaraya-Watson kernel and Imhof method (stated below).

¶ This has been computed by the Imhof method similar to the imhof function in R.

† We will show that relative local elasticity is suitable also for counterparty risk modelling.

† In particular, upon generalising to other tenors, a universal relative elasticity surface can be determined which would then be completely specified with a set of parameters making the operational set-up and regular maintenance relatively straightforward.

‡ Two Gompertz functions $g_1\left(x\right)$ and $g_2\left(x\right)$ are used and joined at a mid-level point.

§ We conducted an additional brief investigation into higher rate currencies covering rates’ levels above 15% (ARS, EGP, GHS, KZT, NGN, UAH). Our observations suggest that on average positive elasticity continue into much higher rate levels – up to 30%’s, however we did not consider the data sufficient to extend regime study to this levels.

¶ Also Burke (2017) reports that for the 1978–2016 period the 1 year UK Libor shows an increase of the 4th moment with rates level, with faster increase after 2.5%. In further communication he stated that if 1-year Libor rates are assumed to follow jump-diffusion process then at lower rates (below about 2.5%) 2nd moment is well explained by diffusion part, whilst at higher rates jumps play stronger role.

‖ In common with literature on elasticity we use ‘positive elasticity’ to describe positive (proportional) dependence of a variable on rates’ level. For elasticity function $E$, this means positive slope. Negative and zero elasticity relate to negative and zero slopes of $E$ correspondingly.

† We use elasticity of 1-day filtered returns.

† Recall that in the absence of stochastic component, volatility is scaled by elasticity function, whilst VAR, for a linear position, is proportional to volatility.

† This is the natural choice if the calibration frequency is daily.

‡ Both in this counterparty risk example, and in market risk discussion earlier, we used the relative elasticity function, which is the result of integration of elasticity gradient function shown on figure 4. For completeness we note that in case of infinitely small Monte Carlo steps $G\left(r\right)$ could be also be evolved from $r_{j,t-1}$ to $r_{j,t}$ using the gradient curve directly.

† As discussed in the context of designing counterparty risk model (Kotecha and Chorniy, 2010), BEI has significantly different dynamics from realised inflation, with market in the past over-predicting the highs and lows of realised values, so in the case of elasticity it is also possible that the expectation of inflation driven by capital markets will show different pattern from realisations driven by country’s economy.

‡ For example, ‘priced volatility’ mechanism (see, for example, Chorniy and Greenberg 2015 for review and detailed discussion) which connects equity and volatility, explains the negative link between volatility and stock price by assuming that volatility is the measure of risk and is ‘priced’ by investors, thus an increase of the volatility (e.g. in response to bad news) makes equity riskier and raises the required return on equity, causing an (immediate) reduction of the stock price. For the volatility to drive the stock price in this way, the volatility has to be persistent as investors need to expect higher future returns over a significant period of time. If this mechanism is to act on government securities and correspondingly on nominal rates as it does on equity, then the correlation of volatility returns with rates returns should move from zero to positive correlation and mean reversion should weaken as volatility increases need to be persistent (see comments on stochastic volatility for rates in the preceding section). In this new world modelling stochastic volatility for rates may acquire higher priority. The general discussion which mechanisms and why govern relations within various asset classes between spot values and its volatilities lies outside the scope of this article.

† https://www.bankofengland.co.uk/statistics/research-datasets.

‡ If the nominal rate is the sum of real rate and inflation, and both are stochastic process, then sum of their variances (proportional to squared elasticity) approximately gives the variance (proportional to squared elasticity) of nominal rate.

† Our preliminary investigations support this view. Shifted lognormal behaviour for short tenors has also been found by Andrei Lyashenko (personal communication) which further supports our hypothesis of a stochastic floor.