The Wilkie Model (1984-1995)

The Wilkie Model was the ubiquitous stochastic asset model for British actuarial use from the mid-1980s to the end of the 1990s—and, to a lesser degree, beyond. Its genesis can be found in the stochastic equity modelling that Wilkie developed and applied as a member of the Maturity Guarantees Working Party. Wilkie further developed this work into a broader stochastic asset model—which quickly became known as the Wilkie model—in a paper presented to the Faculty of Actuaries in 1984 and published in the Transactions of the Faculty of Actuaries in 1986.[1]

The Wilkie model shared many stylistic and methodological similarities with the equity modelling that Wilkie had developed for the Maturity Guarantees Working Party. Equity prices were again modelled as the ratio of dividend pay-outs to dividend yields using a time series model structure. The model was expanded beyond the working party’s asset modelling in two key dimensions: stochastic models for the long-term government bond (consol) yield and inflation were developed; and a ‘cascade’ structure was introduced such that the time series models of dividends, dividend yields and consol yields included dependencies on past and present inflation as well as past values of themselves. The modelling approach was heavily based on time series analysis and postulated some quite convoluted statistical relationships in the joint behaviour of equities, consols and inflation. The model structure and calibrations were driven directly by the relationships Wilkie found in the historical data (based on UK series from 1919 to 1982). He was not concerned with imposing many economic restrictions or null hypotheses on the projected asset behaviour. These characteristics were again evident when Wilkie published an updated version of the model in 1995.[2] This update extended the asset coverage of the model, most notably by including models of shortterm government bond yields and property.

Wilkie’s preference for modelling historical behaviour as he found it, without imposing strong economic prior beliefs, led to some striking implications for the risk/reward profile of different asset classes and, especially, for what could be generated by dynamic asset strategies. The structure and calibration of the Wilkie model implied that most variation in equity dividend yield was due to changes in the expected future return of equities (without any corresponding change in risk) rather than being a signal of a significant change in dividend growth expectations. As Wilkie himself noted in a 1986 paper,[3] this meant that dynamic asset strategies that switched between equities and bonds using simple rules based on their relative yield levels could generate returns in excess of those produced by simply buying and holding equities.

This begged an interesting quasi-philosophical modelling question: should actuaries’ risk and capital assessments include ‘credit’ for investment strategies that take account of projected market ‘inefficiencies’? Should these inefficiencies be assumed to prevail into the future? Are these strategies gaming the asset model or are they reflective of a fundamental reality that ought to be incorporated into the assessment of the risk profile of the business? Such issues remain at large in twenty-first-century principle-based reserving.

These questions generated very little discussion in the 1980s actuarial papers and sessional meetings. They did, however, attract greater attention (and discomfort) amongst actuaries in the 1990s, particularly from those in the vanguard of modernising actuarial thought through the firmer embrace of financial economic principles. Arguably, this is perhaps ironic as Fama and French’s 1988 dividend yield and expected return empirical analysis that emerged between the first publication of the Wilkie model and the profession’s eventual interest in financial economics could provide some intellectual support from financial economics for Wilkie’s approach.

Risk and return anomalies also existed in the relative behaviour of other asset classes—in particular, between equities and property. Wilkie had a fairly limited volume of available property data (28 years from 1967 to 1994) and it implied average property returns significantly in excess of the average return generated by the equity calibration, yet with a very similar risk profile. Such a feature might possibly be rationally explained by illiquidity premiums or related transaction costs, but it was never Wilkie’s objective to try to provide an economic explanation for market behaviour: he simply calibrated to what he found.

The other notable historical feature of the Wilkie model was its approach to modelling the yield curve. In his original 1984/1986 paper, Wilkie only attempted to model a single yield (the consol yield), so the issue of how to consistently model all points on the yield curve never arose. But Wilkie chose a model structure for the consol yield that was quite convoluted. The yield was a function of its value in the previous three years (and hence was nonMarkovian). It was unnecessarily complicated—Wilkie himself noted in his original paper that a first-order autoregressive process could produce very similar results and in the 1995 update of the model he implemented this change. Writing later, Wilkie acknowledged that the original consol yield model may have been an example of ‘over-parameterisation’.[4]

Wilkie never really grasped the nettle of producing a full stochastic yield curve model. In his 1995 paper, he developed a short-term interest rate model to go alongside the consol yield, but he left it to the user to decide how to join the two dots. He referenced polynomial functions in the 1995 paper, and in a later paper of 2003 he suggested a specific polynomial functional form to interpolate between the two yields.[5] He did not consider arbitrage constraints or advocate the use of the arbitrage-free yield curve models that had been developed since Vasicek’s 1977 paper. (Interestingly, Brennan and Schwartz derived the arbitrage-free yield curve pricing function for a two-factor model where the first factor is the short rate and the second factor is the consol yield in 1982[6]). Indeed, Wilkie seemed uninterested in Vasicek’s arbitrage-free logic and its insights for constraints on the term premiums generated across the yield curve. He was fairly dismissive of the entire stochastic yield curve modelling stream of financial economic research, writing in 1995:

Many alternative yield curve models have been proposed in the academic literature ... Unfortunately, to my mind, they are usually based on an assumption about how yield curves ought to behave rather than being based on how they actually do behave.[7]

This comment highlights Wilkie’s deeply empirical modelling philosophy. And there was some truth in his observations: the arbitrage-free yield curve structures of the financial economics literature had features that were deduced from the arbitrage-free condition rather than purely from empirical observation; moreover, most of these models were developed for the purposes of derivative pricing rather than to describe the fullest description of ‘real-world’ yield curve dynamics. Equally, however, Wilkie offered no clear rationale for why these approaches would not be the most appropriate way of jointly modelling the behaviour of, say, a ten-year bond yield alongside his cash and consol yield; and he did not raise or address the potential limitations of using a yield curve model that could permit bond arbitrage and that provided no means of controlling the relative size of the term premiums available across the term structure. Of course, the importance of this point depended on the use to which the model was put. In the 1980s and 1990s, the model was mainly used in the analysis of long-term asset allocation choices between asset classes such as equities, property and long-term government bonds in the context of funding long-term liability cashflows. For these purposes, the absence of a robust yield curve model was relatively unimportant. But for later actuarial applications such as the analysis of alternative derivative hedging strategies for guaranteed annuity options, the model was wholly inadequate. Of course, Wilkie never envisaged or advocated the use of the model in this type of application when he developed it.

For context, it is worth noting, however, that some actuarial research was published that did try to embrace the new yield curve modelling technology of Vasicek et al. and showed how it could be made use of in actuarial practice. In particular, Phelim Boyle, who we met earlier as one of the pioneers of the application of option pricing theory to life assurance guarantees, published a paper in the Journal of the Institute of Actuaries as early as 1978 that showed how Redingtons duration calculations could be generalised to allow for the volatility term structure implied by modern yield curve methods.[8] Redington’s duration approach had assumed parallel yield curve movements, which implied that long rates were as volatile as short rates. Empirical evidence, and arbitrage-free yield curve models, suggested long rate volatility was lower than short rate volatility. This meant that if matching the duration of a set of liability cashflows with a long bond and a short bond, more would really be required in the long bond than was implied by Redington’s duration calculation. This was a fundamentally important point for asset-liability duration management and Boyle showed how the new financial economic modelling approaches could cast fresh light on it.

The Wilkie model’s public domain and wide application meant that it was subject to considerable peer scrutiny—most notably, a working party was established in 1989 to review the model and it published a paper with its findings in 1992.[9] But it focused mainly on the statistical quality of the time series fits, especially for the inflation model, rather than on arguably the more fundamental question of whether such a statistically orientated and empirical approach was desirable and what its economic limitations would be.

The Wilkie model was an important step forward for the British actuarial profession in embracing the probabilistic analysis of the financial market risks that increasingly dominated the balance sheets that actuaries oversaw. More generally, it was also important in motivating the profession to develop its quantitative modelling skills. But it was also a product of the profession as it existed. In particular, Wilkie’s reticence to incorporate any of the insights that could be delivered from the (then-recent) developments in financial economics into his model could be viewed in hindsight as a missed opportunity for the further development of the profession’s financial risk management thinking.

  • [1] Wilkie (1986a).
  • [2] Wilkie (1995).
  • [3] Wilkie (1986; 2).
  • [4] Wilkie et al. (2003), p. 319.
  • [5] Wilkie et al. (2003), pp. 356—58.
  • [6] Brennan and Schwartz (1982).
  • [7] Wilkie (1995), p. 874.
  • [8] Boyle (1978).
  • [9] Geoghegan et al. (1992).
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