# Equity-Linked Maturity Guarantees: Towards Risk-Based Solvency (1971-1982)

Unit trusts and unit-linked business emerged as a major competitor to with- profit policies in the UK long-term savings market in the 1960s. By the early 1970s, unit-linked business had become a significant business line for many British life offices. These products were generally much simpler than with- profit policies, and their financial management required less actuarial input. An interesting complication arose however, when life offices attempted to differentiate from the competition by providing maturity guarantees with their unit-linked funds—typically these would provide a minimum guaranteed pay-out of a return of the premium(s) paid at the specified maturity date. The maturity date typically varied between ten and 25 years. These guarantees were provided on underlying funds that were typically invested entirely in equities.

The pricing and reserving for these guarantees was not initially subject to any actuarial sophistication. In some cases, the reserve was merely retrospective, i.e. an accumulation of the (small) additional premiums charged for the provision of the guarantee. Life actuaries found themselves having to catch up with their products, and by the second half of the 1970s they were further prompted into action by the industry regulator’s stated intention to impose some prescriptive measures on reserving for these guarantees. Somewhat perversely, the relative simplicity of the unit-linked product created an actuarial risk management challenge. With-profit funds’ discretionary levers of investment policy, bonus policy and policyholder cross-subsidy were not available to the actuary to mitigate the guarantee risk embedded in unit-linked business. This meant that the office bore the full exposure of the risks that they wrote in the policy; and the resultant transparency of those risks meant that their measurement and modelling were a lot more straightforward than was the case for with-profit business.

In 1968, at the Staple Inn discussion of Skerman’s with-profit bonus paper, Jim Pegler (who at the time was only months from becoming the Institute President) raised the general topic of how to balance security with surplus distribution. Pegler raised the possibility of using a *probabilistic* basis for setting reserves. This is notable as perhaps the earliest such reference in a British actuarial paper, though Pegler recognised it would be challenging to implement and stopped short of advocating such an approach:

The author drew attention to the over-riding need to ensure security for policyholders ... they had to find a way of arriving at a reasonable degree of security, to which absolute priority should be given ... He imagined that theoretically they could approach the problem from the point of view of the Theory of Risk and estimate a probability of ruin. But even supposing that a reasonably accurate numerical figure could be obtained, he doubted whether it was helpful in practice to consider whether a probability of, say, 1 in 1,000 or 1 in 10,000 should be aimed at; he was choosing figures more or less at random, as he had little idea of the order of magnitude which was relevant.^{[1]}

The application of the probability of ruin concept to a real-life with-profits business and all its complexities was a daunting challenge in 1968. But for unit- linked business, it was much more feasible. As actuarial thinking caught up with unit-linked maturity guarantee practices, it was natural that a probability of ruin approach to setting adequate maturity guarantee reserves should at least be considered. The Prudential actuary Sidney Benjamin pioneered the development of this train of thought with a paper that was first discussed at Staple Inn in 1971. The meeting was described by one of the actuaries present as ‘by far the stormiest I have ever attended’.^{[2]} Indeed, it was so stormy that Skerman, as President, closed the meeting and classed the discussion as confidential so that its content would never be published.^{[3]} The paper was never published by the Institute. It was only formally presented and published five years later in 1976 at the twentieth International Actuarial Congress in Tokyo.

What was it about Benjamin’s work that provoked such a convulsive reaction within the actuarial profession? In stark terms, the British actuarial profession had not been trained or educated to think about risk. Or, to put it more kindly, they had been trained to assume that an insurance office could *diversify away* risk. This worked well, up to a point, for mortality and other insurance risks. But it did *not* work for financial market risk, which was inherently non-diversifiable—all policyholders had guarantees on the performance of funds that would behave similarly. There was an elegant equivalence in the retrospective and prospective reserves produced by charging for expected costs and accumulating net premiums. But this was of no use when a contingency reserve was required for a non-diversifiable risk that could not be fully funded by the premiums received (at least not without a highly sophisticated dynamic investment strategy which was far from the actuarial minds of the time). With non-diversifiable risk, an adequate prudential reserve may be an order of magnitude greater than the value of the premiums accumulated from charging for the risk. Benjamin’s paper put the actuarial profession face-to-face with the realities of non-diversifiable risk, taking it into uncharted waters. It was a seminal moment in the history of the British profession. With the benefit of hindsight, it marked the start of a 30-year struggle to modernise its thinking on the measurement and management of financial market risk.

In the framework of Benjamin’s paper, the required maturity guarantee reserve was calculated as the discounted present value of the maturity guarantee shortfall (i.e. the shortfall in the final underlying fund value relative to the guaranteed maturity proceeds) assessed at a specified percentile level (the accepted probability of ruin). This calculation required two key inputs: an assumption for the acceptable probability of ruin; and a probabilistic model of the behaviour of the underlying equity funds that would allow the probability distribution of the guarantee shortfall to be assessed. This reserve could be funded by the discounted present value of the premiums that would be charged for providing the guarantees. If the reserving requirement exceeded this present value, this excess would have to be funded using the office’s capital.

Benjamin developed his probabilistic equity model by considering the history of annual UK equity returns for the 51 years from 1919 to 1970. He conducted various statistical tests on the time series and concluded that there was not statistically significant evidence to reject the null hypothesis that the annual returns were independent. Given this result, he argued that a probability distribution for the n-year equity index could be generated by randomly sampling (with replacement) *n* points from the 51 historical annual return data points. He then suggested that the probability of ruin be set at 2 %, and estimated the reserve required by generating 50 simulation paths and using the one that produced the largest reserve.

This was all undoubtedly methodologically quick and dirty, but it captured the essence of the problem and it produced insightful results. Benjamin’s calculations implied that the *mean* ten-year guarantee shortfall for annual premium business paying a premium of one and with a maturity guarantee of ten was 0.11 (i.e. 1.1 % of the guarantee), and that the second percentile was 4.5 (i.e. 45 % of the guarantee). Discounting at a risk-free yield of 2.5 % implied a starting reserve of 39 % of the maturity guarantee would be required. Some of this reserve could be funded by accumulating the premiums charged for granting the guarantee, but, at typical charging rates, this still implied that the office would need to fund an additional reserve of over 30 % of the maturity guarantee. Benjamin himself wrote that this reserve was ‘unexpectedly high’ and suggested that it meant ‘the contract is probably not a commercial proposition’.^{[4]}

There was something of the spirit of Bachelier in Benjamin’s research, at least during this stage in his career. He had a talent for taking ideas such as game theory or risk theory and envisioning how they could be newly applied to actuarial problems. But his work needed further development and refinement to be rigorously implemented and made accessible to the wider profession. A couple of papers quickly followed the Tokyo Congress that provided this development: a substantial paper by W.F. Scott^{[5]} and a shorter note by David Wilkie^{[6]} that were both published in 1977.

Scott followed Benjamin’s general probabilistic risk-based framework but his implementation of the reserving assessment differed in some key ways. Most significantly, Scott’s analysis of historical equity returns led him to reject the assumption that annual equity market returns were independent over time. Scott used the same data as Benjamin—UK equity market returns from 1919 to 1970 as calculated from the de Zoete equity index—but Scott identified statistically significant serial correlation of -0.30 at a two-year lag. Benjamin never explicitly tested serial correlation in his analysis of the independence of historical returns but instead used runs tests and a few other forms of statistical test of independence. Scott tested lags of one to four years, and whilst the serial correlation was negative for each of two, three and four years, it was only the two-year lag correlation that was statistically significant from zero. Nonetheless, Scott concluded this was sufficient evidence to reject the independence hypothesis. This analysis is noteworthy for its early documentation of evidence of statistically significant negative serial correlation in long-term empirical equity market returns. It anticipated Fama and French’s work by more than a decade (and there is nothing to suggest Fama and French were aware of this work). Scott’s explanation of this negative serial correlation also anticipated, at least in a heuristic way, Shiller and Roll’s excess short-term volatility arguments. Scott argued that the extraordinary UK equity market volatility of 1973 to 1976 (during which time dividend yields moved from 3 % to 12 % to 5 %) could only be explained by concluding ‘the market panicked and then corrected itself’^{[7]} (though he was not the first actuary or market participant to suggest financial markets could behave in such a way).

The stochastic modelling skills of the profession at this time were still embryonic, and Scott had a preference for a simple model that would not require Monte Carlo simulation methods in its implementation. So, rather than developing a stochastic model of equities with negative serial correlation in returns, he instead used a standard random walk model but with an annual volatility that was lowered to reflect the long-term volatility that would arise in the presence of the assumed negative serial correlation. Specifically, he used an annual equity return volatility assumption of 10 % rather than the 19 % that he found in the annual historical returns data. He did not provide any statistical basis for the substantial size of this adjustment.

Scott’s rejection of the independence of equity returns through time and the consequent assumption that long-term returns would therefore be materially less volatile naturally resulted in significantly reduced reserving requirements for long-term maturity guarantees. Scott’s modelling implied that the ten-year annual premium return-of-premium maturity guarantee required an initial reserve of 15 % of the guarantee rather than the 30 %-or-greater estimate produced by Benjamin. And that was with a 0.5 % probability of ruin instead of Benjamin’s less ambitious 2 % assumption.

Scott’s paper was primarily focused on reserving, but he also considered how to set the premium that should be charged for provision of the guarantee. He argued the guarantee premium should be based on the expected guarantee shortfall discounted at the risk-free yield, appealing to the ‘equivalence principle’. This implied very low premiums for the guarantee—for example, the expected cost of the ten-year guarantee was estimated at 0.5 % of the guarantee

(a Black-Scholes put option price would have implied 4-5 %). He also discussed how the reserving requirement could be mitigated and considered several approaches. None of them, however, recognised that the guarantee was a put option and that a body of theoretical work and trading practice had recently been developed on how to price and hedge such options.

Wilkie’s paper further refined Scott’s analysis. Again, the most significant development was in the assumptions underlying the stochastic modelling of the equity market return. Wilkie confidently grasped the nettle that Scott had avoided: he developed an explicitly auto-regressive equity model that was fitted to the serial correlations observed in the historical returns and that was implemented using simulation. This modelling suggested Scott’s downward adjustment of annualised volatility from 19 % to 10 % was too great. Wilkie also updated the historical dataset to include returns up to 1977, thereby including the exceptional volatility period of 1973-1976. These resulted in reserving estimates that were closer to Benjamin’s than Scott’s.

Wilkie also criticised Scott’s argument that the premium charged for the guarantee should be calculated on the basis of expected cost, arguing that shareholders would require a return on the capital that they needed to use to support the business (Benjamin’s paper had made a similar argument). Wilkie suggested that shareholders would require a return of 2 % in excess of the riskfree rate on their capital on the grounds that the riskiness of their exposure was comparable to reasonable quality corporate bonds. He argued this cost should be added to the expected guarantee shortfall. Whilst such an approach was more sophisticated than Scott’s, it was still far from utilising the available economic insights on option pricing that were published a few years earlier by Black, Scholes and Merton.

Benjamin, Scott and Wilkie differed in their recommended modelling assumptions, but they all shared a substantial common ground that was a radical departure from traditional actuarial thought: maturity guarantees on unit- linked business ought to require a contingency reserve that should be assessed by specifying a probability of ruin, y, and then using a stochastic model of the underlying unit funds to determine how much reserve is required to fund guarantee shortfalls with probability (1-y).

This rapid actuarial research output of 1976 and 1977 had one curious institutional feature: none of it was published by the Institute of Actuaries. The papers of Scott and Wilkie were both Faculty papers, whilst, as we have seen, Benjamin had been published under the auspices of the twentieth International Actuarial Congress. The Institute had not touched maturity guarantees since the Benjamin Staple Inn debacle of 1971. This was to be addressed by an Institute working party led by F.B. Corby, but it never published its findings due to ‘problems in resolution of fundamental points’.^{[8]} So Corby went ahead and published his own paper with his personal views in 1977.^{[9]}

Corby rejected the probability-of-ruin-and-stochastic-model reserving approach advocated by each of Benjamin, Scott and Wilkie. He argued that ‘it was unlikely that it [his working party] would be able to derive a model of stock market behaviour which would be satisfactory for extrapolation into the future and which would be generally acceptable as a basis for reserving’ and that ‘in the United Kingdom ... little attention has been paid to a ruin probability approach’.^{[10]}

Corby explained his proposed alternative: ‘The approach followed is to assume a trend line for the performance of the relevant index together with a spread about that line. To obtain the reserve at inception of a single contract, it is assumed that all purchases are made at the top of the range and all sales (i.e. maturities) at the bottom of the range.’^{[11]}

Thus the reserve for an annual premium product was calculated by assuming that the guarantee shortfall at maturity was:

where *r* is the assumed long-term expected equity return and *k* is some measure of the extent to which the equity market can diverge from this trend.

This naturally led to the question of how to set the parameter values for *k *and r. Corby did not have a clear answer for that. He tabulated results for a range of values for *k* (0.2, 0.3, 0.4) and *r* (5 %, 7.5 %, 10 %) and noted that values of *k* = 0.4 and *r*=7.5 % produced results similar to Benjamin’s. So his approach was simpler and more transparent, especially to actuaries who did not have a familiarity with stochastic modelling, but it was somewhat arbitrary and Corby’s only means of implementing it was to select parameters that were consistent with the stochastic modelling analysis developed elsewhere. His approach still demanded that a choice of stochastic model and calibration be made, even though he believed that it was not possible to make such a choice in a way that would be satisfactory or acceptable.

The Corby paper was also the institute’s first explicit engagement with financial economics’ developments in option pricing theory. Corby noted the existence not of the original Black-Scholes-Merton papers, but of a number of papers that had been written more recently by academics at the University of British Columbia on the application of option pricing theory specifically to unit-linked maturity guarantees.^{[12]} These papers by Brennan, Schwartz and the actuary Phelim Boyle are notable as the earliest work on applying option pricing theory to the pricing of the financial guarantees found in life assurance policies. They directly applied risk-neutral valuation to the guarantees and showed that they could be hedged using portfolios with dynamically rebalanced holdings of risk-free bonds and underlying units. Corby relegated discussion of these concepts to an appendix and delegated the writing of the discussion to another actuary, PJ. Nowell.

Nowell dismissed the option pricing work as a theoretical irrelevance, concluding: ‘Here practice and theory are irreconcilable ... the investment procedure [dynamic hedging] is a perfectly reasonable theoretical concept but as a practical proposition it is one which contains risks greater than the risk which it is designed to eliminate’.^{[13]} Corby and Nowell threw the baby out with the bath water. There was no analytical basis for their assertion that real- life dynamic hedging would result in an overall *increase* in the life office’s risk position. And the fundamental economic insight of the guarantee replication argument and its implications for pricing appeared to be lost on them. However, it is interesting to note that a couple of speakers at the Staple Inn discussion of the paper raised the prospect of ‘immunising’ the office from guarantee losses by holding a negative exposure to the underlying units in the contingency reserve.^{[14]} These actuaries did not refer to option pricing theory or dynamic hedging. To them, delta hedging was an intuitive extension of Redington’s immunisation theory.

Whilst Corby and Nowell’s discussion of option pricing theory and dynamic hedging was rather perfunctory and dismissive, the topic and its application to unit-linked maturity guarantees received further attention from the profession in the following years. In particular, T.P. Collins published an Institute paper in 1982 which discussed in detail the concept of investing the guarantee premiums in a dynamic hedging strategy.^{[15]} Collins’s analysis was intelligent and open-minded. He provided a thorough examination of the challenges of a ‘real-life’ implementation of a dynamic hedging strategy, such as transaction costs and discrete-time rebalancing frequencies. He highlighted that the hedging strategy is exposed to large short-term movements (up or down) in the underlying asset value and that large switches in the hedge portfolio may be required as the guarantee nears maturity when the underlying unit value is close to the guaranteed amount.

Collins modelled what the historical performance of hedging and ‘conventional’ investment strategies would have been over the period of1930-1978 in the UK. He concluded that the dynamic hedging strategy ‘compares unfavourably with the conventional strategy’.^{[16]} However, he caveated this conclusion with the observation that ‘The conclusion ... depends critically on whether the [historical] price series used in the investigations is representative of the underlying distribution . if the price had not risen sharply after the end of 1974 the immunization [dynamic hedging] strategy would have proved superior’.^{[17]}

Collins correctly identified the couple of key features of the historical return series that drove his conclusion. One was the negative serial correlation (mean-reversion) in the returns. This effect reduced the risk of the conventional strategy, but did not reduce the cost of the hedging strategy (which is driven by the short-term volatility experience and is not affected by serial correlation). Secondly, the hedge strategy did not mitigate the impact of exceptionally large short-term movements that could occur between hedge rebalancing actions. This was partly a reflection of the limited sophistication of his modelled hedge strategy implementation, but nonetheless reflected the genuine ‘gap’ risk that is present in a dynamic hedge program. The paper was a perceptive contribution to thinking on the applicability of dynamic hedging of long-term guarantees, though its empirical emphasis missed the pricing insights at the core of option pricing theory.

A working party report on unit-linked maturity guarantees was published in 1980^{[18]} (by which point Collins’s paper had already been submitted but not yet published). The working party tried to establish an actuarial orthodoxy on the vexed topic of how to reserve for these guarantees. Benjamin and Wilkie were both members of the working party, whilst Corby was not. Its conclusions were therefore unsurprisingly broadly consistent with the reserving approach advocated by Benjamin, Scott and Wilkie and did not consider Corby’s approach. The working party recommended that a probability of ruin approach to reserving should be used, and that a stochastic model of equity returns be used to assess the reserve.

Wilkie developed a new version of his stochastic model of equity returns for use by the working party in its reserving analysis. Its notable new feature relative to the model he used in his 1977 paper was that the equity total return index was not modelled directly, but instead as the ratio of dividend pay-outs to dividend yield. Explicit stochastic processes were developed for dividends (its logarithm followed a normally distributed random walk with upward drift) and dividend yields (a mean-reverting lognormal process). The equity price was the ratio of the two. This structure provided the flexibility to capture the volatility patterns implied by Wilkie’s view of material mean-reversion in long-term equity returns—it could be fitted to views for both short-term and long-term equity return volatility that were consistent with significant serial correlation in equity prices. It produced similar results to Wilkie’s 1977 model. That is, it produced a significant long-term mean-reversion effect that resulted in lower long-term price and return volatility than a ‘random walk’ model with the same year-to-year volatility assumption. The reserves estimated in the 1980 working party report were broadly similar to those produced in Wilkie’s 1977 paper.

The working party report discussed dynamic hedging and, whilst noting it was a subject which merited further investigation, they considered that ‘there is no basis for reducing maturity guarantees reserves because a company follows some form of immunization strategy and, in fact, a company that follows such a strategy without fully appreciating the difficulties could well require greater maturity guarantee reserves than would otherwise be the case’.^{[19]}

The working party report brought the British actuarial profession’s ten- year saga with unit-linked maturity guarantees to a close. The quantum of reserves recommended by the working party was not substantially different to the rough-and-ready estimates produced by Benjamin a decade earlier. The working party’s two fundamental recommendations—that appropriate risk- based reserves should be held for unhedged market risk, and that no reduction in reserves can be obtained by hedging the risk—made the maturity guarantee feature of unit-linked policies commercially untenable and it was no longer sold in volume by British life offices. In 1981, Duncan Ferguson, who would go on to be Institute President in 1996, was able to state at a Staple Inn sessional meeting: ‘The considerable expertise which actuaries have now developed to handle linked policies has forced them to accept the logic that capital guarantees are too expensive to be given ... and that in the end their offices can afford to do little more than provide policies under which the customer takes nearly all the risks.’^{[20]}

This story reflects the stresses that arose when life offices underwrote, in a very transparent way, significant volumes of non-diversifiable financial market risk rather than (largely) diversifiable mortality risk. As we have seen, this trend towards taking more market risk had been underway for the previous 50 years. But up until the unit-linked maturity guarantee product, this risktaking had been cloaked in the complexities of with-profits business. The simplicity of the unit-linked maturity guarantee made this risk-taking highly transparent, and it caught the profession unprepared.

Nonetheless, over the ten years following Benjamin’s stormy evening at Staple Inn—the sessional meeting that never was—the profession made real progress, not least because of the precocious young Scots actuary David Wilkie. The profession had accepted the principle that life offices should hold contingency reserves beyond the retrospective accumulation of premiums to support financial market risk exposures. They had adopted a stochastic modelling approach that could be used to provide reasonable estimates of the required reserves. The working party and Collins had produced intelligent and unprejudiced appraisals of the merits of dynamic hedging. Whilst they had significant reservations, they had indicated a desire to constructively engage with the new ideas of financial economics and option pricing theory to further actuarial thought and practices. Significant and permanent developments in actuarial thought on the management of financial risk had been achieved.

- [1] Pegler in Discussion, Skelman (1968), p. 94.
- [2] P. Smith in Discussion, Corby (1977), p. 274.
- [3] S. Benjamin in Discussion, Corby (1977), p. 277.
- [4] Benjamin (1976a), p. 25.
- [5] Scott (1977).
- [6] Wilkie (1977).
- [7] Scott (1977), pp. 374-75.
- [8] Corby (1977), p. 261.
- [9] Corby (1977).
- [10] Corby (1977), p. 262.
- [11] Corby (1977), p. 264.
- [12] Brennan and Schwartz (1976b), Boyle and Schwartz (1977). Also see Brennan and Schwartz (1979b; 2).
- [13] Corby (1977), p. 273.
- [14] Fagan, p. 282, and Seymour, p.285, in Discussion, Corby (1977).
- [15] Collins (1982).
- [16] Collins (1982), p. 280.
- [17] Collins (1982), p. 281.
- [18] Ford et al. (1980).
- [19] Ford et al. (1980), p. 112.
- [20] D.G.R. Ferguson, in Discussion, Redington (1981), p. 389.