Liability Valuation (1800-1952)

The actuarial valuation of liabilities was, and arguably still is, the quintessential actuarial task of life actuaries. It has been a topic of much debate throughout most of the history of actuarial thought in both life assurance and defined benefit pensions. The second half of the nineteenth century was arguably the period of its greatest technical development in life assurance. Throughout this period, the liability valuation was calculated prospectively as the present value of a set of projected cashflows. Technical debates centred around a handful of methodology choices for the projection and discounting of these cashflows. The major (and interrelated) valuation methodology topics were (and, to some extent, still are!):

• • The treatment of future premiums in the liability valuation. All liability valuations could be described as the difference between the present value of the assurance claim plus related expenses; and the present value of the future premiums due over the remaining term of the policy. But there was significant debate over the treatment of future premiums, and, in particular, whether it was appropriate to recognise the profit margins in future premiums in today’s liability valuation.
• • Related to the above treatment of profit margins was the treatment of expenses. In particular, should future expenses be explicitly modelled in the valuation, or be implicitly allowed for in the treatment of future premiums.
• • Should the mortality and interest rate assumptions used in the valuation be best estimates of future experience and market conditions; or should they include margins for prudence; or be tied to the basis that was currently in use for pricing the product; or that was used in pricing the product at the time of the valuation.
• • Should the with-profit liability valuation include an explicit allowance for future bonuses or only the accrued contractual obligation.

As discussed in Chap. 2, the first life office liability valuations were performed by William Morgan in the 1780s. The solvency of the Equitable was never doubted during this period, and the object of his valuations was primarily to limit the pressure for faster distribution of surplus. At this time, Morgan’s liability valuation basis was the same as the Equitable’s premium basis (a 3 % interest rate and the Northampton mortality table). At the time of the liability valuation, the premium basis was arguably already recognised as being significantly more prudent than a best estimate of future experience. The liability valuations included the bonuses already accrued, but made no provision for future bonuses. The value of the future (gross) premiums was deducted from the reserve. He did not explicitly reserve for future expenses. He likely believed that the margins in the liability valuation (premium) basis adequately allowed for them.

Over the course of the first half of the nineteenth century, a number of other prototypical actuaries, both within the emerging group of life offices (Joshua Milne, T.R. Edmonds and others such as David Jones, Charles Babbage and Griffith Davies), and outside it (such as John Finlaison and Augustus De Morgan) were actively considering all aspects of nascent life office actuarial practice. The concept of a net premium valuation emerged as an alternative conceptual approach to Morgan’s liability valuation approach by the middle of the century (though as we discuss below, Morgan’s approach could be cast as a form of net premium valuation). It is difficult to precisely date the first conception of the net premium valuation, especially as there were no professional actuarial journals in the first half of the nineteenth century. However, we know it was established in actuarial thinking by 1853 as it was discussed at length in the testimonies of leading actuaries such as Finlaison and Edmonds to the 1853 Parliamentary Select Committee on Assurance Associations.^[1]

To understand the workings of the net premium valuation and its rationale, it is perhaps easiest to start from a consideration of the more intuitive, ‘gross premium’ valuation. We can loosely define the gross premium valuation as a liability present value that is assessed using best estimate assumptions for all future experience, including an explicit projection of future expenses, and which includes the contractual premiums in the valuation. Consider a new without-profit regular premium whole-of-life policy. The office sets the regular premium for the contract such that it expects to generate a profit. If the actuary performed a gross premium valuation of the liability at the point of inception of the policy, it would therefore have a negative value: the present value of the future regular premiums would exceed the present value of the sum assured. Negative valuations created difficulties. Most obviously, the contract could be surrendered and the surrender value must be non-negative. More generally, it was argued that it was inappropriate to immediately recognise the profit that was expected to be generated over the lifetime of the policy in the current valuation of the liability.

The net premium valuation method aimed to only recognise the profit loading in the premiums at the point at which each premium is received. The liability valuation would be made on the assumption that the future regular premiums would be ‘net premiums’—defined by Bailey in 1878 as ‘the premium which, according to the best judgement that can be formed, will suffice for the risk but leave no margin beyond’.^[2] The net premium would therefore be calculated on best estimate assumptions for the future behaviour of the risks underlying the contract, but would make no allowance for expenses—the assumption being that expenses are adequately covered by the profit margin in the gross premiums that is being discarded in the net premium valuation. By construction, a net premium valuation would produce a liability value of zero at the inception of the policy.

There was much heated debate in the mid-nineteenth century over the relative merits of using Morgan’s method of valuing liabilities using gross premiums and the premium basis or valuing using net premiums and a realistic valuation basis. This was further confused by the observation that Morgan’s method could be cast as a net premium method, in the sense that the gross premiums were the net premiums implied by the premium basis (but not by a ‘best estimate’ basis). Like any net premium valuation, Morgan’s approach would generate a liability value of zero at inception. But it did so by putting the ‘wrong’ parameters into the net premium valuation formula, and this made the progression of the reserves through the life of the policy less natural or transparent than a ‘pure’ net premium valuation (where the reserve progresses by reserving the net premium portion of the gross premiums as they are received and the remainder is used for expenses and released to surplus).

The founding of the Institute of Actuaries in 1848 and Faculty of Actuaries in 1856 provided an important professional infrastructure for the development of British actuarial thought and practice. It is no coincidence that the third quarter of the nineteenth century is one of the most fertile periods of development in actuarial thought. In particular, their professional journals, the Journal of the Institute of Actuaries and the Transactions of the Faculty of Actuaries, and the practice of presenting these papers at formal professional sessional meetings were powerful vehicles in disseminating and debating actuarial theory and practice. A series of Institute papers by Charles Jellicoe in the early 1850s,^[3] followed by a paper by Thomas Sprague in 1863, helped to establish the net premium valuation as the fundamental piece of actuarial machinery for liability valuation.

The net premium valuation quickly became an article of faith for the actuarial profession, but, to the profession’s credit, its limitations were highlighted and debated as early as 1870. The discussions of these limitations produced unusually ill-tempered exchanges amongst actuaries. Sprague, by then a preeminent actuary and vice-president of the Institute, wrote in 1870 of ‘the intolerant manner in which some of the advocates of the net premium method of valuation insist upon its being the one method that should be employed in all cases’.^[4] Clearly, there has always been something about liability reserving methods that ignites actuarial passions like few other things can!

The net premium valuation, in its original form, was parsimonious and elegant. When its assumptions for interest rates and mortality rates held true, surplus emerged from the contract in a most natural way: when a regular premium was received, the liability value would increase by the net premium, whilst the asset value (before the deduction of expenses) would increase by the actual premium. Thus the profit loading in the premium would be recognised with the receipt of each premium. If the net premium portion of the actual premium was reserved when it was received, and mortality and interest rate experience was as assumed in the net premium valuation, then the reserve would be exactly sufficient to meet claims as they fell due. But the net premium valuation had some counterintuitive features. As we have seen, it did not explicitly allow for future expenses—the difference between the actual and net premium was expected to take account of those. But there was nothing fundamental in the method that could capture the impact of unexpected increases in future expectations for expenses (which would be especially problematic if the new expectation for expenses exceeded the margin of the actual premium over the net premium).

Indeed, it might be argued that the elegance of the net premium valuation could quickly unravel if anything unexpected happened. Suppose interest rates fell after the inception of a contract. If the net premium valuation used the old, higher interest rate, this would be imprudent, as assets could not be expected to earn that rate. But if a new lower interest rate was used in the net premium valuation, this resulted in a higher net premium being used in the valuation, even though the terms of the contract and the actual size of the premium payable had not changed (a feature succinctly described by Redington as ‘a technical idiosyncrasy which has no counterpart in the facts’^[5]). Again, there was a risk that this could result in using net premiums in the valuation that exceeded the actual premiums net of expected future expenses.

A number of papers were presented to the Institute around 1870 that raised objections to the net premium valuation concept. Some of these merely highlighted that there were some actuarial tasks where the net premium valuation was clearly inappropriate. Sprague highlighted that in a valuation of liabilities for the purposes of determining a transfer value from business to another, it was ‘manifestly unjust’^[6] to ignore the actual premiums that were a contractual reality of the business. He also highlighted that there were US regulations that required net premium valuations to be made on a prescribed basis that produced higher net premiums than the actual premiums that were being charged in the marketplace, which he described as a ‘manifest absurdity’. Makeham presented a paper that went further than Sprague, arguing that the net premium valuation approach was not even appropriate for its core application of determining surplus distribution.^[7] Sprague also noted that the superintendent of the New York Insurance Department had conceded that ‘valuations for Government purposes will ... ultimately have to be made upon actual premiums received by the different companies’.^[8] A firmer regulatory context was similarly emerging in the UK following the failure of the

Albert Insurance Company in 1869—the 1870 Life Assurance Companies Act created, for the first time, a regulatory requirement for British life offices to assess prospective reserves on life policies. The methodology, however, was not prescribed to the same degree as in the USA. British actuaries retained more freedom to choose how to determine their reserves, but were compelled to disclose their methods. Thus the British life industry’s ‘freedom with publicity’ regulatory regime was born.

The net premium valuation retained a loyal following, despite the above criticisms. In 1878, the influential Arthur Bailey (the pioneer of actuarial thinking on investments) presented a staunch defence of the net premium valuation. As might be expected of Bailey, his position was sophisticated and nuanced. He argued that the net premium valuation should be made only with best estimates of future experience, and that the premium basis should play no direct role. But he also argued that the implementation of the net premium valuation method should include a comparison of the actual premiums and net premiums of the contract ‘in order to ascertain what portion [of the actual premium] represents the real prime cost of the risk [i.e. the net premium]’.^[9] His advocacy of the net premium valuation was explicitly ‘when the object is to determine the amount of divisible surplus’.^[10]

The debate rumbled on, and over the following decades a number of practical complexities emerged that further challenged the implementation of an elegant, parsimonious net premium valuation. The net premium valuation implicitly assumed that the expenses associated with the policy are incurred uniformly over the product’s duration (or, more accurately, over the premium-paying term). But practices in the acquisition of new business were changing such that much of the expense associated with a policy was incurred at its inception in sales commission payments. This meant that a standard net premium valuation could result in a loss being recognised at inception, even though the policy was profitable. A fundamental attraction of the net premium method was that it deferred the recognition of profit, but it was arguably now overstating this deferral by initially stating a loss.

The practice of paying significant up-front commissions for new business sales had first emerged in Germany, and it was the President of the German Life Institute, Dr Zillmer, who first proposed an actuarial adjustment to the net premium valuation that would allow for the change in the expense pat- tern.^[11] His proposal was first made in 1863 and it essentially allowed for an asset to be created that recognised that an initial expense had been incurred that would otherwise be due later in the contract. Zillmer’s work did not have an immediate impact in Britain as the tendency at the time was for sales commissions to be paid with each regular premium received rather than at the policy inception. But by 1870 British commercial practices had increasingly moved towards the up-front commission model, and net premium valuations started to generate losses on new business. Sprague’s 1870 paper introduced Zillmer’s approach to the Institute and commended it as ‘a very able work’.^[12] It was broadly well-received as sound actuarial practice in Britain and the USA, though there were reservations that it could be imprudently misused or abused. US regulators were particularly sceptical and ‘commuted commisions’ were not admissible as assets in the US regulatory returns.

Zillmer’s logic could be shoe-horned into the net premium valuation method. It detracted from one of its fundamental advantages—parsimony— but it did not deliver a fatal blow. A more serious and fundamental challenge to the logic of the net premium valuation arose in its application to with- profit business and its bonus system. The logic of the net premium valuation as the natural method for recognising the emergence of surplus demanded the use of best estimates of future experience for mortality and interest rates in the valuation. This ensured that the surplus that emerged would simply be the difference between the actual and net premiums (providing the experience was consistent with the best estimate used in the valuation basis). By this time, with-profit premium rates typically included ‘bonus loadings’ that set them at significantly more expensive levels than without-profit premium rates. The premium rates were set by using interest rate and mortality assumptions that included substantial margins relative to the experience that was expected. This, in itself, was not a fundamental issue for the use of net premium valuations: if the net premium valuation was performed using best estimate interest rate and mortality assumptions, the net premium would be significantly lower than the actual premium and a surplus would gradually emerge with the receipt of each regular premium. However, assuming the experience assumptions were borne out in practice, such an approach would result in the same cash amount of surplus emerging with each premium. This was not consistent with the way that surplus was distributed to with-profit policies—the approach of adding a reversionary bonus to the sum assured meant that an increasing cash value of surplus would be added as policy duration increased. The cost of bonus was not perfectly aligned with the premium pattern. So, for growing with-profit businesses writing high volumes of new business, this implementation of the net premium valuation could result in a larger surplus being recognised than was desirable. In order to defer the recognition of this surplus, it became standard practice in the final quarter of the nineteenth century to add loadings to the assumptions used in the net premium valuation—the assumed interest rate would be artificially lowered by as much as 1.5 % below the interest rate that was expected to be earned.^[13]

Valuation practice had come full circle! This was very similar to Morgan’s approach of 100 years earlier of using the premium basis to produce a bastardised form of net premium valuation. The approach ‘worked’ in the sense that it produced a slower emergence of surplus that was commensurate with the reversionary bonus system. But it worked by putting the wrong (interest rate) parameters into the wrong (net premium) formula. It was now very hard to explain what the net premium valuation was intended to mean. There was an increasing sense amongst many in the actuarial profession that something more logical and ‘scientific’ ought to be done. This movement arguably began with Sprague’s 1870 paper, but it was not until 1907 that an alternative reserving system was clearly proposed when C.R.V. Coutts, the assistant actuary of the National Mutual, presented his paper ‘Bonus Reserve Valuations’.

Coutts proposed replacing the net premium valuation machinery with a more direct and transparent valuation method. He argued that the actual gross premiums charged by the contract should be used in its valuation, together with explicit estimates of future expenses and best estimates of interest rates and mortality. Of course, for with-profit business the presence of bonus loadings in the premium basis would result in negative liability values, especially at early policy durations. He proposed to address this by a controversial means: he argued that the future reversionary bonuses that the office intended to pay should be explicitly reserved for as a part of the current liability: ‘in the valuation balance sheet, the present value of the bonus at the rate to be declared at a given valuation should be treated as a liability for the future existence of the policies’.^[14] To the layperson, this may seem a very natural approach. But it created a great deal of tension and controversy within the actuarial profession. There were two main concerns—one technical, and one relating to policyholder communication.

The technical concern was that the bonus reserve valuation required several accurate estimates of the various dimensions of future experience, and the valuation result would be highly sensitive to those assumptions. The margins in with-profit premium rates and net premium valuations were large and approximate, and the notion of, for example, having to estimate interest rates at a granularity of 1/16 of 1 % in valuations created some consternation amongst actuaries.

The second concern related to the management of policyholders’ expectations: if actuaries started to explicitly reserve for future bonus as though it was a promise, did that turn it into a promise? In the discussion at Staple Inn following the presentation of Coutts’s paper, S.J.H.W. Allin commented, ‘they should be very careful about doing anything which tended to give policyholders the idea that the whole of the funds of the company belonged to them to do as they liked with ... It was even possible that policyholders, if their bonuses were reduced ... might insist that the contract between them and the company had been broken’.^[15] Of course, the application of a net premium valuation with bonus loadings in the assumptions also reserved for future bonus, but this was an implicit approach buried in the actuarial machinery— it was believed that this opaqueness reduced the risk of encouraging inappropriately demanding expectations from policyholder.

The net premium and gross premium methods went on to live in a somewhat uncomfortable coexistence. The stability of the net premium valuation was attractive to actuaries in measuring the emergence of distributable surplus from one valuation period to the next; gross premium valuations (without future bonuses) would be the preferred approach for testing the solvency or absolute adequacy of the reserves to meet liabilities. Redingtons seminal 1952 paper that introduced his immunisation theory was actually a paper primarily about reviewing these with-profit liability valuation methods. In this review, he emphasised that the distinct questions that actuaries attempt to answer could not be answered by the same valuation:

A valuation has two main purposes, and the fundamental difficulty is that these two purposes are in conflict. The first and primary purpose is to ensure that the office is solvent. The second is to allow the surplus to emerge in an equitable way suited to the bonus system. The solvency criterion leads to a changing valuation basis ... On the other hand, the pursuit of equity of emergence of surplus tends to lead to stable valuation bases, influenced mainly by retrospective considerations.^[16]

Later in the paper, he elaborates further:

Surplus can be estimated only by a passive valuation policy which leaves the valuation basis both of assets and liabilities unchanged ... Yet when we turn to the question of solvency there is no word to be said in favour of a passive policy; it is only an active policy, paying full regard to existing (and estimated future) experience, which has any significance.^[17]

This provided an intellectual basis for performing distinct actuarial valuations in distinct ways for distinct purposes. It was broadly representative of the British actuarial practices that had emerged over the first half of the twentieth century, although there was not universal agreement within the profession on the respective roles of the different valuation methods. The Staple Inn discussion of Redington’s paper makes clear that there remained many life actuaries who had decided that the net premium valuation should be regarded as a piece of actuarial antiquity. But time has proven it to be a resilient idea. At the start of the twenty-first century, it remained enshrined in the regulatory solvency regimes of many countries around the world.

• [1] Parliamentary Report (1853).
• [2] Bailey (1878), p. 118.
• [3] Jellicoe (1851), Jellicoe (1852), Sprague (1863).
• [4] Sprague (1870), p. 411.
• [5] Redington (1952), p. 307.
• [6] Sprague (1870), p. 414.
• [7] Makeham (1870).
• [8] Sprague (1870), p. 417.
• [9] Bailey (1878), p. 119.
• [10] Bailey (1878), pp. 125-26.
• [11] Zillmer (1863).
• [12] Sprague (1870), p. 420.
• [13] Courts (1908), p. 162.
• [14] Coutts (1908), p. 161.
• [15] Courts (1908), p. 170.
• [16] Redington (1952), p. 298.
• [17] Redington (1952), p. 304.