A1. Constructing Discount Curves

A.1.1 It is clear from market prices that different financial instruments exhibit different internal rates of return. The traditional actuarial model of a flat discount rate is no recipe for market consistent valuations, at least not unless the market consists of a single instrument (or in the exceptional case when the yield curve is entirely flat). Furthermore, the discount rate depends on more than the instrument's maturity date, as it is quite possible for instruments with a common maturity date to be priced with different yields. This frustrates any attempt to explain the yields on all instruments with a single yield curve depending only on an instrument's maturity.

A.1.2 The most successful theory for explaining fixed income instrument prices is the “fungibility hypothesis”. This states that the market value of a financial instrument with fixed cash flows can be built up from by valuing each cash flow at a discount rate that reflects the term of that cash flow. The price of an instrument such as a bond is explained using a set of discount rates, one for each cash flow date. The practical problem is how to reconstruct the curve of discount rates from market instruments, given that each instrument references more than one discount rate.

A.1.3 We might attempt to replicate a set of promised cash flows using government bonds, using interbank instruments such as deposits and swaps, or using corporate bonds together with purchased default protection using credit default swaps. These portfolios could produce equivalent promised cash flows, but in recent times the initial market values would have been quite different. This does not necessarily violate the law of one price, because we could argue that the cash flows are not exactly the same in all cases; for example all of these structures involve some risk of counterparty default but the details are different in each case. As no future cash flow promise can be absolutely free of default risk, the question remains as to which of these portfolios should be used to value a set of promised cash flows and what adjustment, if any, should be made for credit risk.

A.2. Mathematical Problem Statement

A.2.1 We can express discount curve estimation as a linear mathematical problem, given the market prices of a set of calibration instruments. The inputs to the curve estimation are:

• • market prices of financial instruments;

• • cash flow amounts and dates for those instruments;

• • a family of curves to be fitted, and

• • a weighting scheme to define goodness of fit.

A.2.2 A weighting scheme requires an assessment of relative reliability for different price inputs. An extreme version of a weighting scheme is to use as many calibration bonds as the formula has parameters so that the formula exactly replicates all calibration bonds. This could be interpreted as giving unit weight to the calibration bonds and zero weight to the rest of the bond universe. One advantage of specifying an exact fit for a small set of bonds is that an auditor can easily verify the fit is achieved. It is much more difficult to demonstrate that a proposed curve truly optimises goodness of fit.

A.2.3 Many different algorithms are in use for this purpose. The book by Anderson et al. (Reference Anderson, Breedon, Deacon, Derry and Murphy1996) gives a good survey of techniques available. Other specific algorithms are published by CEIOPS (2010) and European Central Bank (2004 and later).

A.3. Default Risk

A.3.1 Fixed income instruments are promises of future cash. None of these instruments is certain to be honoured in full, although some are more secure than others. For many developed economies, government bonds are considered the most secure, with collateralised bank instruments second and various other forms of corporate debt falling into third place. In the event of default, investors may still recover some portion of the debt even if it is not paid in full. Although bonds may default there is almost never any provision for a bond to overpay, except in the sense that a bond subject to default risk may trade at a lower price than a risk-free bond and therefore offer higher returns in good outcomes.

A.3.2 There is therefore a difference between the promised cash flows and the expected cash flows. In a mathematical sense, the expected cash flows will be lower because of the contribution of default scenarios.

A.3.3 There are several measures of a bond's (or a bond issuer's) financial strength. Specialist rating agencies collate statistics for the frequency of bond defaults, split according to the previously ascribed rating. Another measure of bond default risk is the internal rate of return (or yield to maturity) on a bond given its promised cash flows and market price. The internal rate of return, however, is only a relative measure of default risk in the sense that a bond with a higher internal rate of return has a higher spread. The prices of some derivatives, e.g. credit default swaps (CDS), are also sensitive to the perceived likelihood of a given issuer defaulting.

A.3.4 There is a rough correspondence between credit risks as measured by credit rating agencies and credit risk as reflected in bond yields or CDS spreads. A strong credit rating usually translates into lower yields, although this is not a hard and fast rule and there are usually exceptional bonds where the market's view and the rating agency views are different.

A.3.5 Rating agencies compile statistics about the historic default rates for bonds with different credit ratings. The surprising fact is that although the better bond grades have lower default rates, differences in historic default frequencies are far smaller than the differences in yields. This implies that, if historic patterns of default repeat themselves, then investors in riskier bonds earn a higher return.

A.3.6 There are several possible explanations for this.

• • Defaults are uncertain, and the higher expected return (net of defaults) compensates investors for that uncertainty.

• • Historic defaults may not be an accurate guide to future defaults. For example, there may be some rare but catastrophic events which are missing from the data set or ignored by the analyst in questionFootnote 15.

• • Riskier bonds may also entail other higher costs, for example wider dealing spreads or higher management expenses, so that the additional return net of costs and defaults is smaller than appears at first sight.

A.4 Adjusting Bond Yields For Default Risk

A.4.1 A yield curve applies to bonds or financial transactions with a defined element of credit risk. These yields may then be applied to a promised cash flow stream. Use of unadjusted bond yields implies that the credit risk of the cash flows to be valued is in some sense consistent with the credit risk of the bond.

A.4.2 If the cash flows are to be valued at a different credit standing to the calibration instruments, then a relative adjustment is required to move from the calibration status to the desired reporting basis. This default adjustment can take several forms: an adjustment for historic defaults, an adjustment for spread differences or an adjustment for the proportion of spreads ascribed to default riskFootnote 16.

A.4.3 An adjustment for historic defaults means subtracting from a bond yield the historic losses for bonds assessed as of similar credit risk. Table 3 shows historic losses that may be used for such an adjustment. The yield remaining after deduction for historic closes then gives an estimate of the prospective return based on expected (rather than promised) cash flows. However, such a calculation relies on a number of assumptions, not all of which appear to be realistic, including:

Table 3

European Issuer-Weighted Cumulative Default Rates, 1985–2009. Source : Moody's Investors Service

• • Consistent application of credit rates between bonds and over time.

• • Past experience is a good estimate of future default losses.

• • An assessment of loss given default. Rating agencies publish statistics on default frequency by monitoring trigger events but often do not keep track of subsequent recoveries. Using the published default frequency as a deduction is equivalent to assuming 100% loss given default.

A.4.4 Just as a default rate can be subtracted from an input yield, a default rate may also be added to the yield to reflect the credit standing of the liabilities to be valued.

A.4.5 An adjustment for expected defaults leaves within the yield the market's required reward for the uncertainty in defaults. The combined effect of expected losses and market require risk is captured in the difference, or “spread” between yields on two instruments, other things such as term and liquidity being equal. This provides a mechanism for adjusting for the total default risk effect, although like any measure based on spreads this is only a relative adjustment rather than an absolute adjustment that entirely removes the effect of default risk.

A.4.6 We note that adjusting discount rates is not always an appropriate tool for reflecting default risk, especially if the underlying cash flows are correlated with the risk of default. For example, let us suppose a reinsurer is assessed to have a ruin probability of 1%, and we wish to value a high layer catastrophe reinsurance policy with a 1% probability of claim (the claim then being likely to be much larger than the reinsurer's available capital resources). If the claim event and the default event coincide then the reinsurance policy is worthless, as the reinsurer is never able to pay a valid claim. The fact that the reinsurer survives in the other 99% of cases is irrelevant for evaluating this particular reinsurance contract.

A.4.7 There is some circularity in the relation between default risk and discount rates. The perceived default risk in a set of promised cash flows will affect how creditors assess that promise. There is also an effect in the reverse direction. A discount rate used in a valuation to determine funding or capital requirements also affects the cash flow security. For example, if a pension funds its liabilities assuming 10% per annum future investment returns, this may result in lower contributions and lower accumulated assets, and hence lower benefit security, compared to assuming 1% per annum investment returns. The use of a low-risk discount rate does not necessarily imply that the benefits are correspondingly secure. For example, if a pension fund determines its contributions by reference to a liability discounted at gilt yields, this does not imply that the benefit promise to members is of the same default quality as government bonds. The pension promise could be less secure, for example if the sponsor allows themselves twenty years to make good any deficits, or could be more secure than government bonds if the sponsor covenant is exceptionally strong.

A.5 Premium For Price And Default Risk

A.5.1 Historic experience shows significant differences in long term returns from different asset classes. Over very long periods of time, equities have returned more than bonds which, in most economies, have earned more than cash. [Dimson et al, Reference Dimson, Marsh and Staunton2002]. These observations can be explained in terms of a risk premium. Investors demand, and therefore capital markets supply, higher expected returns for riskier assets. The higher returns exist only in the sense of expected values and underperformance can also occur. There is no finite time horizon over which we can be sure that risky assets will outperform low risk assets.

A.5.2 Investment risk takes many forms. The easiest to measure is risk associated with price volatility. The prices of long bonds are typically more volatile than those of shorter bonds. Even in the absence of defaults, the prices of credit risky bonds are typically more volatile than prices of bonds with lower risk of default. That is because the market's assessment of future credit risk is, itself, volatile (and not, in general, negatively correlated with bond yields). While historic risk premiums are observable on many asset classes, decomposition of an observed premium into risk types has proved difficult, with both the methodology and numerical values subject to rigorous debate.

A.5.3 How do these risk premiums appear in discount rates? Matching methodologies do not usually require an assessment of risk premiums. For example, under a matching methodology the present value of a 10 year cash flow is calculated with reference to the market price of a 10 year bond. Risk premiums might explain why this ten year bond yield is higher than expected returns on shorter bonds, rolled over, but such decomposition is unnecessary for the matching calculation.

A.5.4 An exception to this rule arises when a cash flow is of a longer term than any available investment. In this case there is no perfect match and one is left to select between various imperfect alternatives. One alternative is to hold the longest available bond, rolling it over on maturity into another bond whose future yield is unknown but whose mean may be estimated for example by time series analysis. A second alternative is to construct a theoretical yield from expected rolled over returns but also extrapolating risk premiums. The latter approach typically gives higher yields (and so lower liabilities) to the extent of the chosen risk premium.

A.6 Allowance For Taxation And Other Expenses

A.6.1 Allowances for tax and other expenses may appear in several different places within a matching calculation. For example, the yield on a bond includes, among other things, an allowance for expected defaults. If that bond yield is then used in a matching calculation, then the defaults might be deducted from the discount rate. Alternatively, discounting may be based on the promised return and defaults modelled separately as an item of expense. These two calculations could give the same answer; the difference is a matter of presentation. However, neither of these two acceptable approaches give the same answer as counting the default risk twice, or as not counting it at all.

A.6.2 Similar issues apply to fund management expenses, where these are incurred as a percentage of the fund value. One has the option of an explicit expense allowance or, alternatively, a deduction from the discount rate. However, here a matching calculation creates a further complexity when the institution chooses not to hold the matching portfolio and when the actual expenses on the current portfolio are different from the expenses on a matching portfolio. Examples of difficult decisions might include:

• • A pension fund who could match liabilities with a government bond portfolio incurring management fees of 0.2% per annum. They choose instead to invest in actively managed equities incurring a management fee of 1.5% per annum. The matching argument suggests a discount rate based on government bonds minus 0.2%.

• • An insurer writes annuities in a currency for which long dated bonds are mostly issued by corporate and not the government. The insurer therefore invests in corporate bonds and uses a liability discount rate based on the bond yields minus an allowance for defaults, which they interpret as a theoretical yield on government bonds. Their expense provision is based on their actual expenses, as the hypothetical government bond does not exist and so there is no way to establish the expenses involved in such a non-existent portfolio.

A.6.3 Considering default losses or percentage management fees, we are at least in the comfortable situation where the cost is proportional to the value assets held. It is less clear how to allow for non-proportional expenses. Non proportional expenses may either be concave (the marginal expense reduces with the asset quantity) or convex (marginal expense increases with asset quantity).

A.6.4 Examples of concave costs include costs of software systems, ALM department, internal and external audit.

A.6.5 The most common example of convex costs is dealing costs, including the cost of forced asset disposal. The convex feature applies for two reasons – firstly because the market price impact of a large trade exceeds that of a small one. Secondly, where asset disposal is forced in order to pay benefits, an institution may retain discretion over the order in which assets are sold, starting with the most liquid assets. The illiquidity costs are then convex in the quantity of illiquid assets held, as the illiquid assets not only incur higher disposal costs but these costs are also more frequent if large illiquid asset holdings squeeze out the cushion of liquid assets that would otherwise be sold first.

A.6.6 Tax is roughly linear but may have concave or convex elements depending on the investor's situation. A particular difficulty with tax is that calculations such as pooling for capital gains assessment purposes cut across particular definitions of liabilities. An example of a difficult situation is as follows:

A.6.7 These dilemmas reach to the heart of a question related to discounting, namely – what, precisely, are the cash flows to be valued? Are we to include only the cash flows promised to the beneficiary, or are we to include other associated costs of meeting the liability? Are those costs to be measured relative to the current portfolio or to the hypothetical costs of managing a theoretical matching portfolio? Are costs to be measured on a marginal or average basis? Are they to be recognised as an explicit cash flow, or included within general expense provisions, or expressed as a reduction to the discount rate?

A.7 Premiums For Illiquidity Or Expenses

A.7.1 It is convenient in investment theory to ignore tax, administrative costs, bid-offer spreads and the market impact of trades. This gives rise to investment optimisation problems in which the probability distribution of gross-of-tax mid-market-to-mid-market returns is the primary input. It also gives rise to the elegant matching theory for pricing which has so successfully been applied to traded financial markets such as derivative exchanges.

A.7.2 While this theory has given us many insights, there is also much that it fails to explain. In particular, investors incur tax, administration and illiquidity costs. These costs also affect market prices; an investor will pay more for an investment that saves on later expenses even if it has no effect on the cash flows promised to beneficiaries of an insurance or pension fund. However, at the current stage of knowledge, we lack an encompassing theory that explains all of these prices simultaneously. We must also recognise that illiquidity itself is uncertain, so a bond illiquidity premium necessarily contains an element of risk premium as well as expected losses. What we can do is can investigate statistically how yields are affected by measures of liquidity (for example, bid-offer spread, issue side or trading volumes). This results in a decomposition which is shown schematically in Figure 2.

Figure 2

Decomposition of a bond gross redemption yield

A.7.3 What we lack is a unifying theory to say how liquidity and expense elements should be reflected in the value of an arbitrary cash flow. We might measure bond liquidity by reference to bid-offer spreads or trade market impact but these are meaningless for liabilities. This gives a degree of arbitrariness to any matching attempt which seeks assets and liabilities of equivalent liquidity. The fact that any such assessment is arbitrary does not prevent actuaries searching for it, nor indeed from announcing discoveries, as the prize is a lower stated liability, see CRO forum and CFO Forum (2010). QIS 5 Technical Specification, Risk-free interest rates. European Commission.

A.7.4 Part of the conceptual difficulty in formulating a unifying theory is the mismatch between linear premiums and non-linear costs. If a bond yield contains a certain reward for illiquidity, then the investor's reward for illiquidity increases linearly with the quantity of this bond held. On the other hand, illiquidity costs are typically convex in the quantity held. An optimising investor then increases the illiquid bond holdings until the varying marginal cost reaches the given marginal reward. At this point, the average cost will be lower than the marginal cost (because of the convex schedule) so the investor can, in some sense, count the difference as a gain. There is a vigorous debate regarding where this gain belongs in accounting terms – is it a higher recorded asset value, a lower recorded liability or should it be excluded altogether from the balance sheet and recognised by investors as franchise value?

A.8. Case Study

A.8.1 Development of the European Solvency II framework for insurance supervision provides an interesting case study for how these elements may be combined in practice. While the final rules have yet to be established, the European Commission has issued a series of quantitative impact studies to test possible specifications for the new rules. At the time of writing (November 2010) European insurers had just submitted their calculations for the fifth quantitative impact study, know as QIS5. These calculations were performed as at the 31 December 2009.

A.8.2 Figure 3 shows some raw data: the mid-market swap rates against 6 month EURIBOR at various terms at 31/12/2009, according to Bloomberg. In this chart, we have used a non-uniform scale for the bond term. If t is the term, then we have used a uniform scale for 6t/(10+t). The purpose of this transformation is to make the short and long limits of the curve more visible.

Figure 3

Mid-market swap rates at 31/12/09

A.8.3 The input swap rates are not used for discounting, as each rate refers to a stream of several cash flows. Instead, QIS 5 published spot rates and a formula (the Smith Wilson formula) for interpolation or extrapolation. These curves are often used as input for economic scenario generators, which are then used for valuing future guarantees in relation to interest rates or to asset returns. To understand these valuations, it is helpful to express the yield curve as a table of forward rates, that is the implied short interest rates in future years. These are show below for the 31/12/2009 Euro data set in Figure 4.

Figure 4

Yield curve at 31/12/09, expressed as a curve of forward rates

A.8.4 Seven adjustments are applied to move from the input data to the final published curve for with-profits business (the adjustments are slightly different for other business lines but the principles are the same). The effect of these adjustments is shown in Figure 5.

Figure 5

Adjustments to swap curve for with-profits business

A.8.5 The adjustments can be described as follows:

• • The 0.3975% adjustment for illiquidity is calculated as follows. A market yield on corporate bonds is supposed to be 1.46% higher than swap yields. Of this, 0.4% corresponds to expected losses, leaving an expected premium of 1.06% net of expected losses. This expected premium is assumed to be half (0.53%) a reward for bearing the uncertainty of defaults, and half (another 0.53%) for illiquidity. With profits business is assumed to be only 75% illiquid, so a provision is made for a quarter of the illiquidity premium (0.1325%) to be incurred as illiquidity costs. Rather than providing for these costs as cash flows, the effect is deducted from the discount rate, leading finally to a net 0.3975% addition to the discount rate.

• • Between 5 and 10 years there is some irregularity in the forward curve; there is an implicit differentiation in going from swap curves to forward curves, and this often gives rise to small oscillations in the forward curve. The curves have been smoothed using a cubic spline algorithm, the precise details of which have not been published.

• • Between 15 and 20 years there is an illiquidity catch up effect. The illiquidity premium is assumed to apply only for terms as far as the existing bond market, deemed to be a maximum term of 15 years. Firms are prevented from taking credit for future investments in illiquid bonds. This is reflected in a requirement that spot rates beyond 20 years are unaffected by the illiquidity premium adjustment. Combining this with a 15 year rat that does include such an adjustment, forces an unwind effect between 15 and 20 years where the effect of illiquidity premiums is to reduce the implied forward rate.

A.8.6 Over the whole term structure, there is a deduction of 0.1% to make a nominal allowance for the credit risk inherent in the use of 6 month EURIBOR. Over most of the term structure, this effect is dwarfed by other adjustments, but it is visible between 20 and 30 years.

A.8.7 At long terms where data is sparse, some sort of interpolation or extrapolation is needed. Within QIS 5, two approaches were used. The published Smith Wilson approach was used for extrapolation while a proprietary spline approach was used for interpolation. This creates a discontinuity at the point where the curve switches from one point to another.

A.8.8 Although swap data exist for the Euro up to terms of 50 years, for the purpose of QIS 5 the data beyond 30 years was considered unreliable as these swaps were illiquid. As a result, these points were discarded and the extrapolation commences after 30 years rather than 50. This could be regarded as another form of illiquidity adjustment.

A.8.9 Finally, Solvency II introduces the idea of unavoidable market risk. The best estimate value of a 50 year liability is determined with respect to an extrapolated curve, but in addition the firm must add a risk margin to the best estimate, in recognition of the uncertainty surrounding the determination of the 50 year rate and the likely imperfection in any attempt to hedge this risk. In the chart above we have re-expressed this risk margin as a deduction from the discount rate.

A.8.10 While each of these building blocks has its own rationale, in combination the effect is to produce QIS 5 yield curves whose shape is dramatically at odds with the input data. Some of the advantages of matching valuation – particularly the consistency with possible hedging instruments, may in this case be lost.

B.1 Single Cash Flows

Consider a future cash flow of 1 at time t. The present value of that cash, discounted at stochastic return, is exp(−μt−σB_t). The p ^th quantile of this present value under the budgeting approach is:

where Φ⁻¹ is the inverse Normal cumulative distribution function.

B.2. Perpetual Annuity

Consider an annuity paying continuous cash flows at a rate exp(γt) per annum, where γ<min{r, μ}. Negative values are usual, in which case the mean future life time is −1/γ.

The present value of this annuity using the matching approach is:

1. Matching value=

2. It is easily seen that this is also the discounted mean term of the annuity.

3. We can identify the integral of the p ^th quantile, by carrying out the integral:

4. Integrated p ^th quantile =

5. We substitute (μ−γ)t = z ²/2 and write

By way of a reality check, we notice that if δ is very large this tends to infinity, while it tends to zero when δ is large and negative.

This calculation does not, however, give the percentile of the underlying present value. Instead, we can use a result of Dufresne (Reference Dufresne1990) who proved that the reciprocal of the present value has a Gamma distribution:

If we discount at the median return μ, the present value is 1/(μ−γ) which is also the discounted mean term. Scaling, the distribution is:

This then depends only on , that is, the volatility divided by the discounted mean term.

C.1 Circumstances When The Choice Of Methodology Is Irrelevant

C.1.1 Both ‘matching’ and ‘budgeting approaches ultimately involve applying ‘time value’ adjustments to future monetary cash flowsFootnote 17. A highly important, if self-evident, corollary is that the two may produce the same answers (or may at least result in application of the same discount rates), if the resulting time value adjustments are the same. It is therefore first helpful to explore the circumstances in which the two give the same answers (and hence choice between them is irrelevant) before moving on to identifying which one to use when choice between them actually makes a difference.

C.1.2 For the two approaches to result in different ‘values’ being placed on future (liability) cash flows we need one or more of the following to apply:

1. a) The assets or liabilities being assessed in different possible exercises need to vary in some manner not itself linked to the discount rate being used. For example, in a pension fund ongoing planning ‘valuation’ the future liabilities might be assumed to include some allowance for discretionary pension increases, whereas in a corresponding discontinuance ‘valuation’ carried out for the same pension scheme we might only wish to include liabilities for which an explicit contractual guarantee of payment exists. There might be similar divergences in liabilities being included in equivalent types of exercises carried out for, say, with-profits books of life insurance companies;

2. b) We need to believe it appropriate to handle un-hedgeableFootnote 18 aspects of the liabilities differently in different possible exercises. If any liability elements fall into this category then this necessarily creates some subjectivity in how a market consistent value might be derived for the liabilities;

3. c) For liability elements that are hedgeable, we need to disagree on what are the market values of assets that best match the liabilities. This highlights another possible element of subjectivity in the computation of market consistent values. Market values are only tightly bound if markets are deep, liquid and transparent and if bid-offer spreads are very small. Few markets come close to this level of ‘perfection’. In practice there will be some blurring between (b) and (c). Most liabilities that actuaries work with are only partly hedgeable. Their market consistent values thus involve a blend incorporating some (objective) market based data as well as some subjective valuer input as per (b), see e.g. Kemp (Reference Kemp2009).

4. d) The ‘expected’ future returns on the assets assumed to be used to fund the liabilities need to differ from the market implied future returns available on the (possibly notional) asset portfolio that best matches or replicates the liabilities. By ‘expected’ we here mean the combination of:

   1. (i) The ‘statistically’ expected return (e.g. arithmetic or geometric mean) of the distribution of returns that the actuary and/or client thinks will apply to the assets in question (loosely speaking corresponding to a ‘best estimate’ in a going concern accounting exercise), and

   2. (ii) Any adjustment for ‘prudence’ overlaid on the (i) as per Section 4Footnote 19

5. e) The treatment of credit risk between the two partiesFootnote 20 involved in the asset or liability needs to differ from that implicit in the market value of the corresponding ‘matched’ assets.

C.1.3 We can view C.1.2(b) and C.1.2(c) as involving subtleties relating to exactly how we might derive a ‘market consistent’ valuation. They are not therefore really issues in relation to choice between a ‘matching’ and a ‘budgeting’ approach, although they do highlight that market consistent valuation techniques may not always be as objective as we might ideally like.

C.1.4 We might also view (a) in a similar vein, given that the differential there arises because of application of discounting to two different liability streams and thus not ostensibly linked to choice of discount rate.

C.1.5 However, such a stance could miss out some of the many different types of exercise potentially carried out by actuaries and some of the meanings conventionally ascribed to the term ‘valuation’ in them. In some types of exercise different elements of the overall assumption set may be formulated in tandem rather than singly. Margins may be deliberately included in one element of the assumption set even though this is known to be unrealistic in isolation, because a corresponding unrealistic assumption is used in another part of the assumption set. For example, Patel & Daykin (Reference Patel and Daykin2010) describe the net premium valuation methodology as applied to with-profit, i.e. participating, life insurance business. This methodology may now be considered rather historic. In its time, however, it was deemed an appropriate one for providing (relatively) smooth emergence of surplus even though individual assumption elements (including the discount rates used) were often obviously unrealistic when considered in isolation. It only ‘worked’ because the liabilities being valued also assumed other unrealistic elements (e.g. unrealistically low bonus rates) that in some appropriate sense could be expected to ‘compensate’ for lack of realism in the discount rates being used.

C.1.6 It would also gloss over some essential differences between:

1. a) future cash flows that already correspond to firm commitments present at the ‘valuation’ date; and

2. b) future cash flows the existence of which is more or less discretionary in nature.

C.1.7 We believe that the more ‘contingent’ nature of liabilities falling into category (b) makes it more justifiable to adopt a budgeting style calculation in relation to them, at least for some purposes. Inherent in their nature is a ‘plan’ that the employer is following, with a corresponding range of outcomes, which the employer can modify if circumstances so dictate. Less attention therefore needs to be paid to any potential ‘fair’ or ‘realistic’ value ascribed by the market to such liabilities (if they are incompatible with other elements of the overall ‘plan’ that the employer has for continuing in business) unless and until the liabilities become more clearly crystallised in nature.

C.1.8 However, there is an implicit assumption in such an assertion. It is that the computation being carried out is solely applicable to the employer itself in its endeavour to trade profitably. More common in practice is for actuarial work to be used or applied, implicitly or explicitly, by more than one party. We consider the additional issues this raises in Section C.6 below.

C.2 Valuation, Utility And The Impact Of Markets

C.2.1 The lack of unanimity in what should be understood by the terms ‘value’ and ‘valuation’ is one reason why we refer to the approaches set out in sections 3 and 4 by the epithets ‘matching’ and ‘budgeting’ respectively.

C.2.2 In broad terms, ‘matching’ and ‘budgeting’ approaches can be viewed as corresponding respectively to the types of technique underlying ‘valuation’ and ‘planning’ exercises as referred to in relevant BAS Standards. Unfortunately, the terms ‘value’ and ‘valuation’ are already used in many different ways in different branches of the actuarial profession (and outside it). We are wary of using terms that readers may think that they understand when in practice they have a different understanding to the writers. When we have had to use terms such as ‘value’ and ‘valuation’ we have generally tried to add additional describing terms to them, e.g. ‘market’, ‘market consistent’ or ‘intrinsic’ value.

C.2.3 Of course, terminological issues can also arise with the use of terms such as ‘matching’ and ‘budgeting’. For example, the word ‘matching’ might unwittingly give the impression that the only exercises to which ‘matching’ approach might apply are ones where the investment strategy being adopted is a ‘matched’ investment strategy. As we saw in Chapter 3, this is not the case. The term ‘budgeting’ is also potentially open to misinterpretation. However, we think that the risk of this happening is less than with the use of ‘matching’ or ‘valuation’. ‘Budgeting’ generally gives the flavour of there being some risk of not meeting the budget (as, to a lesser extent, does ‘planning’). This, as we shall also see below, is a key element in whether a budgeting type approach might be relevant for a particular exercise, although not the only element.

C.2.4 An example of how ‘value’ can be interpreted in a variety of different ways in a financial context is the situation in which an investor is thinking about buying (or selling) some shares in a company (or, indeed, someone thinking about buying or any other type of asset or liability). We might expect the investor/analyst to first carry out an analysis of what ‘value’ he or she thinks a share in the company is worth, based on a subjective analysis and incorporating subjective assumptions about its future business prospects etc. Here the term ‘value’ is really shorthand for the investor's view of the ‘intrinsic’ value or worth (to him) of the security. The investor might then only invest in the company if this subjective value assessment is materially higher than its then current market value (or might sell some or all of his or her existing exposure, if any, if the subjective value assessment is materially lower than its current market value).

C.2.5 Technically speaking, such an exercise is most akin to a ‘budgeting’ or ‘planning’ exercise. The investor has in mind a ‘plan’, here involving possible investment in the company, and wishes to formulate a view as whether to go ahead with the plan. Indeed, it is prototypical of many other budgeting/planning exercises.

C.2.6 For example, it highlights the importance of investment views in choice between ‘matching’ and ‘budgeting’ approaches. Suppose we deem possibilities C.1.2(a), (b) and (c) as not really relevant to this choice and for the moment park to one side possibility C.1.2(e). We then see that differences in practice between ‘matching’ and ‘budgeting’ approaches to determining liability discount rates are driven primarily by differences in the returns assumed to be available in the future on the assets used to fund the liabilities vis-à-vis those that we would expect to be available were we to follow a matched investment strategy.

C.2.7 Whilst not always articulated as such, any such difference in assumed returns ultimately corresponds to expression of an investment view about the different returns potentially available on different assets. This is true even if the view is dressed up as, say, involving, an ‘equity risk premium’ rather than an investment view as such. The assertion that equities will outperform in the future is just that, i.e. an assertion rather than a known fact. This is true even if historic evidence is that they have done so in the past and there is good economic rationale for believing that they might do so in the future, given a long enough time-frame. Even more obviously just an assertion, and hence actually an investment view being expressed by a particular commentator, is that any such equity risk premium should have a particular value.

C.2.8 This example is also prototypical in terms of its interaction with utility theory, see e.g. Kemp (Reference Kemp2010). The underlying premise of utility theory is that different goods or services may have different utilities to different economic participants and may therefore have different intrinsic ‘values’ to those participants.

C.2.9 This logic is difficult to fault but does need some refinement if the good or service being valued is monetary in natureFootnote 21 and there is a deep, liquid and transparent market in it (and such a market is expected to continue to exist in the future).

C.2.10 The reason is simple. The mere existence of such a market can be expected to influence the behaviour of economic participants relative to the situation where no such market existed.

C.2.11 Just because an investor believes that shares in a company are intrinsically ‘worth’ X to him does not by itself mean that he or she will be happy to buy it for X less some small margin. It is not as if the investor will actually ‘consume’ in any physical sense the cash flow stream implicit in the investment. Instead, any benefit or otherwise that he receives from such a holding will ultimately need to involve monetary transactions (this being the essence of a financial good or service)Footnote 22.

C.2.12 In particular, if the market value of the asset is Y and Y is noticeably smaller than X then we can expect rational market participants who are buying modest amounts of shares in the company to be prepared to buy shares at only Y or thereaboutsFootnote 23. A corollary is that sellers will also only be prepared to sell at (approximately) Y, even if, in their opinion, the intrinsic worth of the share to themselves is much lower than this.

C.2.13 Of course, not all market participants may be ‘rational’ in this respect. However, most commentators (and most actuaries) would agree that the greater is the perceived differential between X and Y the more likely it is that the economic participant will behave ‘rationally’ in this regard, and not, in his own opinion, (materially) overpay for (or under-sell) something relative to its market price.

C.2.14 In short, market valuations, to the extent that they exist, may be expected to have a strong influence on how utility theory interacts with the actual behaviour of economic participants. The underlying arguments may still apply but they then have less impact on (now market) price and more impact on quantity of exposure bought or sold.

C.3 Transparency

C.3.1 The existence or otherwise of a ready market in an asset or liability can thus be expected to have a major impact on behaviour patterns of economic participants, including nearly all types of institution advised by or employing actuaries. Indeed, we think it is such an important contributor that we recommend, where practical, that any material divergence between the values placed on contractual asset or liability cash flows and their market or market consistent values should be highlighted in actuarial work, together with an explanation of the main contributors to this divergence.

C.3.2 Such a recommendation ought to provide better transparency, and ought to enable recipients of actuarial advice to understand better the factors contributing and assumptions underlying this advice.

C.3.3 It also implicitly reflects some longer-term trends in actuarial work. Another similarity between ‘matching’ and ‘budgeting’ calculations at least as far as actuaries are concerned is that both are typically now in practice applied primarily to liabilities rather than to assets. This partly reflects the types of organisation that are advised by or employ actuaries and the services that they expect from these actuaries. However, we suspect that this characteristic of modern actuarial advice is also partly presentational and provides some clues as to how non-actuaries typically interpret valuation disciplines.

C.3.4 In general, actuarial work ultimately focuses on interactions between assets and liabilities, rather than on each entirely in isolation. It ought, therefore, in theory to be agnostic between the two sides of the balance sheet, with no particular bias towards application to liabilities rather than assets. Similar overall conclusions can arise even if quite different approaches to discounting future liabilities are used, if the ways in which discounting is applied to future asset cash flows are adjusted accordingly.

C.3.5 However, this does not accord with how actuarial practice has actually evolved. Some years ago, as explained in Patel & Daykin (Reference Patel and Daykin2010), it was relatively common for UK actuaries to value assets in pension budgeting exercises at values other than their market value. The assets in question were then (and still are) typically relatively liquid so it was (and still is) relatively straightforward to determine a reasonably accurate market value for them. More recently, this type of approach has become less usual, and asset valuations used in pension budgeting exercises have become more commonly more market value driven, more in line with approaches adopted in most other actuarial disciplines.

C.3.6 Implicit in this shift seems to be recognition that the more liquid and easily quantifiable is the market value of a set of cash flows, the more difficult it is to explain to clients a calculation in which some different valuation is placed on these cash flowsFootnote 24. Typically, actuaries advise organisations whose liabilities are less liquid and less easily quantifiable in market value terms than their assets. Thus, to the extent that actuaries have diverged from market value based approaches in budgeting calculations, the tendency seems to have been for them to concentrate this divergence solely on the side of the balance sheet that is less liquid and therefore more difficult to value (in a market value based manner).

C.3.7 Whilst such an approach may be desirable from a presentational perspective in one respect, it is potentially undesirable in another respect. As we have seen above, divergences between ‘matching’ and ‘budgeting’ approaches are primarily driven by assumptions concerning return on assets rather than being linked to how liquid or otherwise the assets are relative to the liabilities. We can always re-express a comparison between a ‘matching’ and a ‘budgeting’ approach applied to the same organisation in a manner in which the values placed on the liabilities are identical, but the values placed on the assets differ, see Figure 6.

Figure 6

Comparison of balance sheets arising using example ‘matching’ and ‘budgeting’ approaches, if the liabilities are the same in either case

C.4 Budgeting For Stochastic Models

C.4.1 Deterministic budgeting and matching calculations share the form of a discounted cash flow calculation. Stochastic calculations show greater differences in the methodology.

C.4.2 Ford et al. (Reference Ford, Benjamin, Gillespie, Hager, Loades, Rowe, Ryan, Smith and Wilkie1980), set out a stochastic budgeting framework which has been re-used many times since, notably by Wilkie et al. (Reference Wilkie, Waters and Yang2003) who contrast this to a matching approach. Their approach is to consider the probability that a given quantity of assets is ultimately sufficient to meet the liabilities, with a certain probability of success. Success is defined only with reference to cash flows; there is no need to pass intermediate solvency tests provided at that all cash flows are ultimately met. A “best estimate” basis is sometimes equated to a 50% probability of sufficiency.

C.4.3 This “best estimate” basis has a number of limitations as a valuation measure. In particular, taking the example of Wilkie et al. (Reference Wilkie, Waters and Yang2003), a liability may be valued at zero if it is zero with 50% probability and positive otherwise. This seems intuitively wrong, and various devices have been developed to address this and other shortcomings, most obviously the use of higher (more prudent) percentiles, for example the 75%-ile or 95%-ile.

C.4.4 Deterministic present value calculations satisfy several properties that prudent stochastic valuations do not inherit. For example, to value a stream of cash flows in a deterministic calculation, we can simply add together the present values of each cash flows. This does not work in a stochastic setting. For example, these calculations for a stream of cash flows, are quite different:

1. (a) For each cash flow, determine the initial quantity of assets sufficient to meet that cash flow with 75% confidence. Then add together these asset requirements.

2. (b) Determine the initial quantity of assets sufficient to pay all cash flows as they fall due, with 75% confidence.

C.4.5 At larger percentiles, Method (a) usually produces a higher asset requirement. This is because method (a) is the sum of 75%-iles for several random variables, while method (b) is the 75%-ile of the sum.

C.4.6 To illustrate the ideas, we consider six specific liabilities. These are:

• • A fixed single cash flow, or bullet, at time horizons 1, 5 and 25 years.

• • An annuity, based on a constant force of mortality, with discounted mean terms of 1, 5 and 25 years.

We assume that asset total returns are a lognormal random walk with volatility of 20%. The liabilities are standardised so that the present value, discounted at the median return, is 1. See Figure 7.

Figure 7

Comparison of present values: single payment v. annuity

C.4.7 The calculations in Figure 7were performed used method (b) above, that is, the percentile of the total present value, rather than the total of percentiles. Method (a) gives slightly higher values at the 75% and 95% levels for the annuity. For the bullet cash flows, methods (a) and (b) are of course equivalent.

C.4.8 Unsurprisingly, the effect of prudence is more marked for longer time horizons as there is more time for things to go wrong. We see that the median requirement for annuities is greater than the present value discounted at the median return. This is particularly visible for the 25 year annuity.

C.5 Matching for Stochastic Models

C.5.1 In contrast, matching valuations contain no concept of a prudent valuation. The market value of a matching asset portfolio is a single number, not a distribution. Prudence is more often incorporated as a series of stress tests – for example a requirement that assets should still exceed the value of the matching portfolio after a 40% fall in asset prices. We can therefore re-express a given level of confidence under a budgeting calculation in terms of the size of asset fall which could be sustained and remain solvent on a matching basis. Equivalently, we can express the required assets under a budgeting calculation as a multiple of the matching present value. For example, if assets are required to be sufficient after a 40% fall, this is equivalent to requiring assets equal to 1.67 times the matched present value.

C.5.2 It is common in budgeting to take credit for expected returns on risky assets, as this may be offset by an explicit use of a prudent quantile. In the examples below we assume a 3% per annum risk premium, on a geometric mean basis.

C.5.3 We can now show the budgeting asset requirements as a proportion of the matching valuation for the six example cash flows. See Figure 8.

Figure 8

Some typical patterns comparing budgeting to matching calculations

C.5.4 Firstly, we see that the median requirement falls with the liability term. This reflects the effect of the risk premium, which features in the budgeting calculation but is excluded from the matching calculation. The longer the time horizon, the greater the impact of the risk premium.

C.5.5 At higher levels of confidence, there are two competing effects. Firstly, there is the effect of volatility, which is greater for longer time horizons. Ultimately, however, the risk premium term dominates. Broadly speaking, this is because, in terms of the log total return index, the risk premium effect is proportional to time t while the standard deviation is proportional to the square root of t. This effect is particularly visible in our 25 year annuity example, where the budgeting asset requirement at 75% confidence is only just over half the matching requirement.

C.6 Who Are Actuaries Advising?

C.6.1 The share analyst example introduced in Section C.2.4 is less prototypical of actuarial work in one major respect. Everyone expects a share analyst to be acting only for one side of a possible transaction or legal framework. If the investor buys cheap and sells dear then this is beneficial to the investor following the advice of the good analyst, presumably ultimately at the expense of someone else whose market activities are less profitable.

C.6.2 In contrast, a large fraction of actuarial advice is actually implicitly or explicitly relied upon by a range of interested parties. This is still often true even if the actuary has a single client. For example, a scheme actuary will typically legally have as his or her client the trustee body responsible for the scheme. However, in practice, their work output might still influence a range of different parties in different ways. They might, for example, advise on the cost of augmenting benefits of some but not all scheme members. Even if contributions are provided by the sponsor to meet these benefit improvements, the scheme might subsequently get into difficulties, and their advice might ultimately lead to some beneficiaries being advantaged relative to others, at least relative to the situation in which their advice had been different.

C.6.3 Most types of provision of information for statutory accounting purposes ultimately fit this characteristic. Accounting information will normally be used by many different parties for many different purposes. Most recipients will expect it not to favour unduly one party over another.

C.6.4 Many commentators strongly favour market based approaches when fairness between parties is perceived to be paramount. Market based prices are seen as more objective and less open to bias than potentially any other way of valuing cash flows, even if they are not always easy to identify reliably.

C.6.5 Another important objective of accounting data is to achieve comparability. We can think of this as conceptually expanding ‘fairness’ between participants to include ‘fairness’ in comparisons between the reported positions of different companies. Whilst accounting standards setters and other interested parties are still engaging in (sometimes heated) debate on such topics, the general trend appears to be towards greater use of market based approaches where the assets and liabilities are clearly crystallised, except perhaps when the debate turns to issues of pro-cyclicality.

C.6.6 Assets and liabilities that have less obviously already crystallised are more challenging to handle in such a context. At one extreme would be the valuation of intangibles or other goodwill items. Considerable variation in approach can apply in practice if there is considerable uncertainty about what cash flows might arise.

C.6.7 This differentiation mirrors the one that we introduced in Section C.1.6 between liabilities that were akin to contractual commitments and those that were more discretionary in nature.

C.6.8 Ignoring ‘discretionary’ assets and liabilities in their entirety may be misleading, since it may present an overly optimistic or pessimistic view of the likely costs and benefits of continuing to operate the current business model. More usually, there would be some attempt to formulate approaches that sought a suitable balance between the greater comparability that ought to arise by including standardised assumptions about such cash flows versus the potential for misinformation by doing so. This arguably explains some of the not ‘fully’ market consistent elements present in, say, ‘market consistent’ embedded valuations.

C.6.9 Choice of discount rates can also be strongly linked with the potentially thorny topic of profit recognition. The more favourable are the future return assumptions that we adopt for the assets/liabilities, the greater will be the deemed current ‘profit’.

C.6.10 However, what we assume in this respect does not (we may presume) influence actual outcomes. So, all that may be happening here is that ‘value’ creation may be being front-loaded into the present and away from the future. Over the long term this may not appear to matter very much, but again different people may benefit depending on how the numbers are calculated and incentive arrangements may be altered.

C.7 A Simple Practical Example

C.7.1 John Smith has a house with a mortgage of £250,000. He also has a savings policy, the proceeds of which he intends to use to repay his mortgage in 20 years’ time. The savings policy currently has accumulated funds of £50,000.

In assessing how much John should pay into his savings each month (in addition to his interest costs) John carries out a budgeting calculation, which might look as follows:

Limitations of this budgeting calculation:

1. (a) It gives no information on the value of the outstanding liability.

2. (b) It gives no information on the likelihood that the annual payments will be adequate to meet the mortgage liability after 20 years.

3. (c) It gives no information on what the potential shortfall might become if the investment strategy does not go to plan.

C.7.2 Of course (b) and (c) above can be addressed by further more sophisticated budgeting calculations (looking at different possible investment returns and their likelihood). However to address (a) above, a matching calculation as required. This matching calculation might simply be:

C.7.3 The importance of the matching calculation is obvious when a “solvency” or “transaction” question is being asked, for example:

• • What is the value of the outstanding liability if need to make immediate repayment (e.g. because the house is being sold)?

• • What is the overall value of the estate for inheritance tax purposes?

• • What is the overall value of the marital assets/liabilities in a divorce?

C.7.4 In all of the above situations it is clear that the shortfall calculated on the matching approach (i.e. £200,000) is the correct figure for consideration rather than the shortfall calculated on the budgeting approach (i.e. £56,516).

C.7.5 Both the budgeting and matching calculations have their relevance and importance. The key is to ensure that the correct approach is used depending on the question being asked.