Pension scheme derisiking. An application of prospect theory to pension incentive exercises. June 4, 2018

Transcription

1 Pension scheme derisiking. An application of prospect theory to pension incentive exercises June 4, 218 1

2 Abstract We describe the workings of a pension increase exchange (PIE) exercise. We apply cumulative prospect theory (CPT) to the choices facing a typical pension scheme member (M) offered a PIE. We derive a continuous version of Kahneman and Tversky s CPT formula specific to M. We analyse the implications of this formula to the choices facing M. Finally we derive conditions for a PIE that would be attractive to M and summarise the strengths and weaknesses of our analysis. 1 Introduction One of the major challenges facing the pension industry in the UK is the chronic underfunding of pension schemes. Pension schemes undertake derisking exercises to reduce the level of risk in the scheme; the exact nature of these exercises will depend on the types of risks the schemes attempt to reduce. One of the most common types of these exercises are the so-called incentive exercises. This covers various options presented to scheme members to incentivise them to transfer to an external party, some or all of a scheme s obligations to the members. The objective being to reduce the total obligations to the scheme under favourable terms to the scheme. This should make the scheme more likely to meet the remaining obligations. Terms that are favourable to the scheme may or may not be favourable to members. UK pensions law prevents the scheme from offering terms that would materialy undermine the future financial security of the affected members 1. In all instances the scheme needs to gauge the amounts to offer that would; 1) incentivise a large number of members to take up the option and 2) reduce the level of risk in the scheme without compromising the scheme s solvency. Finding this balance can be very difficult, especially where there are no tools to gauge how best to incentivise take-up by members. Prospect theory describes this exact problem. That is, how to model decision making under risk. As far as the author is aware there has been no application of prospect theory in this context for the UK. We will apply prospect theory to understand the factors that impact the level of take-up of an incentive exercise and how to design an inducement to achieve the schemes dual aims above. There are numerous types of these exercises. For the purpose of this paper, we will limit ourselves to pension increase exchanges (PIE). This is because these tend to be the most common types of these exercises and the most amenable to a prospect theory treatment. We will begin by describing how a pension increase exchange exercise works. 1

3 2 Pension increase exchange exercise 2.1 Impact on pension scheme This gives a scheme member (M) the option to exchange future annual inflation linked increases in their pension for annual fixed increases. We set this out mathematically as follows. Let c n be the probability weighted pension recieved by M in year n where n = represents the present and c n is weighted by the probability of M surviving to year n. Let r n be the n year risk-free spot rate and y n be the n year spot inflation rate based on the UK government inflation rate curve. The present value of the scheme s pension obligation to M is given by P en(ȳ) = n c n (1 + y n ) n (1 + r n ) n (1) where ȳ is a vector of the inflation spot rate and Y is a singleton containing ȳ. The exercise invloves offering M a fixed increase f in place of ȳ. UK regulation 1 requires a lower bound on f based on the calculation P en(ȳ) = n c n (1 + y n ) n (1 + r n ) n = n c n (1 + f) n (1 + r n ) n = P en( f) (2) where f is the vector of a fixed increase f. The obvious question would be how does providing this option to M benefit the scheme? One of the benefits to the scheme of M taking up the option would be less volatility in the value assigned to M and potentially smaller payouts if long-term inflation outpaces the current inflation curve. However, the most significant benefit to the scheme comes through in the manner in which UK schemes are required to value pension liabilities and hold capital against these liabilities. P en(ȳ) is based on what is termed a best estimate basis. That is c n, y n and r n are all based on current market and demographic estimates. UK schemes are required to use prudent assumptions to value pensions (see2). In practice, they apply margins for prudence in most assumptions which results in a higher estimate for the pension in comparison to the best estimate basis. This is what is reported to the industry regulator as the value of the pension obligation and is the minimum capital required for the scheme to be deemed fully funded. This is known as the technical provisions basis (TP basis). Hence, if we assume the TP value of liabilites is given by P en(ŷ) then P en(ŷ) > P en(ȳ). M taking up the option for any F such that P en(ŷ) P en( f) > results in a reduction of the capital required to meet the obligation to M. 2

4 2.2 Risk to M of taking up the option The main risk to M is actual inflation may be higher than inflation implied by F, in which case, M suffers a loss over the period in which the pension is paid. Should actual inflation be lower than that implied by F then M benefits from taking up the option. We can set this out as follows: P en( d) = n c n (1 + d n ) n (1 + r n ) n < P en(ȳ) < n c n (1 + u n ) n (1 + r n ) n = P en(ū) (3) where d D is the set of all inflation curves that result in a lower pension than ȳ and ū U is the set of all inflation curves that result in a higher pension than ȳ. If we take ā to be the actual inflation curve that will be experienced then ā D results in M making a profit from taking up the option whereas ā U results in M suffering a loss from taking up the option. Prospect theory tells us that M will make choices depending on the relative probabilities of P en( d) and P en(ū). Prospect theory also tells us that M will have different aversions to risks for P en( d) and P en(ū). In building a prospect theory formulation for this problem we should consider other individuals that may have a material impact on M s decision making. UK law requires that in providing such an option to M, the scheme must provide (and meet the cost of) M recieving financial advice. This is provided by an Independent Financial Adviser (IFA) whose role is to ensure the option chosen is fair and appropriate for M (see 1). This introduces an interesting aspect to the decision making process. It means decisions are made with an understanding of the quantitative differences of the choices. The IFA themselves do not benefit in any way from M s choices nor do they suffer any losses if M makes the wrong choice. Hence, they are completely independent of the scheme providing the offer, and act as a fail-safe to ensure M does not take up an option that is clearly sub-optimal. This is important as it means we can ignore instances of M accepting best estimate options that are very low as a result of the prevailing market conditions, for example a very low or flat inflation curve compared to historic levels. 3 Increase exchange prospect theory formulation Prospect theory was introduced by Kahneman and Tversky in as a model to desribe inconsistencies between empirical evidence and expected 3

5 utility theory in decision making. The original work done was later improved upon 4 to reflect desirable features of a decision making model. This included capturing stochastic dominance in decision making under risk. This resulted in cumulative prospect theory (CPT). The key results from CPT are that individuals tend to over-weight small probabilities and under-weight large probabilities. Individuals also tend to be more risk averse to losses compared to gains. Thus individuals frame games of chance relative to a reference point. That is, how much they stand to gain or lose relative to their current position. This is at odds with expected utility theory which states that individuals are indifferent to reference points and assess games of chance based on their total expected utility. In order to derive a CPT formulation for our problem we first consider discrete inflation outcomes for M. 3.1 Cumulative prospect theory for M in discrete form We assume a countable set of inflation curves exist that captures the universe of future inflation outcomes. Let Y be as in (1) above. Let ū i U and d i D as in (3) above such that P en( d i ) < P en(ȳ) < P en(ū i ). Let g i, for i Z be such that; P en(ȳ) P en( d i ) >, for i > g i = P en(ȳ) P en(ȳ) =, for i = (4) P en(ȳ) P en(ū i ) <, for i < Let U i U be the set containing all vectors ū i, ū j such that P en(ū i ) = P en(ū j ) and D i D be the set containing all vectors d i, d j such that P en( d i ) = P en( d j ) i, j Z. To maintain stochastic dominance we can assume that M is indifferent between inflation curves that produce the same g i since M is only concerned with gains or losses relative to P en(ȳ). It is therefore posssible to group all curves that produce the same outcome under sets U i and D i for losses and gains respectively. We index the sets so that g i is a monotonically increasing series in i. That is, g i > g k i > k i, k Z. Let D U Y = S R N where n N and Σ is a σ -algebra over S. Assume a probability measure µ : Σ, 1 such that (S, Σ, µ) is a probability space and µ(s) = 1. Let p i, q i, 1 such that, µ(d i ) = p i and µ(u i ) = q i (5) CPT tells us that individuals apply subconcious weights to probabilities they encounter; we can set up a value function for M s choice based on these 4

6 weighted probabilities. Let w + 1 and w 1 be the weight functions applied to the probabilities of gains and losses respectively where w + () = w () = and w + (1) = w (1) = 1. Let v + and v be the utility functions for gains and losses respectively. CPT is structured so as to avoid decisions that would contradict stochastic dominance. Consequently, losses and gains are evaluated cumulatively according to their relative sizes. 4 gives us a formula for M s value function V (G) based on CPT V (G) = + i=1 i= 1 v + (g i ) w + ( v (g i ) w ( p k ) w + ( k=i k=i q k ) w ( k=i+1 k=i 1 p k ) q k ) where G = {g i } and G R. If we take P to denote probability and θ >, φ < where θ, φ R, then P (g i < θ) = k i=1 µ(d i), i > P (g i > φ) = t i= 1 µ(u i) i < We can now define a measurable function H(x) :, ), 1 so that { P (g i < x), for i > H(x) = (7) P ( g i < x), for i < then, for i > p k = k=i µ(d k ) = 1 k=i k=i 1 k=1 (6) µ(d k ) = 1 P (g k < g i ) = 1 H(g i ), (8) and, for i < (bearing in mind that g i > g k i > k i, k Z) k=i q k = k=i µ(u k ) = 1 k=i+1 k= 1 µ(u k ) = 1 P (g k g i+1 ) = 1 P (g k > g i ) = 1 P ( g k < g i ) = 1 H( g i ) Therefore (6) becomes V (G) = v + (g i ) w + (1 H(g i )) w + (1 H(g i+1 ) + i= 1 i=1 v (g i ) w (1 H( g i )) w (1 H( g i 1 )) (9) (1) 5

7 Since all gains are greater than (g i >, i > ) and all losses less than (g i <, i < ), it is easy to see that < w ± (1 H(±g i )) < w ± (1 H()) (11) 3.2 Cumulative prospect theory for M in continuous form In order to generalise the problem to capture all possible inflation outcomes, it is necessary to extend the analysis above for continous gains and losses. We take smaller and smaller partitions of R and include them in G and assume H is a continuous function in its domain. From the definition of an integral and (11); lim V (G) as w ± (1 H(±g i )) w±(1 H(±g i±1 )) = w+ (1 H()) v + (g)dw + (1 H(g))+ w (1 H()) v (g)dw (1 H( g)) (12) where we denote the first integrand I + (g) and the second integrand I (g). For ease of notation we will also write w + = w + () and w = w (1 H( g)). Consider I + (g), by applying a change of variable; I + (g) = w+ (1 H()) v + (g)dw + = k2 v + (g) dw + k 1 dg dg (13) Since H is a measurable function, and differentiable functions are a subset of measurable functions (since differentiable = continuous = measurable). We limit H to a family of differentiable functions on, ). This is a reasonable assumption since it is reasonable to assume the cumulative probability of g is a smooth function as g. That is, there is no reason to suppose ĝ whose probability of occurence is much higher or lower than a neighbourhood ĝ ± ɛ of ĝ for some ɛ >. Consequently, we can define the derivative of H on, ). We define h a continuous function in the domain of H such that H(g) = g h(t)dt = = h(t)dt (that is h is the g derivative of H also known as the probability density function of g) then by the chain rule; dw + dg = d dg ( g ) h(t)dt. dw + dx(g) where x(g) = g h(t)dt, g > Applying Leibniz integral rule to evaluate the derivative of the integral, dw + dg = h( ). (d ) h(g). dg dg dg + h(t) g dt dw +. d(x(g)) 6 g

8 dw + = h(g) +. d(x(g)) = h(g)dw + dx (14) = I + (g) = k2 k 1 v + (g)h(g) dw + dx dg We can determine the limits of the integral by solving for g in w + (1 H(g)) = = w + () = = = H(g) = 1 = g = from the definitions of w + and H(g). When w + (1 H(g)) = w + (1 H()) = g = Therefore k 1 = and k 2 =. = I + (g) = v + (g)h(g) dw + dx dg = I +(g) = v + (g)h(g) dw + dx dg (15) We can follow the same steps to derive an expression for I (g). Equation (13) becomes, I (g) = w (1 H()) v (g)dw = k4 k 3 v (g) dw dg (16) dg dw dg = d ( ) dw h(t)dt. dg dx( g) g where x( g) = g h(t)dt, g < = dw dg = h( ). (d ) h( g). d( g) h(t) + dg dg g g dt dw. d(x( g)) dw = + h( g) +. d(x( g)) = h( g)dw dx (17) = I (g) = k4 k 3 v (g)h( g) dw dx dg When w (1 H( g)) = = 1 H( g) = = H( g) = 1 = g = = g =. When w (1 H( g)) = w (1 H()) = g =. So k 3 = and k 4 =. = V (G) = = I (g) = v + (g)h(g) dw + dx dg + 7 v (g)h( g) dw dg (18) dx v (g)h( g) dw dg (19) dx

9 4 M s value function given utility and probability weighting functions 4.1 Utility function We will assume the same utility functions applied by Kahneman and Tversky in 4 since we have no emprical evidence to the contrary. These are given by; v + (g) = g α, g >, < α < 1 (2) v (g) = λ( g) β for g <, < β < 1, λ > (21) We fix α and β less than one in order to have a concave utility function for gains and a convex utility function for losses in line with evidence from CPT. A power function is ideal as it leads to a flat function as g and the opposite as g. That is, individuals are more sensitive to changes in utility nearer reference point than away from the reference point. λ is a measure of loss aversion. The higher the value of λ the more risk averse to losses M becomes. λ > 1 reflects a higher sensitivity to losses than to gains. 4.2 Probability weighting function There are a few possible choices for probability weighting functions. Our preference is a tractable function that reflects M s personal weighting preferences. Satchell and Davis 5 discuss the links between the level of loss aversion and the shape of the probability weighting function and conclude the two are somewhat interlinked. Some of the more common weighting functions proposed for CPT are Prelec s function w + = w = e ( ln(x)ϕ) for a probability x. The benefit of this function is its mathematical tractability (especially with exponential distributions in gains and losses since the function itself is an exponential). It is also a one parameter model and can easily describe empirical evidence of CPT (for example the so-called fourfold pattern of risk attitudes in 4 and 5. Although Prelec s function is relatively simple and tractable, it models aggregate behaviour as opposed to individual behaviour 5. In order to capture individual probability weightings one may revert to Kahneman and Tversky weighting function. w(x) = x γ x γ + (1 x) γ 1 γ (22) 8

10 With this function we sacrifice mathematical tractability for accuracy (the ability to capture individual attitudes to risk probabilities). We note that setting γ = 1 produces the result w(x) = x. That is, no weight is applied to probabilities in this case. Plotting the function reveals that the over/underweighting of probabilities depends on whether γ > 1 or γ < 1 (see appendix). For example γ =.5 leads to an under-weighting of probabilites close to 1 and over-weighting of probabilites close to with w(x) = x for x.28. γ = 1.5 has the opposite effect with w(x) = x at x.675. Very large or very small choices of γ moves the intersection point closer to 1 or respectively. Evidence of CPT is consistent with γ < 1. Kahneman and Tversky formulated w(x) with two parameters to give different functions for w + and w thus capturing individual attitudes that differ for gains and losses. Evidence suggests that a one parameter model of the function achieves the same goal 6. For the purpose of this study we assume a one parameter model of the formula as this improves the mathematical tractability of the expression. To calculate the value function we will need an expression for the derivative of w as in (19). dw dx = w = γxγ 1 x γ + (1 x) γ 1 γ x γ 1 γ xγ + (1 x) γ 1 γ 1 γx γ 1 γ(1 x) γ 1 x γ + (1 x) γ 2 γ = γx γ 1 x γ + (1 x) γ 1 γ xγ x γ + (1 x) γ 1 x γ 1 (1 x) γ 1 x γ + (1 x) γ 1 γ We can factorise w(x) from the expression above. w = γ x w(x) (1 x) γ 1 w(x)xγ 1 x γ + (1 x) γ γ w = w(x) x K(x) + K(1 x) x γ 1, where K(x) = x γ + (1 x) γ We note that γ = 1 K(x) = K(1 x) = 1. (23) 4.3 Value function We can now substitute (2), (21) and (23) into (19) to produce an expression for V (G). Bearing in mind that x(g) = (19) becomes, V (G) = g α γ h(g)w + () + K() K(H(g)) dg + λ( g) β γ h( g))w (1 H( g)) + K(1 H( g)) K(H( g)) dg 1 H( g) (24) 9

11 Notice if we perfom a change of variable on I (g), let u = g = du = 1, dg (24) becomes V (G) = g α γ h(g)w + () + K() K(H(g)) dg + λ(u) β γ h(u))w (1 H(u)) + K(1 H(u)) K(H(u)) du 1 H(u) (25) Substituting g back into I (u) does not change the value of the integral (ie making the substitution u = g = du = 1). We can also swap the limits of dg integration by multiplying by -1. This gives = g α γ h(g)w + () + K() K(H(g)) dg λ(g) β γ h(g))w () + K() K(H(g)) dg (26) We also notice that w + () = w () = w() as in (22); this produces the expression V (G) = (g α λg β γ )h(g)w(1 H(g)) + K() K(H(g)) dg (27) We can summarise this expression as an expectation under the distribution of g. That is V (G) = γ E (g α λg β ) w() E (g α λg β )w()(k(h(g)) K()) (28) Notice when γ = 1, w(1 H(g)) = 1 H(g) and K(H(g)) = K(1 H(g)) = V (G) = E (g α λg β ). This is the result one would expect from expected utility theory where no weightings are applied to the probabilities. However in our CPT formulation we can see that the expected utility is weighted by the ratio w(x)/x and the difference between the over and under-weighting of probabilities captured in K(H(g) K() over all possible values of g. (28) is what we would expect in terms of risk decisions from a CPT perspective. V (G) depends on the expected difference between the utility from losses and that from gains (E(g α λg β )) weighted by M s approximation of 1

12 probabilities w(). If α = β (if the gains and losses utility functions are reflections of each other on the line g = 1) then (28) becomes α w() V (G) = (1 λ)γ E g (29) (1 λ)e g α w((k(h(g)) K() In which case V (G) is directly dependent on the difference 1 λ; a measure of the aversion to losses relative to gains. This is as expected since CPT tells us that M would make decisions that reflect his/her aversion to losses. For λ = 1 (29) produces V (G) =. This is an expected result since, in this case, M has the same sensitivity to gains and losses. Since the distribution of g is symmetric about g = (H(g) = H( g)), then its expected value is. In this case, M is satisfied with the minimum f with no additional inducement required since f is the expected result. That is, M is satisfied with the expected utility theory result. However, this is not the experience of most schemes that run these types of inducement exercises as the level of take-up tends to be low for options offered at the best estimate (2% according to Willis Towers Watson 7). We can explain this using CPT. From the perspective of M exchanging guaranteed inflation linked increases for a fixed increase exposes M to potential gains but more importantly potential losses. That is, in the current state M is sure to get at least inflation, regardless of what happens in the future. If M takes up the option there is the potential that inflation is much higher than the increase accepted. In which case M would see a fall in real income. Scheme members tend to favour guaranteed inflation linked increases (that is, λ > )which is inline with CPT and goes against utility theory that says members should be indifferent between inflation linked increases and the expected equivalent. 5 Designing a pension increase exchange for M Given an expression for h, (27) may be difficult or impossible to solve analytically. However, (28) may be solved numerically or estimated using various techniques for example monte-carlo simulation. 11

13 5.1 Case 1: V (G) = If V (G) = then, theoretically, no further inducement is required and f from (2) may be acceptable. Offering an additional increase δ slightly above f would then produce the result V (G) = (P en( f + δ) P en(ȳ)) α > where δ is the vector of the new increase. This may be much more attractive to M. The scheme should calculate δ so as not to give an offer above the TP basis, that is P en(ŷ) P en( f + δ) > (see section 2.1 for details). 5.2 Case 2: V (G) > If V (G) > then the minimum offer f should be a sufficient inducement for M to take up the option without the need for any additional increases. This would maximise the benefit to the scheme of providing the option. 5.3 Case 3: V (G) < Finally if V (G) < the scheme would need to solve for the minimum δ that would lead to a positive V (G). This problem can be solved by finding δ such that P en( f + δ) P en(ȳ) α = V (G), then increasing the offer above this value of δ. That is solving; P en( f + δ) P en(ȳ) α = E (g α λg β )w()(k(h(g)) K() γ E (g α λg β w() ) (3) Simplifying (3), we solve for δ such that ( P en( f + δ) = E (g α λg β )w()(k(h(g)) K() ) 1 α γ E (g α λg β w() ) P en(ȳ). (31) If P en( f + δ) > P en(ŷ) then the exercise is of no value to the scheme as M is not inclined to take an inducement that would benefit the scheme. In this case, at least for M, alternative incentive exercises may be more beneficial to the scheme. 12

14 5.4 Benefits of CPT for schemes considering pension increase exercises This type of insight can be of great value to the scheme as it may prevent spending time and resources designing exercises that may never be taken up by M and/or members similar to M. Given the high administrative cost of running such an exercise (the scheme needs to fund actuarial expertise to calculate f, pay for IFA s as well as communication to the members, meet the cost of updating records and so on) being able to determine, beforehand, the conditions most likely to result in high take-up is likely to result in significant savings for the scheme. 6 Conclusion We have derived a CPT formulation for PIE and described how this can be tailored to a member M to optimise take-up. However, there remain some questions that this study has not fully tackled. Firstly, we have assumed a symmetric distribution of g. The author has performed some analysis comparing actual rolling annual inflation to the inflation curve over the last 2 years (see appendix). Charting these differences does imply a symmetric distribution centred at, however, a more detailed analysis is welcome to clarify this point. These differences have historically appeared in clusters. That is, periods of higher/lower than expected inflation are followed by periods of the same position with smaller magnitudes. This implies a mean-reverting semi-markov process underlying g. This should be reflected in any H(g) used to model inflation. Secondly, a less technical issue with applying the analysis above is the practicality of the CPT approach. For example, equation (31) is a fairly complex problem to solve, especially, if this is performed for a large number of individuals with different attitudes to risk (different parameters for (31)). With enough computing power this could be run as a large single modelling exercise. The objective being to provide confidence intervals for take-up at various levels of inducement as part of an investigative study of the potential of a PIE exercise. If the PIE exercise is expected to be large-scale, savings from performing such an investigation are likely to outweigh the cost of implementing such a model. Finally, it may prove challenging to accurately determine the parameters γ, α, β and λ, especially if scheme members are unwilling to participate in studies to estimate the parameters. Estimates from existing CPT studies may be used to provide approximate results with the understanding of any 13

15 existing margins of error introduced. CPT is a powerful predictive tool that has not really been used to solve practical problems in the pension management and actuarial field. This, inspite of the proliferation of problems framed in a similar manner to prospect theory experiments. More and more PIE exercises are likely to be done as UK schemes come under increasing pressure to repair existing deficits. CPT could prove to be a useful tool in modelling the likely success of such exercises and saving schemes both time and money in designing and implementing PIE exercises. Schemes may benefit most if they have a clear understanding of the strengths and weaknessess of a CPT approach to the problem and implement CPT accordingly. More work should be done to apply CPT to other inducement exercises offered to scheme members. References 1 The Pensions Regulator. Incentive exercises, accessed (3 June 218). 2 The Pensions Regulator. Funding defined benefits, Code of practice 3, accessed (3 June 218). 3 Kahneman, D. Tversky, A. Prospect Theory: An Analysis of Decision under Risk, Econometrica, (47)(2), pp Kahneman, D. Tversky, A. Advances in Prospect Theory: Cumulative Representation of Uncertainty, Jourrnal of Risk and Uncertainty, (5), pp , Davies, G. Satchell, S. Continuous Cumulative Prospect Theory and Individual Asset Allocation, Cambridge Working Papers in Economics, 467, Gonzalez, R. Wu, G. On the shape of the probability weighting function, Cognitive Psychology, 38, pp , Willis Towers Watson Pension increase exchange options, Corporate-and-Trustee-Briefing/216/1/Pension-increase-exchangeoptions, Corporate and Trustee Briefing, Pension increase exchange options, accessed (3 June 218). 14

16 7 Appendix Probability weighting function =1.5, =.5, =1.7.6 w(x) x 15