Tail risk in pension funds: an analysis using ARCH models and bilinear processes

Abstract

Pension funding rules and practice contain implicit smoothing and counter-cyclical mechanisms. We set up a stylized model to investigate whether this may give rise to tail risk, in the form of large but rare losses, when pension liabilities are imperfectly but optimally hedged by pension fund assets. We find that pension losses follow a nonlinear dynamic process, and we derive a complete description of the stochastic properties of this process using Markov chain and bilinear stochastic process theory. The resulting pension dynamics resembles that of a modified ARCH model, which suggests that bursts in volatility may occur, and tail risk may be present. Simulations confirm that pension losses exhibit skewness, leptokurtosis and heavy tails, specially when cash flow smoothing is pronounced. Regulators and investors should be aware of the total amount of smoothing in pension funds as this may contribute to extreme losses, which may adversely affect the security of employee benefits as well as the valuation of firms with corporate pension plans.

Appendices

Appendix: 1 Proof of approximation in Eq. (5)

Recall that Y is the liability when it is calculated at the term-independent discount rate δ. From Eq. (2),

$$ Y = \sum_{x=E}^R m_x \ell_x \widetilde{\alpha}(x), $$

(27)

where $\widetilde{\alpha}(x)$ is the present value, at rate δ and at time t, of a deferred annuity paying $1 yearly in advance from retirement age R till death, for an individual aged x ≤ R at time t:

$$ \widetilde{\alpha}(x) = \sum_{j=0}^{\infty} e^{-(R-x+j)\delta} \ell_{R+j}/\ell_x. $$

(28)

The normal cash flow Z _t is Z _t  = Z ⁽¹⁾ _t  − Z ⁽²⁾ _t  + Z ⁽³⁾ _t , where each component is defined in Sect. 3.2.

$$ Z_t = m_R \ell_R \alpha(t,R) - \sum_{x=E}^{R-1} (m_{x+1}- m_x) \ell_x \alpha(t,x) + e^{-\delta} (\delta_t - \delta)Y_t. $$

(29)

By analogy to Y in Eq. (27), we may define Z to be the normal cash flow Z _t when it is calculated using the term-independent discount rate δ.

$$ Z = m_R \ell_R \widetilde{\alpha}(R) - \sum_{x=E}^{R-1} (m_{x+1}- m_x) \ell_x \widetilde{\alpha}(x). $$

(30)

Noting that m _E  = 0 by definition (no benefit right endowed at entry into the plan), the above may be rewritten as

$$ Z = \sum_{x=E}^{R} m_x \left[ \ell_x \widetilde{\alpha}(x) - \ell_{x-1} \widetilde{\alpha}(x-1) \right]. $$

(31)

The term in square brackets above may be further simplified using Eq. (28):

$$ \ell_x \widetilde{\alpha}(x) - \ell_{x-1} \widetilde{\alpha}(x-1) = \sum_{j=0}^{\infty} e^{-(R-x+j) \delta } \ell_{R+j} (1 - e^{-\delta}) = \ell_x \widetilde{\alpha}(x) (1 - e^{-\delta}). $$

(32)

Finally, substituting Eq. (32) in Eq. (31) and using Y as defined in Eq. (27) gives

$$ Z = \sum_{x=E}^{R} m_x \ell_x \widetilde{\alpha}(x) (1 - e^{-\delta}) = Y(1-e^{-\delta}). $$

(33)

As in Assumption 1, where the liability Y _t is linearized around δ, we may linearize Z _t in Eq. (29):

$$ Z_t = Z e^{ -D_Z(\delta_t - \delta) } + e^{-\delta} (\delta_t - \delta)Y_t, $$

(34)

where D _Z  =  − Z ⁻¹∂Z/∂δ. Using Eq. (3) from Assumption 1 as well as Eq. (33), we obtain

$$ \frac{Z_t}{Y_t} = \frac{Z}{Y} \frac{Y}{Y_t} e^{-D_Z(\delta_t - \delta) } + e^{-\delta} (\delta_t - \delta) = (1-e^{-\delta}) e^{ -(D_Z - D_Y)(\delta_t - \delta) } + e^{-\delta} (\delta_t - \delta). $$

(35)

The difference between the modified durations of Z and Y, which appears in Eq. (35), is easily derived from Eq. (33). We note that ∂Z/∂δ = (∂Y/∂δ)(1 − e ^−δ) + Ye ^−δ. Upon dividing by both sides of Eq. (33), we find that Z ⁻¹(∂Z/∂δ) = Y ⁻¹(∂Y/∂δ) + (e ^δ − 1)⁻¹. Hence, D _Z  − D _Y  =  −(e ^δ − 1)⁻¹. Substituting this into Eq. (35) results in

$$ \frac{Z_t}{Y_t} = (1-e^{-\delta}) \exp\left[ -\frac{\delta_t - \delta}{e^{\delta}-1} \right] + e^{-\delta} (\delta_t - \delta) \approx (1-e^{-\delta}) \left[ 1 - \frac{\delta_t - \delta}{e^{\delta}-1} \right] + e^{-\delta} (\delta_t - \delta) = 1-e^{-\delta}. $$

(36)

This concludes the proof of the approximation in Eq. (5). $\square$

Appendix: 2 Proof of proposition 1

It immediately follows from Eq. (7) and Assumption 2 that

$$ {{\mathbb{E}}} \left[ {F_t} | {{{\mathcal{F}}}_{t-1}} \right] = \left( F_{t-1} + C_{t-1} - Z \right) {{\mathbb{E}}}\left[ {\exp\left( r_t^A - r_t^L + \delta \right)} \right] = e^{\delta}\left( F_{t-1} + C_{t-1} - Z \right). $$

(37)

From the definition of the loss L _t in Eq. (8), we therefore obtain

$$ L_t = e^{\delta}(F_{t-1} + C_{t-1} - Z) - F_t. $$

(38)

It is convenient to work in terms of the unfunded liability, $\overline{F}_t = 1-F_t,$ and replace Z using Eq. (5), yielding $\overline{F}_t - e^{\delta}\overline{F}_{t-1} = L_t - e^{\delta} C_{t-1}.$ Upon substituting the supplementary contribution from Eq. (9), we get $\overline{F}_t - e^{\delta} \overline{F}_{t-1} = L_t - e^{\delta} {\varvec{\pi}}' {\bf L}_{t-1}.$ This is a linear difference Eq. in $\overline{F}_t$ which may be solved using standard techniques. It is easily verified that its solution is

$$ \overline{F}_t = {\varvec{\lambda}}' {\bf L}_t, $$

(39)

where ${\varvec{\lambda}} = (\lambda_1, \ldots, \lambda_p)'$ and λ _j  = e ^(j-1)δ + π _j  − ∑ ^j _k=1 e ^(j-k)δπ _k for $j \in [1, p]$. Equation (39) shows that the unfunded liability is the accumulated value of unpaid losses.

Now, the main recurrence relation may be obtained by substituting F _t from the budget constraint in Eq. (7) directly into the r.h.s. of Eq. (38). Simplifying with the help of Eq. (5) and using the unfunded liability $\overline{F}_t = 1 - F_t$ results in

$$ L_t = \left[ \exp(r_t^A - r_t^L) - 1 \right] \left[ e^{\delta}(\overline{F}_{t-1}-C_{t-1})- 1 \right]. $$

(40)

Replacing C _t from Eq. (9) and $\overline{F}_t$ from Eq. (39) then yields the recurrence relation in Eq. (10) in Proposition 1. We note that ${\varvec{\beta}} = e^{\delta}({\varvec{\lambda}} - {\varvec{\pi}})$ and that β _p  = 0. $\square$

Appendix: 3 Proof of proposition 2

Part (i). ${{\mathbb{E}}{L_t} = {\mathbb{E}}\left[ {\varepsilon_t} \right]{\mathbb{E}}\left[ {v_t} \right] = 0}$, by independence of $\varepsilon_t$ and v _t . ${{\mathbb{E}}{L_t}}$ exists ${\Leftrightarrow\, {\mathbb{E}}{|L_t|}<\infty}$ (Feller 1971, p. 136). {L _t } is evidently adapted to ${\{\mathcal{F}_t\}}.$ For $\tau < t,\, \varepsilon_t$ is independent of ${{\mathcal{F}}_\tau}$ and ${{\mathbb{E}} \left[ {L_t} \, | \, {{\mathcal{F}}_\tau} \right] = {\mathbb{E}}\left[ {\varepsilon_t} \right] {\mathbb{E}} \left[ {v_t} \; | \; {\mathcal{F}_\tau} \right] = 0}.$ Hence, {L _t } is a martingale difference sequence (m.d.s.) (Davidson, 1994, p. 230). The covariance result is a property of m.d.s.: see e.g. Davidson (1994, Corollary 15.4). It is easily verified: ${{\mathbb{E}}\left[ {L_t L_\tau} \right] = {\mathbb{E}}\left[ {\varepsilon_t v_t L_\tau} \right] = {\mathbb{E}}\left[ {\varepsilon_t} \right] {\mathbb{E}}\left[ {v_t L_\tau} \right] = 0}$ when t > τ.

Part (ii). Given the results in part (i), we need only consider VarL _t to establish covariance-stationarity of {L _t }. Since ${{\mathbb{E}}{\varepsilon_t} = 0}$ and ${{\mathbb{E}}\left[ {L_t L_\tau} \right]=0}$ for τ ≠ t, it follows that ${{\mathbb{E}}{v_t^2} = \sum \beta_j^2 {\mathbb{E}}{L_{t-j}^2} + 1}.$ Now, ${{\rm Var}{L_t} = {\mathbb{E}}{L_t^2} = {\mathbb{E}}\left[ {\varepsilon_t^2} \right]{\mathbb{E}}\left[ {v_t^2} \right]},$ giving a pth-order linear difference equation: VarL _t  = ∑ ^p _j=1 σ² β ² _j VarL _t-j  + σ². It is well-known that stability depends on the zeros of the characteristic polynomial $z^{p-1}-\sigma^2 \beta_1^2 z^{p-2} - \cdots - \sigma^2 \beta_p^2$ satisfying |z| < 1. Since the coefficients $\sigma^2 \beta_j^2,\, j=1, \ldots, p$ are non-negative, a necessary and sufficient condition for stability is that $\sum \sigma^2 \beta_j^2 = \sigma^2 {\varvec{\beta}}' {\varvec{\beta}} < 1.$ The initial values $L_l, \ldots, L_{l-p+1}$ are known at some time l < 0 and, letting $l\rightarrow -\infty,$ all second moment terms containing the initial values $L_l, \ldots, L_{l-p+1}$ vanish geometrically iff σ² b′ b < 1. Finally, from the pth-order linear difference equation in VarL _t above, VarL _t converges to $\sigma^2 /\left( 1 - \sigma^2 \sum \beta_j^2 \right).$ $\square$

Appendix: 4 proof of proposition 3

First, note that $L_t=\varepsilon_t v_t$ (Eq. (11a)) with v _t independent of $\varepsilon_t$ and $\{\varepsilon_t\}$ being i.i.d. (Assumption 2) and hence trivially stationary and ergodic, so that ergodicity and stationarity carry across from {v _t } to {L _t } and vice-versa.

Part (i). To apply Theorem 1, we first obtain the conditions under which {L _t } is irreducible.

{L _t } in Eq. (10) is not a bilinear process, according to Definition 1 because k ≠ 0 in Eq. (18). However, using Eqs. (11a) and (b), we can write

$$ v_t = \sum_{j=1}^{p} b_j v_{t-j} \epsilon_{t-j} - 1, $$

(41)

$$ v_t + 1 = \sum_{j=1}^{p} b_j (v_{t-j} + 1) \epsilon_{t-j} - \sum_{j=1}^{p} b_j \epsilon_{t-j}. $$

(42)

This conforms to the class of bilinear processes in Eq. (20) and Theorem 4 is therefore applicable.

The state-space representation of {v _t } consists of the state Eq. (13a), with A _t and B _t defined in Eqs. (14) and (15) respectively, together with the observation equation $v_t = {\varvec{\beta}}' {\bf L}_t -1.$ In accordance with Theorem 4, the irreducibility of the Markov chain {L _t } in Eq. (13a) may be guaranteed by insisting on the condition on the density of $\varepsilon_t,$ which does not violate Assumption 2, specifically ${{\mathbb{E}}{\varepsilon_t}=0}.$ Controllability of {v _t } follows by comparing Eqs. (41) and (20) and noting that there is no “autoregressive” element and no possibility of coincident roots in the polynomials of Eq. (21).

To conclude the application of Theorem 1, we proceed to show that the technical conditions ${{\mathbb{E}}{\log^+\|{\bf A}_t\|} < \infty}$ and ${{\mathbb{E}}{\log^+\|{\bf B}_t\|} < \infty}$ do hold. Using the maximum row sum matrix norm induced by the max vector norm (Horn and Johnson 1985, p. 295), $\|{\bf A}_t\| = \max(1, |\varepsilon_t|\sum|\beta_j|)$ and $\|{\bf B}_t\| = |-\varepsilon_t| = |\varepsilon_t|.$ Hence, $\log^+\|{\bf A}_t\| = \log\left[ \max( 1, \max( 1, |\varepsilon_t|\sum|\beta_j| ) ) \right] = \log^+\left( |\varepsilon_t|\sum|\beta_j| \right).$

Now, $0 \leq \log^+(k|\varepsilon_t|) \leq k|\varepsilon_t|,$ for any k ≥ 0, and hence ${{\mathbb{E}}{ \log^+(k|\varepsilon_t|) } \leq {\mathbb{E}}\left[ { k|\varepsilon_t| } \right]}.$ By assumption, ${{\mathbb{E}}{\varepsilon_t} = 0}.$ Therefore, ${{\mathbb{E}}\left[ {k \varepsilon_t} \right]}$ exists and is finite, which is equivalent to ${{\mathbb{E}}{|k \varepsilon_t|} < \infty}$ (Feller 1971, p. 136). Hence, ${{\mathbb{E}}{ \log^+(k|\varepsilon_t|) } < \infty}.$ Replacing k by ∑|β _j | yields ${{\mathbb{E}}{\log^+\|{\bf A}_t\|} < \infty}.$ Replacing k by 1 yields ${{\mathbb{E}}{\log^+\|{\bf B}_t\|} < \infty}.$ This may also be confirmed by a direct application of Theorem 5 of Kristensen (2009).

This concludes the proof of part (i).

Part (ii). This is an application of Theorem 2. In Eq. (41), {v _t } conforms to the bilinear process in Definition 1 as required in Theorem 2. Theorem 2 is therefore applicable to {v _t } with $\widetilde{{\bf A}}_t = {\bf A}_t$ and $\widetilde{{\bf B}}_t = {\bf B}_t.$ A straightforward inspection of the proof of Theorem 2 (Pham 1985, Theorem 4.1) shows that, because B _t in Eq. (15) does not contain a quadratic in $\varepsilon_t$ (as would be the case for a general bilinear process under Theorem 2), we need not insist that ${{\mathbb{E}}{\varepsilon_t^4}<\infty}$ and merely require that ${{\mathbb{E}}{\varepsilon_t^2}<\infty}.$

We claim that Q = Q I _p , where Q is the stationary variance of {L _t }, given on the r.h.s. of Eq. (12), is a positive solution of the matrix Eq. (19), thereby satisfying the condition of Theorem 2. This may be seen by noting that ${{\mathbb{E}}{{\bf L}_t}={\bf 0}_p}$ from part (i) of Proposition 2 and using the state Eq. (13a) to obtain ${{\rm Var}{{\bf L}_{t+1}} = {\mathbb{E}}\left[ {{\rm Var} \left[ {{\bf L}_{t+1}} \, | \, {\varepsilon_{t+1}} \right]} \right] + {\rm Var}\left[ { {\mathbb{E}} \left[ { {\bf L}_{t+1} } \, | \, { \varepsilon_{t+1} } \right] } \right] = {\mathbb{E}}\left[ { {\bf A}_{t+1} ({\rm Var}{{\bf L}_{t}}) {\bf A}_{t+1}' } \right] + {\rm Var}{{\bf B}_{t+1}}}.$ In the covariance-stationary state, this equation becomes ${{\rm Var}{{\bf L}_{t}} = {\mathbb{E}}\left[ { {\bf A}_{t} ({\rm Var}{{\bf L}_{t}}) {\bf A}_{t}' } \right] + {\rm Var}{{\bf B}_{t}}}$ which corresponds to Eq. (19), and is satisfied by VarL _t  = Q I _p in accordance with Proposition 2.

This concludes the proof of part (ii).

Part (iii). This is an application of Theorem 3. In Eq. (41), {v _t } conforms to the bilinear process in Definition 1 as required in Theorem 3, which is therefore applicable to {v _t } with $\widetilde{{\bf A}}_t = {\bf A}_t$ and $\widetilde{{\bf B}}_t = {\bf B}_t.$ Note, from Eq. (11a), that ${{\mathbb{E}}\left[ {L_t^{2m}} \right] = {\mathbb{E}}\left[ {\varepsilon_t^{2m}} \right] {\mathbb{E}}\left[ {v^{2m}} \right]}.$

Part (iv). From Proposition 2, {L _t } is a martingale difference sequence (m.d.s.). Therefore, {C _t } and $\{\overline{F}_t\}$ are weighted sums of m.d.s. from Eqs. (9) and (39). Moments of $C_t,\, \overline{F}_t$ and L _t follow from Eqs. (9), (39), (10) respectively. {L _t } is serially uncorrelated but {C _t } and $\{\overline{F}_t\}$ have zero autocorrelation at lags p and greater. Parts (iv) and (v) are consistent for lag τ = 0 since, e.g., $\sum\nolimits_{j=1}^p ({\varvec{\pi}} {\varvec{\pi}}')_{(j,j)} = \hbox{tr}({\varvec{\pi}} {\varvec{\pi}}') = {\varvec{\pi}}' {\varvec{\pi}}.$ $\square$

Appendix: 5 Averaging and amortization

Actuarial standards of practice discuss asset value averaging in detail: see for example ASB (2009). A taxonomy of practical methods is presented by Committee on Retirement Systems Research (2001) and described mathematically by Owadally and Haberman (2004b). This results in averaging losses, say over ${n\in{\mathbb{N}}}$ years, with allowance for interest:

$$ L_t^A = \frac{1}{n} \sum_{j=0}^{n-1} e^{j\delta} L_{t-j}. $$

(43)

Smoothed losses are then paid off, usually in equal tranches, over a number of years (say ${m\in{\mathbb{N}}}$) through the device of amortization (ASB 2007a; McGill et al. 2004). A supplementary contribution is paid at time t by amortizing every previous “averaged” loss that has not yet been paid in full over, say, m years:

$$ C_t = \frac{1}{\alpha_m} \sum_{j=0}^{m-1} L_{t-j}^A, $$

(44)

where $\alpha_m = 1 + e^{-\delta} + \cdots + e^{-(m-1)\delta}$ is the present value of an annuity over m years.

Working through the double summation in Eqs. (43) and (44), one may express the supplementary contribution in terms of the filter specified in Eq. (9), with p = m + n − 1. Letting $\overline{p}=\max(m,n)$ and $\underline {p}=\min(m,n)$ and using some algebra returns the following filter weights π _j .

$$ \pi_j = \left\{ \begin{array}{ll} (n\alpha_m)^{-1} e^{(j-1)\delta} \alpha_j & \hbox{if}\, j\in[1,\underline{p}] \\ (n\alpha_m)^{-1} e^{[\min(j,n)-1]\delta} \alpha_{\underline{p}} & \hbox{if}\, j\in[\underline{p},\overline{p}] \\ (n\alpha_m)^{-1} e^{(n-1)\delta} \alpha_{p-j+1} & \hbox{if}\, j\in[\overline{p},p] \\ 0 & \hbox{otherwise}\\ \end{array}\right. $$

(45)

The weights {π _j } above do satisfy the requirement that ∑ ^p _j=1 e ^−(j-1)δ π _j  = 1 as set out under Eq. (9). Equation (9) shows that the supplementary contribution includes a proportion π _j of the loss that occurred j years ago, $j\in[1,p].$