Does surplus/deficit sharing increase risk-taking in a corporate defined benefit pension plan?

Abstract

This paper studies the surplus-/deficit-sharing effects on the risk-taking of a corporate defined benefit pension plan. Our analytical results show that when a surplus-/deficit-sharing rule is introduced, the participants’ risk-taking increases, while the direction of the surplus-/deficit-sharing effect on the equityholders’ risk-taking is ambiguous. The numerical analysis reveals that for plausible parameter values, the equityholders’ risk-taking increases due to the introduction of surplus/deficit sharing. The participants’ risk-taking increases much more substantially than the equityholders’ risk-taking when introducing surplus/deficit sharing. The participants’ risk-taking is more sensitive to the level of funding than the equityholders’ risk-taking: The participants’ risk-taking can become extremely high for low funding levels. This high sensitivity of the participants’ risk-taking to low funding levels is reduced by introducing deficit sharing. Risk-taking is independent of the funding level when the surplus and deficit proportions are equal.

Appendices

Appendix A: Proof of the speculative fund with program (2)

With optimization program (2), the utility function argument, denoted by X, is defined by:

$$\begin{aligned} X\equiv A-L \end{aligned}$$

(21)

One differentiates Eq. (21) and divides by X. The following is obtained:

$$\begin{aligned} \frac{\hbox {d}X}{X}=\frac{A}{X}\frac{\hbox {d}A}{A}-\frac{L}{X}\frac{\hbox {d}L}{L} \end{aligned}$$

(22)

Let us derive the A dynamics. One introduces the S and $\eta$ dynamics, as given by Eqs. (7) and (8), respectively, in Eq. (6), and takes account of the relationship $x_{S}+x_{\eta }=1$. One obtains:

$$\begin{aligned} \frac{\hbox {d}A}{A}=\left( x_{S}\left( \mu _{S}-r\right) +r\right) \hbox {d}t+x_{S}\sigma _{S}\hbox {d}W^{S} \end{aligned}$$

(23)

Replacing the A and L dynamics, as given by Eqs. (23) and (9), respectively, in Eq. (22), yields:

$$\begin{aligned} \frac{\hbox {d}X}{X}= & {} \left[ \frac{A}{X}\left( x_{S}\left( \mu _{S}-r\right) +r\right) -\frac{L}{X}\mu _{L}\right] \hbox {d}t \nonumber \\&+\,\frac{A}{X}x_{S}\sigma _{S}\hbox {d}W^{S}-\frac{L}{X}\sigma _{L}\hbox {d}W^{L} \end{aligned}$$

(24)

Let the indirect utility function J be defined as:

$$\begin{aligned} J(X(t),K(t),t)\equiv \max _{x_{S}}E_{t}\left[ U(X(T))\right] \end{aligned}$$

(25)

with J increasing, strictly concave in X, once differentiable with respect to t and twice differentiable with respect to X and K.

The Hamilton–Jacobi–Bellman optimality condition is:

$$\begin{aligned} 0=\max _{x_{S}}DJ(X(t),K(t),t) \end{aligned}$$

(26)

where D denotes the Dynkin operator.

The Dynkin of J is defined by:

$$\begin{aligned} DJ= & {} J_{t}+J_{X}X\mu _{X}+\frac{1}{2}J_{XX}X^{2}\sigma _{X}^{2} \nonumber \\&+\,\mathop \sum \limits _{i}J_{K_{i}}K_{i}\mu _{K_{i}}+\frac{1}{2}\mathop \sum \limits _{i} \mathop \sum \limits _{j}J_{K_{i}K_{j}}K_{i}K_{j}\sigma _{K_{i}K_{j}} \nonumber \\&+\,\mathop \sum \limits _{i}J_{XK_{i}}XK_{i}\sigma _{XK_{i}} \end{aligned}$$

(27)

The subscripts on J denote partial derivatives. $\sigma _{kl}$ stands for the covariance between any variables k and l. The variable X dynamics is written under its general form $\frac{\hbox {d}X}{X}=\mu _{X}\hbox {d}t+\sigma _{X}\hbox {d}W^{X}$ .

One replaces the parameters of the X dynamics, as given by Eq. (24), in Eq. (27), and derives with respect to $x_{S}$ to obtain the speculative fund shown in Table 1.

Appendix B: Proof of the speculative fund with program (3)

With optimization program (3), the utility function argument, denoted by Y, is defined by:

$$\begin{aligned} Y\equiv \beta A+(1-\beta )L+(\alpha -\beta )C \end{aligned}$$

(28)

One differentiates Eq. (28) and divides by Y. One obtains:

$$\begin{aligned} \frac{\hbox {d}Y}{Y}=\frac{\beta A}{Y}\frac{\hbox {d}A}{A}+\frac{(1-\beta )L}{Y}\frac{\hbox {d}L}{L}+ \frac{(\alpha -\beta )}{Y}\hbox {d}C \end{aligned}$$

(29)

Let us derive the dynamics of the call C. One applies Ito’s lemma to the function $C(t,A,L,K_{1},K_{2},\ldots ,K_{N})$. This yields:

$$\begin{aligned} \hbox {d}C= & {} C_{t}\hbox {d}t+C_{A}\hbox {d}A+C_{L}\hbox {d}L+\sum _{i}C_{K_{i}}\hbox {d}K_{i} \nonumber \\&+\,\frac{1}{2}C_{AA}\hbox {d}A\hbox {d}A+\frac{1}{2}C_{LL}\hbox {d}L\hbox {d}L+\frac{1}{2} \sum _{i}\sum _{j}C_{K_{i}K_{j}}\hbox {d}K_{i}\hbox {d}K_{j} \nonumber \\&+\,C_{AL}\hbox {d}A\hbox {d}L+\sum _{i}C_{AK_{i}}\hbox {d}A\hbox {d}K_{i}+\sum _{i}C_{LK_{i}}\hbox {d}L\hbox {d}K_{i} \end{aligned}$$

(30)

where subscripts on C denote partial derivatives.

One replaces the A, L and $K_{i}$ dynamics, as given by Eqs. (23), (9) and (10), respectively in Eq. (30). The obtained C dynamics are replaced in Eq. (29). The following Y dynamics is obtained:

$$\begin{aligned} \frac{\hbox {d}Y}{Y}= & {} \frac{\beta A}{Y}\left[ \left( x_{S}\left( \mu _{S}-r\right) +r\right) \hbox {d}t+x_{S}\sigma _{S}\hbox {d}W^{S}\right] \nonumber \\&+\,\frac{(1-\beta )L}{Y}\left[ \mu _{L}\hbox {d}t+\sigma _{L}\hbox {d}W^{L}\right] \nonumber \\&+\,\frac{(\alpha -\beta )}{Y}\left[ \begin{array}{c} \left[ \begin{array}{c} C_{t}+C_{A}A\left( x_{S}\left( \mu _{S}-r\right) +r\right) +C_{L}L\mu _{L}+\sum _{i}C_{K_{i}}K_{i}\mu _{K_{i}} \\ +\frac{1}{2}C_{AA}\left( Ax_{S}\sigma _{S}\right) ^{2}+\frac{1}{2} C_{LL}\left( L\sigma _{L}\right) ^{2}+\frac{1}{2}\sum _{i} \sum _{j}C_{K_{i}K_{j}}K_{i}K_{j}\sigma _{K_{i}K_{j}} \\ +C_{AL}ALx_{S}\sigma _{SL}+\sum _{i}C_{AK_{i}}AK_{i}x_{S}\sigma _{SK_{i}}+\sum _{i}C_{LK_{i}}LK_{i}\sigma _{LK_{i}} \end{array} \right] \hbox {d}t \\ +C_{A}Ax_{S}\sigma _{S}\hbox {d}W^{S}+C_{L}L\sigma _{L}\hbox {d}W^{L}+\sum _{i}C_{K_{i}}K_{i}\sigma _{K_{i}}\hbox {d}W^{K_{i}} \end{array} \right] \nonumber \\ \end{aligned}$$

(31)

The indirect utility function J, the optimality condition and the Dynkin of J are as defined when deriving the speculative fund for program (2), except that X is replaced with Y in Eqs. (25), (26) and (27), respectively.

One replaces the parameters of the Y dynamics, as defined by Eq. (31), in the Dynkin of J, and derives with respect to $x_{S}$. This yields the speculative fund shown in Table 1.

Appendix C: Proof of the speculative fund with program (4)

With optimization program (4), the utility function argument, denoted by Z, is defined by:

$$\begin{aligned} Z\equiv \left( 1-\beta \right) (A-L)+(\beta -\alpha )C \end{aligned}$$

(32)

Differentiating Eq. (32) and dividing by Z, one obtains:

$$\begin{aligned} \frac{\hbox {d}Z}{Z}=\frac{\left( 1-\beta \right) A}{Z}\frac{\hbox {d}A}{A}-\frac{(1-\beta )L }{Z}\frac{\hbox {d}L}{L}+\frac{(\beta -\alpha )}{Z}\hbox {d}C \end{aligned}$$

(33)

One replaces the dynamics of A, L and C, as given by Eqs. (23), (9) and (30) respectively, in Eq. (33). This yields:

$$\begin{aligned} \frac{\hbox {d}Z}{Z}= & {} \frac{\left( 1-\beta \right) A}{Z}\left( \left( x_{S}\left( \mu _{S}-r\right) +r\right) \hbox {d}t+x_{S}\sigma _{S}\hbox {d}W^{S}\right) \nonumber \\&-\,\frac{(1-\beta )L}{Z}\left[ \mu _{L}\hbox {d}t+\sigma _{L}\hbox {d}W^{L}\right] \nonumber \\&+\,\frac{(\beta -\alpha )}{Z}\left[ \begin{array}{c} \left[ \begin{array}{c} C_{t}+C_{A}A\left( x_{S}\left( \mu _{S}-r\right) +r\right) +C_{L}L\mu _{L}+\sum _{i}C_{K_{i}}K_{i}\mu _{K_{i}} \\ +\frac{1}{2}C_{AA}\left( Ax_{S}\sigma _{S}\right) ^{2}+\frac{1}{2} C_{LL}\left( L\sigma _{L}\right) ^{2}+\frac{1}{2}\sum _{i} \sum _{j}C_{K_{i}K_{j}}K_{i}K_{j}\sigma _{K_{i}K_{j}} \\ +C_{AL}ALx_{S}\sigma _{SL}+\sum _{i}C_{AK_{i}}AK_{i}x_{S}\sigma _{SK_{i}}+\sum _{i}C_{LK_{i}}LK_{i}\sigma _{LK_{i}} \end{array} \right] \hbox {d}t \\ +C_{A}Ax_{S}\sigma _{S}dW^{S}+C_{L}L\sigma _{L}dW^{L}+\sum _{i}C_{K_{i}}K_{i}\sigma _{K_{i}}dW^{K_{i}} \end{array} \right] \nonumber \\ \end{aligned}$$

(34)

Applying the method described above, one obtains the speculative fund shown in Table 1.