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Minimum return guarantees, investment caps, and investment flexibility


We study the merits of capped retirement products with guarantee for investors who have the flexibility to dynamically adjust their investment strategy. All contracts under consideration are fairly priced such that the net profit of the provider is zero. Without the rider, an expected utility maximizing CRRA investor does not want an investment cap. Here, she commits herself to a strategy a priori. With the flexibility rider, the optimization problem changes and the optimal strategy is a response to an exogenously set price. A fair pricing then anticipates the optimal response of the investor. We show that the maximum expected utility of the investor can, for anticipated fairly priced products, be obtained for a finite cap. Thus, a capped product design can give a Pareto improvement to the otherwise uncapped contract version.

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  1. One example for a capped product design is the Index Select offered by Allianz Life Insurance in Germany.

  2. There is no incentive for the investor to reveal her preferences. In addition, it seems not possible to explain to the customer that there are different prices for the same contract (all investors can decide dynamically on their investments!).

  3. For different contract specifications within GMAB contracts we refer to

  4. Notice that the financial market model described by (1) and (2) is arbitrage free and complete, i.e., there is a uniquely defined equivalent martingale measure.

  5. Other justifications of guarantees are e.g., discussed in Døskeland and Nordahl (2008).

  6. The interested reader is referred to Mahayni and Schneider (2012).

  7. Notice that \(h(0)<0\) can only be replicated by a short position in a bond, i.e., borrowing. In addition, recall that \(P=S_0=100\). For \(h(0)=0\), accounting for borrowing constraints means that admissible investment payoffs are bounded by the buy and hold payoff \(h(x)=x\). In particular, \(h'(x)\in [0,1]\), and \(h''(x)\le 0\) do not allow for leverage strategies which give a convex payoff profile.

  8. Recall that for the commitment solution (first best solution), the cap does not help (cf. Sect. 2). The CRRA investor does not want a cap (and/or guarantee), cf. Remark 1.

  9. Notice that it does not matter here if the investor caps her payoff herself or if the provider caps the Merton solution and uses the freed amount to finance the guarantee, i.e., all contracts are fairly priced.


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Correspondence to Judith C. Schneider.

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The authors thank seminar and conference audiences in Hannover, Munich, Wuppertal, and Zurich as well as Nikolaus Schweizer and Sven Balder. We further wish to thank the anonymous referee for his valuable comments which significantly improved the paper.


Appendix 1: Proofs of Section 2

1.1 Useful expectations

Lemma 1

(Useful change of measure) Let \(W_t^*\) denote a \(P^*\)-Brownian motion where \(P^*\) is the risk neutral measure. Then, \(\hat{W}_t=W_t^*-m\sigma t\) is a \(\hat{P}\)-Brownian motion where

$$\begin{aligned} \left( \frac{\, {d}\hat{P}}{\,{ dP}^*}\right) _t:= e^{m\sigma W_t^*-\frac{1}{2}m^2\sigma ^2 t}. \end{aligned}$$

Lemma 2

Let \(h(t,x)=h^{\text {Merton}}(t,x;v)\) where \(h^{\text {Merton}}\) is defined by Eq. (4). In addition, let \(d_1(t,K,m)\) be defined by Eq. (6). Then it holds

$$\begin{aligned} E_{*,t}\left[ e^{-r (T-t)}h(T,S_T)\right]&= h(t,S_t)\\ E_{*,t}\left[ e^{- r (T-t)}h(T,S_T){1}_{\{S_T\le K_1\}}\right]&=h(t,S_t) {\mathcal {N}}\left( -d_1(t,K_1,m)\right) \\ E_{*,t}\left[ e^{-r (T-t)}\left( h(T,S_T)-C_T\right) {1}_{\{S_T>K_2\}}\right]&= h(t,S_t) {\mathcal {N}}\left( d_1(t,K_2,m)\right) \\&\quad -e^{-r(T-t)}C_T {\mathcal {N}}\left( d_1(t,K_2,0)\right) . \end{aligned}$$


With Lemma 24, it immediately follows

$$\begin{aligned}&E_{*,t}\left[ e^{- r (T-t)}h(T,S_T){1}_{\{S_T\le K_1\}}\right] \\&\quad =h(t,S_t) \hat{P}\left( S_T\le K_1\right) \\&\quad = h(t,S_t) {\mathcal {N}}\left( -\frac{ \ln \frac{S_t}{K_1}+(r+\left( m-\frac{1}{2}\right) \sigma ^2)(T-t)}{\sigma \sqrt{T-t}}\right) \end{aligned}$$


$$\begin{aligned}&E_{*,t}\left[ e^{-r (T-t)}\left( h(T,S_T)-C_T\right) {1}_{\{S_T>K_2\}}\right] \\&\quad = h(t,S_t)\hat{P}(S_T>K_2)-e^{-r(T-t)}C_T P^*(S_T>K_2)\\&\quad = h(t,S_t) {\mathcal {N}}\left( \frac{ \ln \frac{S_t}{K_2}+(r+\left( m-\frac{1}{2}\right) \sigma ^2)(T-t)}{\sigma \sqrt{T-t}}\right) \\&\qquad -e^{-r (T-t)} C_T {\mathcal {N}}\left( \frac{ \ln \frac{S_t}{K_2}+(r-\frac{1}{2}\sigma ^2)T}{\sigma \sqrt{T}}\right) . \end{aligned}$$

1.2 Proof of Proposition 1

For a fairly priced floored (and capped) contract, the optimality of the Merton solution follows straightforwardly with the results given in El Karoui et al. (2005). Since the optimal investment fractions of a CRRA are independent of her initial wealth (investment), a fairly priced guarantee (and cap) only results in an adjustment of the initial investment (i.e., from the contribution P to the investment premium \(P^{\text {Inv.}}\)). In particular, the fair pricing condition is

$$\begin{aligned} E_*[e^{-r T}\min \{\max \{ V_T,G_T\}, C_T\}]=P. \end{aligned}$$

Notice that

$$\begin{aligned}&E_*[e^{-r T}\min \{\max \{ V^*_T,G_T\}, C_T\}]\\&\quad = E_*\left[ e^{- r T}h^{\text {Merton}}(T,S_T;\hat{V}_0)\right] -E_*\left[ e^{-r T}(h^{\text {Merton}}(T,S_T;\hat{V}_0)-G_T){1}_{\{S_T<K_1\}}\right] \\&\qquad - E_*\left[ e^{-r T}(h^{\text {Merton}}(T,S_T;\hat{V}_0)-C_T){1}_{\{S_T>K_2\}}\right] \end{aligned}$$

where \(K_1\) and \(K_2\) are defined by the conditions

$$\begin{aligned} h^{\text {Merton}}(T,K_1;\hat{V}_0)&=G_T \quad \text { and }\;\;h^{\text {Merton}}(T,K_2;\hat{V}_0) =C_T\\ \text {i.e., } K_1&=S_0\left( \frac{G_T}{\hat{V}_0 M_T}\right) ^{1/m}\quad \text { and }\;\; K_2 =S_0 \left( \frac{C_T}{\hat{V}_0 M_T}\right) ^{1/m}\\ \text {where } M_T&:= e^{(1-m)(r+0.5 m \sigma ^2)T}\quad \text { and }\;\;m=\frac{\mu -r}{\gamma \sigma ^2}. \end{aligned}$$

With Lemma 2, we immediately have

$$\begin{aligned}&E_*[e^{-r T}\min \{\max \{ V^*_T,G_T\}, C_T\}]\\&\quad = \hat{V}_0-\left( \hat{V}_0{\mathcal {N}}\left( -d_1(0,K_1,m)\right) - e^{-r T} G_T{\mathcal {N}}\left( -d_1(0,K_1,0)\right) \right) \\&\qquad -\left( \hat{V}_0 {\mathcal {N}}\left( d_1(0,K_2,m)\right) -e^{-r T}C_T {\mathcal {N}}\left( d_1(0,K_2,0)\right) \right) . \end{aligned}$$

In particular, the budget constraint then implies

$$\begin{aligned} \hat{V}_0=\frac{P-e^{-r T} G_T{\mathcal {N}}\left( -d_1(0,K_1,0)\right) -e^{-r T}C_T {\mathcal {N}}\left( d_1(0,K_2,0)\right) }{1-{\mathcal {N}}\left( -d_1(0,K_1,m)\right) -{\mathcal {N}}\left( d_1(0,K_2,m)\right) }. \end{aligned}$$

Appendix 2: Proofs of Section 3

1.1 Proof of Proposition 3

Notice that the t-value \(V_t^{\text {N B}}\) of the optimal payoff \(h^{*, \text { Exo.}}(S_T)\) (cf. Eq. 9) without borrowing constraints is determined by

$$\begin{aligned} V_t^{\text {NB}}= E_{*,t}\left[ e^{-r(T-t)}h^{*, \text { Exo.}}(S_T)\right] . \end{aligned}$$

With Lemma 2 of the Appendix 2 and the shortcut notation \(h(t,S_t)=h^{\text {Merton}}(t,S_t;\hat{V}_0)\), it immediately follows

$$\begin{aligned} V_t^{\text {NB}}&= h(t,S_t) - E_{t,*}\left[ e^{- r (T-t)}\left( h(T,S_T){1}_{\{S_T\le K_1\}}+(h(T,S_T)-C_T){1}_{\{S_T>K^*_2\}}\right) \right] \\&= h(t,S_t)\left[ 1-{\mathcal {N}}\left( -d_1(t,K_1,m)\right) - {\mathcal {N}}\left( d_1(t,K_2,m)\right) +\frac{C_T}{h(t,S_t)}{\mathcal {N}}\left( d_1(t,K_2,0)\right) \right] . \end{aligned}$$

Now, consider the budget constraint and the determination of \(\hat{V}_0\). Recall that \(\hat{V}_0\) is implicitly defined by the condition \(V_0 = E_*\left[ e^{-r T}h^{*,\text { Exo.}}(T,S_T;\hat{V}_0)\right] \), i.e., \(V_0=V_0^{\text {NB}}(\hat{V}_0)\). The rest of the proof follows with Proposition 3 and \(h(0,S_0)=\hat{V}_0\).

1.2 Proof of Proposition 4

Notice that the number of assets \(\Delta ^{(S)}\) of the replication strategy is

$$\begin{aligned} \Delta _t^{(S)}=\frac{\partial }{\partial S_t}V_t^{\text {NB}}=f^{\text {NB}}(t,S_t)\frac{\partial }{\partial S_t}h(t,S_t)+h(t,S_t) \frac{\partial }{\partial S_t}f^{\text {NB}}(t,S_t). \end{aligned}$$


$$\begin{aligned} \frac{\partial }{\partial S_t}h(t,S_t)&= m \frac{h(t,S_t)}{S_t} \text { implies } \Delta _t^{(S)}\\&=\frac{h(t,S_t)f^{\text {NB}}(t,S_t)}{S_t}\left( m+ S_t \frac{\frac{\partial }{\partial S_t}f^{\text {NB}}(t,S_t)}{f^{\text {NB}}(t,S_t)} \right) . \end{aligned}$$

For \(C_T<\infty \), we have

$$\begin{aligned} f^{\text {NB, No Cap}}(t,S_t)&= {\mathcal {N}}\left( d_1(t,K_1,m)\right) - {\mathcal {N}}\left( d_1(t,K_2,m)\right) \\&\quad +\frac{e^{-r(T-t)}C_T}{h(t,S_t)}{\mathcal {N}}\left( d_1(t,K_2,0)\right) \end{aligned}$$

and it holds

$$\begin{aligned}&\frac{\partial }{\partial S_t}f^{\text {NB, No Cap}}(t,S_t) = \frac{1}{S_t\sigma \sqrt{T-t}} \left[ {\mathcal {N}}'\left( d_1(t,K_1,m)\right) - {\mathcal {N}}'\left( d_1(t,K_2,m)\right) \right. \\&\qquad + \left. e^{-r (T-t)}C_T\; \frac{ {\mathcal {N}}'\left( d_1(t,K_2,0)\right) -m\sigma \sqrt{T-t}{\mathcal {N}}\left( d_1(t,K_2,0)\right) }{h(t,S_t)}\right] . \end{aligned}$$

In the limiting case \(C_T\rightarrow \infty \), this simplifies to

$$\begin{aligned} \frac{\partial }{\partial S_t}f^{\text {NB, No Cap}}(t,S_t)&= {\mathcal {N}}'\left( d_1(t,K_1,m)\right) \frac{\partial d_1(t,K_1,m)}{\partial S_t}\\&= {\mathcal {N}}'\left( d_1(t,K_1,m)\right) \frac{1}{S_t\sigma \sqrt{T-t}}. \end{aligned}$$

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Mahayni, A., Schneider, J.C. Minimum return guarantees, investment caps, and investment flexibility. Rev Deriv Res 19, 85–111 (2016).

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  • Minimum return guarantees
  • Investment caps
  • Investment flexibility
  • Pareto efficient contract design

JEL Classification

  • G11
  • G22