Minimum return guarantees, investment caps, and investment flexibility

Abstract

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.

Appendices

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

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

(24)

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

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

Proof

With Lemma 24, it immediately follows

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

and

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

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

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

Using

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

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

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