Robust Solutions to the Life-Cycle Consumption Problem

Abstract

This paper demonstrates how the well-known discrete life-cycle consumption problem (LCP) can be solved using the Robust Counterpart (RC) formulation technique, as defined in Ben-Tal and Nemirovski (Math Oper Res 23(4):769–805, 1998). To do this, we propose a methodology that involves applying a change of variables over the original consumption before deriving the RC. These transformations allow deriving a closed solution to the inner problem, and thus to solve the LCP without facing the curse of dimensionality and without needing to specify the prior distribution for the investment opportunity set. We generalize the methodology and illustrate how it can be used to solve other type of problems. The results show that our methodology enables solving long-term instances of the LCP (30 years). We also show it provides an alternative consumption pattern as to the one provided by a benchmark that uses a dynamic programming approach. Rather than finding a consumption that maximizes the expected lifetime utility, our solution delivers higher utilities for worst-case scenarios of future returns.

Appendices

Appendix: Proof of Lemma 1

• 1. With the logarithm utility function, $LCP^{*}$ is the following:

  $$\begin{aligned}&\max \limits _{\begin{array}{c} \tilde{c}_t\ge 0 \end{array}} \log (c_0)+\sum _{t=1}^{N-1}\beta _t \log (\tilde{c}_t)+\beta _{N}\log \left( w_0-\sum _{k=1}^{N-1}\tilde{c}_k\right) \\&\quad + \min \limits _{r \in \mathcal {S}} \sum _{t=1}^{N-1}\beta _t\log (R_t)+\beta _{N}\log (R_N)\\&\text {s.t.}\sum _{k=0}^{N-1}\tilde{c}_k \le w_0 \end{aligned}$$

  The inner problem does not depend on the decision variable, so it can be omitted. Consider now the unconstrained problem. The optimal conditions are:

  $$\begin{aligned}&\beta _t\tilde{c}_t^{-1}-\beta _{N}\left( w_0-\sum _{k=0}^{N-1} \tilde{c}_k\right) ^{-1}=0 \quad \forall t:1\ldots N-1\nonumber \\&c_0^{-1}-\beta _{N}\left( w_0-\sum _{k=0}^{N-1}\tilde{c}_k\right) ^{-1}=0 \end{aligned}$$

  (A.1)

  Then

  $$\begin{aligned} \tilde{c}_t=\beta _tc_0 \end{aligned}$$

  (A.2)

  Plugging (A.2) into (A.1) we have

  $$\begin{aligned}&\beta _{N} c_0 =\left( w_0-\sum _{k=0}^{N-1}\tilde{c}_k\right) =w_0-c_0 -\sum _{t=1}^{N-1}\beta _t\\&\implies c_0=\frac{w_0}{1+\sum _{t=1}^{N}\beta _t} \end{aligned}$$

  This solution is feasible. Plugging (A.2)

  $$\begin{aligned} \sum _{t=0}^{N-1}\tilde{c}_t&= \tilde{c}_0 +w_0\sum _{t=0}^{N-1}\frac{\beta _t}{1+\sum _{k=1}^{N}\beta _k}\\&=\frac{w_0}{1+\sum _{t=1}^{N}\beta _t}+w_0\sum _{t=0}^{N-1} \frac{\beta _t}{1+\sum _{k=1}^{N}\beta _k}=w_0 \end{aligned}$$

• 2. When $\theta =0$, the $LCP^{*}$ equals

  $$\begin{aligned}&\max \limits _{\begin{array}{c} \tilde{c}_t\ge 0 \end{array}} \tilde{c}_0^{\gamma }+\sum _{t=1}^{N-1}\beta _t\tilde{c}_t^{\gamma } \hat{a}_t+\beta _{N}\left( w_0-\sum _{k=0}^{N-1}\tilde{c}_k\right) ^{\gamma }\hat{a}_{N}\\&\text {s.t.}\quad \sum _{t=0}^{N-1}\tilde{c}_t \le w_0 \end{aligned}$$

  The optimal conditions for the unconstrained are:

  $$\begin{aligned}&\beta _t\hat{a}_t\tilde{c}_t^{\gamma -1}-\beta _{N} \hat{a}_N\left( w_0-\sum _{k=0}^{N-1}\tilde{c}_k\right) ^{\gamma -1}=0 \quad \forall t:1\ldots N-1\\&c_0^{\gamma -1}-\beta _{N}\hat{a}_N\left( _0-\sum _{k=0}^{N-1} \tilde{c}_k\right) ^{\gamma -1}=0 \end{aligned}$$

  Hence we obtain that

  $$\begin{aligned} \tilde{c}_t=(\beta _t\hat{a}_t)^{\frac{1}{1-\gamma }}c_0 \end{aligned}$$

  (A.3)

  Plugging (A.3) into the first optimal conditions we have:

  $$\begin{aligned}&(\beta _{N}\hat{a}_N)^{\frac{1}{1-\gamma }} c_0=w_0-\sum _{t=0}^{N-1}\tilde{c}_t\\&\quad =w_0-c_0-\sum _{t=1}^{N}(\beta _t\hat{a}_t)^{\frac{1}{1-\gamma }}c_0\\&\quad \implies c_0=\frac{w_0}{1+\sum _{t=1}^{N}(\beta _{t} \hat{a}_t)^{\frac{1}{1-\gamma }}} \end{aligned}$$

  This solution is feasible. Plugging (A.3)

  $$\begin{aligned}&\sum _{t=0}^{N-1}\tilde{c}_t = \tilde{c}_0 +w_0\sum _{t=0}^{N-1}\frac{(\beta _t\hat{a}_t)^{\frac{1}{1-\gamma }}}{1+\sum _{k=1}^{N}(\beta _{k}\hat{a}_k)^{\frac{1}{1-\gamma }}}\\&\quad =\frac{w_0}{1+\sum _{t=1}^{N}(\beta _{t}\hat{a}_t)^{\frac{1}{1-\gamma }}}+\sum _{t=0}^{N-1}\frac{w_0(\beta _t\hat{a}_t)^{\frac{1}{1-\gamma }}}{1+\sum _{k=1}^{N}(\beta _{k}\hat{a}_k)^{\frac{1}{1-\gamma }}}=w_0 \end{aligned}$$

• 3. When $N=1$, then the $LCP^{*}$ equals

  $$\begin{aligned} \max \limits _{\begin{array}{c} 0 \le \tilde{c}_t\le 0 \end{array}} \tilde{c}_0^{\gamma } +\beta _{1}(w_0-\tilde{c}_0)^{\gamma }(\hat{a}-\theta \sqrt{V}) \end{aligned}$$

  The optimal condition is

  $$\begin{aligned} c_0^{\gamma -1}-\beta _{1}(w_0-\tilde{c}_0)^{\gamma -1}(\hat{a}-\theta \sqrt{V})=0 \end{aligned}$$

  which leads to

  $$\begin{aligned} c_0= \frac{w_0}{1+(\beta _1(\hat{a}-\theta \sqrt{V}))^{\frac{1}{1-\gamma }}} \end{aligned}$$

Appendix: Proof of Eq. (11)

By Eq. (10) and the definition of $R_t=\prod _{k=1}^t r_k$,

$$\begin{aligned}&\hat{a}_t =E(R_t^{\gamma })=E\left( \left[ \prod _{k=1}^t e^{\mu +L_{k}z}\right] ^\gamma \right) \\&\quad = e^{\gamma \mu t} E\left( e^{\gamma \sum _{k=1}^t \sum _{j=1}^{N}L_{k,j}z_j}\right) =e^{\gamma \mu t} E\left( e^{\gamma \sum _{j=1}^N z_j \sum _{k=1}^{t} L_{k,j}}\right) \end{aligned}$$

Since z is a standard normal i.i.d. vector and that $LL^T=C$, this is equal to

$$\begin{aligned}&= e^{\gamma \mu t} \prod _{j=1}^N E\left( e^{\gamma z_j \sum _{k=1}^{t} L_{k,j}}\right) = e^{\gamma \mu t} \prod _{j=1}^N e^{\frac{\gamma ^2}{2}\left[ \sum _{k=1}^{t} L_{k,j}\right] ^2}\\&=e^{\gamma \mu t} e^{\frac{\gamma ^2}{2}\sum _{j=1}^N\left[ \sum _{k=1}^{t} L_{k,j}\right] ^2}=e^{\gamma \mu t}e^{\frac{\gamma ^2}{2}\sum _{k=1}^t\sum _{l=1}^{t} C_{k,l}}\\&=e^{\gamma \mu t+\frac{\gamma ^2}{2}A(t,t)} \end{aligned}$$

with

$$\begin{aligned}&A(t,t) = \sum _{k=1}^t\sum _{l=1}^{t} C_{k,l}\\&\quad =\sigma ^2\sum _{k=1}^t\sum _{l=1}^{t}\rho ^{\left| k-l\right| } =\frac{t-t\rho ^2-2\rho +2\rho ^{t+1}}{(1-p)^2}\\&Cov(R_s^{\gamma }, R_t^{\gamma })\\&\quad = e^{\gamma \mu (t+s)} Cov\left( e^{\gamma \sum _{k=1}^t L_{k}z}, e^{\gamma \sum _{l=1}^s L_{l}z}\right) = e^{\gamma \mu (t+s)} \\&\qquad \left[ E(e^{\gamma \left( \sum _{k=1}^t L_{k}z+\sum _{l=1}^s L_{l}z\right) }-E\left( e^{\gamma \sum _{k=1}^t L_{k}z}\right) E\left( e^{\gamma \sum _{l=1}^s L_{l}z}\right) \right] \end{aligned}$$

Denoting $A(t,s):=\sum _{k=1}^t\sum _{l=s}^t C_{t,s}$, we have shown that $E(e^{\gamma \sum _{k=1}^t L_{k}z})=e^{\frac{\gamma ^2}{2}B(t,t)}$. z is a standard normal i.i.d. vector and that $LL^T=C$, thus

$$\begin{aligned}&E\left( e^{\gamma \left( \sum _{k=1}^t L_{k}z+\sum _{l=1}^s L_{l}z\right) }\right) \\&\quad =E(e^{\gamma \sum _{j=1}^N z_j \left( \sum _{k=1}^t L_{k,j}+\sum _{l=1}^s L_{l,j}\right) }\\&\quad =\prod _{j=1}^N E\left( e^{\gamma z_j \left( \sum _{k=1}^t L_{k,j}+\sum _{l=1}^s L_{l,j}\right) }\right) \\&\qquad \prod _{j=1}^N e^{\frac{\gamma ^2}{2}\left( \sum _{k=1}^t L_{k,j}+\sum _{l=1}^s L_{l,j}\right) ^2}=e^{\frac{\gamma ^2}{2}\sum _{j=1}^N\left( \sum _{k=1}^t L_{k,j}+\sum _{l=1}^s L_{l,j}\right) ^2}\\&\quad =e^{\frac{\gamma ^2}{2}[A(t,t)+A(s,s)+2A(s,t)]} \end{aligned}$$

Thus $Cov(R_s^{\gamma }, R_t^{\gamma })$ equals

$$\begin{aligned}&e^{\gamma \mu (t+s)} \left[ e^{\frac{\gamma ^2}{2}[A(t,t)+A(s,s)+2A(s,t)]} - e^{\frac{\gamma ^2}{2}[A(t,t)+A(s,s)]}\right] \\&\quad = e^{\gamma \mu (t+s)}e^{\frac{\gamma ^2}{2}[A(t,t)+A(s,s)]}[ e^{\gamma ^2 A(s,t)}-1 ]\\&\quad =\hat{a}_s\hat{a}_t[ e^{\gamma ^2 A(s,t)}-1] \end{aligned}$$