Theory

To understand how the package works, we treat the algorithm proposed by [1] as our foundation. While the original authors introduced the use of Taylor expansions to approximate the value function, most standard implementations are limited to a restrictive second-order (mean-variance) approximation, and they assume that wealth transitions are purely multiplicative.This package extends the foundational Brandt logic in two major ways:

• Generalized High-Order Expansions: It provides the exact mathematical expressions and algorithmic implementation for a Taylor expansion of any arbitrary order $k$. By deriving the multinomial expansion of the budget constraint and the associated high-order derivatives of the value function, this tool allows for extreme precision when capturing non-normalities (skewness, kurtosis) and higher-order moments in asset returns.
• Wealth-Dependent Returns (Exogenous Cash Flows): It breaks the standard assumption of wealth independence in parts of the budget constraint by natively supporting affine wealth transitions. This allows the algorithm to seamlessly incorporate non-tradeable income, labor income, or fixed consumption costs across time.

Goal of the algorithm

In this package we are treating a terminal wealth optimization problem. More specifically, in this package we consider portfolio choice problems at timesteps $n = 1, 2, \ldots, M$, where $M + 1$ is some terminal timestep. This portfolio choice problem at timestep $n$ is defined by an investor who maximizes the expected utility of their wealth at the terminal timestep $M + 1$ by trading $N$ risky assets and a risk-free asset (cash). Formally the investor's problem at timestep $n$ is

\[ V_n(W_n, Z_n) = \max_{\{\omega_m\}_{m = n}^{M}} \mathbb{E}_n[u(W_{M + 1})]\]

subject to the sequence of budget constraints

\[ W_{m + 1} = W_m (\omega_m^\top R^e_{m + 1} + R_{m + 1})\]

for all $m \geq n$. Here $R^e_{m + 1}$ can be interpreted as the excess return of the risky assets over the risk-free asset, and $R_{m + 1}$ is the gross return of other processes that may depend on wealth $W_m$. Furthermore, $\{\omega_s\}_{m=n}^{M}$ is the sequence of portfolio weights chosen at times $m = n, \ldots, M$ and $u$ is the investor's utility function. The process $Z_n$ is a vector of state variables that are relevant for the investor's decision making. The goal of this package is to find $\{\omega_m\}_{m=1}^{M}$.

Extension: Wealth-Dependent Returns

A key feature of this implementation is that the gross return $R_{m+1}$ is not restricted to be exogenous. We allow $R_{m+1}$ to depend on the current level of wealth $W_m$ through the following structure:

\[ R_{m+1} = X_m + \frac{Y_m}{W_m}\]

where $X_m$ and $Y_m$ are functions of state variables contained in $Z_m$. This formulation is particularly powerful as it allows the algorithm to incorporate non-tradeable income or fixed costs. For instance, if $Y_m$ represents labor income, the budget constraint correctly captures that income is added to wealth regardless of the portfolio choice $\omega_m$.

Example

Consider an investor who receives a stochastic labor income $O_n$ at each timestep and saves a proportion $p \in [0,1]$ of that income. Let $R^f_n$ be the gross risk-free rate. The wealth at time $n+1$ is:

\[ W_{n+1} = W_n (\omega_n^\top R^e_{n+1} + R^f_n) + p O_n\]

By setting $X_n = R^f_n$ and $Y_n = p O_n$, this matches our budget constraint $W_{n+1} = W_n (\omega_n^\top R^e_{n+1} + R_{n+1})$.

Solution of the algorithm

The goal of this section is to give the final solution needed to set up the algorithm. We acknowledge that the result can look quite daunting and so it is not the expectation that the reader can immediately make sense of why the solution looks like it does. Despite this, it can help to first see where we are working towards before explaining how it is derived. Let us first start by specifying the notation.

Notation

Let $\omega_m = (\omega^1_m, \ldots, \omega^N_m)$ be the components of the portfolio weights at some timestep $m$ and write $R^e_{m + 1} = (R^{e, 1}_{m + 1}, \ldots, R^{e, N}_{m + 1})$ for the components for the excess return vector corresponding to each asset that is traded. If $\omega_m$ is chosen optimally, we decorate it with a $\star$ in the superscript, i.e. $\omega_m^{\star}$ is the vector of optimal portfolio weights at timestep $m$. Furthermore, let us assume that the current timestep is denoted by $n$ and let us also define

\[ \hat W_{M + 1} = W_n R_{n + 1} \prod_{m = n + 1}^{M} ((\omega_m^{\star})^\top R^e_{m + 1} + R_{m + 1}).\]

Result

We assume that the current timestep is denoted by $n$ and that all optimal portfolio weights at times $m > n$ are known, i.e. $\{\omega_m^\star\}_{m = n + 1}^{M}$ are known. Assuming the value function $V$ is infinitely differentiable in the first argument, the optimal portfolio weights $\omega_n^\star$ is the limit of the solutions of the equation indexed by $k$

\[\begin{aligned} \sum_{r = 1}^{k} \frac{W_n^{r - 1}}{(r - 1)!} &\Biggl(\sum_{k_1 + \ldots + k_N = r - 1} \binom{r - 1}{k_1, \ldots, k_N} \prod_{i=1}^N (\omega_n^i)^{k_i} \times \\ &\mathbb{E}_n \left[\partial^{r} u(\hat W_{M + 1}) \prod_{m = n + 1}^{M} ((\omega_m^\star)^\top R^e_{m + 1} + X_{m})^r \prod_{i=1}^N (R^{e,i}_{n + 1})^{k_i} R^e_{n+1}\right]\Biggr) = 0, \end{aligned}\]

where $\binom{p}{k_1,\ldots, k_N}$ is the multinomial given by

\[ \binom{p}{k_1,\ldots, k_N} = \frac{p!}{k_1 ! k_2 ! \cdots k_N !}.\]

Given this result, the pragmatic reader may come up with some reasonable questions:

• How should one choose $k$?
• How should one calculate the conditional expectation value $\mathbb{E}_n[\cdot]$?
• How should one find the optimal portfolio weights $\omega_m^\star$ for $m > n$?

The short and concise answers are:

• The higher $k$, the better. In our experience $4$ suffices for most applications.
• This is done by means of regressions. This can be unstable however. Dropping part of the data generated by the integrand $\cdot$ in $\mathbb{E}_n[\cdot]$ seems to partly solve this issue.
• This is done by first solving for $\omega_M^\star$. You can then recursively find the solution for any time.

If the above does not make sense, do not fret. This will soon all make sense once we treat the derivation of the above.

Deriving the solution: so, how do we get there?

The (rough) outline of the derivation is as follows:

1. We prove the value function satisfies the Bellman equation

\[V_n(W_n, Z_n) = \max_{\omega_n} \mathbb{E}_n[V_{n + 1}(W_{n + 1}, Z_{n + 1})]\]

1. We derive the Taylor expansion the value function $V_{n + 1}$ in the conditional expectation. This gives a new expression for $V_n$. Note: This is where the $k$ came from in the solution.
2. Using the new expression for $V_n$ we derive the first order conditions of finding the optimal $\omega_n$.
3. We manipulate the first order conditions so that all unknown quantities are replaced by known quantities.

The Bellman equation

The value function satisfies the Bellman equation

\[ V_n(W_n, Z_n) = \max_{\omega_n} \mathbb{E}_n[V_{n + 1}(W_{n + 1}, Z_{n + 1})]\]

with terminal condition $V_{M + 1} (W_{M + 1}, Z_{M + 1}) = u(W_{M + 1})$.

Proof

We see that

\[\begin{aligned} V_n(W_n, Z_n) &= \max_{\{\omega_m\}_{m = n}^{M}} \mathbb{E}_n\left[ u(W_{M + 1}) \right]\\ &= \max_{\omega_n} \max_{\{\omega_m\}_{m = n + 1}^{M}} \mathbb{E}_n\left[ u(W_{M + 1}) \right] \\ &= \max_{\omega_n} \max_{\{\omega_m\}_{m = n + 1}^{M}} \mathbb{E}_n\left[ \mathbb{E}_{n + 1}\left[ u(W_{M + 1}) \right] \right] \\ &= \max_{\omega_n} \mathbb{E}_n\left[ \max_{\{\omega_m\}_{m = n + 1}^{M}} \mathbb{E}_{n + 1}\left[ u(W_{M + 1}) \right] \right] \\ &= \max_{\omega_n} \mathbb{E}_n\left[ V_{n + 1}(W_{n + 1}, Z_{n + 1}) \right]. \end{aligned}\]

The first equality follows from the definition of the value function. The second equality follows from separating the maximization over $\omega_n$ and the remaining maximization over $\{\omega_m\}_{m = n + 1}^{M}$. The third equality follows from the law of iterated expectations. The fourth equality follows from the fact that the maximization over $\{\omega_m\}_{m = n + 1}^{M}$ is independent of the choice of $\omega_n$ and can thus be moved inside the expectation. And the last equality follows from the definition of the value function at time $t + 1$.

Taylor expanding the value function

Let us assume that the value function $V$ is $C^{k + 1}$ in the first argument for $k \geq 2$. Then the value function (using the Bellman equation) satisfies

\[\begin{aligned} V_n(W_n, Z_n) &= \max_{\omega_n} \Biggl\{ \sum_{r = 0}^{k} \frac{W_n^r}{r!} \mathbb{E}_n \left[ \partial_1^r V_{n+1}(W_n R_{n+1}, Z_{n+1}) (\omega_n^\top R^e_{n+1})^r \right] \\ &+ \frac{W_n^{k+1}}{(k+1)!} \mathbb{E}_n \left[ \partial_1^{k+1} V_{n+1}(\xi, Z_{n+1}) (\omega_n^\top R^e_{n+1})^{k+1} \right] \Biggr\} \end{aligned}\]

for some $\xi$ between $W_n R_{n + 1}$ and $W_n (\omega_n^\top R^e_{n + 1} + R_{n + 1})$. Here we assume that all moments exist and are finite.

Proof

Let us consider the mapping $f(x) = V_{n + 1} (x, Z_{n + 1})$ for fixed $n$ and $Z_{n + 1}$. Using Taylor's theorem, we have that

\[ f(x) = \sum_{r = 0}^k \frac{1}{r!} f^{(r)}(x_0)(x - x_0)^r + \frac{f^{(k + 1)}(\xi)}{(k+1)!}(x - x_0)^{k + 1}\]

for some $\xi$ between $x$ and $x_0$. Taking $x_0 = W_n R_{n + 1}$ and $x = W_{n + 1}$, and noting that $W_{n + 1} - W_n R_{n + 1} = W_n \omega_n^\top R^e_{n + 1}$ by the budget constraint, the result follows.

Finding the first order conditions

The next is to solve the static maximization problem in the Taylor expansion. The next proposition provides the final equation for this.

Let $\omega_n = (\omega_n^1, \dots, \omega_n^N)$ be the components of the portfolio weights at time $n$. Similarly, write $R^e_{n+1} = (R_{n+1}^{e,1}, \dots, R_{n+1}^{e,N})$. The first order conditions (FOC) are given by:

\[ \sum_{r = 1}^{k} \frac{W_n^{r - 1}}{(r - 1)!} \mathbb{E}_n \left[ \partial_1^r V_{n + 1}(W_n R_{n + 1}, Z_{n + 1}) (\omega_n^\top R^e_{n + 1})^{r - 1} R^e_{n + 1} \right] + \frac{W_n^k}{k!} \mathbb{E}_n \left[ \partial_1^{k + 1} V_{n + 1}(\xi, Z_{n + 1}) (\omega_n^\top R^e_{n + 1})^{k} R^e_{n + 1} \right] = 0\]

or alternatively:

\[\begin{aligned} \sum_{r = 1}^{k} \frac{W_n^{r - 1}}{(r - 1)!} &\Biggl(\sum_{k_1 + \dots + k_N = r - 1} \binom{r - 1}{k_1, \dots, k_N} \prod_{i=1}^N (\omega_n^i)^{k_i} \times \\ &\mathbb{E}_n \left[ \partial_1^r V_{n + 1}(W_n R_{n + 1}, Z_{n + 1}) \prod_{i=1}^N (R^{e, i}_{n + 1})^{k_i} R^e_{n+1} \right]\Biggr) \\ &+ \frac{W_n^k}{k!} \mathbb{E}_n \left[ \partial_1^{k + 1} V_{n + 1}(\xi, Z_{n + 1}) (\omega_n^\top R^e_{n + 1})^{k} R^e_{n + 1} \right] = 0. \end{aligned}\]

Proof

To derive the FOC, we take the static maximization problem from the Taylor expansion, take the derivative with respect to $\omega_n$, and set it to $0$. After this, we divide both sides by $W_n$. This yields the first result.

We now note that for arbitrary $r \in \mathbb{N}$, it holds by the multinomial theorem that:

\[(\omega_n^\top R^e_{n + 1})^{r-1} = \sum_{k_1 + \dots + k_N = r-1} \binom{r-1}{k_1, \dots, k_N} \prod_{i=1}^N (\omega_n^i R^{e, i}_{n + 1})^{k_i}\]

where $\binom{p}{k_1, \dots, k_N}$ is the multinomial coefficient. Using this identity and the linearity of the expectation, we find the second result.

Finding an alternative way of writing the value function

The last problem we ought to solve is the problem of having no expression for the derivatives of the value function $\partial^r_1 V_{n + 1}(W_n R_{n+1}, Z_{n + 1})$. To that end, we note the following lemma.

Let $\{\omega^\star_m\}_{m = n + 1}^{M}$ be the optimal sequence of portfolio weights at times $m = n + 1, \dots, M$, and denote

\[\hat{W}_{M+1} = W_n R_{n + 1} \prod_{m = n+1}^{M} ((\omega_m^\star)^\top R^e_{m + 1} + R_{m + 1}).\]

Then

\[\partial^{r}_1 V_{n + 1}(W_n R_{n + 1}, Z_{n + 1}) = \mathbb{E}_{n + 1} \left[ u^{(r)}(\hat{W}_{M+1}) \prod_{m = n + 1}^{M} ((\omega_m^\star)^\top R^e_{m + 1} + X_m)^r \right].\]

Proof

Since we are interested in taking the derivative of the first argument of the value function $V_{n + 1}$, we need to make the dependence of $W_{n + 1}$ (the first argument) as clear as possible. For that reason we will first rewrite the budget constraint after which we show the above.

Recall that the budget constraint is given by

\[ W_{m + 1} = W_m (\omega_m^\top R^e_{m + 1} + R_{m + 1})\]

where $R_{m + 1} = X_{m} + Y_m / W_m$. Defining $G_{m + 1} = (\omega_m^\top R^e_{m + 1} + X_m)$ and rewriting the budget constraint to

\[ W_{m + 1} = W_m (\omega_m^\top R^e_{m + 1} + X_m) + Y_m,\]

we can write

\[ W_{M + 1} = W_{n + 1} \left(\prod_{m = n + 1}^{M} G_{m + 1}\right) + \sum_{m = n + 1}^{M} Y_{m} \left(\prod_{p = m + 1}^{M} G_{p + 1}\right)\]

where $n < M$ is some arbitrary timestep. The critical reader might demand a proof of this. This can be found in the appendix. Using this, we now note that

\[\frac{\partial W_{M + 1}}{\partial W_{n + 1}} = \prod_{m = n + 1}^{M} G_{m + 1}.\]

Using the definition of the value function and assuming $\{\omega^\star_m\}$ is the optimal sequence, we have

\[ V_{n + 1}(x, Z_{n + 1}) = \mathbb{E}_{n + 1} \left[ u \left( W_{M + 1}\right) \right],\]

where

\[ W_{M + 1} = x \left(\prod_{m = n + 1}^{M} G_{m + 1}\right) + \sum_{m = n + 1}^{M} Y_{m} \left(\prod_{p = m + 1}^{M} G_{p + 1}\right)\]

Using the Leibniz rule to differentiate under the expectation sign, we compute the $r$-th order derivative of the value function with respect to its first argument $x$ which is

\[ \partial_1^{r} V_{n + 1}(x, Z_{n + 1}) = \mathbb{E}_{n + 1} \left[ \frac{\partial^r}{\partial x^r} u(W_{M + 1}) \right].\]

By the chain rule, and noting that $W_{M+1}$ is an affine function of $x$, the first derivative is:$\frac{\partial}{\partial x} u(W_{M+1}) = u'(W_{M+1}) \frac{\partial W_{M+1}}{\partial x} = u'(W_{M+1}) \left( \prod_{m = n + 1}^{M} G_{m+1} \right)$Since the term $\prod G_{m+1}$ does not depend on $x$, subsequent derivatives follow a power-rule pattern for the marginal growth term. For any $r \geq 1$, we have

\[\frac{\partial^r}{\partial x^r} u(W_{M+1}) = u^{(r)}(W_{M+1}) \left( \prod_{m = n + 1}^{M} G_{m+1} \right)^r.\]

Substituting the expression for the terminal wealth evaluated at the specific point $x = W_n R_{n+1}$, which we denote as $\hat{W}_{M+1}$, gives

\[\partial_1^{r} V_{n + 1}(W_n R_{n+1}, Z_{n + 1}) = \mathbb{E}_{n + 1} \left[ u^{(r)}(\hat{W}_{M+1}) \left( \prod_{m = n + 1}^{M} G_{m+1} \right)^r \right].\]

This yields the desired result, expressing the derivative of the value function solely in terms of the utility function's derivatives, the realized future wealth, and the marginal returns on investment.

Final result

Using this, we can now reformulate the FOC.

The first order conditions (FOC) are given by:

\[\begin{aligned} \sum_{r = 1}^{k} \frac{W_n^{r - 1}}{(r - 1)!} &\Biggl(\sum_{k_1 + \dots + k_N = r - 1} \binom{r - 1}{k_1, \dots, k_N} \prod_{i=1}^N (\omega_n^i)^{k_i} \times \\ &\mathbb{E}_n \left[ u^{(r)}(\hat{W}_{M+1}) \prod_{m = n + 1}^{M} ((\omega_m^\star)^\top R^e_{m + 1} + X_{m})^r \prod_{i=1}^N (R^{e, i}_{n + 1})^{k_i} R^e_{n+1} \right]\Biggr) \\ &+ \text{Remainder} = 0. \end{aligned}\]

High level details on implementation

With the mathematical framework established, the implementation follows a backward-recursive simulation procedure. The process can be summarized in the following five steps:

Data generation

First, the stochastic processes for asset returns and state variables are generated across $S$ simulation paths. This includes the excess returns $R^e_{n+1}$, and the wealth-dependent return components $X_n$ and $Y_n$ embedded within the state vector $Z_n$.

Backward Induction

The algorithm proceeds backward in time from the penultimate step $M$. At any current timestep $n$, we assume that the optimal portfolio choices for all future periods, $\{\omega_m^\star\}_{m=n+1}^{M}$, have already been determined in previous iterations of the recursion.

Computation of Integrands

We now choose a grid of $\mathcal{W}_n = \{W_1, \ldots, W_K\}$ at timestep $n$.

[!NOTE] The current package only implements a static grid that is the same for each timestep.

For each simulation path and current time $W_n \in \mathcal{W}_n$, we compute the realized values of the "integrands" required for the FOC. We define these realizations as

\[\mathcal{Y}_{n+1} = u^{(r)}(\hat{W}_{M+1}) \prod_{m=n+1}^{M} ((\omega_m^\star)^\top R^e_{m+1} + X_{m+1})^r \prod_{i=1}^N (R^{e,i}_{n+1})^{k_i} R^e_{n+1}\]

These values represent the pathwise realizations of the marginal utility of wealth compounded by the marginal growth of the portfolio.

Estimation of Conditional Expectations

The conditional expectations $\mathbb{E}_n[\mathcal{Y}_{n+1}]$ are estimated using cross-sectional regressions on the state variables $Z_n$. We assume the expectation can be approximated by a basis of the state variables

\[\mathbb{E}_n[\mathcal{Y}_{n + 1}] = \Phi_n \theta_n,\]

where $\Phi_n$ is the design matrix and $\theta_n$ are the parameters to be estimated.

To be explicit, let $Z_n = (Z_n^{(1)}, \ldots, Z_n^{(N)})$ be the vector of $N$ state variables, and let $p$ be the chosen polynomial degree. For $S$ simulation paths, the design matrix $\Phi_n$, the response vector $\mathcal{Y}_{n + 1}$, and the coefficient vector $\theta_n$ are structured as follows:

The Design Matrix ($\Phi_n$):

This matrix, of size $S \times (N \cdot p + 1)$, contains the intercept and the powers of each state variable for every path. Writing $Z_{n} = (Z_{n}^{(1)}, \ldots, Z_{n}^{(N)})$ for the different state variables, and $Z_{n, i}$ for the $i$-th simulation path of state variables of $Z_n$ at time $n$, the design matrix is given by

\[\Phi_n = \begin{pmatrix} 1 & (Z_{n,1}^{(1)})^1 & \cdots & (Z_{n,1}^{(1)})^p & \cdots & (Z_{n,1}^{(N)})^1 & \cdots & (Z_{n,1}^{(N)})^p \\ 1 & (Z_{n,2}^{(1)})^1 & \cdots & (Z_{n,2}^{(1)})^p & \cdots & (Z_{n,2}^{(N)})^1 & \cdots & (Z_{n,2}^{(N)})^p \\ \vdots & \vdots & \ddots & \vdots & \ddots & \vdots & \ddots & \vdots \\ 1 & (Z_{n,S}^{(1)})^1 & \cdots & (Z_{n,S}^{(1)})^p & \cdots & (Z_{n,S}^{(N)})^1 & \cdots & (Z_{n,S}^{(N)})^p \end{pmatrix}.\]

The Response Vector ($\mathcal{Y}$):

This vector contains the $S$ realized values of the integrand for the specific derivative order $r$ and multinomial combination $k_i$.

\[\mathcal{Y}_{n + 1} = \begin{pmatrix} \mathcal{Y}_{n+1, 1} \\ \mathcal{Y}_{n+1, 2} \\ \vdots \\ \mathcal{Y}_{n+1, S} \end{pmatrix}.\]

The Coefficient Vector ($\theta$):

Solving the regression yields the coefficients that map the state variables to the expected value.

\[\theta_n = \begin{pmatrix} \theta^0_n & \theta^{1,1}_n & \cdots & \theta^{1,p}_n & \cdots & \theta^{N,p}_n \end{pmatrix}^\top.\]

Regression Strategies

To find the coefficients $\theta_n$, the package provides two strategies implemented in the estimate_coefficients functions:

Standard OLS:

Solves the system using a pre-computed QR factorization of $\Phi_n$. This is highly efficient, as the factorization $\Phi_n = QR$ is performed once per timestep and reused for all $r$ derivatives and multinomial combinations.

$\alpha$-Trimmed OLS:

To prevent extreme wealth paths (outliers) from dominating the regression, this strategy identifies the "body" of the distribution. It calculates the indices $k_{low} = \lfloor \alpha S \rfloor + 1$ and $k_{high} = \lceil (1-\alpha)S \rceil$, sorts the vector $\mathcal{Y}_{n + 1}$, and performs the regression using only the rows of $\Phi_n$ and elements of $\mathcal{Y}_{n + 1}$ corresponding to the sorted indices between $k_{low}$ and $k_{high}$. This "brute force" stabilization method involves:

• Sorting the realized $\mathcal{Y}_{n+1}$ across all paths.
• Discarding the extreme $\alpha\%$ tails (e.g., the top and bottom 1%) to remove the influence of outliers.
• Running the OLS regression on the remaining data to estimate the coefficients $\theta_n$.

Numerical Optimization

Finally, we substitute the estimated expectations back into the Taylor-expanded FOC to solve for $\omega_n^\star$

\[\sum_{r = 1}^{k} \frac{W_n^{r - 1}}{(r - 1)!} \left( \sum_{k_1 + \dots + k_N = r - 1} \binom{r - 1}{k_1, \dots, k_N} \prod_{i=1}^N (\omega_n^i)^{k_i} \times \mathbb{E}_n \left[ \mathcal{Y}_{n + 1}\right] \right) = 0\]

This polynomial equation is solved numerically. To ensure convergence, we provide an initial guess based on the second-order ($k=2$) approximation.

Initial guess for polynomial equation

The second-order expansion yields a linear FOC, providing a closed-form starting point for the solver:

\[a_n + W_n B_n \omega_n = 0 \implies \omega_n^{(0)} = -W_n B_n^{-1} a_n,\]

where $a_n$ is a vector and $B_n$ is a matrix with columns $b_{i, n}$ varying $i$ defined by:

\[\begin{aligned} a_{n} &= \mathbb{E}_n \left[ u'(\hat{W}_{M+1}) \prod_{m = n + 1}^{M} ((\omega_m^\star)^\top R^e_{m + 1} + X_m) R^e_{n+1} \right], \\ b_{i, n} &= \mathbb{E}_n \left[ u''(\hat{W}_{M+1}) \prod_{m = n + 1}^{M} ((\omega_m^\star)^\top R^e_{m + 1} + X_m)^2 R^{e, i}_{n + 1} R^e_{n+1} \right]. \end{aligned}\]

Appendix

Proof of the rewritten budget constraint

In this section we will show

\[ W_{M + 1} = W_{n + 1} \left(\prod_{m = n + 1}^{M} G_{m + 1}\right) + \sum_{m = n + 1}^{M} Y_{m} \left(\prod_{p = m + 1}^{M} G_{p + 1}\right).\]

holds true for any time $n$, where $G_{m + 1} = (\omega_m^\top R^e_{m + 1} + X_m)$.

Base Case:

Let $M = n + 1$. From the budget constraint we have $W_{n+2} = W_{n+1} G_{n+2} + Y_{n+1}$. Using the formula for $M = n+1$, the first term yields

\[W_{n+1} \prod_{j=n+1}^{n+1} G_{j+1} = W_{n+1} G_{n+2},\]

while the second term yields

\[\sum_{j=n+1}^{n+1} Y_j \prod_{k=j+1}^{n+1} G_{k+1} = Y_{n+1} \cdot (1) = Y_{n+1}.\]

where we used that the product $\prod_{k=n+2}^{n+1}$ is empty, thus equals 1. Hence, the base case holds.

Inductive Step:

Assume the formula holds for some $M = K$. That is:

\[W_{K+1} = W_{n+1} \left( \prod_{j=n+1}^{K} G_{j+1} \right) + \sum_{j=n+1}^{K} Y_j \left( \prod_{k=j+1}^{K} G_{k+1} \right)\]

We examine the wealth at $M = K + 1$ (which is $W_{K+2}$) using the budget constraint:

\[W_{K+2} = W_{K+1} G_{K+2} + Y_{K+1}\]

Substitute our inductive hypothesis for $W_{K+1}$

\[ W_{K+2} = \left[ W_{n+1} \left( \prod_{j=n+1}^{K} G_{j+1} \right) + \sum_{j=n+1}^{K} Y_j \left( \prod_{k=j+1}^{K} G_{k+1} \right) \right] G_{K+2} + Y_{K+1}\]

Distribute $G_{K+2}$ into both terms

\[W_{K+2} = W_{n+1} \left( \prod_{j=n+1}^{K} G_{j+1} \cdot G_{K+2} \right) + \sum_{j=n+1}^{K} Y_j \left( \prod_{k=j+1}^{K} G_{k+1} \cdot G_{K+2} \right) + Y_{K+1}\]

The first part merges into the product: $W_{n+1} \prod_{j=n+1}^{K+1} G_{j+1}$. The $G_{K+2}$ term enters the summation's product, updating the upper bound to $K+1$. The $Y_{K+1}$ term is equivalent to $Y_{K+1} \prod_{k=K+2}^{K+1} G_{k+1}$, which allows us to fold it into the summation as the $j = K+1$ term. Combining these:

\[W_{K+2} = W_{n+1} \left( \prod_{j=n+1}^{K+1} G_{j+1} \right) + \sum_{j=n+1}^{K+1} Y_j \left( \prod_{k=j+1}^{K+1} G_{k+1} \right)\]

The formula holds for $M = K+1$.

References