A Budget-Feasible Reformulation of SmartDCA

Abstract

Calvet, Herranz-Celotti, and Valimamode propose SmartDCA, a family of rules that adapt periodic purchases to the current price level and prove, without market assumptions, that SmartDCA improves the (unit-weighted) average acquisition price relative to standard Dollar-Cost Averaging (DCA) [1]. In this note we introduce a budget-feasible framework that enforces a basic constraint inherent to DCA-style investing: the amount invested at each date cannot exceed the cash available from scheduled contributions and accumulated uninvested balances. We formalize two variants: a static cap (no reallocation of withheld cash) and a dynamic cash-account model (carry-over). These definitions are intended to separate (i) improvements in average acquisition price from (ii) statements about terminal wealth when uninvested cash is explicitly accounted for.

1. Background: DCA and SmartDCA

Fix a discrete time grid $t = 1, \ldots, T$. Let $(p_t)_{t=1}^T$ be the strictly positive price process of a single risky asset. A DCA investor contributes a constant cash flow $c_b > 0$ at each date and invests it immediately, i.e.,

purchasing $q_t^{\text{DCA}} := c_b/p_t$ units at time $t$.

In [1], SmartDCA modifies the invested amount as a function of the relative price level. Let $p_r > 0$ denote a reference price (e.g., a fixed anchor or a moving average), and define $r_t := p_r/p_t$. A generic (possibly unbounded) SmartDCA rule can be written as

including the $\rho$-SmartDCA class $\phi(r) = f(r)^\rho$ with $\rho \geq 0$ as studied in [1]. The paper focuses on the average acquisition price

and establishes (under mild conditions on $\phi$) that SmartDCA attains a lower $\mu$ than DCA without any probabilistic assumption on the path $(p_t)$ [1].

Motivation for a budget-feasible reformulation. Rule (1) allows $c_t$ to exceed the periodic contribution $c_b$, which implicitly assumes access to additional liquidity (borrowing or an external cash reserve not modeled). To align with the DCA principle "invest at most what is available from contributions and accumulated cash", we introduce explicit cash accounting and admissibility constraints.

2. Model: Terminal wealth with explicit cash accounting

We define the number of units purchased at time $t$ by

and the cumulative risky holdings by $Q_T := \sum_{t=1}^T q_t$. We track uninvested cash through a (nonnegative) cash balance process $(B_t)_{t=1}^{T+1}$. Throughout this note (unless stated otherwise) we take the cash account to have zero interest; an extension with a deterministic risk-free rate is straightforward.

The terminal wealth is defined as

This definition makes the comparison between strategies well-posed even when investment schedules differ: the uninvested portion of contributions remains part of wealth via $B_{T+1}$.

3. Budget-feasible SmartDCA: no carry-over

The simplest way to enforce "do not invest more than the periodic contribution" is the pointwise constraint

A capped SmartDCA rule can be represented by choosing $\phi$ in (1) such that $\phi(r) \in [0, 1]$ for all $r > 0$, yielding

In this section, "no carry-over" means that any withheld amount $(c_b - c_t)$ is not reinvested at later dates and remains in cash until $T$ (in the zero-rate case). Thus,

By contrast, DCA satisfies $c_t = c_b$ and hence $B_{T+1} = 0$ and

3.1 SmartDCA does not guarantee higher terminal wealth

We now show that, under the static cap (3), capped SmartDCA does not dominate DCA in terms of terminal wealth in a pathwise sense.

Theorem 3.1 (No pathwise wealth dominance under a static cap). Assume the static budget cap (3) and consider the capped schedule $c_t = c_b \, \phi(r_t)$ with $\phi(r_t) \in [0, 1]$ for all $t$. Then capped SmartDCA does not guarantee a higher terminal wealth than DCA: there exist price paths $(p_t)_{t=1}^T$ for which

and there also exist price paths for which $V_T^{\text{cap}} > V_T^{\text{DCA}}$.

Proof. Define the terminal-wealth difference

Rearranging terms yields

Using $c_t = c_b \, \phi(r_t)$, we obtain the factorization

Since $\phi(r_t) \in [0, 1]$, we have $\bigl(1 - \phi(r_t)\bigr) \geq 0$ for every $t$, hence the sign of each summand in (6) is entirely determined by the factor $\left(1 - \frac{p_T}{p_t}\right)$.

If we select a price path such that the terminal price is the maximum along the path, i.e., $p_T = \max_{1 \leq t \leq T} p_t$, then $\frac{p_T}{p_t} \geq 1$ for all $t$, hence $\left(1 - \frac{p_T}{p_t}\right) \leq 0$ for all $t$, and therefore $\Delta V_T \leq 0$. Moreover, the inequality is strict as soon as there exists at least one date $t$ for which $\phi(r_t) < 1$ and $p_t < p_T$, in which case $V_T^{\text{cap}} < V_T^{\text{DCA}}$.

Conversely, if we select a price path such that the terminal price is the minimum along the path, i.e., $p_T = \min_{1 \leq t \leq T} p_t$, then $\frac{p_T}{p_t} \leq 1$ for all $t$, hence $\left(1 - \frac{p_T}{p_t}\right) \geq 0$ for all $t$, and thus $\Delta V_T \geq 0$, with strict inequality under the analogous non-degeneracy condition.

Hence there exist price paths for which capped SmartDCA underperforms DCA in terminal wealth, and price paths for which it outperforms, proving the claim. $\square$

Interpretation. Under the static cap, capped SmartDCA systematically holds back a nonnegative cash amount $c_b - c_t$ relative to DCA. Equation (6) shows that this cash retention helps precisely when the terminal price is relatively low compared to past prices (so that delaying purchases is beneficial), and hurts when the terminal price is relatively high (so that earlier exposure would have been preferable). In particular, the improvement in average acquisition price established in [1] does not translate into a pathwise guarantee of higher terminal wealth once the cash component is made explicit.

This result already shows that improvements in average acquisition prices do not imply pathwise dominance in terminal wealth under a static budget cap.

4. Budget-feasible SmartDCA: carry-over

A more faithful representation of periodic investing with liquidity constraints is to allow cash carry-over. Let $B_t$ denote the cash balance just before investing at time $t$, with initial condition $B_1 = 0$. After receiving the contribution $c_b$, the investor allocates $c_t$ to the asset, subject to the admissibility constraint

and updates the cash balance as

The pair (7)-(8) rules out leverage and explicitly models the funding source of purchases. Standard DCA corresponds to $c_t = c_b$ for all $t$ and yields $B_t \equiv 0$.

Given a target SmartDCA schedule $\tilde{c}_t := c_b \, \phi(r_t)$, a canonical budget-feasible implementation is the projection

where $B_t$ is generated by (8). The resulting wealth is still given by (2).

4.1 A structural decomposition of terminal wealth

We first derive an identity that makes explicit what the carry-over constraint can and cannot achieve.

Theorem 4.1 (Carry-over SmartDCA is a rescheduling rule). Assume (7)-(8) with $B_1 = 0$ and a zero-interest cash account. Then the terminal wealth admits the decomposition

and satisfies the pathwise bound

Proof. Starting from the definition of terminal wealth,

and using the cash dynamics (8) with $B_1 = 0$, we obtain

Hence

which yields (10). To emphasize the economics of the decomposition, one may view each summand as

We now prove the bound (11). By the admissibility constraint (7),

Moreover, since $B_1 = 0$ and $B_{t+1} = B_t + c_b - c_t$ with $c_t \geq 0$, we have the crude pathwise upper bound

because the cash balance cannot exceed the cumulative contributions received up to time $t - 1$. Therefore

Taking absolute values in (10) gives

which is (11). $\square$

Interpretation. Identity (10) shows that, once the contribution stream $(c_b)_{t=1}^T$ is fixed and leverage is excluded, any budget-feasible SmartDCA rule can only affect terminal wealth through the timing of purchases encoded by $(c_t)_{t=1}^T$. The baseline $c_b T$ corresponds to the terminal value of "keeping all contributions in cash" (at zero rate), while the second term captures the incremental gain (or loss) from converting part of the cash into the risky asset at different dates and holding it to $T$.

The bound (11) provides a pathwise control on how far terminal wealth can deviate from the full-cash baseline under (7)-(8). In particular, the deviation is jointly limited by (i) the liquidity constraint, which implies $c_t \leq t \, c_b$, and (ii) the realized relative price changes $p_T/p_t$. In this sense, carry-over SmartDCA should be interpreted as a rescheduling mechanism for exposure rather than a construction that can guarantee systematic outperformance in terminal wealth without additional assumptions on the price path.

Remark (risk-free rate). The preceding constructions extend immediately to the case where uninvested cash earns a deterministic per-period risk-free rate $r \geq 0$. In the carry-over (dynamic) setting, (8) is replaced by

while in the static-cap setting the cash term in (4) becomes the time-$T$ value of the retained amounts, e.g.

under discrete compounding. All identities and comparisons remain of the same nature: introducing $r$ changes the algebraic form of the cash component but does not alter the qualitative message of this note—once cash is explicitly accounted for, capped/budget-feasible SmartDCA primarily modifies investment timing rather than providing a pathwise guarantee of higher terminal wealth relative to DCA.

5. Conclusion

This note revisits SmartDCA through the lens of budget feasibility, i.e. under the basic operational constraint that the invested amount at each date cannot exceed the investor's available cash. We formalized two natural implementations: a static cap (no reallocation of withheld cash) and a dynamic cash-account model (carry-over). In both settings, making the cash component explicit leads to simple decompositions of terminal wealth that clarify what SmartDCA can and cannot guarantee. In particular, while SmartDCA can improve the average acquisition price as established in [1], this improvement does not translate into a pathwise dominance of terminal wealth over standard DCA once liquidity constraints and retained cash are accounted for. Under budget feasibility, SmartDCA should therefore be interpreted primarily as a rescheduling rule that changes the timing of exposure rather than a construction ensuring systematic wealth outperformance without additional assumptions on the price path.

Acknowledgements

I am grateful to Vincent Tena (Universite Paris Dauphine-PSL) and Vincent Danos (CNRS, ENS-PSL) for carefully reading earlier versions of this note and for their helpful comments and corrections.

References

[1] Emmanuel Calvet, Luca Herranz-Celotti, and Karim Valimamode. SmartDCA superiority. arXiv:2308.05200, 2023. doi:10.48550/arXiv.2308.05200.