Discount Rate Formulation

Purpose

This spec defines how discount rates are incorporated into the Cobre SDDP solver: the discounted Bellman equation, stage-dependent discount factors, effect on the future cost variable $\theta$, cumulative discounting, and the effect on lower/upper bound computation.

For the cyclic-mode formulation (where discounting is required for convergence), see Horizon Modes.

For notation conventions (index sets, parameters, decision variables, dual variables), see Notation Conventions.

Symbol convention

This spec uses $d$ for the discount factor. The deficit variable $\delta_{b,k,s}$ uses a different symbol (lowercase delta), so there is no conflict.

1 Motivation

The discount factor $d \in (0, 1]$ captures the time value of money or risk preference, where future costs are valued less than present costs. This is essential for:

1. Infinite horizon problems: Ensuring convergence of the value function
2. Economic consistency: Reflecting opportunity cost of capital
3. Risk adjustment: Implicitly reducing weight of distant uncertain outcomes

2 Discounted Bellman Equation

The standard risk-neutral Bellman recursion with discount factor $d$ is:

$$V_t(x_{t-1}) = \mathbb{E}_{\omega_t}\left[\min_{x_t \in \mathcal{X}_t(\omega_t)} \left\{ c_t(x_t, u_t) + d_{t \to t+1} \cdot V_{t+1}(x_t) \right\}\right]$$

where:

• $c_t(x_t, u_t)$ is the immediate cost at stage $t$
• $V_{t+1}(x_t)$ is the future cost function (cost-to-go)
• $d_{t \to t+1}$ is the discount factor for the transition from stage $t$ to $t+1$

Formulation Note

The discount factor $d$ multiplies only the future cost $V_{t+1}$, not the immediate cost $c_t$. This is the standard SDDP convention:

• Immediate cost $c_t$: Not discounted (incurred “now” at stage $t$)
• Future cost $d \cdot V_{t+1}$: Discounted to present value at stage $t$

This is mathematically equivalent to computing all costs at “time 0” (stage 1) present value, where stage $t$ costs are multiplied by $\prod_{s=1}^{t-1} d_{s \to s+1}$ in the objective. The Bellman formulation above is the recursive form that SDDP exploits.

3 Input Specification

Discount rates are specified as an annual rate in stages.json, within the policy_graph section:

{
  "policy_graph": {
    "type": "finite_horizon",
    "annual_discount_rate": 0.06,
    "transitions": [
      { "source_id": 0, "target_id": 1, "probability": 1.0 },
      { "source_id": 1, "target_id": 2, "probability": 1.0 }
    ]
  }
}

The system converts the annual rate to a per-transition discount factor based on each stage’s duration:

• Stage duration $\Delta t$ is derived from end_date - start_date, expressed in years
• Transition discount factor: $d_{t \to t+1} = \frac{1}{(1 + r_{annual})^{\Delta t}}$
• The duration used is that of the source stage (the stage whose future cost is being discounted)

A value of 0.0 means no discounting ($d = 1.0$ for all transitions).

Individual transitions may override the global rate:

{
  "source_id": 59,
  "target_id": 48,
  "probability": 1.0,
  "annual_discount_rate": 0.1
}

Transition-specific discount rates are configured per transition in the case-level policy_graph.

4 Discount Factor in the Stage Subproblem

The discount factor is applied to the future cost variable $\theta$ in the stage $t$ objective, not to the cut coefficients:

$$Q_t(x_{t-1}, \omega_t) = \min_{x_t, u_t, \theta} \left\{ c_t(x_t, u_t) + d_{t \to t+1} \cdot \theta \right\}$$

subject to all standard constraints (load balance, hydro balance, etc.) and Benders cuts:

$$\theta \geq \alpha_i + \sum_{h} \pi^v_{i,h} \cdot v_h + \sum_{h,\ell} \pi^{lag}_{i,h,\ell} \cdot a_{h,\ell} \quad \forall i$$

The cut coefficients $(\alpha_i, \pi^v_{i,h}, \pi^{lag}_{i,h,\ell})$ are the undiscounted values from the backward pass. Cuts are stored and managed in undiscounted form — the discount factor appears only in the objective coefficient of $\theta$. See Cut Management for cut generation and aggregation details.

Why discount on θ, not on cuts

Applying the discount factor to the $\theta$ variable rather than scaling each cut individually is simpler and avoids modifying cut coefficients. The mathematical result is identical — if $\theta \geq \alpha + \pi^\top x$ and $d \cdot \theta$ appears in the objective, it is equivalent to having $\theta' \geq d \cdot (\alpha + \pi^\top x)$ with $\theta'$ in the objective.

5 Cumulative Discounting

For a path from stage 1 to stage $T$, the cumulative discount factor is:

$$d_{1 \to T} = \prod_{t=1}^{T-1} d_{t \to t+1}$$

The present value at stage 1 of costs incurred at stage $T$ is:

$$\text{PV}_1[c_T] = d_{1 \to T} \cdot c_T$$

where $d_{1 \to 1} = 1$.

6 Lower Bound with Discounting

The deterministic lower bound at iteration $k$ is:

$$\underline{z}^k = c_1(\hat{x}_1^k) + d_{1 \to 2} \cdot \theta_1^k$$

where $\theta_1^k$ is the optimal value of the future cost variable at stage 1. Because the discount factor cascades through $\theta$ at each stage, the lower bound already reflects cumulative discounting to stage 1 present value.

7 Upper Bound (Simulation) with Discounting

When simulating the policy to estimate the upper bound:

$$\bar{z}^k = \frac{1}{M} \sum_{m=1}^{M} \sum_{t=1}^{T} d_{1 \to t} \cdot c_t(\hat{x}_t^{k,m})$$

Each stage’s immediate cost is explicitly discounted to stage 1 present value using the cumulative discount factor.

For stopping rules that use these bounds, see Stopping Rules.

8 Reporting

Both the lower bound $\underline{z}$ and upper bound $\bar{z}$ represent total expected cost expressed in present value at stage 1:

• A lower bound of $100M means the optimal policy costs at least $100M in stage-1 dollars
• Future costs are already discounted: a $1M cost at stage 12 with $d_{1 \to 12} = 0.95$ contributes $0.95M to the bounds
• Comparisons between bounds and between iterations are valid because they use consistent discounting
• When reporting per-stage costs in simulation outputs, Cobre reports both nominal (undiscounted) and present value costs

9 Infinite Horizon Considerations

For cyclic policy graphs (infinite periodic horizon), discounting is required for convergence. The cumulative discount around one full cycle must satisfy:

$$d_{cycle} = \prod_{t \in \text{cycle}} d_{t \to t+1} < 1$$

This ensures the value function remains finite: $\lim_{n \to \infty} d_{cycle}^n \cdot V_t(x) = 0$.

Typical setup: An annual discount rate of 6% gives $d_{cycle} \approx 0.94$ per 12-month cycle.

For the complete cyclic-mode formulation — including the season-indexed cut pool, the cut-sharing equation, and the cycle convergence criterion — see Horizon Modes.

Cross-References

• SDDP Algorithm — Core Bellman equation and forward/backward pass structure that discount rates modify
• Cut Management — Cut coefficients remain undiscounted; discount applied to $\theta$ in objective
• Stopping Rules — Convergence criteria using discounted lower/upper bounds
• Upper Bound Evaluation — Inner approximation uses discounted vertex values
• Horizon Modes — Finite vs cyclic policy graphs; cyclic-mode formal structure (season function, cycle convergence inequality, season-indexed cut pool, fixed-point Bellman operator)
• Notation Conventions — Index sets, parameters, decision variables, and dual variables used throughout