Worked Example: Staking Rewards (1-month epoch)

This document walks through a concrete, numbers-first example of how UTIL staking rewards are budgeted and allocated over a one-month epoch. We apply a time-weighted stake multiplier and a fee-first funding policy, and show the effect of early unstaking (forfeiture of unvested rewards only; principal is never slashed). Use this as a baseline you can parameterize for your community; formal equations and an appendix table follow below.

Parameters

• Total staked at epoch start: $`S = 100{,}000`$ UTIL

• APR target: $`6\%`$ ⇒ monthly base rate $`r_{\text{mo}} = 0.06/12 = 0.005`$

• Time multiplier: $`f(T_i) = 1 + \alpha T_i`$ with $`\alpha = 0.01`$ per month (commitment length $`T_i`$ in months)

• Stakers:

  • Alice: $`s_A = 10{,}000`$ UTIL, $`T_A = 6`$ → $`f_A = 1.06`$
  • Bob: $`s_B = 20{,}000`$ UTIL, $`T_B = 3`$ → $`f_B = 1.03`$
  • Carol: $`s_C = 70{,}000`$ UTIL, $`T_C = 12`$ → $`f_C = 1.12`$

Step 1 — Weighted stake & reward pool

Weight each stake by its time multiplier:

• $`w_A = s_A f_A = 10{,}000 \times 1.06 = 10{,}600`$
• $`w_B = s_B f_B = 20{,}000 \times 1.03 = 20{,}600`$
• $`w_C = s_C f_C = 70{,}000 \times 1.12 = 78{,}400`$
• $`\sum w_i = 109{,}600`$

Monthly rewards to distribute (mint side) follow the base rate on total stake:

$`R_{\text{epoch}} = r_{\text{mo}} \cdot S = 0.005 \times 100{,}000 = 500 \\text{ UTIL}`$

Step 2 — Pro-rata allocation by weighted share

$`R_i = R_{\text{epoch}} \times \\frac{w_i}{\\sum w_j}`$

• Alice: $`500 \times \frac{10{,}600}{109{,}600} \approx 48.36`$ UTIL
• Bob: $`500 \times \frac{20{,}600}{109{,}600} \approx 94.16`$ UTIL
• Carol: $`500 \times \frac{78{,}400}{109{,}600} \approx 357.48`$ UTIL
• (Rounding ⇒ total ≈ 500 UTIL)

Intuition: bigger and longer commitments earn proportionally more via $`w_i=s_if(T_i)`$.

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Early Unstaking Penalty (loss of unvested reward only)

Assume Bob committed to $`T_B=3`$ months but exits after $`t=2`$ months. His term reward (sum across the first two months) is multiplied by a vesting factor $`t/T_B`$.

For a single month’s reward $`R_B`$ (≈ 94.16 UTIL as above), his kept amount at month 2 is:

$`R_{B,\\text{kept}} = \\frac{t}{T_B} \\times R_B = \\frac{2}{3}\\,R_B \\approx 62.77 \\text{ UTIL}`$

The forfeited portion $`(1-\\tfrac{t}{T_B})R_B \\approx 31.39`$ UTIL returns to the Treasury reward pool for redistribution in the next epoch.

Principal stake is never slashed; only the unvested reward is forfeited.

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Fee-based replenishment (reduces reliance on minting)

Suppose this month the platform collects 2,000 UTIL in fees. Policy:

1. Use fees to fund the target reward budget (here, 500 UTIL) before minting.
2. Route any surplus fees to the Contributor Pool (or keep a % to top up the reward buffer).

• Fees cover the 500 UTIL rewards → Mint = 0 for this epoch.
• Remaining fees $`2{,}000 - 500 = 1{,}500`$ UTIL → Contributor Pool (to pay verified work), or per policy split (e.g., 1,000 to Contributor Pool, 500 retained in a reward buffer).

This creates a closed loop: communities spend UTIL on services → Treasury receives fees → fees finance staking rewards and contributor rewards → contributors earn UTIL for building/operating the platform that communities use.

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Optional: Multi-epoch glimpse

• If Bob exits before his 3-month term ends, next epoch’s S drops (from 100k to 80k if he fully leaves), and his forfeited rewards (≈ 31.39 UTIL from month 2) go back into the next epoch reward pool—slightly boosting remaining stakers’ pro-rata outcomes or lowering mint needs.

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Equations summary (for the paper)

• Time multiplier: $`f(T_i)=1+\\alpha T_i`$

• Weights: $`w_i=s_i f(T_i)`$, $`\\;W=\\sum_i w_i`$

• Epoch reward budget (mint target): $`R_{\\text{epoch}}=r_{\\text{mo}}\\cdot S`$

  • Fee-first rule: $`R_{\\text{funded}}=\\min(\\text{fees},R_{\\text{epoch}})`$; $`R_{\\text{mint}}=\\max(0,R_{\\text{epoch}}-\\text{fees})`$

• Allocations: $`R_i = (R_{\\text{funded}}+R_{\\text{mint}})\\cdot \\frac{w_i}{W}`$

• Early exit (loss of unvested reward only): $`R_{i,\\text{kept}}=\\frac{t}{T_i}R_i,\\;\\;R_{i,\\text{forfeit}}=\\left(1-\\frac{t}{T_i}\\right)R_i`$

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Awesome — here’s the appendix table for the staking example you approved, followed by a first draft of the Sell pressure risk & liquidity analysis section (parameterized + a clean baseline scenario you can tweak).

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Appendix A — Worked Example Table (1-month epoch)

Staker	Stake (UTIL)	Term (mo)	Time Mult. f(T)	Weighted Stake wᵢ	Share wᵢ/Σw	Reward (UTIL)	Early Unstake?	Kept Reward	Forfeited Reward
Alice	10,000	6	1.06	10,600	9.68%	48.36	No	48.36	0.00
Bob	20,000	3	1.03	20,600	18.80%	94.16	Exit @ 2/3	62.77	31.39
Carol	70,000	12	1.12	78,400	71.52%	357.48	No	357.48	0.00
Totals	100,000	—	—	109,600	100%	500.00	—	468.61	31.39

Parameters used: APR 6% ⇒ monthly base rate $`r_{mo}=0.005`$; $`f(T)=1+0.01T`$; epoch reward budget $`R_{epoch}=r_{mo}\\cdot S=500`$ UTIL; fee-first funding (fees cover rewards before mint). Bob’s early exit forfeits unvested rewards only (principal never slashed).

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