# Pool Interest Rates
## Pool-Level APY Dynamics
The aggregate variable rate earned by the pool is:
$$
\text{IR}*{\text{Pool}}
\= W*{\text{Aave}};\text{IR}\_{\text{SOFR}}
* \sum\_{n=1}^{N} W\_{n},\text{IR}\_{n}
$$
where
$$
\begin{aligned}
W\_{\text{Aave}} &: \text{ idle liquidity in Aave (earns }\text{IR}*{\text{SOFR}}\text{)} \\\[2pt]
W*{n} &: \text{ weight of merchant }n \\
&: \bigl(\sum\_{n} W\_{n}=1-W\_{\text{Aave}}\bigr)
\end{aligned}
$$
Cash-flows are then split between the senior and junior tranches:
$$
W\_{\text{USD3}};\text{IR}\_{\text{USD3}}
* W\_{\text{sUSD3}};\text{IR}*{\text{sUSD3}}
\= \text{IR}*{\text{Pool}},\qquad
W\_{\text{USD3}} = 0.85,;
W\_{\text{sUSD3}} = 0.15
$$
## USD3 and sUSD3 APY
$$
\text{IR}*{\text{USD3}}
\= \frac{\text{IR}*{\text{Pool}}
\- W\_{\text{sUSD3}};\text{IR}*{\text{sUSD3}}}
{W*{\text{USD3}}}
$$
USD3 receives interest first and benefits from a real-time cash reserve that enables most redemptions to clear instantly.
$$
\text{IR}*{\text{sUSD3}}
\= \text{IR}*{\text{USD3}}
\+ \frac{\bigl(\text{Excess Spread} - \text{Reserve Accrual}\bigr)}
{W\_{\text{sUSD3}}}
$$
sUSD3 earns every remaining basis-point of spread after senior distributions and reserve top-ups, but it also absorbs first losses.
## Merchant Discount Factor Rate
3Jane gives you an upfront advance and, in return, buys a fixed specified amount of your future yield. *The Discount Factor Rate (DFR) is a discount that reduces the specified amount if you repay the advance amount early, effectively charging you less. The discount shrinks towards zero over time as the days since funding increases.*
$$
\text{RP}(N) = A \times \left( 1 + \sum\_{n=1}^{N} \text{P}\_{n} \right)\\\[2em]
\text{DFR}(N) = 1 - \frac{\mathrm{RP}(N) - A}{\mathrm{A} \times (F-1)}
$$
$$
\begin{aligned}
N &: \text{days since funding} \\\[2pt]
A &: \text{advance amount at funding} \\\[2pt]
\text{F} &: \text{fixed Factor (set at funding)} \\\[2pt]
\text{RP} &: \text{repurchase amount by day N} \\\[2pt]
\text{DFR} &: \text{discount factor rate by day N} \\\[2pt]
\text{P}\_{n} &: \text{daily pacing increment} \\\[2pt]
\end{aligned}
$$
Each day, a tiny pacing increment is applied:
(1) base factor: the pool’s baseline conditions (utilization)
(2) credit risk factor: your credit risk profile
(3) urgency factor: whether you were late that day
Those three slices add up to your daily fraction Pn.
$$
\begin{aligned}
\text{P}*{n}
&: \text{B}*{n}
\+ \text{C}*{n}
\+ \text{L}*{n} \\\[2pt]
\text{B}*{n} &: \text{implied base apy, expressed as a (1) day factor, pool-wide. Derived from the pool’s utilization curve} \\\[2pt]
\text{C}*{n} &: \text{implied credit risk apy, expressed as a (1) day factor, per-user. Derived from the 3CA algorithm} \\\[2pt]
\text{L}\_{n} &: \text{implied urgency apy, expressed as a (1) day factor, per-user. Applied on days flagged late} \\\[2pt]
\end{aligned}
$$
Note: DFR according to the formula is updated daily (as of 11:59 p.m. UTC). Your real-time DFR (and hence early payoff amount) will be updated to include time elapsed from last day DFR prior to payment.
**Example:**
* Advance: **A = 100,000**
* Factor: **F = 1.15**
* Specified Amount: **A⋅F = 115,000**
| bn | | | | | | |
|---|
| D | Bn | Cn | Ln | Pn | Sum Pn | RP(N) | DFR(N) |
| 1 | 0.000115 | 0.000049 | 0 | 0.000164 | 0.000164 | 100,016 | 99.8904% |
| 2 | 0.000112 | 0.000052 | 0 | 0.000164 | 0.000328 | 100,032 | 99.7808% |
| 3 | 0.000117 | 0.000046 | 0.000136 | 0.000301 | 0.000630 | 100,063 | 99.5799% |
| 4 | 0.000109 | 0.000050 | 0 | 0.000160 | 0.000790 | 100,079 | 99.4731% |
| 5 | 0.000120 | 0.000053 | 0 | 0.000173 | 0.000964 | 100,096 | 99.3571% |