> For the complete documentation index, see [llms.txt](https://docs.ibis.app/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.ibis.app/platform/rate-and-apy-figures.md).

# Rates and APY

A fixed-rate facility accrues at its configured rate. A variable-rate Epochs facility applies a yield declared for each epoch within its locked range. Every rate is a nominal annual rate in basis points (bps), and 10,000 bps is 100%.

Realized and historical APY (annual percentage yield) measure past share-price growth. A linear yield estimate projects yield at an assumed rate. These reporting conventions create no entitlement to yield: the facility credits yield under its [accrual laws](/platform/yield-accrual.md).

## Configured and declared rates

| Term                 | Symbol                     | Meaning                                                               | Units               | Products                                    |
| -------------------- | -------------------------- | --------------------------------------------------------------------- | ------------------- | ------------------------------------------- |
| Fixed rate           | $$r$$                      | The rate at which the share price accrues                             | bps, nominal annual | Fixed-rate Epochs and Continuous facilities |
| Variable-yield range | $$y\_{min}$$, $$y\_{max}$$ | The range locked for an epoch; the declared yield must fall within it | bps, nominal annual | Variable-rate Epochs facilities             |
| Declared yield       | $$y$$                      | The yield declared for one epoch and applied at its settlement        | bps, nominal annual | Variable-rate Epochs facilities             |

The fixed rate and variable-yield range are set per facility; declared yield is set for each epoch. [Facility terms and bounds](/platform/facility-terms.md) lists the limits, who sets them and when changes apply, including which terms lock when an epoch opens.

Compounding separates a nominal annual rate from effective annual yield. A Continuous facility compounds once per compounding period: 730 bps (7.30%) compounded daily gives an effective annual yield of 7.572%, above the nominal rate. An Epochs facility compounds at each settlement. [How yield accrues](/platform/yield-accrual.md) gives the effective annual yield formula.

## Realized APY

Realized APY annualizes share-price growth over a measured interval, using wall-clock time and a year of 365 days. Current values come from the chain; historical figures are derived from recorded history.

### General form

**Realized APY**

$$
\text{realized APY} = \left(\frac{I\_{end}}{I\_{start}} - 1\right) \times \frac{Y}{\Delta t\_w}
$$

Where:

$$I\_{start}$$ is the share price at the start of the measured interval, in assets per share to 18 decimals. $$I\_{end}$$ is the share price at the end of the measured interval, in assets per share to 18 decimals. $$\Delta t\_w$$ is the wall-clock length of the interval in seconds, with nothing excluded. $$Y$$ is one year of 365 days, 31,536,000 seconds.

For display, the annual fraction is multiplied by 10,000 to express it in basis points and rounded to a whole basis point. Each figure below states its reporting convention for rounding. The underlying rates have [bounds](/platform/facility-terms.md), but realized APY is unbounded. It is negative when the share price ends below its starting level.

Illustrative example: a share price of 1.000000000000000000 at the start of a 73-day interval and 1.012000000000000000 at its end give a realized APY of 600 bps (6.00%).

Late settlement lowers a variable-rate epoch's realized APY because the declared yield covers only the locked epoch length, while realized APY includes all wall-clock time. Realized APY can therefore fall below the configured or declared rate.

### Per epoch

**Per-epoch realized APY**

$$
APY\_n = \left(\frac{I\_{close,n}}{I\_{open,n}} - 1\right) \times \frac{Y}{\Delta t\_{w,n}} \times 10{,}000
$$

Where:

* $$APY\_n$$ is the realized APY of epoch $$n$$, in basis points.
* $$n$$ is the index of the epoch.
* $$I\_{open,n}$$ is the share price recorded for the opening of epoch $$n$$: the continuing-holder price pinned at the settlement of the previous epoch, or the share price at activation for the first epoch, to 18 decimals.
* $$I\_{close,n}$$ is the continuing-holder price pinned at the settlement of epoch $$n$$, to 18 decimals.
* $$\Delta t\_{w,n}$$ is the wall-clock time in seconds from the opening of epoch $$n$$ to the completion of its settlement, not the locked epoch length.
* $$Y$$ is 31,536,000 seconds.

Per-epoch realized APY is rounded to the nearest whole basis point for display. The underlying rates have [bounds](/platform/facility-terms.md); this figure is unbounded.

Illustrative example: a variable-rate epoch opens at a share price of 1.002000000000000000, with a locked length of 10 days and a declared yield of 900 bps (9.00%). Its settlement pins a continuing-holder price of 1.004470684931506849 and completes 10.5 days after the epoch opened. The realized APY is 857 bps (8.57%); the half day of late settlement lowers the realized figure below the declared yield.

When a fixed-rate settlement spans several transactions, growth during Settling appears in the next epoch's figure. The time spent Settling counts toward the settling epoch's wall-clock length.

## Historical APY

### Epochs facility

An Epochs facility's historical APY is a principal-weighted average of realized APY for epochs whose settlement completed within the trailing 30 days. The weighted average applies when the window contains at least two settled epochs.

Otherwise, historical APY uses the latest settled epoch's realized APY. A facility with no settled epoch has no figure; declared yield is never substituted.

**Historical APY on an Epochs facility**

$$
\text{historical APY} = \frac{\sum\_{n} V\_n \times APY\_n}{\sum\_{n} V\_n}
$$

Where:

$$n$$ is the index of an epoch settled within the window; the sums run over those epochs. $$V\_n$$ is the principal the facility recorded at the settlement of epoch $$n$$, which is the amount of pending deposits converted into shares at that settlement, in assets to 6 decimals (bound). When any epoch in the window records no principal, every weight is 1. $$APY\_n$$ is the per-epoch realized APY above, in basis points, before display rounding.

The average is rounded to the nearest whole basis point and lies between the lowest and highest of its per-epoch figures.

Illustrative example: two epochs settled in the window, one with a realized APY of 857 bps (8.57%) on 4,000,000.000000 USDC of principal and one with 880 bps (8.80%) on 5,000,000.000000 USDC, give a historical APY of 870 bps (8.70%).

### Continuous facility

A Continuous facility's historical APY annualizes share-price growth over the trailing 30 days. It starts from the latest daily reading at or before the window's start, or, if none is that old, the earliest daily reading inside the window. There is no figure without a daily reading or when the current share price is below the starting reading.

**Historical APY on a Continuous facility**

$$
\text{historical APY} = \left\lfloor \frac{(I\_{live} - I\_{start}) \times Y \times 10{,}000}{I\_{start} \times \Delta t\_w} \right\rfloor
$$

Where:

$$I\_{start}$$ is the share price at the starting reading, to 18 decimals. $$I\_{live}$$ is the current share price, to 18 decimals. $$\Delta t\_w$$ is the wall-clock time in seconds from the starting reading to the current reading. $$Y$$ is 31,536,000 seconds.

Continuous historical APY is rounded down to a whole basis point. Excluding intervals where the current share price is below the starting reading gives the figure a lower bound of zero.

Illustrative example: a share price that moves from 1.000000000000000000 to 1.006017432523893634 over exactly 30 days gives 732 bps (7.32%). The figure is above the nominal rate of 730 bps (7.30%) because the facility compounds daily.

## Linear yield estimate

The linear yield estimate applies an assumed constant rate to an asset amount over a horizon, with no compounding. It assumes the amount earns for the whole horizon. The result is a reporting convention, not credited yield.

**Linear yield estimate**

$$
y\_h = \left\lfloor \frac{a \times r \times h}{Y \times 10{,}000} \right\rfloor
$$

Where:

$$y\_h$$ is the estimated yield over the horizon, in assets to 6 decimals. $$a$$ is the asset amount, in assets to 6 decimals. $$r$$ is the assumed annual rate in basis points. For a variable-yield range, $$y\_{min}$$ and $$y\_{max}$$ are substituted for $$r$$ in turn to give two estimates. $$h$$ is the horizon in seconds. $$Y$$ is 31,536,000 seconds.

The estimate is rounded down to the asset's smallest unit, 0.000001 (bound). The assumed rate is subject to the fixed-rate and variable-yield-range [bounds](/platform/facility-terms.md).

Illustrative example: 10,000.000000 USDC at 730 bps (7.30%) over 7 days gives an estimate of 14.000000 USDC.

An Epochs deposit starts earning only when settlement converts it into shares. The estimate ignores its time as a pending deposit; credited yield depends on that start time and the [accrual rules](/platform/yield-accrual.md).