> ## Documentation Index
> Fetch the complete documentation index at: https://docs.momentrates.xyz/llms.txt
> Use this file to discover all available pages before exploring further.

# Strategy maths

> Fixed, Leverage, Forward start (a later Start), Close and fills, period by period.

For builders and auditors.

Every Moment action is a chain of per-period trades between the principal side and the interest side of each maturity; this page walks through each one, period by period.

> [!NOTE]
>
> **Illustrative, before fees.** The first four walk-throughs use round numbers: three periods $M_1, M_2, M_3$, each at a 10% period rate ($r\,t = 0.10$). Real periods are much shorter and rates differ by period. The fee section below shows how the 0.5% taker fee changes each one.

Notation: $\text{TBT}[a, b)$ is TBT over the window from $a$ to $b$, in USDS. See [How it works](https://docs.momentrates.xyz/for-builders/how-it-works.md).

## Fixed

Give up the balance from Now to $M_3$; receive a bigger flat balance from $M_3$ on. Each step fills a **bid** in one period, earliest first, and each step's output feeds the next.

| Step | Period | Gives up | Receives | Holding after |
| - | - | - | - | - |
| 0 | Start | — | — | 100 $\text{TBT}[\text{Now}, \infty)$ |
| 1 | $[\text{Now}, M_1)$ | 100 $\text{TBT}[\text{Now}, M_1)$ | 10 $\text{TBT}[M_1, \infty)$ | 110 $\text{TBT}[M_1, \infty)$ |
| 2 | $[M_1, M_2)$ | 110 $\text{TBT}[M_1, M_2)$ | 11 $\text{TBT}[M_2, \infty)$ | 121 $\text{TBT}[M_2, \infty)$ |
| 3 | $[M_2, M_3)$ | 121 $\text{TBT}[M_2, M_3)$ | 12.1 $\text{TBT}[M_3, \infty)$ | 133.1 $\text{TBT}[M_3, \infty)$ |

$$
100 \times 1.10 \times 1.10 \times 1.10 = 133.1
$$

**Result:** 133.1 $\text{TBT}[M_3, \infty)$. The fixed yield is 33.1, locked at trade time. At $M_3$ it becomes a plain flat balance of 133.1.

## Leverage

Give up the balance after $M_3$; receive a much bigger balance from Now to $M_3$. Each step fills an **ask** in one period. It runs from the far period back to Now.

| Step | Period | Gives up | Receives |
| - | - | - | - |
| 0 | Start | — | Holding 33.1 $\text{TBT}[\text{Now}, \infty)$ |
| 1 | $[M_2, M_3)$ | 12.1 $\text{TBT}[M_3, \infty)$ | 121 $\text{TBT}[M_2, M_3)$ |
| 2 | $[M_1, M_2)$ | 11 $\text{TBT}[M_2, \infty)$ | 110 $\text{TBT}[M_1, M_2)$ |
| 3 | $[\text{Now}, M_1)$ | 10 $\text{TBT}[M_1, \infty)$ | 100 $\text{TBT}[\text{Now}, M_1)$ |

In total it gives up $12.1 + 11 + 10 = 33.1$ of the balance after $M_3$, which is all of it.

**Result:** 133.1 $\text{TBT}[\text{Now}, M_3)$ and nothing after $M_3$. That is about **4.0×** the yield exposure of the 33.1 paid, over the window.

Break-even: the vault's realised yield over the window must average the rate paid (10% per period here). If it earns 0%, the position ends with nothing after $M_3$: a 100% loss of what was paid. See [Leverage](https://docs.momentrates.xyz/use-the-app/leverage.md).

## Forward start

Fixed until Start, then Leverage over the window. Here the window is $[M_2, M_5)$, also at 10% per period. Both phases run in one atomic transaction.

| Phase | Mechanic | Gives up | Receives |
| - | - | - | - |
| 1 | Fixed to $M_2$ (2 periods) | 27.36 $\text{TBT}[\text{Now}, M_2)$ | Holding grows to 33.1 $\text{TBT}[M_2, \infty)$ |
| 2 | Leverage over $[M_2, M_5)$ (3 periods) | 33.1 $\text{TBT}[M_5, \infty)$, stepped back as above | 133.1 $\text{TBT}[M_2, M_5)$ |

$$
27.36 \times 1.10^2 \approx 33.1
$$

**Result:** nothing held before $M_2$, about 4.0× exposure over $[M_2, M_5)$, nothing after $M_5$. The same 133.1-wide window costs 27.36 instead of 33.1, because the balance grows at a fixed rate first. If only part of a balance is used, the rest stays flat.

## Close

Close removes exposure from chosen periods and returns the value as a flat balance from Now (or as vault shares). It is priced at the market rates of that moment.

Before the close (illustrative):

$$
133\ \text{TBT}[M_1, M_2) + 173\ \text{TBT}[M_2, \infty)
$$

Close 60 from every period from $M_1$ on, that is 60 $\text{TBT}[M_1, \infty)$. To turn it into a flat amount $y$ from Now, keep $y$ of it and trade the rest across $[\text{Now}, M_1)$:

$$
(60 - y)\ \text{TBT}[M_1, \infty) \to \frac{60 - y}{r\,t}\ \text{TBT}[\text{Now}, M_1) = y \quad\Rightarrow\quad y = \frac{60}{1 + r\,t} = \frac{60}{1.10} \approx 54.55
$$

| | Before | After |
| - | - | - |
| $[M_1, M_2)$ | 133 | 73 |
| $[M_2, \infty)$ | 173 | 113 |
| Proceeds | — | 54.55 flat $\text{TBT}[\text{Now}, \infty)$, or 54.55 USDS of sUSDS shares |

The proceeds are the **present value** of what was closed, not its raw unit count. The TimeBalance shrinks but stays one shape. See [Close](https://docs.momentrates.xyz/use-the-app/close.md).

## How the 0.5% taker fee changes each rate

The taker fee is multiplicative on the rate: $r(1 - f)$ when a step fills a bid, $r / (1 - f)$ when it fills an ask, with $f = 0.5\%$. Each step of a walk-through pays it on its own period.

| Strategy | Steps fill | Rate per step | Effect in the example |
| - | - | - | - |
| Fixed | Bids | $10\% \times 0.995 = 9.95\%$ | 100 → 109.95 → 120.89 → **132.92** (not 133.1) |
| Leverage | Asks | $10\% / 0.995 \approx 10.05\%$ | 33.1 buys about 132.55 over the window, about **4.00×** (4.02× before the fee) |
| Forward start | Bids, then asks | 9.95% to Start, then about 10.05% | Reaching 33.1 at $M_2$ costs about 27.38 instead of 27.36, and that 33.1 then buys about 132.55 of window instead of 133.1 |
| Close of a Fixed or future window | Asks | About 10.05% | Proceeds about 54.52 instead of 54.55 |
| Close of a Leverage window | Bids | 9.95% | The interest side bought back is 0.5% smaller |

Makers on the other side of each step trade at their full quoted rate. The app quotes every rate net of fees.

## Limit order fill

A bid and an ask in the period $[\text{Oct 2}, \text{Oct 17})$ of the example market (15 days, $t = 15/365$). Rates are illustrative.

> [tab] Bid filled by a Fixed taker
>
> **Maker** places a bid at 5.26%. It deposits 100 $\text{TBT}[\text{Oct 17}, \infty)$, the interest side.
>
> A **taker**'s Fixed trade to Oct 17 fills this bid on its $[\text{Oct 2}, \text{Oct 17})$ leg and receives the 100 interest side. At the maker's rate, that is worth:
>
> $$
> \frac{100}{0.0526 \times 15/365} \approx 46{,}261\ \text{TBT}[\text{Oct 2}, \text{Oct 17})
> $$
>
> The taker hands over about 46,494 principal side. The fee is 0.5% of it, about 232, taken in kind. So the taker's rate is $5.26\% \times 0.995 \approx 5.23\%$.
>
> **At the maker's next sync**, 46,261 $\text{TBT}[\text{Oct 2}, \text{Oct 17})$ lands in its TimeBalance. The maker now holds leverage over that period, bought at exactly 5.26%.

> [tab] Ask filled by a Leverage taker
>
> **Maker** places an ask at 5.30%. It deposits 10,000 $\text{TBT}[\text{Oct 2}, \text{Oct 17})$, the principal side. While it waits, the window earns nothing until Oct 2 (its present value rises as Oct 2 approaches), then earns the vault's yield until it fills or ends.
>
> **Taker** does Leverage over the period and receives the 10,000 principal side. At the maker's rate, that costs:
>
> $$
> 10{,}000 \times 0.053 \times 15/365 \approx 21.78\ \text{TBT}[\text{Oct 17}, \infty)
> $$
>
> The taker hands over about 21.89 interest side. The fee is 0.5% of it, about 0.11, taken in kind. So the taker's rate is $5.30\% / 0.995 \approx 5.33\%$.
>
> **At the maker's next sync**, 21.78 $\text{TBT}[\text{Oct 17}, \infty)$ lands in its TimeBalance. The maker has fixed its rate at exactly 5.30%.

Fill order inside the bin follows the cohort rules. See [Liquidity mechanics](https://docs.momentrates.xyz/for-builders/liquidity-mechanics.md).

## Recursive round trip

A recursive bin in the same period quotes a bid at 5.05% and an ask at 5.30% ($k \approx 1.05$). Take a fill of 10,000 principal side.

| Step | Taker | Bin gives | Bin receives |
| - | - | - | - |
| 1 | Fixed fills the bid | 20.75 $\text{TBT}[\text{Oct 17}, \infty)$ | 10,000 $\text{TBT}[\text{Oct 2}, \text{Oct 17})$ |
| 2 | Leverage fills the ask | 10,000 $\text{TBT}[\text{Oct 2}, \text{Oct 17})$ | 21.78 $\text{TBT}[\text{Oct 17}, \infty)$ |
| Net | | | **+1.03** $\text{TBT}[\text{Oct 17}, \infty)$ |

$$
10{,}000 \times (5.30\% - 5.05\%) \times 15/365 \approx 1.03
$$

The principal side is back where it started; the bin has earned the spread on the interest side. The two takers each paid the 0.5% fee on top: the Fixed taker got about 5.02%, the Leverage taker paid about 5.33%. The bin kept its full spread. See [Liquidity mechanics](https://docs.momentrates.xyz/for-builders/liquidity-mechanics.md).


## Related topics

- [How it works](https://docs.momentrates.xyz/for-builders/how-it-works.md)
- [For traders](https://docs.momentrates.xyz/background/for-traders.md)
- [Integrating with Moment](https://docs.momentrates.xyz/for-builders/integrating-with-moment.md)


This documentation is built and hosted on [Mintlify](https://mintlify.com), a developer documentation platform.