Strategy maths

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. Notation: $\text{TBT}[a, b)$ is TBT over the window from $a$ to $b$, in USDS. See How it works.

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

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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. 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.

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

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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 $[\text{Now}, M_1)$: $$(60 - y)\ \text{TBT}[M_1, \infty) \to \frac{60 - y}{r\,t}\ \text{TBT}[\text{Now}, M_1) = y \quad\Rightarrow\quad y = \frac{60}{1 + r\,t} = \frac{60}{1.10} \approx 54.55$$ The proceeds are the present value of what was closed, not its raw unit count. The TimeBalance shrinks but stays one shape. See Close.

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

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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. Fill order inside the bin follows the cohort rules. See Liquidity mechanics.

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