📈Take on Leverage and Forget

APE: The Math Behind Holdable Leverage

APE is SIR's leveraged token designed to magnify returns through directional price exposure. Unlike conventional leveraged products that erode returns with time-based fees (e.g., daily funding costs or volatility decay), APE decouples profitability from holding duration. Gains and losses are determined exclusively by two factors: the ratio between entry and exit prices, and a single upfront fee.

The Formula

Within the convex zone, APE's returns follow:

$$\textrm{Exit value} = x(1-f)\left(\frac{p'}{p}\right)^l$$

where

• $x$ is the initial investment

• $f$ is the initial fee (e.g., ~9% for ^1.5, ~17% for ^2 — higher leverage means higher fee)

• $p$ is the entry price

• $p'$ is the exit price

Key Implications

• Fixed Initial Fee: A fixed fee is deducted upfront, meaning the traders experience an immediate loss on opening their position.

• Convex Returns: The exponent amplifies gains and losses non-linearly:

  • Upside: Profits accelerate faster than linear growth as prices rise.

  • Downside: Losses are mitigated compared to normal leverage, and the trader is never liquidated.

• No Time Penalty: Returns depend purely on price movement, not holding duration.

Example Scenario

Assume an initial investment $x=\$1000$ in the pair (ETH/USD)^1.5, and the fee is $f=9\%$.

Price Change	Calculation	Gain Multiple	Exit Value	Net Return
-50%	$0.91\cdot0.5^{1.5}$	0.32x	$322	-68%
-25%	$0.91\cdot0.75^{1.5}$	0.59x	$591	-41%
0%	$0.91\cdot1^{1.5}$	0.91x	$910	-9%
+50%	$0.91\cdot1.5^{1.5}$	1.67x	$1,672	+67%
+100%	$0.91\cdot2^{1.5}$	2.57x	$2,574	+157%
+150%	$0.91\cdot2.5^{1.5}$	3.60x	$3,597	+260%

Critical Considerations

1. Liquidity Limits: The power-law formula holds within the convex zone. Beyond the saturation price, where demand for leverage exceeds available LP liquidity, gains become path dependent and volatility decay can occur — similar to traditional leveraged tokens. The size of the convex zone depends on the ratio of LP inventory to trader notional. See Liquidity and Leverage for details.

2. Fee Structure: The initial fee necessitates a minimum price recovery to breakeven. For example, to offset a 9% fee at ^1.5, the price must rise by ~7%.

Conclusion

APE offers a unique balance of amplified upside and defined risk, making it a powerful tool for investors confident in directional price movements. By decoupling returns from time and focusing purely on price action, SIR ensures transparency and simplicity in profit generation — provided the vault operates within the convex zone.