Adaptive Yield Split

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The Adaptive Yield Split is a unique feature of YTs that manages the return distribution dynamically conditional to the liquidity deposited on each side (Senior/Junior) of the Tranche.
Mathematically, the formulas behind this mechanism consider mainly Senior and Junior liquidity ratios to compute Senior and Junior returns.
Liquidity ratios
First, we define the Senior and Junior TVL ratios as
TVL ratioSr=LiquiditySeniorLiquiditySenior+Junior\text{TVL ratio}_{Sr} = \frac{\text{Liquidity}_{Senior}}{\text{Liquidity}_{Senior + Junior}}TVL ratioSr​=LiquiditySenior+Junior​LiquiditySenior​​
TVL ratioJr=LiquidityJuniorLiquiditySenior+Junior\text{TVL ratio}_{Jr} = \frac{\text{Liquidity}_{Junior}}{\text{Liquidity}_{Senior + Junior}}TVL ratioJr​=LiquiditySenior+Junior​LiquidityJunior​​
Senior and Junior yields
The Senior return can be calculated as
APYSr=Base APY×Yield shareSr(1)\text{APY}_{Sr} = \text{Base APY} \times \text{Yield share}_{Sr} \qquad \tag{1}APYSr​=Base APY×Yield shareSr​(1)
where the Base APY is the underlying Tranches yield and the Yield share of the Senior side is a piecewise function conditional to the liquidity on the Senior tranche. 
Yield shareSr={99%if TVL ratioSr≥99%LiquiditySeniorLiquiditySenior+Juniorif TVL ratioSr>50%50%if TVL ratioSr≤50%\text{Yield share}_{Sr} = 
    \begin{dcases}
        99\% & \text{if } \text{TVL ratio}_{Sr} \geq 99\%  \\
        \\
         \dfrac{\text{Liquidity}_{Senior}}{\text{Liquidity}_{Senior + Junior}} & \text{if } \text{TVL ratio}_{Sr} > 50\%  \\
        \\
        50\% & \text{if } \text{TVL ratio}_{Sr} \leq 50\%  \\
    \end{dcases}Yield shareSr​=⎩⎨⎧​99%LiquiditySenior+Junior​LiquiditySenior​​50%​if TVL ratioSr​≥99%if TVL ratioSr​>50%if TVL ratioSr​≤50%​
The Junior return can be calculated as
APYJr=(Base APY−APYSr)×TVL ratioSrTVL ratioJr+Base APY(2)\text{APY}_{Jr} = \frac{(\text{Base APY} - \text{APY}_{Sr}) \times \text{TVL ratio}_{Sr}}{\text{TVL ratio}_{Jr}} + \text{Base APY} \qquad \tag{2}APYJr​=TVL ratioJr​(Base APY−APYSr​)×TVL ratioSr​​+Base APY(2)
Normal case
When Senior liquidity represents 50-99% of the funds in the Tranche, we use Equation (1) to compute the Yield share of the Senior side.
Alternatively, we use some fixed percentages. There are two hedge cases:
1.The majority of total Tranche's liquidity lying on the Senior side (more than 99%)
2.Less than half of total Tranche's liquidity lying on the Senior side (less than 50%)
Yield shareSr={99%if LiquiditySeniorLiquiditySenior+Junior≥99%50%if LiquiditySeniorLiquiditySenior+Junior≤50%\text{Yield share}_{Sr} = 
    \begin{dcases}
        99\% & \text{if } \dfrac{\text{Liquidity}_{Senior}}{\text{Liquidity}_{Senior + Junior}} \geq 99\%  \\
        \\
        50\% & \text{if } \dfrac{\text{Liquidity}_{Senior}}{\text{Liquidity}_{Senior + Junior}} \leq 50\%  \\
    \end{dcases}Yield shareSr​=⎩⎨⎧​99%50%​if LiquiditySenior+Junior​LiquiditySenior​​≥99%if LiquiditySenior+Junior​LiquiditySenior​​≤50%​
In the first case, we set the Yield share of Senior Tranches equal to 99% while in the second case, we set it equal to 50%. These two hedge cases link to the principle that
Senior Tranche receives most of the underlying yield when liquidity is low on the Junior side (i.e. low coverage on Senior funds), or receives a guaranteed minimum portion of the underlying yield when Junior liquidity is high (i.e. high coverage on Senior funds);
Junior Tranche receives outperforming APYs on the Junior Tranches, no matter what the amount of deposited liquidity on the Senior is.
The guaranteed minimum portion, aka the Yield share of the Senior Tranches, has been set to half the Base APY (see HC#2) when the Senior liquidity is smaller than the Junior one.
Senior coverage and Junior overperformance
The formulas of the Senior coverage provided by the Junior counterparty and the Junior boosted yield vs the underlying return are
Funds coverageSr=LiquidityJuniorLiquiditySenior\text{Funds coverage}_{Sr} = \frac{\text{Liquidity}_{Junior}}{\text{Liquidity}_{Senior}}Funds coverageSr​=LiquiditySenior​LiquidityJunior​​
OverperformanceJr=APYJr−Base APYBase APY\text{Overperformance}_{Jr} = \frac{\text{APY}_{Jr} - \text{Base APY}}{\text{Base APY}}OverperformanceJr​=Base APYAPYJr​−Base APY​
Examples
We compute the returns of the Senior and the Junior sides using the formulas listed previously, assuming
An average underlying APY of 10%
Total liquidity of the Tranches equal to $10,000,000
Standard case: between 50 and 99% of the total Tranche's liquidity lying on the Senior side
Text
Liquidity
Expected APY
Senior side
$8m
 8%
Junior side
$2m
18%
The Senior Yield share is equal to 80%. 
Senior funds coverage is 25% and the Junior overperformance vs base APY is 1.8x.
Hedge case 1: the majority of total Tranche's liquidity lying on the Senior side (≥\geq≥
99%)
Text
Liquidity
Expected APY
Senior side
$9.9m
10%
Junior side
$100
20%
The Senior Yield share is set to 99% (HC#1). 
Senior funds coverage is 0% and the Junior overperformance vs base APY is 1.99x.
Hedge case 2: less than half of total Tranche's liquidity lying on the Senior side (≤\leq≤
50%)
Text
Liquidity
Expected APY
Senior side
$4m
 5%
Junior side
$6m
13%
The Senior Yield share is set to 50% (HC#2). 
Senior funds coverage is 150% and the Junior overperformance vs base APY is 1.33x.
Text	Liquidity	Expected APY
Senior side	$8m	8%
Junior side	$2m	18%
Text	Liquidity	Expected APY
Senior side	$9.9m	10%
Junior side	$100	20%
Text	Liquidity	Expected APY
Senior side	$4m	5%
Junior side	$6m	13%
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