Murabaha Fee Rate Curve

The Murabaha Fee or more specifically the 'Profit Rate' the pool charges every time a Murabaha is executed is designed to be a function of utilisation vs target utilisation. We take a ‘market rate (MR) ’ as a benchmark for what this fee should be when the pool is within its optimal parameters (ie Utilisation = Target Utilisation). The Market Rate will be based on market analysis of the Murabaha rates offered by traditional Islamic finance and fixed borrow rates offered by DeFi protocols.

Murabaha Fee Rate Formula

The Murabaha fee increases/decreases linearly when utilisation is below the target utilisation. There is a Minimum Murabaha Rate when utilisation is 0%. If Utilisation increases beyond the target utilisation the increase in the Murabaha fee is more steep.

Formula: When Utilisation < Target Utilisation

$\text{Murabaha Fee} = \min_{\text{rate}} \text{Murabaha} + \left( \frac{\text{Market}_{\text{rate}} - \min_{\text{rate}} \text{Murabaha}}{\text{Target Utilization}} \right) \times \text{Utilization ratio}$

There is a Minimum Murabaha Rate when utilisation is 0%

Formula: When Utilisation ≥ Target Utilisation

$\text{Murabaha Fee} = \text{Market}_{\text{rate}} + \left( \frac{\max_{\text{rate}} - \text{Market}_{\text{rate}}}{1 - \text{Target}_{\text{Utilization}}} \right) \times (\text{Utilization} - \text{Target}_{\text{Utilization}})$

We set an Max Rate for the Murabaha Fee which would be achieved if the utilisation ratio was 100%:

The Market Rate, Minimum Fee and Upper Bound are all parameters than can be adjusted in real time by governance to manage liquidity risk and market dynamics.

The Protocol Fee charged by AZKA is constant at 1% when the Murbaha Fee is between the Lower and Upper Ranges set by Governance. The protocol Fee is equal to the 'Minimum Murabaha Rate' when below the Lower Range and then increases to a fixed percentage of the Murabaha Fee if it is greater than the Upper Range. These parameters are all calibrated by Governance.

Formula: Protocol Fee

The Sum % Fee is essentially the 'Annual Murabaha Fee (%)' displayed on the UI for each pool:

Sum % Fee = Murabaha Fee + Protocol Fee

The Protocol Fee is a flat fee of 1% when the Murabaha Fee is between the lowerRange and upperRange.

When the Murabaha Fee is less than ‘Lower Range’ the protocol Fee is equal to:

Protocol Fee_lower = Minimum Murabaha Fee Rate

When the Murabaha Fee is greater than ‘Upper_Range’ the protocol fee is equal to:

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Protocol Fee _upper = upper_protocolFee_Bound * Murabaha Fee

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Where the upper_protocolFee_Bound is a value that increases the protocol fee when the pool utilisation is high in a manner that is proportional to the change in Murabaha Fee Rate.

For example, assume the pool is at optimal Utilisation and the Murabaha Fee is 5%. As long as the pool's utilisation stays between say 10% and the Target Utilisation (ie, 50%) the protocol fee would be 1%, resulting in a Sum Fee of 6% for any Murabaha's with a yearly duration (this assumes the lower range is 10% Utilisation and the uppper range is 50% utilisation). Now let's further assume the Murabaha Fee increases to 50% due to the pools utilisation becoming 80%, an Upper_ProtocolFee_Bound of 0.1 would result in a protocol fee of 5%. Hence in such a utilisation scenario, the Sum Fee a Murabaha Taker can incur for a debt with a yearly duration would be 55%.