Interest rate model

The Apollo Protocol's interest rate strategy is calibrated to handle liquidity risks and optimize asset utilization. Borrow interest rates are closely linked to the Utilization Rate (U). The U value signifies the availability of an asset in the protocol's reserves, and the interest rate model is employed to manage associated liquidity risks through user incentives:

• When an asset is available: a low-interest rate is applied to encourage borrowers.

• When an asset is scarce: a high-interest rate is applied to encourage borrowers to repay and depositors to add more assets.

A liquidity risk becomes apparent when utilization is high. It turns problematic as $U$ gets closer to 100%. To adjust the model to this constraint, the interest rate curve is split into two parts around an optimal utilization rate $U_{optimal}$. Before the point $U_{optimal}$, the slope is gradual. After passing that point, it starts becoming steep.

The interest rate$R_t$is as follows:

$if \hspace{1mm} U < U_{optimal}: \hspace{1cm} R_t = R_0 + \frac{U_t}{U_{optimal}} R_{slope1}$

$if \hspace{1mm} U \geq U_{optimal}: \hspace{1cm} R_t = R_0 + R_{slope1} + \frac{U_t-U_{optimal}}{1-U_{optimal}}R_{slope2}$

In the technical implementation of borrowing rates, the calculation method relies primarily on approximations that affect higher interest rates. The actual borrow rate can be resulted as:

$Actual APY = (1+Theoretical APY/secsperyear)^{secsperyear}-1$

Both variable and stable interest models are derived from the formula above with different parameters for each asset.

• When $U < U_{optimal}$ the borrowing interest rates increase slowly for the utilization.
• When $U \geq U_{optimal}$ the borrow interest rates increase rapidly for the utilization resulting in over 50% APY if the liquidity is fully utilized.

Model Parameters

Interest rate parameters have been calibrated per cluster of assets that share similar risk profiles. First, it's crucial to distinguish assets that are used predominantly as collateral (volatile assets) which need liquidity at all times to enable liquidations. These assets require a low Optimal Utilization rate typically calibrated around 45%. Secondly, liquidity on Apollo Protocol is an important factor as the more liquidity, the more stable the utilization becomes: interest rates of assets with lower liquidity should be more conservative. For example, low liquidity stablecoins have a low Optimal Utilization Ratio than those with higher liquidity.

It's also key to consider market conditions: how can the asset be used in the current market? A cost to borrow on Apollo Protocol must be aligned with market yield opportunities. Alternatively, there will be a rate arbitrage with incentives for users to borrow all the liquidity on Apollo Protocol to obtain higher yield opportunities.

If market conditions change, the parameters of the interest rate can be adapted. These changes would have to adapt not only to Apollo Protocol's market availability but also to incentives across DeFi.

Interest Rate Model Parameters

Asset	$U_{optimal}$	Base	Slope 1	Slope 2
ETH	65%	0%	8%	100%
DAI	90%	0%	4%	60%

Interest Rates vs Utilizations for each asset are below:

Interest Rates/Utilization	$XOR	$DAI	$ETH
1%	0.16%	0.04%	0.12%
2%	0.31%	0.09%	0.25%
3%	0.47%	0.13%	0.37%
4%	0.62%	0.18%	0.49%
5%	0.78%	0.22%	0.62%
6%	0.93%	0.27%	0.74%
7%	1.09%	0.31%	0.86%
8%	1.24%	0.36%	0.98%
9%	1.40%	0.40%	1.11%
10%	1.56%	0.44%	1.23%
11%	1.71%	0.49%	1.35%
12%	1.87%	0.53%	1.48%
13%	2.02%	0.58%	1.60%
14%	2.18%	0.62%	1.72%
15%	2.33%	0.67%	1.85%
16%	2.49%	0.71%	1.97%
17%	2.64%	0.76%	2.09%
18%	2.80%	0.80%	2.22%
19%	2.96%	0.84%	2.34%
20%	3.11%	0.89%	2.46%
21%	3.27%	0.93%	2.58%
22%	3.42%	0.98%	2.71%
23%	3.58%	1.02%	2.83%
24%	3.73%	1.07%	2.95%
25%	3.89%	1.11%	3.08%
26%	4.04%	1.16%	3.20%
27%	4.20%	1.20%	3.32%
28%	4.36%	1.24%	3.45%
29%	4.51%	1.29%	3.57%
30%	4.67%	1.33%	3.69%
31%	4.82%	1.38%	3.82%
32%	4.98%	1.42%	3.94%
33%	5.13%	1.47%	4.06%
34%	5.29%	1.51%	4.18%
35%	5.44%	1.56%	4.31%
36%	5.60%	1.60%	4.43%
37%	5.76%	1.64%	4.55%
38%	5.91%	1.69%	4.68%
39%	6.07%	1.73%	4.80%
40%	6.22%	1.78%	4.92%
41%	6.38%	1.82%	5.05%
42%	6.53%	1.87%	5.17%
43%	6.69%	1.91%	5.29%
44%	6.84%	1.96%	5.42%
45%	7.00%	2.00%	5.54%
46%	12.45%	2.04%	5.66%
47%	17.91%	2.09%	5.78%
48%	23.36%	2.13%	5.91%
49%	28.82%	2.18%	6.03%
50%	34.27%	2.22%	6.15%
51%	39.73%	2.27%	6.28%
52%	45.18%	2.31%	6.40%
53%	50.64%	2.36%	6.52%
54%	56.09%	2.40%	6.65%
55%	61.55%	2.44%	6.77%
56%	67.00%	2.49%	6.89%
57%	72.45%	2.53%	7.02%
58%	77.91%	2.58%	7.14%
59%	83.36%	2.62%	7.26%
60%	88.82%	2.67%	7.38%
61%	94.27%	2.71%	7.51%
62%	99.73%	2.76%	7.63%
63%	105.18%	2.80%	7.75%
64%	110.64%	2.84%	7.88%
65%	116.09%	2.89%	8.00%
66%	121.55%	2.93%	10.86%
67%	127.00%	2.98%	13.71%
68%	132.45%	3.02%	16.57%
69%	137.91%	3.07%	19.43%
70%	143.36%	3.11%	22.29%
71%	148.82%	3.16%	25.14%
72%	154.27%	3.20%	28.00%
73%	159.73%	3.24%	30.86%
74%	165.18%	3.29%	33.71%
75%	170.64%	3.33%	36.57%
76%	176.09%	3.38%	39.43%
77%	181.55%	3.42%	42.29%
78%	187.00%	3.47%	45.14%
79%	192.45%	3.51%	48.00%
80%	197.91%	3.56%	50.86%
81%	203.36%	3.60%	53.71%
82%	208.82%	3.64%	56.57%
83%	214.27%	3.69%	59.43%
84%	219.73%	3.73%	62.29%
85%	225.18%	3.78%	65.14%
86%	230.64%	3.82%	68.00%
87%	236.09%	3.87%	70.86%
88%	241.55%	3.91%	73.71%
89%	247.00%	3.96%	76.57%
90%	252.45%	4.00%	79.43%
91%	257.91%	10.00%	82.29%
92%	263.36%	16.00%	85.14%
93%	268.82%	22.00%	88.00%
94%	274.27%	28.00%	90.86%
95%	279.73%	34.00%	93.71%
96%	285.18%	40.00%	96.57%
97%	290.64%	46.00%	99.43%
98%	296.09%	52.00%	102.29%
99%	301.55%	58.00%	105.14%
100%	307.00%	64.00%	108.00%

Deposit APY

The borrowing interest paid is distributed as yield for lenders of the protocol, except for the share of yields sent to the ecosystem reserve defined by Reserve Factor. The interest rate is paid on the borrowed asset, which is then used for APOLLO buyback, and afterward, those APOLLO tokens are distributed among all lenders. The deposit APY, $D_t$, is:

$D_t = U_t V_t(1-R_t)+FR_t$

• $U_t$, Utilization Rate
• $V_t$, the variable rate
• $R_t$, Reserve Factor
• $FR_t$, Fixed rate

​Net APY

For your reference, it is calculated at frontend as follows:

Net APY = sum of PL by Asset in USD / Total Deposited in USD

PL by Asset in USD = Interest PL + Reward

Interest PL = deposited * APY in USD - borrowed * APY in USD

Reward = deposited * depositAPR in USD + borrowed * borrowAPR in USD