Interest-Rate Models: Where the Yield Comes From

In a bank, your interest rate is the end of a long, opaque chain: a central-bank target, a risk committee, a pricing desk, and a fee schedule nobody reads. In a DeFi pool there is no committee, no desk, and no human at all. The rate is a public formula anyone can read on-chain, and it stares at exactly one number: how busy the pool is right now. That’s it. Deposits flow in, loans flow out, and a curve recomputes the rate every block. This lesson is about that curve — where the yield you earn (and the cost you pay) actually comes from.

Utilization: how busy the pool is

Picture the pool as a reservoir. Suppliers pour water in; borrowers pump water out. Utilization ($U$) is simply how much of the reservoir is currently pumped out:

$U = \frac{\text{total borrowed}}{\text{total supplied}}$

Worked example: suppliers have deposited $10,000,000 and borrowers have taken $8,000,000 of it. Then

$U = \frac{8{,}000{,}000}{10{,}000{,}000} = 0.8 = 80\%.$

That leftover $2,000,000 is the free liquidity — the part suppliers could still withdraw on demand. As $U$ climbs toward 100%, that free slice shrinks toward zero: the pool is fully lent out, and a supplier who wants their money back may find the cupboard bare. Utilization is the single dial every other number in this lesson turns on.

The kinked (two-slope) model

Here’s the clever part. A naive pool would just let the rate rise in a straight line as $U$ goes up. DeFi pools instead use a kinked, two-slope curve, and the kink is the whole point.

Below an optimal utilization $U^{*}$ (the kink — say 80%), the borrow rate rises on a gentle slope: borrowing is cheap and the protocol is happy to lend out more. Cross $U^{*}$, though, and the rate jumps onto a much steeper slope. The message to borrowers is blunt: the pool is running low, so each extra dollar you borrow now costs a lot more. That steep section is a deliberate brake to keep some liquidity free so suppliers can still withdraw.

The model, in two pieces. For $U \le U^{*}$:

$r_{borrow} = r_0 + \frac{U}{U^{*}}\,s_1$

and for $U > U^{*}$:

$r_{borrow} = r_0 + s_1 + \frac{U - U^{*}}{1 - U^{*}}\,s_2$

Here $r_0$ is the base rate at zero utilization, $s_1$ is the gentle first slope, and $s_2$ is the steep second slope. Take the common example parameters $r_0 = 0$, $s_1 = 4\%$, $s_2 = 75\%$, $U^{*} = 80\%$.

Borrow APR at $U = 50\%$ (below the kink, so the first formula):

$r_{borrow} = 0 + \frac{0.50}{0.80}\times 4\% = 0.625 \times 4\% = 2.5\%.$

Borrow APR at $U = 90\%$ (above the kink, so the second formula):

$r_{borrow} = 0 + 4\% + \frac{0.90 - 0.80}{1 - 0.80}\times 75\% = 4\% + \frac{0.10}{0.20}\times 75\% = 4\% + 0.5 \times 75\% = 4\% + 37.5\% = 41.5\%.$

Notice the violence of that jump: going from 50% to 80% busy only takes the rate from 2.5% to 4%, but pushing past the kink to 90% rockets it to 41.5%. That asymmetry is the kink doing its job.

See the curve move

Drag the slider below. Watch the borrow curve bend sharply upward at the kink, and watch the supply curve (the next section) track underneath it.

The supply rate falls out of it

Suppliers don’t get their own formula — their rate is derived from the borrow rate:

$r_{supply} = r_{borrow}\times U \times (1 - \text{reserveFactor}).$

The intuition is fair, once you see it. Suppliers only earn interest on the fraction that’s actually lent out — that’s the $\times\,U$. Money sitting idle in the reservoir earns nothing. Then the protocol skims a cut for itself before paying the rest to suppliers — that’s the $\times\,(1 - \text{reserveFactor})$.

Worked example at $U = 80\%$ (right at the kink, so $r_{borrow} = 4\%$) with a reserve factor of 10%:

$r_{supply} = 4\% \times 0.80 \times (1 - 0.10) = 4\% \times 0.80 \times 0.90 = 2.88\%.$

A side-by-side at our example parameters:

The reserve factor

The reserve factor is the slice of borrower interest the protocol keeps for itself instead of forwarding to suppliers. It funds a safety buffer — a treasury that can absorb bad debt if a liquidation goes wrong, or backstop the pool in a shortfall. A reserve factor of 10% means suppliers receive 90% of the interest borrowers pay (after the utilization adjustment); 10% accrues to the protocol. Riskier assets often carry higher reserve factors, because they’re more likely to need that buffer.

Why the kink exists

The deepest reason for the kink is a bank-run problem. A bank promises you instant withdrawals but lends most of your deposit out; if everyone shows up at once, there isn’t enough cash, and you get a run. A DeFi pool faces the same trap: if $U$ hits 100%, every dollar is lent out and no supplier can withdraw until a borrower repays.

The steep second slope is the pool’s automatic defense — a pure price signal. As utilization creeps past the kink, borrowing gets punishingly expensive, which prices borrowers out (some repay to escape the high rate) and the soaring supply rate lures fresh suppliers in. Both forces push $U$ back down toward $U^{*}$, restoring free liquidity — no committee required, just incentives doing the steering. (The flip side, what happens to your loan when liquidity or prices move against you, is the liquidation story we tackle in lesson 5.)

Variable vs stable rates

Most DeFi borrow rates are variable: they float every single block as $U$ moves, so the cost of your loan can change minute to minute. Some protocols also offered a stable rate — typically set higher than the variable rate at borrow time, but steadier, so you trade a premium for predictability (in practice these can still be re-balanced under extreme conditions, so “stable” means “smoother,” not “fixed forever”).

Key Takeaways

Next up in lesson 4: when prices move against a borrower, the health factor decides whether their position survives or gets liquidated — and how that keeps the whole pool solvent.