Premium Pricing

see how InsureDAO calculates premium fee

Quick Summary

Yearly premium rate is determined by the equation below.

Graph p(x). y:premium%, x:available insurance%

$$p(x) = 365\Bigl(\dfrac{k}{x-a}+a\Bigr)+b$$

When user wants to buy certain amount of insurance that decrease available insurance from u1 to u2, premium amount P(x) will be

$$P(x) = \int_{u_2}^{u_1}p(x) dx$$

Then this will be scaled based on the length of period of the insurance.

If user buys 7days insurance, P(x)/365*7 will be the premium amount he has to pay.

Detail

Dynamic Pricing

InsureDAO adopts the methods of AMM to implement dynamic pricing.

We take formula from uniswap

$$x*y=k$$

y axis: premium%

x axis: available insurance%

add variable "a" to let the line pass on the axis.

$$(x+a)(y+a)=k$$

$$f(x)=\frac{k}{x+a}-a$$

Change it yearly premium, and add variable "b" which represent base fee.

$$p(x)=f(x)*365+b$$

Calculate premium fee rate

$$P(x) = \int{g(x)}dx$$

Blue area on the graph shows an example of P(x)

Additional feature

Lower base fee is applied for an insurance pool that has high TVL and low utilization rate.

BondingPremiumV2

Since calculating integral is complicated to perform on the code, and there is no way to do with solidity, above calculation is implemented after simplification.

setup

• take three decimals. 1000000 = 100.000%

• f(x) goes through (1000000, 0) and (0, 1000000)

$$(x+a)(y+a) = k$$

$$(1000000 + a)(0 + a) = k$$

$$a^2+1000000a - k = 0$$

solve for "a" using quadratic formula

$$a = \frac{-1000000 + \sqrt{1000000^2+4k}}{2}$$

$$f(x)=\frac{k}{x+a}-a$$

$$p(x)=f(x)*365+b$$

$$P(x) = \int_{u_2}^{u_1}p(x) dx$$

screenshot from desmos

P(x) simplification

$$P(x) = \int_{u_2}^{u_1}{365(\frac{k}{x+a}-a)+b}\ dx$$

$$[365k\ln\left(\left|x+a\right|\right)+\left(b-365a\right)x]\begin{smallmatrix} u_1 \\ u_2 \end{smallmatrix}$$

Above simplified integral is implemented with the additional feature of low base fee.

Implemented formula

$$a = \frac{-1000000 + \sqrt{1000000^2+4k}}{2}$$

$$[365k\ln\left(\left|x+a\right|\right)+\left(b-365a\right)x]\begin{smallmatrix} u_1 \\ u_2 \end{smallmatrix}$$