# Protocol Math

The Deepr protocol contracts use a system of exponential math, ExponentialNoError.sol, in order to represent fractional quantities with sufficient precision.

Most numbers are represented as a *mantissa*, an unsigned integer scaled by `1 * 10 ^ 18`

, in order to perform basic math at a high level of precision.

## Technical information

### cToken and Underlying Decimals

Prices and exchange rates are scaled by the decimals unique to each asset; cTokens are ERC-20 tokens with 8 decimals, while their underlying tokens vary, and have a public member named *decimals*.

cToken | cToken Decimals | Underlying | Underlying Decimals |
---|---|---|---|

dWMIOTA | 8 | WMIOTA | 6 |

dSMR | 8 | SMR | 18 |

dETH | 8 | WETH | 18 |

dUSDC | 8 | USDC | 6 |

dUSDT | 8 | USDT | 6 |

dWBTC | 8 | WBTC | 8 |

*{: .decimals-events-table }*

### Interpreting Exchange Rates

The cToken Exchange Rate is scaled by the difference in decimals between the cToken and the underlying asset.

`oneCTokenInUnderlying = exchangeRateCurrent / (1 * 10 ^ (18 + underlyingDecimals - cTokenDecimals))`

Here is an example of finding the value of 1 cWMIOTA in WMIOTA with Web3.js JavaScript.

`const cTokenDecimals = 8; // all cTokens have 8 decimal places`

const underlying = new web3.eth.Contract(erc20Abi, wmiotaAddress);

const cToken = new web3.eth.Contract(cTokenAbi, cWmiotaAddress);

const underlyingDecimals = await underlying.methods.decimals().call();

const exchangeRateCurrent = await cToken.methods.exchangeRateCurrent().call();

const mantissa = 18 + parseInt(underlyingDecimals) - cTokenDecimals;

const oneCTokenInUnderlying = exchangeRateCurrent / Math.pow(10, mantissa);

console.log(`1 cWMIOTA can be redeemed for ${oneCTokenInUnderlying} WMIOTA`);

There is no underlying contract for SMR, so to do this with cSMR, set `underlyingDecimals`

to 18.

To find the number of underlying tokens that can be redeemed for cTokens, multiply the number of cTokens by the above value `oneCTokenInUnderlying`

.

`underlyingTokens = cTokenAmount * oneCTokenInUnderlying`

### Calculating Accrued Interest

Interest rates for each market update on any block in which the ratio of borrowed assets to supplied assets in the market has changed. The amount interest rates are changed depends on the interest rate model smart contract implemented for the market, and the amount of change in the ratio of borrowed assets to supplied assets in the market.

Interest accrues to all suppliers and borrowers in a market when any Shimmer address interacts with the market’s cToken contract, calling one of these functions: mint, redeem, borrow, or repay. Successful execution of one of these functions triggers the `accrueInterest`

method, which causes interest to be added to the underlying balance of every supplier and borrower in the market. Interest accrues for the current block, as well as each prior block in which the `accrueInterest`

method was not triggered (no user interacted with the cToken contract). Interest compounds only during blocks in which the cToken contract has one of the aforementioned methods invoked.

Here is an example of supply interest accrual:

Alice supplies 1 SMR to the Deepr protocol. At the time of supply, the `supplyRatePerBlock`

is 37893605 Wei, or 0.000000000037893605 SMR per block. No one interacts with the cSMR contract for 3 blocks. On the subsequent 4th block, Bob borrows some SMR. Alice’s underlying balance is now 1.000000000151574420 SMR (which is 37893605 Wei times 4 blocks, plus the original 1 SMR). Alice’s underlying SMR balance in subsequent blocks will have interest accrued based on the new value of 1.000000000151574420 SMR instead of the initial 1 SMR. Note that the `supplyRatePerBlock`

value may change at any time.

### Calculating the APY Using Rate Per Block

The Annual Percentage Yield (APY) for supplying or borrowing in each market can be calculated using the value of `supplyRatePerBlock`

(for supply APY) or `borrowRatePerBlock`

(for borrow APY) in this formula:

`Rate = cToken.supplyRatePerBlock(); // Integer`

Rate = 37893566

SMR Mantissa = 1 * 10 ^ 18 (SMR has 18 decimal places)

Blocks Per Day = 6570 (13.15 seconds per block)

Days Per Year = 365

APY = ((((Rate / SMR Mantissa * Blocks Per Day + 1) ^ Days Per Year)) - 1) * 100

Here is an example of calculating the supply and borrow APY with Web3.js JavaScript:

`const smrMantissa = 1e18;`

const blocksPerDay = 6570; // 13.15 seconds per block

const daysPerYear = 365;

const cToken = new web3.eth.Contract(cSmrAbi, cSmrAddress);

const supplyRatePerBlock = await cToken.methods.supplyRatePerBlock().call();

const borrowRatePerBlock = await cToken.methods.borrowRatePerBlock().call();

const supplyApy =

(Math.pow(

(supplyRatePerBlock / smrMantissa) * blocksPerDay + 1,

daysPerYear

) -

1) *

100;

const borrowApy =

(Math.pow(

(borrowRatePerBlock / smrMantissa) * blocksPerDay + 1,

daysPerYear

) -

1) *

100;

console.log(`Supply APY for SMR ${supplyApy} %`);

console.log(`Borrow APY for SMR ${borrowApy} %`);