# Lending Trade

Consider a Notional liquidity pool between DAI and 1-year DAI (fDAI maturing in one year), 6-month DAI (fDAI maturing in 0.5 years), 3-month DAI (fDAI maturing in 0.25 years):

<figure><img src="https://1148337737-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FVd4z5s2l5q7DqJhx3Eqg%2Fuploads%2F3Qq5KDCzzmseLn8ScU9n%2Fimage.png?alt=media&#x26;token=ffafe8d9-84fb-4979-9505-b296e5dc59f5" alt=""><figcaption></figcaption></figure>

`Proportion (P) = fCash / (DAI + fCash)`\
`Exchange rate = ((1 / Scalar) * (ln(P / (1-P)))) + ((Anchor * time)+1) - (liquidityFee * time)`\
`Interest rate = (Exchange rate - 1) / Time`\
Time = tenor span i.e. 1 = 1 year, 0.5 = 6 months and so on

Now let us assume you make a trade to buy 1000 fCash tokens

<figure><img src="https://1148337737-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FVd4z5s2l5q7DqJhx3Eqg%2Fuploads%2FyR5hYSUBGVK9MxqWFM6O%2Fimage.png?alt=media&#x26;token=755ad1c6-ddfd-4f36-bdc7-9260f7ff2e04" alt=""><figcaption></figcaption></figure>

`Proportion (P_n) = new fCash balance / (old DAI balance + old fCash balance)`\
`Exchange rate = ((1 / Scalar) * (ln(P_n / (1-P_n)))) + ((Anchor * time)+1) - (liquidityFee * time)`\
`DAI spent = fCash to be received / Exchnage rate = 1000 / Exchnage rate`\
`Interest rate = (Exchange rate - 1) / Time`\
Time = tenor span i.e. 1 = 1 year, 0.5 = 6 months and so on

The pool and rates would look like this after the trade:

<figure><img src="https://1148337737-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FVd4z5s2l5q7DqJhx3Eqg%2Fuploads%2FR1LY7nJ5z4bDPwMHydRx%2Fimage.png?alt=media&#x26;token=7d3644bb-dcfb-4c0b-86c1-b708762591dc" alt=""><figcaption></figcaption></figure>


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