Simple Numerical Example

This page illustrates an example of how interest rates for a given loan is derived, and the managers' and passive liquidity provider's payoff for different scenarios.

As an example, a borrower(utilizer) would request a loan with the following parameters, denominated in USD:

Proposed Principal: 100k

Proposed Interest: 10k

Proposed Duration 1 Year

From the equations outlined in section 7 of the whitepaper, the initial parameters of the AMM, namely its inverse liquidity coefficient(which determines how fast price increases in relation to longZCB bought) and the initial price of longZCB would be constructed.

Here the initial price (b) would be 2P/(P+I)-1 = 0.818 USD. The inverse liquidity coefficient (a) is (1-b)/(P+I) = 0.0000016545.

The p := price of longZCB can then be numerically represented as a function of c:= (longZCB bought - shortZCB bought )

p = 0.0000016545c + 0.818

Say a manager(with a high reputation) deems this borrower creditworthy or thinks that his collateral is sound. He would then start buying its longZCB at this initial price for a total of, say, 10000 USD worth. Then the average price he would pay is calculated by the area under the price curve(10K) divided by the quantity he would buy(which can be derived from this area as input). In this case, it would be around 0.83USD, for a total of 12050 longZCB. When the loan successfully matures, this would be redeemable for 12050 USD.

Now, say there are only two more managers who believe this borrower is creditworthy, and each of them buys 5000 USD worth, for a total of 10000+5000+5000 = 20000 USD worth of longZCB bought, which equates to 11350+ 12050 = 23400 total longZCB bought.

Then the actual principal and interest the borrower will be able to get will be computed using the leverage parameter L, which is a parameter that scales with how much leverage longZCB token holders are incurring(and equally, how little protection junior holders are getting). If L = 3, then the actual principal and interest(using equations in the whitepaper) is

Actual Principal: 60K

Actual Interest: 7.2K

Now, note that the principal is lower and the interest rate higher being 7.2/60 = 12% as opposed to 10/100 = 10% from the borrower's proposal. If there are more managers who buylongZCB, the actual principal will cap at 100K, and interest at 10K.

The capital used to lend out to the borrower is supplied 20K from these managers and 60K - 20K = 40K from the VT holders.

At maturity, let's go through examples of three possible scenarios:

1. The borrower pays back 33.8K. Including the interest as debt, the loss is 67.2K-33.8K = 33.8K. Recall that the total number of longZCB tokens sold is 12050 + 11350 = 23400. The redemption price of longZCB tokens is then set at max(1 - 33800/23400,0) = 0. This means all of the managers' collateral would be used as first loss capital. The VT holders would have a protection of 20K, so their loss would be 33.8K - 20K = 13.8K, which is shared pro-rata.

2. The borrower pays back 55K. The loss is 67.2k - 55K = 12.2K. longZCB redemption price would be at max(1 - 12200/23400, 0) = 0.48. No loss is incurred by VT holders.

3. The borrower pays back the full amount of 67.2K. The redemption price of longZCB tokens is then set at max(1-0/23400, 0) = 1. The first manager would have made a profit of 12050-10000 = 2050 USD, and the second/third manager would have made a combined profit of 11350-10000 = 1350 USD. The VT holders' profit would be 7200 - 2050-1350 = 3800 USD.

Note that the specific payouts from the examples above might vary for different parameters. Also, the AMM parameter computation ignores rewards for validators.

Sub Example 1: Manager with Leverage

Now let's assume that the first manager has a great track record and is able to buy longZCB with leverage. This essentially means he will be able to borrow capital inside VT, and use it to margin long longZCB. When he redeems his longZCB at maturity, he will have to pay back the borrowed amount to VT.

Under the same example as illustrated above, say the manager(with a high reputation) deems this borrower creditworthy and starts buying its longZCB at this initial price for a total of 10000USD worth. If his allowed leverage coefficient is 2, he would instead be able to borrow 5000 USD from VT, and only pay 5000 USD of his own capital to purchase 12050 longZCB.

At maturity, let's go through the same three examples:

1. The borrower pays back 33.8K. The redemption price of longZCB tokens is set at max(1 - 33800/23400,0) = 0. This means all of the managers' collateral would be used as first loss capital. The VT holders would have a protection of 15K, so their loss would be 33.8K - 15K = 18.8K, which is shared pro-rata.

2. The borrower pays back 55K. The loss is 67.2k - 55K = 12.2K. longZCB redemption price would be at max(1 - 12200/23400, 0) = 0.48. No loss is incurred by VT holders, since 15K> 12.2K. The first manager who bought with leverage would have to pay back the vault 5000 USD after redeeming his longZCB tokens at a value 12050*0.48 = 5784, which will leave him with only 784 USD.

3. The borrower pays back the full amount of 67.2K. The redemption price of longZCB tokens is then set at max(1-0/23400, 0) = 1. The first manager would have made a profit of 12050-10000 = 2050 USD. His returns percentage terms would instead be 2050/5000 = 41% as opposed to 20.5% without leverage.

Sub Example 2: ShortZCB buyer

Now, let's assume that there is a VT holder who disagrees with the managers. He wants to hedge potential exposure from this borrower by buying some shortZCB. Under the same example, let's say the VT holder buys shortZCB after all three managers have purchased theirlongZCB, when the final price is 0.87USD. Then the spot price of each shortZCB would be 1-0.87 = 0.13 USD. He then buys 11350 shortZCB, at a total price of 1470USD(as selling would also incur symmetric slippage). This would decrease the net longZCB and lower the actual principal/interest approved for the borrower. A reduction of 11350 net longZCB results in only 12050 net longZCB left, and with the same L = 3,

Actual Principal: 60K

Actual Interest: 7.2K

At maturity, let's go through the same three examples:

1. The borrower pays back 33.8K. Including the interest as debt, the loss is also at 67.2K-33.8K = 33.8K. Recall that the number of longZCB tokens sold is 12050 + 11350 = 23400, and the number of shortZCB tokens sold is 11350. The redemption price of longZCB tokens is then set at max(1 - 33800/23400,0) = 0, and the redemption price of shortZCB would be 1 - 0 = 1. The VT holders would have a protection of 20K - 11.35K + 0.147K.

2. The borrower pays back 55K. The loss is 67.2k - 55K = 12.2K. longZCB redemption price would be at max(1 - 12200/23400, 0) = 0.48, and the redemption price of shortZCB would be 1-0.48 = 0.52. Some loss is incurred by VT holders, since 20K - 11.35K + 0.147K < 12.2K.

3. The borrower pays back the full amount of 67.2K. The redemption price of longZCB tokens is then set at max(1-0/23400, 0) = 1, and the shortZCB redemption price is 0.

In actuality, the amount of shortZCB that can be bought by an individual would not be this high, unless he/she owns a very large amount of VT.

All these processes could be applied to any loan-based instrument.