Specs - BuyMarketCredit() and SellMarketCredit()

[MotokoKusanagi-aka-Major]

• Explaining Formulas

  • Notation
    • $A$ : aexact amount of credit to be sold
    • $r$ : absolute rate (not the APR) obtained by applying credit.debtPos.dueDate to the buyer yield curve
    • $\Delta T$ : the remaining time to maturity i.e. deltaT = credit.debtPos.dueDate - block.timestamp
    • $f$ : Conditional Fragmentation Fee, it can be zero o non-zero depending on specific conditions
    • $f^{\star}$ : Actual Fragmentation Fee, equal to fragmFee in the protocol
    • $k$ : Swap Fee defined as APR (so need to be multiplied by a time to become an absolute rate)
    • $V$ : amount of cash swapped
    • $A_{max}$ : the max amount of credit that can be traded which is equivalent to credit.amount the amount the credit is entitled to

• The sellMarketCredit()

  • In this case, the amountIn is credit
  • With exactAmuontIn=true
    • Formula
      • $V = \frac{A}{(1+r)} (1 - k \Delta T) - f$
    • Require
      • $A \le A_{max}$ since it is not possible to sell more credit than the credit.amount
      • $0 \le k \Delta T < 1$ since of course we need to avoid the swap fee to eat the full or even exceed it, even though that's more of a theoretical issue since
        • the break even is at $\Delta T = \frac{1}{k}$
        • the order of magnitude of the fee we are considering is 0.5% APR so the break even would be for a credit whose remaining lifetime is 200Y
    • Fragmentation Fee
      • if $A = A_{max}$ then $f = 0$ since there is no fragmentation as the full credit is swapped
      • if $A < A_{max}$ then $f = f^{\star}$
    • Explanation
      • The $\frac{A}{1+r}$ is the amount of cash the credit buyer values the amount of credit he is buying, before fees
      • Then fees are applied and
        • $k \Delta T$ is charged since swap fee is a function of the remaining lifetime of the loan and is applied to the cash side of the trade
          • The higher $k$ or $\Delta T$ and the lower the cash returned to the credit seller
        • $f$ is charged to cover the gas costs related to credit fragmentation
          • The higher $f$ and the lower the cash returned to the credit seller
  • With exactAmountIn=false
    • Formula
      • $A = \frac{(V + f) (1+r)}{(1 - k \Delta T)}$
    • Require
      • Recalling the formula for exactAmountIn=true let's compute
        • $V_{max, max} = \frac{A_{max}}{(1+r)} (1 - k \Delta T)$
          • which is the max amount of cash possible, obtained by selling the full credit and therefore assuming no fragmentation fee
        • Let's define $K = \frac{1+r}{1 - k \Delta T}$ so we see immediately that
          • $V_{max,max} = \frac{A_{max}}{K}$ and that plugging it into the formula here we obtain that by submitting $V_{max,max}$ we sell $A_{max}$ and therefore there is no fragmentation and it is correct not to charge the fragmentation fee
        • Now we see in general $A = VK + fK$ so what happens if $V = V_{max,max} - \epsilon$ i.e. the seller wants to get slightly less than the max possible?
        • It happens that charging the fragmentation fee would make $A > A_{max}$ and therefore make the trading impossible
        • The turning point is for $A_{max} = VK + fK$ meaning
        • $V_{max, min} = \frac{A_{max}}{(1+r)} (1 - k \Delta T) - f^{\star}$
        • for which the trader can still sell the full amount, but the amount of cash it gets is less than the previous case since $V_{max,min} < V_{max,max}$ but still both require selling the entire $A_{max}$ amount
      • This is to say that the require are
        • $V = V_{max,max}$ or $V < V_{max,min}$
    • Fragmentation Fee
      • Starting from $V$ they are
        • If $V = V_{max,max}$ then $f = 0$
        • If $V < V_{max,min}$ then $f = f^{\star}$
      • Equivalently starting from $A$ simplythey are
        • if $A = A_{max}$ then $f = 0$ since there is no fragmentation as the full credit is swapped
        • if $A < A_{max}$ then $f = f^{\star}$
    • Explanation
      • The $\frac{A}{1+r}$ is the amount of cash the credit buyer values the amount of credit he is buying, before fees
      • Then fees are applied and
        • $k \Delta T$ is charged since swap fee is a function of the remaining lifetime of the loan and is applied to the cash side of the trade
          • The higher $k$ and $\Delta T$ and the higher the amount $A$ to be sold to get the desired amount out
        • $f$ is charged to cover the gas costs related to credit fragmentation
          • The higher $f$ and the higher the amount $A$ to be sold to get the desired amount out

• The buyMarketCredit()

  • In this case, the amountIn is cash
  • with exactAmountIn=false
    • Formula
      • $V = \frac{A}{(1+r)} + f$
      • We can see that,
        • the amount of cash to send in is equal to the credit desired (exact amount out) converted into cash discounting by the rate and increased by the fragmentation fee and then
    • Require
      • $A \le A_{max}$ since it is not possible to buy more than the full credit amount
    • Fragmentation Fee
      • if $A = A_{max}$ then $f=0$
      • if $A < A_{max}$ then $f = f^{\star}$
  • with exactAmountIn=true
    • Formula
      • $A = (V-f)(1+r)$
      • We can see that,
        • the amount of credit that is obtained is equal to the amount of cash sent in (exact amount in) first subtracted of the fragmentation fee and then converted into credit using the rate
    • Require
      • $A \le A_{max}$
      • We need to compute the $V_{max, max}$ and $V_{max,min}$ as previously explained in the sellMarketCredit() case, but this time we the buyMarketCredit() formula so
        • $V_{max, max} = \frac{A_{max}}{(1+r)} (1 + k \Delta T)$
        • $V_{max, max} = \frac{A_{max}}{(1+r)} (1 + k \Delta T) - f^{\star}$
      • So the conditions for the swap to happen are
        • $V = V_{max,max}$ or $V < V_{max,min}$
    • Fragmentation Fee
      • Starting from $V$ they are
        • If $V = V_{max,max}$ then $f = 0$
        • If $V < V_{max,min}$ then $f = f^{\star}$
      • Equivalently starting from $A$ they are
        • if $A = A_{max}$ then $f = 0$ since there is no fragmentation as the full credit is swapped
        • if $A < A_{max}$ then $f = f^{\star}$

In the case of buyMarketCredit() there is no swap fee in the formula since it is not paid by the credit buyer (even though he is the active side of the trade) but by the credit seller, so the amount of cash received by the credit seller is netCash = cash * (1 - k * \Delta T)