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Weighted Shares
Weighted shares (WS) for a bond is calculated by multiplying Unweighted Shares by the NFT level weight index. This metrics is utilized to issue rewards to NFT bond holders.
To be clear:
Unweighted shares --> Reinvested revenue
Weighted shares --> Payed revenue
This means that at the time of issuing, the WS of a Frank NFT Share will simply be its US multiplied by its weight.
Share level
Weight
US
WS
1
1.00
10
10
2
1.05
100
105
3
1.10
1000
1100
4
1.15
5000
5750
When compounding reinvested revenue as explained in the previous page, also WS increase.
Let's build on the example found on the previous page
$4L_1 + 4L_2 = \begin{cases} \text{US} = 4 \cdot 10 + 4 \cdot 100 = 440\\ \text{WS} = 4 \cdot 10 + 4 \cdot 105 = 460\\ \text{TVL} = 4 \cdot 10 + 4 \cdot 100 = 440 \end{cases}$
Let's reconsider step (3). Instead of reinvesting all revenue, we will be splitting equally the amount we reward (Reward Issuance
$I_R$
) and the amount we reinvest (Share Issuance
$I_S$
).
$I_S = 50\% \text{ Revenue} = 110, \quad I_R = 50\% \text { Revenue} = 110$
Shares issuance
$I_S$
The Shares Reward will now be:
$I_{S, L=1} = 2.5 \text{ Shares}, \quad I_{S, L=2} = 25 \text{ Shares}$
Reward issuance
$I_R$
- Part of the revenue rewarded to bond holders. Rewards issued are divided between bonds according to their weighted shares.
To calculate how many JOE tokens a bond holder is entitled to claim per bond, we must calculate that bond's percentage of ownership over all weighted shares.
$I_{R, L} = \frac{\text{WS}_{L}}{\text{WS}} \cdot I_R$
• $I_{R, L}$
: Rewards issued to a bond of level
$L$
.
• $\text{WS}_L$
: Unweighted shares of a bond of level
$L$
.
Thus:
• $I_{R, L=1} = 2.39 \text{ JOE}$
• $I_{R, L=2} = 25.11 \text{ JOE}$
As you can see, this time Level 2 NFT holders are at an advantage as they gain slightly more than what they would've gained if rewards were distributed proportionally (with US), gaining 25.21 instead of 25. On the other hand, Level 1 NFT holders are at a disadvantage as they gain 2.29 instead of 2.5.
Furthermore, as US increase when rewards get compounded so do WS.

Why creating a two share system?

If we had kept solely the weighted shares, when compounding rewards we were giving higher-level bond holders an exponential advantage compared to lower level bond holders. This would occur because perpetually compounding rewards taking into account bond level weight would eventually lead to an unsustainable gap between low level bonds and high level bonds.
By taking into account weight only when distributing rewards (and not when auto-compounding) we still give an advantage to higher level bond holders, without putting an unbearable burden on low level bonds.