# FIL on FIL vs. Piece of the Pie ROI

#### FiL on FIL ROI
One important measure for quantifying the quality of SP's investments when providing storage for the network, is the FIL on FIL return (we will label as $R_{FoF}$).

This takes into account the absolute number of FIL the SP will invest to be able to start earning FIL rewards. We define it as the Net FIL earnings over a sector lifetime (could also be annualized, or any other time period), over the amount of collateral needed to seal the sector. We note that this definition does not include any hardware investment costs. Also for now we exclude any related gas or penalty costs to maintaining a sector.

We then define
$$R_{FoF}=\frac{Block\,\,reward+Deal\,\,reward_{FIL}}{Collateral}$$
with all these quantities being denominated in FIL.

#### Piece of the Pie ROI

We here want to introduce the alternate concept of Piece of the Pie ROI ($R_{PoP}$), as a more accurate measure of the quality of the SP's investment.

The starting hypothesis is that the absolute number of FIL that one has is not a relevant quantity. The value of 1 FIL will depend on how many FIL are there in the circulating supply.

What is important to token holder is how many FIL they have, in relation to the circulating supply.

The main idea is basically to consider what would be the ROI in terms of Fiat currency. We assue Deal reward is something that is set in fiat terms, regardless of token price, so we simply. replace in the formula with $Deal\,\,reward_{Fiat}$$.

The Fiat ROI is then given by
$$R_{Fiat}=\frac{Block\,\,reward*Token\,\,price+Deal\,\,reward_{Fiat}}{Collateral*Token \,\,Price}$$
where $Token\,\,price$ is in units of Fiat units over FIL units.

We further break down
$$Token\,\,price= \frac{M}{S}$$
where $M$ is the total market cap for Filecoin, and $S$ is the circulating supply.

So the definition of $R_{PoP}$ is simply
$$R_{PoP}=\frac{Block\,\,reward/S+Deal\,\,reward_{fiat}/M}{Collateral/S}$$

## Relevance to FIP 0036 calculations

One of the main questions to address with FIP 0036 is if there are any important changes to SP ROI.

We will illustrate how the two definitions of ROI respond differently to one aspect of FIP 0036, which is the proposed increased in collateral target.

The consensus pledge targets 30% of the circulating supply to have locked, while an increase to possibly 50% has been proposed.

We examine what happens to $R_{FoF}$ and $R_{PoP}$ as the target is increased,

Let us evaluate the quantities involved in $R_{FoF}$:
1) Block reward is unchanged, $Block\,\,reward\rightarrow Block\,\,reward$.
2) Deal reward is unchanged, $Deal\,\,reward\rightarrow\,\,Deal \,\,reward$
3) Collateral is increased, $Collateral\rightarrow Collateral^+$.

Therefore $R_{FoF}$ is always reduced by increasing collateral target,

$$R_{FoF}\rightarrow R_{FoF}^-=\frac{Block\,\,reward+Deal\,\,reward}{Collateral^+}$$

Now lets consider that the relevant quantity is actually $R_{PoP}$, what happens to it under an increase in the collateral target.

Let us break down the Collateral into its components, and *for simplicity, we will assume this is equivalent only to the consensus pledge, which is the majority of the collateral*.

$$Collateral=0.3*S*p_i/P$$
where $p_i$ is the individual SP's quality adjusted power and $P$ is the total network quality adjusted power.

We then have
$$R_{PoP}=\frac{Block\,\,reward/S+Deal\,\,reward_{fiat}/M}{0.3*p_i/P}$$

#### Changing the locking target
Let us now consider the case where we change the locking target from 0.3 to a general $\tau\in (0,1)$

For simplicity in this subsection let us only consider the block reward, considering the deal reward is in fiat terms and should not be affected by FIP 0036, so let's assume
$$R_{PoP}=\frac{Block\,\,reward/S}{0.3*p_i/P}$$

What happens to the circulating supply given a locking target is a complex question that requires more modelling, but here we will take the simple assumption that the locking target works roughly as expected. Mainly if $\tau$ of the circulating supply was targeted for locking, then $(1-\tau)$ of the circulating supply should relmain unlocked. This means we will make the assumption that if the target is changed from $0.3$ to $\tau$, then the circulating supply would roughly change to
$$S\rightarrow \frac{(1-\tau)}{0.7}S$$

So now the question is what happens to the SP's ROI as the target is changed from 0.3 to $\tau$? Plugging back our assumptions we see

$$R_{PoP}\rightarrow \frac{\frac{Block\,\,reward}{S}\frac{0.7}{(1-\tau)}}{\frac{\tau}{0.3}0.3*p_i/P}$$
or
$$R_{PoP}\rightarrow\frac{0.21}{\tau(1-\tau)}R_{PoP}.$$

Then, the question is *is it good for SP ROI to change the locking target?* The answer is that it depends on the new chosen value of $\tau$. Changing the target is good for SP's (the ROI is larger) if it is satisfied that
$$\frac{0.21}{\tau(1-\tau)}>1$$
or
$$\tau^2-\tau+0.21>0.$$
This is a simple quadratic inequality which we can solve. First we see that the ROI will always increase if the new target is close to the edges $\tau\approx 0$ or $\tau\approx 1$, since the denominator blows up in either case.

The solutions to the inequality are
$$\tau_1=0.3$$
and
$$\tau_2=0.7$$

This means that if the target were reduced from its current $0.3$, this would be good for SP ROI. But also if the target is increased, this would be bad for SP ROI, up to the point where it is increased to 0.7. If the target is increased to 0.7 or higher, this is again favorable for SP ROI.

#### Note that only the locking target change of FIP 0036 have been addressed here, we have not analyzed here the effect of the duration multiplier itself, but we believe analyzing this in terms of the $R_{PoP}$ would still be a more meaningful analysis than in terms of $R_{FoF}$.