# Valuing long-term sector commitments
Here we derive a formula for how much the network values longer sector commitments, which can be used to give the sectors an appropriate quality multiplier.

The principle is to first understand what is the value of a sector commited to a minimum sector lifetime. We will use the convention that the minimum lifetime is $T$.

We assume the probability of a sector being extended/renewed after it expires, is given by $p$. Similarly after after the sector is extended once and expires again a second time, it has a chance of renewal $p$.

## Value of a sector commitment, with possible extensions.

What is the value added to the network from a sector being onboarded for a period $T$?

There are two sources of value to the network. We define $a$ as the **storage value**, the amount of value a sector of minimum duration $T$ adds to the network by virtue of it increasing the network's storage power, and therefore making the network more valuable for a period of time $T$. We define $b$ as the **collateral value**, the amount of value a sector of minimum duration $T$ adds to the network by virtue of simply having locked an amount of tokens for a period of time $T$, therefore reducing effective circulating upply.

When a sector is commited, the value to the network is guaranteed for the duration $T$. After that, however, there is a probability $p$ that the sector will be extended.

Here we calculate the expected network profit from a commited sector.

The sector has a probability $(1-p)$ of only lasting for one life time, and a probability $(1-p)p$ of lasting 2 liftimes, and generally $(1-p)p^{n-1}$ for lasting $n$ lifetimes.

If the sector lasts for one lifetime, the profit made is
$$(a+b)(r-1).$$
For two lifetimes, there is a profit $(a+b)(r^2-1)$ is made but of those, $(a+b)(r^2-r)$ are made only after the second lifetime, while $(a+b)(r-1)$ are made already after the first lifetime. Profits that are made sooner are more valuable than profits made later, as they can be used to earn interests. Discounting the later profits with a factor of $1/r$, the profit if the sector lasts two lifetimes is $(a+b)[(r^2-r)/r+(r-1)]$=2(a+b)(r-1)$.

More generally the profit made if the sector lasts $n$ lifetimes is then $n(a+b)(r-1)$.

The expected profit to be made from one sector commitment is then
$$\langle{\rm network\,profit\,from\,sector}\rangle=(a+b)(r-1)(1-p)\sum_{n=1}^\infty (n+1)p^n$$
$$=(a+b)\frac{(r-1)}{(1-p)}$$

## Expected profit from a long term commited sector

When a sector is commited for $N$ lifetimes, there is value added in two ways. First there is certainty that the loan made to the network will last at least for a time $NT$, which means more interest can be earned with certainty. After those initial $N$ periods, there is again probability $p$ of renewal, and so on.

There is also a second benefit associated. Assuming we are going to assign a higher quality multiplier to sectors with longer commitments, that means the collateral locked will actually be higher, which means the profit will be higher. We denote the quality multiplier as $m_t$

The network profit is then given by
$$\langle{\rm network\,profit\,from\,N\,long\,sector}\rangle=(a+bm_t)\left\{[(r-1)+(r^2-r)/r+\dots+(r^{N}-r^{N-1})/r^{N-1}](1-p)\right.$$
$$+(1-p)\left.\sum_{n=1}^{\infty}(n+1)p^n\right\}$$
$$=(a+bm_t)(r-1)\left[\frac{t}{T}(1-p)+\frac{p(2-p)}{1-p}\right],$$
where we expressed back in terms of continuous time $t=NT$.

## Value multiplier
We define the quality multiplier as
$$m_t=\frac{\langle{\rm network\,profit\,from\,N\,long\,sector}\rangle}{\langle{\rm network\,profit\,from\,sector}\rangle},$$
so that
$$m_t=\frac{(a+bm_t)}{(a+b)}\left[\frac{t}{T}(1-p)+\frac{p(2-p)}{1-p}\right](1-p)$$
Notice that this equation still involves $m_t$ in the right hand side, so we still need to solve for $m_t$:
$$m_t=\frac{a}{\frac{a+b}{\left[\frac{t}{T}(1-p)+\frac{p(2-p)}{1-p}\right](1-p)}-b}$$

It is easy to check that $m_1=1$, as we expect.

An interesting feature is that the multiplier grows very fast with $t$ and in fact diverges at some point, when the denominator goes to zero. This time commitment where the multiplier blows up is
$$t^*=\frac{T\left[a+b(1-p(2-p))\right]}{b(1-p)^2}$$

The reasoning why the quality multiplier would blow up is that at the point where the profit from collateral becomes dominant over the profit from storage, then having a higher multiplier will only make the profit larger, having a higher and higher multiplier will end up locking more collateral, which adds more value to the network.

### No blowing up in real life

In real life this exploding multiplier can never be reached, as there exist only a finite amount of tokens out there. Even if only one SP owned all the tokens in the network they cannot pay an infinite collateral.

This gives us a bound on the maximum time commitment that can be theoretically made: The locked collateral must be less than the circulating supply,
$$m_tL\le S.$$

The maximum locking time $t_{\rm Max}$ is the one that saturates this bound:
$$t_{\rm max}=T\left[\frac{S(a+b)}{(aL+Sb)(1-p)^2}-\frac{p(2-p)}{(1-p)^2}\right].$$

## How much do we value adding storage capacity vs how much do we value collateral?

Our formula for the quality multiplier involves both sources of value to the network for commiting a sector, the value of adding a unit of storage, $a$, and the value of locking the correponding collateral $b$.

These actually appear only as a ratio in the final formula, so their absolute magnitude doesn't matter, the only thing that matters is how much do we value storage power vs how much do we value collateral.

To illustrate this we define the ratio $\xi=\frac{a}{b}$, wher $\xi$ represents **how much more valuable it is to add a unit of storage than it is to add a unit of collateral**. **$\xi$ is a measure of how much Filecoin is dedicated to being a data storage network, as with very small $\xi$, it effectively becomes a proof of stake network.** Ultimately $\xi$ can be a parameter we choose to express our goals for the network. The quality multiplier is then
$$m_t=\frac{\xi}{\frac{\xi+1}{\left[\frac{t}{T}(1-p)+\frac{p(2-p)}{1-p}\right](1-p)}-1},$$

with parameters $\xi$ and $p$.