A CLAIM-FEE token is composed of a notional value and a duration. For example, suppose I draw 100 DAI from an ETH vault for two weeks and then pay it back. I am obliged to pay the stability fee during those two weeks while the loan is outstanding. If I buy a two week CLAIM-FEE token for the notional amount of 100 DAI then that CLAIM-FEE token will mint exactly the amount of DAI that I will need to repay the stability fee of my vault (regardless of changes to the stability fee by Maker governance).
- notional value = 100 DAI
- duration = two weeks
However, the CLAIM-FEE token does not have to match the value and duration of my loan. If I hold my 100 DAI loan for more than two weeks, the CLAIM-FEE token will expire; I will be back to paying the floating stability fee out of my pocket, the stability fee which is a floating rate determined by governance. Or alternately, suppose I take out a 200 DAI loan instead of 100 DAI for two weeks. In this case, the CLAIM-FEE token will only pay for half of the stability fee.
Now to answer your question. CLAIM-FEE tokens are purchased by vault owners and are sold by the Maker protocol. Once purchased, the CLAIM-FEE token mints DAI to pay the stability fee (whatever the current rate is) for the duration of its existence. If the stability fee increases after the CLAIM-FEE token is issued then the vault owner will save some money on the stability fee and Maker will collect less DAI then it would have had the CLAIM-FEE token not been purchased. If the stability fee decreases after the CLAIM-FEE token is issued then the vault owner overpaid and Maker will collect more DAI in fees then it would have otherwise. The other thing to keep in mind is that the fees are collected at a different time: with CLAIM-FEE tokens, the fees are collected by Maker when the CLAIM-FEE token is sold. In contrast, without the CLAIM-FEE token, stability fees are collected continuously by the Maker protocol.
The worse possible situation for the Maker protocol is to sell long duration, high notional CLAIM-FEE tokens cheap while the stability fee is zero and then for governance to aggressively raise the stability fee very high. Suppose the stability fee is raised to 20%. In this case, fortunate vault owners who purchased CLAIM-FEE tokens will not be affected by the new, much higher stability fee until the CLAIM-FEE tokens expire. So DAI remains fully backed, but Maker is not able to change the price of risk on these vaults until the CLAIM-FEE tokens expire. Presumably we moved from a low-risk environment to a high-risk environment. The CLAIM-FEE tokens delay the change in fees.
Although it may seem like it would be hard to manage hundreds of CLAIM-FEE tokens with different durations and notional values, they are all priced the same way. There is some mathematical formula that takes CLAIM-FEE token parameters and the stability fee, and outputs price. Another ingredient to this formula is some measure of expected stability fee volatility.
Governance can look at the total duration and notional value of issued CLAIM-FEE tokens to understand how much inertia exists in anticipation of changing the stability fee. If all outstanding DAI loans are 100% covered by CLAIM-FEE tokens for the next two years then stability fees have been effectively fixed for two years. Governance can change the stability fee, but it will not have any effect on loans covered by CLAIM-FEE tokens.
It is the same, but how do you choose time t? With CLAIM-FEE tokens, any arbitrary duration can be selected by the buyer. As CLAIM-FEE tokens are issued, every one will have a different duration. I don’t really see how to offer this kind of duration flexibility via a regular vault.