Deep Dive into Tempus Economics

Josh Kelly
12 min readFeb 17, 2022

This deep dive was written by Cristiano, Tempus’ quantitative guru.

Introduction to Tempus

Tempus is a multi-chain fixed income protocol that enables users to:

  1. Fix their future yield accrued on a wide variety of yield-bearing tokens (like stETH, aUSDC, and yvDAI).
  2. Speculate and leverage up on future yield.
  3. Provide liquidity to the available liquidity pools to earn trading fees on top of existing variable yield on the yield-bearing token.

How does Tempus make this possible?

Tempus provides users with the opportunity to choose different maturities on their investment strategies.

Once users choose a maturity and deposit yield-bearing tokens, they are decomposed into an equal number of:

  • Capital tokens that allow users to collect an equivalent amount of the deposited underlying yield-bearing token at maturity.
  • Yield tokens that allow users to collect the variable yield accrued on the yield-bearing token throughout the duration of the contract.

Note: A yield-bearing token mints the same number of Capital and Yield tokens when a pool is created. As the contract approaches the maturity date, the same yield-bearing token will mint a lesser number of Capital and Yield tokens to compensate for the yield accrued since the initiation of the pool.

Holding both tokens until maturity will result in the same payoff as holding the underlying yield-bearing token. The real benefit of this decomposition can be obtained by swapping Capital or Yield tokens in exchange for the other.

Tempus provides liquidity pools where users can trade Capital and Yield tokens against each other. There are two main purposes of swapping Capital and Yield tokens:

  • Fix the future yield: Users can swap their Yield tokens in exchange for Capital tokens to eliminate the uncertainty about future yields and lock in a fixed yield they can earn at maturity.
  • Speculate on future yield: Users can swap their Capital tokens in exchange for Yield tokens to give up their fixed yield in exchange for participating in the upside of the future variable yield.

Dissecting Capital and Yield tokens

Capital tokens

Capital tokens give tokenholders the right to receive one equivalent unit of the underlying token at maturity.

Why would someone want to hold Capital tokens? Because they trade at a discount with respect to the underlying token price. The price discount represents the fixed yield that allows Capital tokenholders to earn a predetermined rate regardless of how variable yields move over time. This means that the price of 1 Capital token would be less than 1 unit of the yield-bearing token at any time before maturity.

How is the fixed yield determined? Once a pool goes live, the initial fixed yield is set by the Tempus Labs team based on a multi-factor analysis. From then on, the fixed yield is determined by the market participants trading Capital and Yield tokens against each other

How is the Capital token price determined? The Capital token price can be computed as that of a zero-coupon bond with a rate equal to fixed yield, a maturity (in days) equal to T and time lapsed (in days) equal to t:

The current yield variable is the exogenous yield accrued on the underlying yield-bearing token since inception. The Capital token discount with respect to the redemption value can be computed as the difference between the redemption value (one unit of the underlying token) and the Capital token price:

Is there any relationship between Capital token price and fixed yield? The relationship between Capital token price and fixed yield is inverse: the higher the price, the lower the yield and vice versa.

Why? As more people buy the Capital tokens to fix their yield, the price of the token rises due to increase in demand and as a result the fixed yield reduces for the users. As maturity approaches, the Capital token price will converge to that of the underlying token. The discount will then decrease over time as well as the fixed yield one can earn by purchasing the Capital token at that particular point in time.

Yield tokens

Yield tokens give tokenholders the right to receive the variable yield accrued on the underlying token over the duration of the pool.

Why should someone want to hold Yield tokens? If investors expect variable yields to increase in the future, they could benefit by purchasing Yield tokens by selling their Capital tokens and collecting the higher variable yields the underlying token will accrue.

How is the variable yield determined? Variable yield is exogenous as it depends on the demand and supply dynamics of that underlying token. If the lending rate for the yield-bearing token falls then the yield accrued on the underlying tokens will also fall.

How is the Yield token price determined? The Yield token price is computed according to the following formula:

Is there any relationship between Yield token price and variable yields? Since variable yields are exogenous, market expectations on future variable yields will drive the Yield token price. If investors expect future yields to increase, they will purchase more Yield tokens by swapping Capital tokens in exchange and vice versa in case future yields are expected to decrease.

The relationship between Capital token and Yield token

Capital and Yield tokens are tied together by the fixed yield.

Why? Because the fixed yield represents the rate at which investors are willing to forego a unit of variable yields to lock their future yields in advance. The fixed yield can be thought of as the exchange rate between Yield and Capital tokens.

At a first glance, this formula may seem to contradict what we explained above: why should the fixed yield increase when the Yield token price increases? Should it be the opposite?

Let’s go through it in four simple steps:

  1. Market view: Investors formulate their view on future variable yields.
  2. Price action: Investors swap Capital tokens for Yield tokens if they expect a variable yield increase. This would induce a Yield token price appreciation at the expense of the Capital token price. Yield tokens become more in demand and more users want to buy it by swapping their Capital tokens for it. The opposite would happen in case of an expected decline in future variable yields. The Capital token price would increase at the expense of the Yield token price. Expecting the future yield to fall would make a rational trader fix their yield to prevent any downside risks. More Capital tokens are purchased and therefore the Capital token price appreciates.
  3. Impact on fixed yield: The fixed yield would increase as a result of the Yield/Capital exchange rate movements. Why? Let’s recall that the fixed yield depends on the price discount of the Capital token with respect to the underlying token price. The lower (higher) the Capital token price, the higher (lower) the discount and the fixed yield collected at maturity.
  4. Relative value opportunities: If the fixed yield increases, more investors would find it convenient to fix yields in advance. On the other hand, if fixed yield decreases, the option to participate in future yield appreciation becomes more attractive in relative terms.

Liquidity providing and capital efficiency: how Tempus achieves it

Who is a Liquidity Provider (LP) on Tempus?

Liquidity providers (LPs) deposit Capital and Yield tokens of equal value within a certain pool. An equal number of Capital and Yield tokens are minted when yield-bearing tokens are deposited in a pool on Tempus. The proportion of Yield and Capital tokens that can be used to provide liquidity to the Tempus pools depends on the exchange rate in the pool at that moment.

LPs on Tempus earn an income from three different sources:

  • Swap fees: Every time a swap is executed, liquidity providers earn a fee in proportion to the share of liquidity they have provided to the pool.
  • Variable yield on Yield token: Since LPs provide Yield tokens to the liquidity pool, they earn variable yield on these tokens.
  • Fixed yield on Capital token: Since LPs provide Capital tokens to the liquidity pool, they earn a fixed yield on these tokens. Capital leftovers not deposited in the liquidity pool earn a fixed rate too and they will soon have other use cases. For instance, they could be used as collateral to mint stablecoins and earn an additional reward on top of the fixed yield.

To fully understand the added value that Tempus provides, it is worth illustrating a brief view of the current fixed-income market landscape.

Contrary to other fixed-income protocols in DeFi, Tempus allows Capital and Yield tokens to be traded against each other.

The way liquidity pools are structured in Tempus allows for high efficiency for both traders and liquidity providers. The former can save one transaction to fix their yields (or to speculate on future yields as well) compared to peers. The latter do not need any additional capital to provide liquidity in Tempus pools since Capital and Yield tokens trade against each other with no need for an intermediary asset. In fact, providing liquidity on Tempus requires half of the capital needed as compared to other protocols, resulting in a twofold increase in capital efficiency.

Risks associated with adding liquidity on Tempus

Liquidity providers are nonetheless exposed to the risk that the Yield / Capital exchange rate deviates significantly over time. In that case, their holding would be more concentrated on the least valuable token of the two with a resulting loss with respect to the capital initially provided. This phenomenon is known as impermanent loss.

Liquidity pools with non-maturing tokens see the exchange rate driven by market forces. However, Capital token price changes naturally over time since it converges to the value of the underlying yield-bearing token. This would make the Yield / Capital exchange rate change even if no trades occur in the pool.

Tempus provides a solution to this issue by using an AMM function with a so-called “amplification coefficient” that changes the curvature of the AMM function over time.

Amplification Coefficient: TempusAMM

Looking at the chart above, the blue curve shows how Capital and Yield tokens are traded with a high amplification coefficient. The curve looks very similar to the traditional constant-product AMM curve of the form xy=k employed by the likes of Uniswap. In this setting, the Yield / Capital exchange rate has room to fluctuate without making the pool imbalanced. There are two main reasons why this property is very important:

  • The ability for the exchange rate to fluctuate allows the market to execute price discovery in the search for the “fair” price.
  • The ability for the exchange rate to fluctuate without creating pool imbalance provides protection to liquidity providers against impermanent loss.

This setting is particularly useful at the early stage of a pool where the expected volatility is higher as more trades take place. As maturity approaches, the volatility is expected to decrease as the fixed yield that users can lock fades out. Lower volatility means less trading activity, and this implies lower fees earned by liquidity providers. To overcome this issue, the amplification coefficient increases over time.

The increase in the amplification coefficient provides “leverage” to liquidity providers because it allows traders to swap higher volumes with the same price impact. Higher volumes translate into higher fees that allow liquidity providers to maintain a good remuneration throughout the whole duration of the pool.

Because of the lower price impact, Capital and Yield tokens trade close to a 1:1 ratio regardless of their relative share in the pool. This resembles a constant-sum AMM curve of the form x+y=k employed by Curve.

Early redemption

Users may choose to withdraw funds before the maturity of pools in the form of the yield-bearing tokens or backing tokens. Just like a yield-bearing token is locked to mint an equal number of Capital and Yield tokens, the withdrawal of funds requires the merge of an equal number of Capitals and Yields tokens to give the yield-bearing tokens or backing tokens.

Investors holding a fixed income position will swap some of their Capitals for Yields from the TempusAMM. Investors holding mainly Yield tokens will swap some of these for Capital tokens until they become equal in quantity for redeeming the yield bearing token and vice versa in case they hold more Capital tokens.

One of the risks associated with early redemption is impermanent loss due to the change in the price of Capitals and Yields on the TempusAMM, especially when market conditions are volatile.

How to profit from yield fluctuations

To sum it up, we provide a visual representation of the possible scenarios that investors may face when holding a yield-bearing token and how Tempus can help them benefit from each of them.

Let’s go through the details of what each strategy entails:

  • Swap Yield for Capital tokens: This strategy allows users to lock in a fixed yield in advance to be collected at maturity. Fixing a certain return upfront comes at the cost of reducing the exposure to potentially higher variable yield. When determining the number of Yield tokens to swap for Capital tokens, users must strike a balance between the fixed return they want to lock and the variable yield income they are willing to accumulate in exchange for the fixed yield. This strategy plays well in volatile markets where variable yields will likely decrease over time.
  • Provide liquidity: This strategy allows users to earn swap fees from all trading activity taking place on TempusAMM. When yields are rising, people will trade Capital for Yield tokens and vice versa in case of falling yields. Regardless of where yields go, liquidity providers will earn fees as long as people trade. However, the more yields move over time, the higher the risk of impermanent loss for liquidity providers. This strategy is ideal when volatility is low and yields are expected to stay flat until maturity
  • Swap Capital for Yield tokens: This strategy allows users to amplify their exposures to variable yields and, in the case of an increase in interest rates, earn significantly more than the underlying yield-bearing token APR. This strategy is suited for volatile market phases with rising variable yields. Since users need to give up Capital tokens to increase their exposure to variable yields, their fixed return will decrease proportionally. They could suffer capital losses in the case of a fall in yields.

If you want to dig deeper into the nitty and gritty of these strategies, stay tuned for the next Tempus Deep Dive post!

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