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tags: general
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# Tokenized yield: fixed rate or future yield?

[Element Finance](https://www.element.fi/) is an Ethereum-based finance protocol which enables users to deposit a base asset, e.g. ETH or DAI, and mint two derivative assets: a principal asset and a yield asset. By buying or selling the principal token in the market, investors can lock a fixed rate or speculate on future yield. Investors lock a fixed rate by buying and holding the principal token until maturity, like a [zero-coupon bond](https://www.investopedia.com/terms/z/zero-couponbond.asp). If they speculate they could obtain more than the fixed rate, they can sell the principal token. The variable rate may quote at a premium as sellers insure buyers against interest rate uncertainty.

This post explains how to choose between the two strategies.

## Buying or selling the principal token?

Suppose you hold one unit of an interest-bearing asset X. If you expect the yield to decrease in the future, you deposit your asset in the protocol, mint $1$ pT and $1$ yT, sell the principal token, which price is $p$ and stake the proceeds. At maturity, you get
$$
(1+r)p + \text{yT} = (1+r)p + r
$$

which is profitable if
$$
(1+r)p + r > 1+r
$$

or if
$$
(1+r)p > 1
$$

Back at the time of your decision, minting and selling PT is profitable if your expectation of the yield $E(r)$ is greater than the return you get from the alterntative strategy, consisting in buying PT and waiting to be paid back with one unit of X at maturity:
$$
E(r) > \dfrac{1-p}{p}
$$

The investment rule is therefore:

1. elicit your view about future yield $E(r)$
1. if it is greater than the ratio $(1-p)/p$, mint and sell the principal token; if it is lower, buy the principal token.

In the background, what matters is how your expectation differs from the average market expectation. At market equilibrium, the price $p$ of PT is determined by the arbitrage-free condition, which says that the market should be indifferent between buying and selling PT:
$$
E_m(r) = \dfrac{1-p}{p}
$$

with $E_m(r)$ the market expectation of future yield. In other words, those who are optimistic about future yield $\big( E(r)>E_m(r) \big)$ sell the principal token and those who are pessimistic buy it.

## Compounding

The yield strategy can be compounded by repeatedly minting and selling the principal token. The $n$-loop strategy is as follows:

1. deposit 1 X, mint 1 PT and 1 YT
2. sell PT against $p$ X
3. deposit $p$ X, mint $p$ PT and $p$ YT
4. sell $p$ PT against $p^2$ X
5. ...

You expect to get at maturity:
$$
\big( 1+E(r) \big) p^n + E(r) (1+p+...+p^{n-1})
$$

With an infinite number of loops (provided $p<1$):
$$
\dfrac{E(r)}{1-p}
$$

Looping is profitable if
$$
\dfrac{E(r)}{1-p} > 1+E(r)
$$

or:
$$
E(r) > \dfrac{1-p}{p}
$$

which gives the same condition than without compounding. Note that the expectation-based investment rule presented here does not take into account the increased risk associated with leverage. Always use leverage with caution.

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That said, let's finish with a meme.


*Note: this post is the follow up of a [first post](https://hackmd.io/@pre-vert/element1) which presented the first version of Element which allowed the yield token to be tradable as the principal token.*
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## Comparison with selling the yield-bearing token at discount

Instead of splitting the base asset in two components and selling the principal token, the base asset could directly be sold at price $q$. Suppose investors lock their assets until a date forward.

You expect the strategy to be profitable if
$$
\big( 1+E(r) \big)q > 1+E(r)
$$

or if
$$
(1+r)p > 1
$$