---
# System prepended metadata

title: Yield Coupon Compounding Formulation

---

# Yield Coupon Compounding Formulation

The Element Liquidity Pool bootstraps the liquidity of newly issued yield tokens by rolling over funds into each newly minted tranche.  To calculate the value of these yield tokens on any particular day, we can rearrange the terms from annualized yield formula presented in the [yield paper](https://yield.is/Yield.pdf) to solve for present value:

  \begin{equation}
P=\frac{F}{(Y+1)^{T}}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\;\;\;(1)
  \end{equation}
  
The next three sections will describe how we can determine the present value of the Fixed Yield Token (FYT), Yield Coupon (YC) and the total redeemable value.


### FYT Present Value

Given an interest rate of 3%, weekly rollovers, and 1 year maturity, the present value of a FYT can be calculated as follows:

\begin{equation}
Day\:0=\frac{1}{\left(1.03\right)^{1-\frac{0}{365}}}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\end{equation}

\begin{equation}
Day\:1=\frac{1}{\left(1.03\right)^{1-\frac{1}{365}}}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\end{equation}

\begin{equation}
. \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\\
. \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\\
. \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\\
\end{equation}

\begin{equation}
Day\:6=\frac{1}{\left(1.03\right)^{1-\frac{6}{365}}}\cdot\frac{1}{\left(1.03\right)^{1-\frac{0}{365}}}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\end{equation}


More generally, we can say that the present value of the fixed yield token on day $d$ with interest rate $Y_{f}$ rolled over every $l$ days into a new tranche with maturity day $m$ can be calculated with the following product:

  \begin{equation}
\prod_{t=1}^{\lfloor{\frac{d}{l}}\rfloor+1}\frac{1}{(Y_{f}+1)^{\frac{m-d\bmod l}{365}}}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\;\;\;\;(2)
  \end{equation}
  
and rewritten as the following equation:

  \begin{equation}
  Z\left(d\right)=\frac{1}{\left(Y_{f}+1\right)^{\frac{m- d\bmod l}{365}}}^{\lfloor{\frac{d}{l}}\rfloor+1}\qquad\qquad\qquad\qquad\qquad\qquad\qquad (3)
  \end{equation}

The following plot shows us the present value of FYTs at a fixed interest rate and weekly rollovers:

![](https://i.imgur.com/hak0z6A.png)
> Each “stair step” represents the present value of the remaining FYT balance.  Each time the current week’s FYTs are sold and rolled over into the new week’s FYT, the graph takes a “step down” due to the reduced purchasing power.

### YC Present Value

Given a FYT interest rate of 3%, YC interest rate of 5%, weekly rollovers, and 1 year maturity, the present value of a YC can be calculated as follows:

\begin{equation}
Day\:0=\frac{.05}{(1.05)^{1-\frac{0}{365}}}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\end{equation}
\begin{equation}
Day\:1=\frac{.05}{(1.05)^{1-\frac{1}{365}}}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\end{equation}
\begin{equation}
. \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\\
. \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\\
. \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\\
\end{equation}

\begin{equation}
Day\:6=\frac{.05}{(1.05)^{1-\frac{6}{365}}}+\frac{1}{\left(1.03\right)^{1-\frac{6}{365}}}\left(\frac{.05}{(1.05)^{1-\frac{0}{365}}}\right)\qquad\qquad\qquad\qquad
\end{equation}


Again, this can be represented more generally as follows. The present value of the interest coupon on day $d$ with interest rate $Y_{c}$ rolled over every $l$ days into a new tranche with maturity day $m$ can be calculated with the following sum:

  \begin{equation}
\frac{Y_{c}}{\left(Y_{c}+1\right)}\sum_{j=0}^{\lfloor{\frac{d}{l}}\rfloor}Z\left(j\cdot l-1\right)(Y_{c}+1)^{\frac{d-\left(j\cdot l-1\right)}{365}}\qquad\qquad\qquad\qquad\qquad\qquad(4)
  \end{equation}

This plot shows us the present value of YCs at any point in time given a fixed FYT interest rate, various YC interest rates, and weekly rollovers:
![](https://i.imgur.com/9EL7cHt.png)
>Present value is shown for 3 different YC interest rates.  Notice how the stair step is in the positive direction due to the additional YCs received each time the current week’s FYTs are rolled over into the new week’s FYT.  
 
### Total Redeemable Value

The total redeemable value on day $d$ is just the sum of equations *2* & *4*.

  \begin{equation}
\prod_{t=1}^{\lfloor{\frac{d}{l}}\rfloor+1}\frac{1}{(Y_{f}+1)^{\frac{m-d\bmod l}{365}}}+\frac{Y_{c}}{\left(Y_{c}+1\right)}\sum_{j=0}^{\lfloor{\frac{d}{l}}\rfloor}Z\left(j\cdot l-1\right)(Y_{c}+1)^{\frac{d-\left(j\cdot l-1\right)}{365}}\qquad(5)
  \end{equation}

The following plot shows us the total redeemable value of the yield at any point in time given various interest rates and weekly rollovers:

![](https://i.imgur.com/hbiszkP.png)
> The plot illustrates how weekly rollovers are profitable for all 3 YC interest rates. 





