> ## Documentation Index
> Fetch the complete documentation index at: https://docs.exponent.finance/llms.txt
> Use this file to discover all available pages before exploring further.

# Pricing & Math

> How PT and YT prices are derived from implied APY, and the formulas behind the protocol

This page covers the formulas that drive PT/YT pricing across Exponent. Developers don't need to implement pricing themselves — the SDK handles all calculations — but understanding the relationships helps when evaluating positions.

## The Core Invariant

The value of one [PT](/home/learn/emitted-tokens#pt-principal-token) plus one [YT](/developers/learn/emitted-tokens#yt-yield-token) always equals the value of one [SY](/developers/learn/emitted-tokens#sy-standardized-yield):

$$
P_{PT} + P_{YT} = P_{SY}
$$

This is the foundation of the entire system. If market prices diverge from this relationship, arbitrage restores the balance.

## PT Price & Implied APY

[PT](/developers/learn/emitted-tokens#pt-principal-token) trades at a discount to face value. The size of that discount is your fixed return — and the market expresses it as an **implied APY** using continuous compounding:

$$
P_{PT} = e^{-\text{APY} \times t}
$$

Where $t$ is time to maturity as a fraction of a year. Higher implied APY means a bigger discount on PT — and a higher fixed return for the buyer.

| Implied APY | 180 days to maturity | 90 days | 30 days |
| :---------: | :------------------: | :-----: | :-----: |
|      5%     |        0.9753        |  0.9876 |  0.9959 |
|     10%     |        0.9512        |  0.9753 |  0.9918 |
|     20%     |        0.9048        |  0.9512 |  0.9836 |

This works in both directions — given a PT price from the market, you can derive the implied APY: $\text{APY} = -\ln(P_{PT}) / t$.

A quick way to estimate your annualized return without the exponential math: $\frac{1 - P_{PT}}{P_{PT}} \times \frac{365}{d}$. For moderate rates and short durations both methods give nearly the same number — they only diverge meaningfully at high APYs or long maturities.

When PT is priced against SY rather than the base asset, divide by the [exchange rate](/developers/learn/emitted-tokens#the-exchange-rate): $P_{PT}^{SY} = P_{PT} / r$. Because SY appreciates over time, PT always appears cheaper in SY terms.

## YT Pricing & Leverage

[YT](/developers/learn/emitted-tokens#yt-yield-token) price is the complement of PT, adjusted by the [exchange rate](/developers/learn/emitted-tokens#the-exchange-rate):

$$
P_{YT} = \frac{1 - P_{PT}}{r}
$$

Because YT costs a fraction of the full asset, it creates inherent leverage on yield. The leverage multiplier is simply $1 / P_{YT}$:

| PT Price | YT Price (rate 1.05) | Yield Leverage |
| :------: | :------------------: | :------------: |
|   0.90   |        0.0952        |      \~10x     |
|   0.95   |        0.0476        |      \~21x     |
|   0.98   |        0.0190        |      \~53x     |

As maturity approaches, PT converges to 1.00 and YT converges to 0 — this is **time decay**. YT loses value over time even if the underlying rate hasn't changed, because less time remains to earn yield.

## How Yield Accrues

YT holders earn yield based on the growth of the [exchange rate](/developers/learn/emitted-tokens#the-exchange-rate) since their position was created. The larger the rate increase, the more SY becomes claimable. Yield accrues continuously and can be collected at any time — you don't have to wait for maturity.

The earned SY for a YT position is calculated from the change in the inverse of the SY exchange rate:

$$
\text{earned\_SY} = \text{YT\_balance} \times \left(\frac{1}{r_{\text{last}}} - \frac{1}{r_{\text{current}}}\right)
$$

For example, if a user holds 2,000 YT and the exchange rate moves from 2.0 to 6.0:

```
earned_SY = 2000 × (1/2.0 − 1/6.0) = 2000 × (0.5 − 0.1667) = 2000 × 0.3333 ≈ 666 SY
```

After a vault matures, the exchange rate is frozen. Any further appreciation in the underlying protocol does not affect YT yield.

## Fee Decay Model

Both the CLMM and Orderbook use a time-weighted fee model. Fees are highest far from maturity and decay toward zero as the vault approaches expiration:

$$
\text{fee\_rate} = e^{\text{ln\_fee\_rate\_root} \times \tau}
$$

Where $\tau$ is the normalized time remaining (0 at maturity, 1 at one year out). This reflects the decreasing price uncertainty of PT as it nears redemption — closer to maturity means less risk, so lower fees. The protocol also takes a treasury cut from each trade.