# Yield To Maturity

## Contents

## Definition

Let $\xi = \left(z_{T_{1}},z_{T_{2}},...,z_{T_{N}} \right)$ be a stream of cash flows and $T = \left(T_{1}, T_{2}, ..., T_{N}\left)$ the instances of time when these payments occur. Let $p(t;T,\xi)$ be the payment required to obtain the stream of cash flows at time $t < T_{i}, i=1,...,N$. The yield to maturity is the single interest rate used for discounting such that the discounted cash flows are equal to the initial pay-out. The level of the yield to maturity depends on the compounding method.

## Compounding

#### Simple compounding

$\displaystyle{ p(t;T,\xi}) \equiv \sum_{i=1}^{N} \frac{z_{T{i}}}{(1+(T_{i} - t))y_{s}} }$

#### Periodic compounding

$\displaystyle{ p(t;T,\xi}) \equiv \sum_{i=1}^{N} \frac{z_{T{i}}}{\left(1+y_{1}\right)^{T_{i} - t}} }$

#### Continuous compounding

$\displaystyle{ p(t;T,\xi}) \equiv \sum_{i=1}^{N} z_{T{i}}\exp(-y_{c}(T_{i} - t)) }$