# Par yield

Today’s market yield on a coupon paying bond trading at par and redeemable at par

= the fixed coupon rate payable on such a ‘par bond’.

## Contents

## Par yield cash flows

The following example looks at par yield cash flows.

**Example: Calculating repayments**

The par yield for the maturity 0-3 periods is 1.90% per period.

This means that a deposit of £1,000,000 at Time 0 periods on these terms would return:

- Interest at a rate of 1.90% per period on the original £1,000,000, at Times 1, 2 and 3 periods, and
- The principal of £1,000,000 at Time 3 periods

The interest payments will be £1,000,000 x 0.019 = £19,000 per period

The total repaid at Time 3 periods will be: principal £1,000,000 + £19,000 interest = £1,019,000.

An application of par yields is the pricing of new coupon paying bonds.

The par yield is known as the Par rate, Swap rate or Swap yield.

## Conversion

If we know the par yield, we can calculate both the zero coupon yield and the forward yield for the same maturities and risk class.

This is illustrated on the page Converting from par rates.

## Notation

Notation varies between practitioners and contexts.

The yield conversion pages in this wiki use the following notation:

#### Periodic par yields (p)

p_{0-1} or p_{1}: the rate per period for the maturity starting now and ending one period in the future.

p_{0-2} or p_{2}: the rate per period for the maturity starting now, and ending two periods in the future, with periodic interest - calculated at the same periodic par rate for 0-2 periods maturity - paid at the ends of both period 1 and period 2.

p_{0-3} or p_{3}: the rate per period for the maturity starting now, and ending three periods in the future, with periodic interest - calculated at the same periodic par rate for 0-3 periods maturity - paid at the ends of each of periods 1, 2 and 3.

And so on.

#### Periodic zero coupon yields (z)

z_{0-1}: the rate per period for the maturity starting now and ending one period in the future.

z_{0-2}: the rate per period for the maturity starting now, and ending two periods in the future, with all of the rolled up compounded interest paid at the end of period 2.

And so on.

It is best always to spell out expressly what cash flow pattern, maturity and quotation basis you intend, rather than assuming or hoping that others are familiar with your particular organisation's preferred notation.