# Forward yield

The fixed interest rate in the market today for an investment or borrowing commitment:

- Starting at a fixed future date; and
- Ending on a later fixed future date.

The commitment can relate to a physical deposit or borrowing, or - more commonly - a derivative contract to be settled by reference to a notional deposit or borrowing.

For example a forward rate might be quoted for a forward rate agreement for the maturity 2-5 months in the future.

### Example

Taking a simpler example, say the forward rate (or forward yield) for the maturity 2-3 periods is 3% per period.

2 is the time from today (into the future) when the investment or borrowing will start.

3 is the time from today when the investment or borrowing will end.

The difference between Time 3 periods and Time 2 periods is the length of the investment or borrowing.

In this case the length of the investment or borrowing is 3 - 2 = 1 period.

Assuming a deposit, 3% is the rate payable for period 3 only - a single period - which is pre-agreed today, 2 periods before the deposit is contracted to change hands.

This means a mutually binding agreement can be made today, for a deposit of £1,000,000 to be made at Time 2 periods into the future, which will return:

£1,000,000 x 1.03

= £1,030,000 at Time 3 (periods into the future).

### Applications

A common application of forward yields is the pricing of forward rate agreements.

The forward yield is also known as the Forward rate or (sometimes) the Forward forward rate.

(The forward forward rate is technically slightly different, strictly referring to physical borrowings or deposits, rather than to derivative contracts.)

### Quotation basis

Rates are generally quoted in wholesale markets as nominal annual rates.

### Conversion

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

The conversion process and calculation stems from the 'no-arbitrage' relationship between the related yield curves.

This is illustrated on the page Converting from forward rates.

### Notation

Notation varies between practitioners and contexts.

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

*Periodic forward yields ( f):*

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

f_{1-2}: the rate per period for the maturity starting one period in the future, and ending two periods in the future.

f_{2-3}: the rate per period for the maturity starting two periods in the future, and ending three periods in the future.

f_{1-3}: the rate per period for the maturity starting one period in the future, and ending three periods in the future.

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.