Forward Curve

Also known as Forward Rate Curve, forward yield curve, implied forward rates, nominal forward rates

A forward rate is the implied interest rate that would be used for an investment in the future.
Implies the interest rate that would be used on a date in the future

rate between two future dates

Theoretical Curve

These are "implied" forward rates.
The forward yield is the interest rate implied by a zero coupon rate.
Forward rates are a type of market view on where interest rates will be (or should be) in the future
Forward rates are the markets expectation of future rates.
Forward rates are not a prediction of future rates.

The forward yield curve is a plot of forward rates against maturity.
The forward yield curve is the interest rate implied by the zero coupon rates for period of time in the future.

These are used to work out the cash flows in an interest rate swap

If the spot curve is upward sloping the forward curve is above the spot curve
If the spot curve is downward sloping the forward curve is below the spot curve

Assumptions

1) Markets should be arbitrage free (ie you cant make money unless there is a risk involved)
2) A 1 year investment in treasury bills should produce the same return as 2 consecutive 6-month investments in treasury bills.
The 6-month rate you would get four 6-months periods from now is know as the 2 year forward rate denoted:

notation = 4f1

The forward (or forward-forward) yield curve is a plot of forward rates against term to maturity.

Relationship between Spot and Forward

There is a mathematical equivalence between spot rates and forward rates.
We can see that the spot yield is the geometric mean of the forward rates.

Calculate from Spot Rates

Supposing that a bank assesses and quotes the following rates to a company, based on the annual spot yield curve for that company's risk class:

One-year: 3.50%
Two-year: 4.60%
Three-year: 5.40%
Four-year: 6.10%
Five-year: 6.30%

This indicates that the company would have to: pay interest at 3.50% if it wants to borrow a sum of money for one year; pay interest at 4.60% per year if it wants to borrow a sum of money for two years; pay interest at 5.40% per year if it wants borrow a sum of money for three years; and so on.

Alternatively, for a two-year loan, the company could opt to borrow a sum of money for only one year, at an interest rate of 3.50%, and then again for another year, commencing in one year's time, instead of borrowing the money for a total of two years.

Although the company would be uncertain about the interest rate in one year's time, it could request a forward rate from the bank that is fixed today - for example, through a 12v24 forward rate agreement (FRA). The question then arises: how may the value of the 12v24 FRA be determined?

A forward rate commencing in one year for a borrowed sum lasting a year can be calculated as follows:

In summary:

Supposing the company wants to borrow a sum of money for three years on the basis of the above rates:
i. it could pay annual interest at a rate of 5.40% in each of the three years, or
ii. it could pay interest at a rate 3.50% in the first year, 5.71% in the second year and 7.02% in the third year, or
iii. it could pay annual interest at a rate of 4.60% in each of the first two years and 7.02% in the third year.