# Metrics of The Forward Rate

Under the forward rate lies the interest rate, whose term does not begin immediately, but at some point in the future and also matures at a given stage. It is the rate for a budget system or borrowing on the grounds of the time agreed denoted as period s ts (interest rate s to t months).

The forward rate can be calculated from the spot rates at different maturities, and the interest rates are included in the current term structure. They are therefore called implicit interest rates, and the basis of the calculation is the principle of arbitrage.

The listing of the forward interest rate is usually as follows: \ R_ (s, t) or \ r (s, t), where S = start time or duration,  T = maturity. In principle, the forward rate represents the future yield on a bond, and is calculated by applying the yield curve.

For example,  \ R_ (2.7), the rate in force for a five-year investment that begins to run in two years. The spot interest rate is cited as a exceptional case of the forward rate: \ R_ (0.5). This means the interest rate that applies to a five-year investment with an immediate start date.

If RS is the zero-rate for a zero-duration bond to S, so the interest rate of zero coupon bonds, is according to RT, the zero-rate for a zero bond until maturity T, and R (S, T) of the desired current forward rate in S to T, then R (S, T) = \ frac (R_T \ times T-R_S \ S) (TS) cdot.

RF depends on the chosen interest rate method and the selected count convention. Simple forward rates can be calculated with continuous compounding.

To derive the forward rate, the term structure of interest rates is required, and the the elementary formula employed for computing the forward rate is: r_{t_1,t_2} = \left( \left(\frac{1+r_2 d_2}{1+r_1 d_1} \right) -1\right)\left( \frac{1}{d_2-d_1} \right).

Where r1,2 is the forward rate between term t1 and term t2, while d1 is the time length between time 0 and term t1 (in years), d2 is the time length between time 0 and term t2 (in years), r1 is the interest rate for the period time 0 to term t1, and r2 is the interest rate for the period time 0 to term t2.

In the case of a realization of the forward interest rates in period s, considered as spot interest rates: in period t the interest rate r0 (s, t) as the spot rate. In general, the appointment rate is not identical with the spot interest rate in s up for a borrowing or investment t.

Empirically r0 (s, t) is usually not a good estimator for this future spot rate. The single period forward rate r0 (0.1) is equal to the cash interest rate for one period.

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