Forward rate

The forward rate (including forward rate ) denotes an interest rate that applies to a future date. The opposite of the forward rate is the spot rate , which is effective immediately for a specific term.

In general, the forward interest rate is not identical to the spot interest rate in s for a borrowing or investment up to t . In addition, the forward rate does not have to be a good estimate of this future spot rate.

Preliminary remarks

The formulas listed here for the interest calculation use the following symbols:

Spot rate (interest rate for the period from today to time t ): ${\ displaystyle r_ {t}}$
Forward interest from s to t : ${\ displaystyle r_ {s, t}}$
Discount factor at time t : ${\ displaystyle d (t)}$

Thus, the interest rate describes: the interest rate that applies to a five-year investment that begins in two years to run. The spot rate as a special case of the forward rate is noted . ${\ displaystyle \ r_ {2,7}}$ ${\ displaystyle r_ {t} = r_ {0, t}}$

Calculation from spot interest

The forward interest can be calculated clearly from the spot interest at different terms ( interest structure ). The forward interest rates are in the current yield and implicit. They are therefore also called implicit interest rates. Since a yield curve can also be displayed using its discount factors , the forward interest rates can also be calculated from the discount curves . The calculation is based on the principle of no arbitrage . The forward interest rate is generated synthetically ( duplication ). ${\ displaystyle r_ {s}}$ ${\ displaystyle r_ {t}}$

Please note that the forward rate naturally depends on the selected interest rate method and the selected day counting method .

Discrete interest

The following applies to discrete interest (specified in zero rates ):

{\ displaystyle r_ {0} (s, t) = {\ sqrt [{ts}] {\ frac {(1 + r_ {t}) ^ {t}} {(1 + r_ {s}) ^ {s }}}} - 1 {\ text {or}} r_ {0} (s, t) = {\ sqrt [{ts}] {\ frac {d (s)} {d (t)}}} - 1 }

Continuous interest

The following applies to continuous interest (specified in zero rates ):

{\ displaystyle r_ {0} (s, t) = {\ frac {r_ {t} \ cdot t-r_ {s} \ cdot s} {ts}}}

.

example

Let the following zero yield curve be given:

running time	Zero interest rate
1	2.0%
2	3.0%
3	3.7%
4th	4.2%
5	4.5%

In order to prevent arbitrage , the forward interest rate for the period [1,2] - the period of one year beginning in a year - must be so large that at the present time it does not matter whether the first year is 2.0%, invest the second year at the forward rate, or whether you pay 3.0% interest in both years.

So: so , so R (1,2) = 4.0%. ${\ displaystyle (1 \ cdot 2) + (1 \ cdot R (1 {,} 2)) = 2 \ cdot 3 {,} 0}$ ${\ displaystyle R (1 {,} 2) = 6-2 = 4}$

R (2,3) is calculated analogously:, so , so R (2,3) = 5.1%. ${\ displaystyle (2 \ cdot 3.0) + (1 \ cdot R (2 {,} 3)) = 3 \ cdot 3 {,} 7}$ ${\ displaystyle R (2,3) = 11 {,} 1-6 = 5 {,} 1}$

R (3,4):, therefore , so R (3,4) = 5.7%. ${\ displaystyle (3 \ cdot 3 {,} 7) + (1 \ cdot R (3 {,} 4)) = 4 \ cdot 4 {,} 2}$ ${\ displaystyle R (3 {,} 4) = 16 {,} 8-11 {,} 1 = 5 {,} 7}$

R (4,5):, therefore , so R (4,5) = 5.7%. ${\ displaystyle (4 \ cdot 4 {,} 2) + (1 \ cdot R (4 {,} 5)) = 5 \ cdot 4 {,} 5}$ ${\ displaystyle R (4 {,} 5) = 22 {,} 5-16 {,} 8 = 5 {,} 7}$

Overall, then

running time	Zero interest rate	Forward interest rate for one year
1	2.0%
2	3.0%	4.0%
3	3.7%	5.1%
4th	4.2%	5.7%
5	4.5%	5.7%

Relationship between zero yield curve and forward interest rates

The general formula can be transformed to . From this one sees: it holds between s and t that (rising curve - normal case), then it holds , d. H. the forward rate is greater than both zero rates. If, on the other hand, one has a falling curve, then it is also true that the forward interest rate is smaller than both zero rates. ${\ displaystyle r_ {s, t} = r_ {t} + (r_ {t} -r_ {s}) {\ frac {s} {ts}}}$ ${\ displaystyle r_ {t}> r_ {s}}$ ${\ displaystyle r_ {s, t}> r_ {t}}$ ${\ displaystyle r_ {t} <r_ {s}}$ ${\ displaystyle r_ {s, t} <r_ {t}}$