Forward rate

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title: "Forward rate" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["financial-economics", "swaps-(finance)", "fixed-income-analysis", "interest-rates"] description: "Future yield on a bond" topic_path: "economics" source: "https://en.wikipedia.org/wiki/Forward_rate" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Future yield on a bond ::

The forward rate is the future yield on a bond. It is calculated using the yield curve. For example, the yield on a three-month Treasury bill six months from now is a forward rate.

Forward rate calculation

To extract the forward rate, we need the zero-coupon yield curve.

We are trying to find the future interest rate r_{1,2} for time period (t_1, t_2), t_1 and t_2 expressed in years, given the rate r_1 for time period (0, t_1) and rate r_2 for time period (0, t_2). To do this, we use the property, following from the arbitrage-free pricing of bonds, that the proceeds from investing at rate r_1 for time period (0, t_1) and then reinvesting those proceeds at rate r_{1,2} for time period (t_1, t_2) is equal to the proceeds from investing at rate r_2 for time period (0, t_2).

r_{1,2} depends on the rate calculation mode (simple, yearly compounded or continuously compounded), which yields three different results.

Mathematically it reads as follows:

Simple rate

: (1+r_1t_1)(1+ r_{1,2}(t_2-t_1)) = 1+r_2t_2

Solving for r_{1,2} yields:

Thus r_{1,2} = \frac{1}{t_2-t_1}\left(\frac{1+r_2t_2}{1+r_1t_1}-1\right)

The discount factor formula for period (0, t) \Delta_t expressed in years, and rate r_t for this period being DF(0, t)=\frac{1}{(1+r_t , \Delta_t)}, the forward rate can be expressed in terms of discount factors: r_{1,2} = \frac{1}{t_2-t_1}\left(\frac{DF(0, t_1)}{DF(0, t_2)}-1\right)

Yearly compounded rate

: (1+r_1)^{t_1}(1+r_{1,2})^{t_2-t_1} = (1+r_2)^{t_2}

Solving for r_{1,2} yields :

: r_{1,2} = \left(\frac{(1+r_2)^{t_2}}{(1+r_1)^{t_1}}\right)^{1/(t_2-t_1)} - 1

The discount factor formula for period (0,t) \Delta_t expressed in years, and rate r_t for this period being DF(0, t)=\frac{1}{(1+r_t)^{\Delta_t}}, the forward rate can be expressed in terms of discount factors:

: r_{1,2}=\left(\frac{DF(0, t_1)}{DF(0, t_2)}\right)^{1/(t_2-t_1)}-1

Continuously compounded rate

:e^{r_2 \cdot t_2} = e^{r_1 \cdot t_1} \cdot \ e^{r_{1,2} \cdot \left(t_2 - t_1 \right)}

Solving for r_{1,2} yields:

:STEP 1→ e^{r_2 \cdot t_2} = e^{r_1 \cdot t_1 + r_{1,2} \cdot \left(t_2 - t_1 \right)}

:STEP 2→ \ln \left(e^{r_2 \cdot t_2} \right) = \ln \left(e^{r_1 \cdot t_1 + r_{1,2} \cdot \left(t_2 - t_1 \right)}\right)

:STEP 3→ r_2 \cdot t_2 = r_1 \cdot t_1 + r_{1,2} \cdot \left(t_2 - t_1 \right)

:STEP 4→ r_{1,2} \cdot \left(t_2 - t_1 \right) = r_2 \cdot t_2 - r_1 \cdot t_1

:STEP 5→ r_{1,2} = \frac{ r_2 \cdot t_2 - r_1 \cdot t_1}{t_2 - t_1}

The discount factor formula for period (0,t) \Delta_t expressed in years, and rate r_t for this period being DF(0, t)=e^{-r_t,\Delta_t}, the forward rate can be expressed in terms of discount factors:

: r_{1,2} = \frac{\ln \left(DF \left(0, t_1 \right)\right) - \ln \left(DF \left(0, t_2 \right)\right)}{t_2 - t_1} = \frac{- \ln \left( \frac{ DF \left(0, t_2 \right)}{ DF \left(0, t_1 \right)} \right)}{t_2 - t_1}

r_{1,2} is the forward rate between time t_1 and time t_2 ,

r_k is the zero-coupon yield for the time period (0, t_k) , (k = 1,2).

Related instruments

• Forward rate agreement
• Floating rate note

References

References

1. Fabozzi, Vamsi.K. (2012). "The Handbook of Fixed Income Securities". kvrv.

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::