Forward rate

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.

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}

Text derived from https://en.wikipedia.org/w/index.php?oldid=1364991040 under CC BY-SA 4.0; modifications have been made.