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.^[1]

Forward rate calculation

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

We are trying to find the future interest rate for time period $(t_1, t_2)$, 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 solve for the interest rate $r_{t_1,t_2}$ for time period $(t_1, t_2)$ for which the proceeds from investing at rate $r_1$ for time period $(0, t_1)$ and then reinvesting those proceeds at rate $r_{t_1,t_2}$ for time period $(t_1, t_2)$ is equal to the proceeds from investing at rate $r_2$ for time period $(0, t_2)$. Or, mathematically:

Simple rate

$(1+r_1d_1)(1+r_{t_1,t_2}(d_2-d_1)) = 1+r_2d_2$

Solving for $r_{t_1,t_2}$ yields: $r_{t_1,t_2} = \frac{1}{d_2-d_1}\left(\frac{1+r_2d_2}{1+r_1d_1}-1\right)$

Yearly compounded rate

$(1+r_1)^{d_1}(1+r_{t_1,t_2})^{d_2-d_1} = (1+r_2)^{d_2}$

Solving for $r_{t_1,t_2}$ yields : $r_{t_1,t_2} = \left(\frac{(1+r_2)^{d_2}}{(1+r_1)^{d_1}}\right)^{\frac{1}{d_2-d_1}} - 1$

Also, the discount factor formula for period t and rate $r_t$ being: $DF(t)=\frac{1}{(1+r_t)^{t}}$, the forward rate can be expressed in terms of discount factors: $r_{t_1,t_2}=\frac{DF(t_1)}{DF(t_2)}-1$

Continuously compounded rate

$e^{r_1d_1}e^{r_{t_1,t_2}(d_2-d_1)} = e^{r_2d_2}$

Solving for $r_{t_1,t_2}$ yields : $r_{t_1,t_2} = \frac{r_2d_2-r_1d_1}{d_2-d_1}$

$r_{t_1,t_2}$ is the forward rate between term $t_1$ and term $t_2$,

$d_1$ is the time length between time 0 and term $t_1$ (in years),

$d_2$ is the time length between time 0 and term $t_2$ (in years),

$r_1$ is the zero-coupon yield for the time period $(0, t_1)$,

$r_2$ is the zero-coupon yield for the time period $(0, t_2)$,

Related instruments

• Forward rate agreement
• Floating rate note

See also

• Forward price

References

1. Fabozzi, Vamsi.K (2012), The Handbook of Fixed Income Securities (Seventh ed.), New York: kvrv, p. 148, ISBN 0-07-144099-2.

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