Forward rate

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Not to be confused with forward price or forward exchange 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[edit]

To extract the forward rate, one needs the zero-coupon yield curve. The general formula used to calculate the forward rate is:

Simple rate[edit]

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

Compound rate[edit]

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

Exponential rate[edit]

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) ,

Derivation[edit]

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:

(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 the above formula.

Related instruments[edit]

A forward discount is when the forward rate of one currency relative to another currency is higher than the spot rate.

A forward premium is when the forward rate of one currency relative to another currency is lower than the spot rate.

See also[edit]

References[edit]

  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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