101 Concepts for the Level I Exam

Essential Concept 62: Forward Pricing and Forward Rate Models

Forward pricing model: It is expressed as:

$\mathrm{P\ }\left({\mathrm{T}}^{\mathrm{*}}\mathrm{+\ T}\right)\mathrm{=\ P\ }\left({\mathrm{T}}^{\mathrm{*}}\right)\mathrm{\ F\ (}{\mathrm{T}}^{\mathrm{*}}\mathrm{,T)}$

If T* is 1 and T is 2, the present value of $1 to be received 3 years from today, P(3), is given by P(3) = P(1)F(1, 2).

Forward rate model: If we express the forward pricing model in terms of rates, we get the forward rate model.

$\left[\mathrm{1+r\ (}{\mathrm{T}}^{\mathrm{*}}\mathrm{+T)}\right]}^{\mathrm{(}{\mathrm{T}}^{\mathrm{*}}\mathrm{+T)}}\mathrm{=\ }{\left[\mathrm{1+r\ (}{\mathrm{T}}^{\mathrm{*}}\mathrm{)}\right]}^{{\mathrm{T}}^{\mathrm{*}}}{\left[\mathrm{1+f\ (}{\mathrm{T}}^{\mathrm{*}}\mathrm{,T)}\right]}^{\mathrm{T}}$

If T* is 1 and T is 2, then (1 + r(3))³ = (1 + r(1))¹(1 + f(1,2))²

Two interpretations of forward rates: Suppose f(1,2), the rate agreed on today for a two-year loan to be made one year from today, is 12%.

Interpretation 1: 12% is the reinvestment rate that would make an investor indifferent between buying a three-year zero-coupon bond or investing in a one-year zero-coupon bond and at maturity reinvesting the proceeds for two years. In this sense, the forward rate can be viewed as a type of breakeven interest rate.
Interpretation 2: 12% is the two-year rate that can be locked in today by buying a three-year zero-coupon bond rather than investing in a one-year zero-coupon bond and, when it matures, reinvesting the proceeds in a zero-coupon instrument that matures in two years. In this sense, the forward rate can be viewed as a rate that can be locked in by extending maturity by two years.

Implication of the forward rate model

If the spot curve is upward sloping then the forward curve lies above the spot curve.
If the spot curve is downward sloping then the forward curve lies below the spot curve.