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IAI Actuarial Core Principles · Economic Modelling · Models of the term structure of interest rates

In a continuous-time term structure model, the instantaneous forward rate f(0,t) is defined from the zero-coupon bond price P(0,t). Which expression correctly gives f(0,t)?

The instantaneous forward rate equals minus the derivative of the log of the zero-coupon bond price with respect to maturity. The expression -ln P(0,t)/t is only the continuously compounded spot rate, which averages the forward rates over the period.

  1. Af(0,t) = -d/dt ln P(0,t)Correct
  2. Bf(0,t) = d/dt P(0,t)
  3. Cf(0,t) = -ln P(0,t) / t
  4. Df(0,t) = 1/P(0,t) - 1
  5. f(0,t) = ln P(0,t) / t

Explanation

The forward rate is the negative of the log-derivative of the discount function: f(0,t) = -d ln P(0,t)/dt. Option C (-ln P/t) is the spot rate (average of forwards), not the instantaneous forward rate.

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