Question 12.1: The present value of a payment of £1 due in 8 years’ time is......
The present value of a payment of £1 due in 8 years’ time is given by the formula
PV = \frac{1}{(1 + i)^8}
where i is the given interest rate. What is the rate of change of PV with respect to i?
Step-by-Step
If we let
z = 1 + i (1)
then we can write
PV =\frac{1}{z^8} = z^{-8} (2)
Differentiating (1) and (2) gives
\frac{dz}{d i} = 1 \frac{dPV}{dz} = -8z^{-9}
Therefore, using the chain rule, the rate of change of PV with respect to i will be
\frac{dPV}{d i} = \frac{dPV}{d z}\frac{dz}{d i} = -8z^{-9} = \frac{-8}{(1 + i)^{9}}
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