Taylor Series Approximations (Solved Example)

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1. Context

In this solved example, we explore how Taylor Series approximations work and how they are applied to the world of finance. We do not do a rigorous and formal proof of Taylor Series expansions (approximations). Rather, we follow a step-wise solved example that helps us rationalize how these approximations (and their functional form) are arrived at, so that we do not have to memorize anything.

2. Solved Example

A Zero Coupon Bond (ZCB) has the foll. features:

Face Value\$100
Yield (Continuously Compounded)5%
Maturity30 years

If the fair value of this ZCB at any time $t: 0 \leq t \leq 30$ is given by $f(t)$, based on the above:

  1. Calculate $𝑓(16), 𝑓’ (16), 𝑓” (16)$.
  2. A linear approximation to $𝑓(16.5)$.
  3. A quadratic approximation to $𝑓(16.5)$.

3. Video