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IAI Actuarial Core Principles · Actuarial Mathematics for Modelling · Interest rates over different time periods

Over a period of n years, the same principal is invested at simple interest rate i per annum in one account and at effective compound rate i per annum in another, with i > 0. For which values of n do the two accumulations differ in a way that simple interest gives the larger amount?

Simple interest gives the larger accumulation for periods shorter than one year (0 < n < 1). The two are equal at n = 1, and beyond one year compound interest grows faster because interest earns interest.

  1. AFor all n > 1
  2. BFor all n < 1Correct
  3. COnly when n is an integer
  4. DFor all n > 0
  5. Never, they are always equal

Explanation

Compare 1 + ni with (1+i)^n. At n = 0 and n = 1 they are equal. For 0 < n < 1, (1+i)^n < 1 + ni by concavity, so simple interest is larger. For n > 1 compound is larger. So simple interest exceeds compound only for 0 < n < 1; the option given is n < 1 (with positive n).

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