Question

The use of compound interest for completed periods and simple interest for a final fractional period can be analyzed as follows. Consider the investment of 1 for n + k periods, where n is a non-negative integer and 0 < k < 1. Then, the use of interpolation between (1 + i)ⁿ and (1 + i)ⁿ⁺¹, where n is a non-negative integer. To see this, we start with the linear interpolation

(1 + i)^n+k = (1 − k)(1 + i)ⁿ + k(1 + i)ⁿ⁺¹

= (1 + i)ⁿ [(1 − k) + k(1 + i)]

= (1 + i)ⁿ (1 + ki), which is the formula if simple interest is used over the final fractional period. This technique will produce a slightly higher accumulated value than using compound interest throughout. Derive a formula analogous to formula (1) for computing present values using compound discount for completed periods and simple discount for a final fractional period.

Answer 1