Quick questions on Financial Mathematics - TCE Mathematics Applications (Tasmania)

A recurrence relation defines each term from the previous one. This is the natural language of a spreadsheet or financial calculator.

To compare investments with different compounding frequencies, convert the nominal rate to an effective annual rate: $$r_{\text{eff}} = \left(1 + \frac{r_{\text{nom}}}{k}\right)^k - 1$$where$k$is the number of compounds per year. For the$6\%$monthly example,$r_{\text{eff}} = (1.005)^{12} - 1 = 0.061678$, or about$6.17\%$. This is why monthly compounding beats annual at the same nominal rate.

TASC problems often ask you to compare straight-line (flat-rate) and reducing-balance depreciation, or to choose a unit-cost model. Straight-line subtracts a fixed dollar amount each year, so the book value falls along a straight line and can in principle reach zero. Reducing-balance multiplies by a fixed factor less than one each year, so it falls quickly at first then flattens and never quite reaches zero. Unit-cost depreciation links the loss to usage: the rate per unit is $R = \dfrac{\text{cost} - \text{scrap value}}{\text{total units of life}}$, and the book value after $u$ units is $V = \text{cost} - Ru$.