8. What is the domain and range of the following functions? Note that the domain may include real numbers that would not make sense in a finance application.

(a) Annuity present value: VðrÞ ¼ F Pn j¼1ð1 þ rÞj (If this is written in the equivalent form VðrÞ ¼ F 1ð1þrÞn r , the domain initially looks di¤erent. Convince yourself by numerical calculation, or analysis, that r ¼ 0 is not really a problem for this function even in the second form, since the r in the denominator ‘‘cancels’’ an r in the numerator, much like 3r=r:Þ

(b) Bond price: PðiÞ ¼ F r 2 a2n; i=2 þ Fv2n i=2

(c) Loan repayment: Pðr; nÞ ¼ Lðr=12Þ

1v12n r=12