APPENDIX A

It is shown that the parameter i of the Anton distribution equals the internal rate of return x. In other words it is shown that the root of

Image: Formula (A1)

is x = i. (A1) results from (3) and (5). (A1) can be rewritten as

Image: Formula (A2)

Denote

Image: Formula (A3)

Since (A3) is a geometric series if follows that

Image: Formula (A4)

provided that x=!0. Differentiating (A3) with respect to x gives

Image: Formula (A5)

Image: Formula (A6)

On the other hand differentiating (A4) with respect to x gives

Image: Formula (A7)

Image: Formula (A8)

Substituting (A8) into (A6), and the result and (A4) into (A2) gives after some algebra

Image: Formula (A9)

Provided that x=!0 and x=!1 it follows from (A9) after some simple, but lengthy algebra that

Image: Formula (A10)

It is easy to see from (A10) that x = i is the root of equation (A1).

Other roots of equation (A10) are not relevant. It can be proved that no other root exists when N is odd, and that the other root is less than - 1 when N is even.


Goto: The next section (Appendix B)
Goto: The previous section (References)
Goto: The contents section of Ruuhela, Salmi, Luoma and Laakkonen (1982)
Goto: Other scientific publications by Timo Salmi in WWW format

Department of Accounting and Finance, University of Vaasa,
Finland

rru@uwasa.fi
mjl@uwasa.fi
[ts@uwasa.fi] [Photo] [Programs] [Research] [Lectures] [Department] [Faculty] [University]