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Unformatted text preview: Chapter 3. Annuities (Certain) ACTSC231 — Mathematics of Finance Department of Statistics and Actuarial Science University of Waterloo Fall 2010 Instructor: Chengguo Weng C. Weng ([email protected]) – p. 1/3 5 Definition of general annuities (p109110) • An annuity is a regular series of payments XBox e.g. loan payments (Chapter 5) XBox e.g. pension, etc. • An annuitycertain is one annuity that is not contingence on any other factors XBox It is the objective in this course. We might simply say “annuity" to mean the “annuitycertain". XBox A life annuity depends on the survival of a life; see ACTSC331(life contingencies) • Annuity types: Due versus immediate XBox An annuity with payments at the start of each time unit is an annuity due ; paid in advance XBox An annuity with payments at the end of each time unit is an annuity immediate ; paid in arrear Note: An annuityimmediate does not make the first payment immediately. C. Weng ([email protected]) – p. 2/3 5 Annuities: level versus nonlevel (p110) • Level annuities: the amount of payments on each time unit is the same. • nonlevel annuities • Payment period is the interval between two consecutive payments. • Example 3.1: A loan of $10,000 is repaid by monthly payments of $600 over 2 years, payable in advance. i) What is the annual effective rate on the transaction? ii) Does it exceed 35%? C. Weng ([email protected]) – p. 3/3 5 Present value of level annuities • (p111) Consider an annuity of $1 per year payable in advance for n years. The present value is denoted by ä n ⌉ and ä n ⌉ = 1 + v + v 2 + ··· + v n − 1 = 1 − v n 1 − v = 1 − v n d . XBox So the present value of $ P per year payable in advance for n years is P · ä n ⌉ . • (p123) Consider an annuity of $1 per year payable in arrear for n years. The present value is denoted by a n ⌉ and a n ⌉ = v + v 2 + v 3 ··· + v n = v (1 − v n ) 1 − v = 1 − v n (1 − v ) /v = 1 − v n i . XBox So the present value of $ P per year payable in arrear for n years is P · a n ⌉ . C. Weng ([email protected]) – p. 4/3 5 Accumulated value of level annuities • Accumulated value (time n value) of a level annuity of $1 per year: XBox (p113) annuitydue: ¨s n ⌉ = (1 + i ) n + (1 + i ) n − 1 + ··· + (1 + i ) 2 + (1 + i ) = (1 + i ) [(1 + i ) n − 1] (1 + i ) − 1 = (1 + i ) n − 1 i/ (1 + i ) = (1 + i ) n − 1 d . XBox (p124) annuityimmediate: s n ⌉ = (1 + i ) n − 1 + (1 + i ) n − 2 + ··· + (1 + i ) + 1 = 1 · [(1 + i ) n − 1] (1 + i ) − 1 = (1 + i ) n − 1 i . C. Weng ([email protected]) – p. 5/3 5 Some facts • Recall: ä n ⌉ = 1 − v n d , a n ⌉ = 1 − v n i ¨s n ⌉ = (1 + i ) n − 1 d , s n ⌉ = (1 + i ) n − 1 i • Some facts: ä n ⌉ = (1 + i ) · a n ⌉ ⇐⇒ a n ⌉ = v ä n ⌉ ä n ⌉ = 1 + a n − 1 ⌉ ¨s n ⌉ = (1 + i ) · s n ⌉ ⇐⇒ s n ⌉ = v · ¨s n ⌉ s n ⌉ = 1 + ¨s n − 1 ⌉ ¨s n ⌉ = (1 + i ) n · ä n ⌉...
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This note was uploaded on 01/21/2011 for the course ACTSC 231 taught by Professor Chisholm during the Fall '09 term at Waterloo.
 Fall '09
 Chisholm

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