Question

rob has a pension plan and will receive a pension annuity of $19,000 for 19 years starting next year. After 19 payments, the following year (t=20) she receives $10,000, which will then grow at 4% per year thereafter. His life expectancy from today is 40 years (he will receive 40 total payments). The interest rate is 8%. What are these two annuity payments worth today?

Answer 1

First annuity
	PV of annuity
	P = PMT x (((1-(1 + r) ^- n)) / r)
	Where:
	P = the present value of an annuity stream	To be computed
	PMT = the dollar amount of each annuity payment	$                                            19,000
	r = the effective interest rate (also known as the discount rate)	8%
	n = the number of periods in which payments will be made	19
	PV of annuity=	PMT x (((1-(1 + r) ^- n)) / r)
	PV of annuity=	19000* (((1-(1 + 8%) ^- 19)) / 8%)
	PV of annuity=	$                                    182,468.38
2nd annuity
	PV of annuity for growing annuity
	P = (PMT/(r-g)) x (1-((1+g)/(1 + r)) ^n)
	Where:
	P = the present value of an annuity stream	To be computed
	PMT = the dollar amount of each annuity payment	$                                            10,000
	r = the effective interest rate (also known as the discount rate)	8%
	n = the number of periods in which payments will be made	21	(40-19)
	g= Growth rate	4%
	PV of 2nd annuity at T 19=	(PMT/(r-g)) x (1-((1+g)/(1 + r)) ^n)
	PV of 2nd annuity at T 19=	(10000/(8%-4%)) * (1-((1+4%)/(1 + 8%)) ^21)
	PV of 2nd annuity at T 19=	$                                    136,827.41
	PV of this annuity at T0=	136827.41/(1+8%)^19
	PV of this annuity at T0=	$                                       31,704.56
Final answer
	PV of 1st annuity=	$                                    182,468.38
	PV of 2nd annuity=	$                                       31,704.56
	Total Present value	$                                    214,172.95