Question

A 18-year annuity pays $1,300 per month, and payments are made at the end of each month. The interest rate is 12 percent compounded monthly for the first six years and 10 percent compounded monthly thereafter. What is the present value of the annuity?

Answer 1

Step-1:Present value of annuity of first 6 years
Present value of annuity for first 6 years			=	Monthly payment	*	Present value of annuity of 1
			=	$             1,300	*	51.1503915
			=	$     66,495.51
Working:
Present value of annuity of 1			=	(1-(1+i)^-n)/i		Where,
			=	(1-(1+0.01)^-72)/0.01		i	12%/12	=	0.01
			=	51.15039148		n	6*12	=	72
Step-2:Present value of annuity of after 6 years
Present value of annuity for first 6 years			=	Monthly payment	*	Present value of annuity of 1	*	Present value of 1
			=	$             1,300	*	83.6765282	*	0.488496
			=	$     53,138.35
Working:
Present value of annuity of 1			=	(1-(1+i)^-n)/i		Where,
			=	(1-(1+0.00833)^-144)/0.00833		i	10%/12	=	0.008333
			=	83.67652823		n	12*12	=	144
Present value of 1			=	(1+0.01)^-72		Where,
			=	0.488496085		i	12%/12	=	0.01
						n	6*12	=	72
Step-3:Present value of annuity of 18 years
Present value of annuity of 18 years			=	Present value of annuity of 6 years	+	Present value of annuity of 12 years
			=	$     66,495.51	+	$ 53,138.35
			=	$ 1,19,633.86
	So, present value annuity is			$ 1,19,633.86