Question

You take out a 40 year loan with effective annual interest i = 5% and yearly payments at the end of each year being $10,000 for the first 20 years, and then $15,000 for the next 20 years. Find the outstanding balance after the fourth payment. Also find the outstanding balance after the 37th payment.

Answer 1

a.	Outstanding balance after 4th payment	$ 1,94,013.93
Working:
	Outstanding balance is the present value of remianing future payments.
	Outstanding balance after 4th payment	=	Present value of 10,000 for first 16 years	+	Present value of 10,000 for next 20 years
		=	$                                                         1,08,377.70	+	$                                                                85,636.23
		=	$                                                         1,94,013.93
	Present Value of annuity of 1 for 16 years	=	(1-(1+i)^-n)/i		Where,
		=	(1-(1+0.05)^-16)/0.05		i	=	5%
		=	10.83776956		n	=	16
	Present Value of 1 for 16 years	=	(1+i)^-n
		=	(1+0.05)^-16
		=	0.458111522
	Present Value of annuity of 1 for 20 years	=	(1-(1+i)^-n)/i		Where,
		=	(1-(1+0.05)^-20)/0.05		i	=	5%
		=	12.46221034		n	=	20
	Present value of 10,000 for first 16 years	=	Annual payment for 16 years	*	Present Value of annuity of 1 for 16 years
		=	$                                                             10,000.00	*	10.83776956
		=	$                                                         1,08,377.70
	Present value of 10,000 for next 20 years	=	Annual payment for 16 years	*	Present Value of annuity of 1 for 20 years	*	Present Value of 1 for 16 years
		=	$                                                             15,000.00	*	12.46221034	*	0.458111522
		=	$                                                             85,636.23
b.	Outstanding balance after 37th payment		$ 40,848.72
Working:
	Outstanding balance is the present value of remianing future payments.
	Outstanding balance after 37th payment	=	Present value of 15,000 for 3 years
		=	$                                                             15,000.00	*	2.72325
		=	$                                                             40,848.72
	Present Value of annuity of 1 for 3 years	=	(1-(1+i)^-n)/i		Where,
		=	(1-(1+0.05)^-3)/0.05		i	=	5%
		=	2.723248029		n	=	3