Question

A 20 year loan of $50, 000 is taken out at effective annual interest i = 6% for the first 10 years and then i = 7% for the next 10 years. Payments are constant at the end of each year. Find the outstanding balance after the 16th payment.

Answer 1

Step-1:Calculation of annual payment
Annual payment	=	Loan amount	/	Cumulative discount factor
	=	$ 50,000.00	/	11.2820183
	=	$ 4,431.83
Working:
Present Value of annuity of 1 for 10 years	=	(1-(1+i)^-n)/i		Where,
	=	(1-(1+0.06)^-10)/0.06		i	=	6%
	=	7.360087051		n	=	10
Present Value of annuity of 1 for 10 years	=	(1-(1+i)^-n)/i		Where,
	=	(1-(1+0.07)^-10)/0.07		i	=	7%
	=	7.023581541		n	=	10
Present value of 1 received 10 years from now	=	(1+i)^-n		Where,
	=	(1+0.06)^-10		i	=	6%
	=	0.558394777		n	=	10
Cumulative discount factor	=	Present Value of annuity of 1 for first 10 years	+	Present Value of annuity of 1 for next 10 years
	=	7.360087051	+	7.023581541	*	0.558394777
	=	11.2820183
Step-2:Outstanding balance after 16h payment
Loan value is the present value of remaining future cash flows.
So,
Outstanding balance after 16th payment	=	Annual payment	*	Present Value of annuity of 1 for 4 years
	=	$ 4,431.83	*	3.387211256
	=	$ 15,011.55
Working:
Present Value of annuity of 1 for 4 years	=	(1-(1+i)^-n)/i		Where,
	=	(1-(1+0.07)^-4)/0.07		i	=	7%
	=	3.387211256		n	=	4