Question

1. Sam borrows $1,000,000 by a mortgage with annual payments over 30 years at a rate of 9.75% per annum interest. What are his annual payments? what is the remaining balance on his loan after 5 years? 15 years?

Suppose that 10 years after Sam takes out the loan, the mortgage is sold to an investor who requires a 10.5% rate of return on investments, how much is the investor willing to pay for the loan?

Answer 1

1)
	(a)	Annual payment	=	Loan amount	/	Present value of annuity of 1 for 30 years
			=	$       10,00,000	/	9.627108
			=	$    1,03,873.36
		Working:
		Present value of annuity of 1 for 30 years	=	(1-(1+i)^-n)/i		Where,
			=	(1-(1+0.0975)^-30)/0.0975		i	9.75%
			=	9.62710767		n	30
	(b)	Loan balance after 5 years	=	Annual payment	*	Present value of annuity of 1 for 25 years remaining
			=	$    1,03,873.36	*	9.254377
			=	$    9,61,283.21
		Working:
		Present value of annuity of 1 for 25 years remaining	=	(1-(1+i)^-n)/i		Where,
			=	(1-(1+0.0975)^-25)/0.0975		i	9.75%
			=	9.25437694		n	25
	(c)	Loan balance after 5 years	=	Annual payment	*	Present value of annuity of 1 for 25 years remaining
			=	$    1,03,873.36	*	7.715862
			=	$    8,01,472.49
		Working:
		Present value of annuity of 1 for 25 years remaining	=	(1-(1+i)^-n)/i		Where,
			=	(1-(1+0.0975)^-15)/0.0975		i	9.75%
			=	7.715861988		n	15
2)	Value of loan	=	Annual payment	*	Present value of annuity of 1 for 20 years
		=	$ 1,03,873.36	*	8.230909
		=	$ 8,54,972.15
	So, at 10.50% rate of return,investor is willing to pay			$    8,54,972.15
	Working;
	Present value of annuity of 1 for 20 years	=	(1-(1+i)^-n)/i		Where,
		=	(1-(1+0.1050)^-20)/0.1050		i	10.50%
		=	8.230908913		n	20