Question

Joe negotiates an 8-year loan which requires him to pay $1,200 per month for the first 4 years and $1,500 per month for the remaining years. The interest rate is 13%, convertible monthly, and the first payment is due in one month. Determine the amount of the principal in the 17th payment.

The answer is 420.963, but I'm not sure how to get there.

Answer 1

Principal amount of loan is always present value of future cash flows.
Step-1:Present value of monthly payment of $1200 at the end of 16th payment
Principal amount at the end of 16th period		=	Monthly Payment of $ 1200	*	Present value of annuity of 1 for first 32 period
		=	$   1,200.00	*	26.9205828
		=	$ 32,304.70
Working:
Present value of annuity of 1 for first 32 period		=	(1-(1+i)^-n)/i		Where,
		=	26.9205828		i	=	13%/12	=	0.010833
					n			=	32
Step-2:Present value of monthly payment of $1500 at the end of 16th payment
Principal amount at the end of 16th period		=	Monthly Payment of $ 1500	*	Present value of annuity of 1 for next 48 period	*	Present value of 1 for first 32 period
		=	$   1,500.00	*	37.2751898	*	0.70836
		=	$ 39,606.40
Working:
Present value of annuity of 1 for first 48 period		=	(1-(1+i)^-n)/i		Where,
		=	37.2751898		i	=	13%/12	=	0.010833
					n			=	48
Present value of 1 for first 32 period		=	(1+i)^-n		Where,
		=	0.70836035		i	=	13%/12	=	0.010833
					n			=	32
Step-3:Sum of present value of monthly payment at the end of 16th payment
Sum of present value of monthly payment at the end of 16th period		=	$ 32,304.70	+	$ 39,606.40
		=	$ 71,911.10
Step-4:Principal amount in the 17th payment
Payment no.		Beginning principal	Interest	Monthly payment	Principal repayment	Ending Principal amount
		a	b=a*13%*1/12	c	d=c-b
17		$ 71,911.10	$       779.04	$ 1,200.00	$     420.963	$ 71,490.14
So, amount of the principal in the 17th payment is $ 420.963