A woman purchased a new car today for $16,000. She paid $2,000 down and agreed to...

A woman purchased a new car today for $16,000. She paid $2,000 down and agreed to make 47 monthly installments of $250 and a nal balloon payment at the end of the 48th month. The interest rate is 15 percent compounded monthly. What is the amount of the final balloon payment?

Solutions

Expert Solution

Solution:
	Amount of the final balloon payment is $9,357.87

Working Notes:
	Car value today = down payment + Present value of annuity of $250 monthly payment of 47 months + Present value of 48th period balloon payment



	Present value of annuity of $250 monthly payment of 47 months

	present value of annuity = Px[ 1-1 /(1 + i)^n)]/ i

	P=monthly payment=$250
	i= interest rate per period = 15%/12
	n= no. Of period = 47

	PV of annuity= Present value of annuity of $250 monthly payment of 47 months


	present value of annuity = Px[ 1-1 /(1 + i)^n)]/ i

	present value of annuity= 250 x (1-1/(1+(15%/12))^47)/(15%/12)
	present value of annuity= $8,845.156105

	Car value today = down payment + Present value of annuity of $250 monthly payment of 47 months + Present value of 48th period balloon payment


	16,000 = 2,000 + $8,845.156105 + Present value of 48th period balloon payment

	Present value of 48th period balloon payment
	=16000 - 2000 - $8,845.156105
	=$5,154.843895

	Present value of 48th period balloon payment
	=Balloon payment/( 1 + (15%/12))^48
	=Balloon payment/( 1 + (15%/12))^48

	Present value of 48th period balloon payment = Balloon payment/( 1 + (15%/12))^48

	$5,154.843895 = Balloon payment/( 1 + (15%/12))^48
	Balloon payment = $5,154.843895 x ( 1 + (15%/12))^48
	Balloon payment = $9357.870882
	Balloon payment = $9,357.87


	Car value today = down payment + Present value of annuity of $250 monthly payment of 47 months + Present value of 48th period balloon payment


	= 2000 + 250 x (1-1/(1+(15%/12))^47)/(15%/12) + $9357.870882/( 1 + (15%/12))^48

	=16,000
	=16,000
Please feel free to ask if anything about above solution in comment section of the question.

jeff jeffy answered 2 hours ago

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