Question

You are considering whether to buy or lease a car.

If you lease, you have to pay a refundable security deposit of $5 hundred, and a monthly lease payment of $424 for 3 years, with payments due at the beginning of the month.

If you buy, you will pay a downpayment of $21 hundred, and a monthly loan payment of $565, over the same period of time, with payments due at the end of the month. The car is estimated to have a residual value of $6 thousand at the end of this time.

If the appropriate discount rate is 7 percent, what is the relative benefit of buying as opposed to leasing?

Hint: find the present value of all the buy cash flows, and subtract the present value of all the lease cash flows. Pay attention to the signs!

Answer 1

Net benefit in buying =PV of lease payment-PV of buying
Present value of lease payments=Security deposit+PV of annuity due of lease payments-PV of security deposit refundable
PV of refundable security deposit			=	Security deposit/(1+r/12)^t*12
			=	$500/(1+0.07/12)^3*12
			=	$500/(1+0.005833)^36
			=	$500/1.232926
			=	$          405.54
Present Value of annuity			=	P+P[{1-(1+r)^-(n-1)}/r]
		where P	=	Lease payment = $424
		t	=	3*12=36 months
		r	=	7%/12=0.5833% or0.005833
PV of annuity			=	424*[{1-(1+0.005833)^-36}/0.005833]
			=	424+424*[{1-(1+0.005833)^-35}/0.005833]
				= 424+424*[{1-(1.005833)^-35}/0.005833]
				= 424+424*[{1-1.005833^-35}/0.005833]
				=424+424*[{1-0.815819}/0.005833]
				= 424+424*[{0.184180}/0.005833]
				= 424+424*[0.184180/0.005833]
				= 424+424*[31.575568]
				=$424+$13388.04
				13812.0407
PV of leasing the car			=	$500+$13812.04-$405.54
			=	$                     13,906.50
PV of buying the car			=	Downpayment+PV of monthly payment-PV of residualvalue
Downpayment			=	$2,100
PV of residual value			=	Residualvalue/(1+r/12)^t*12
			=	$6,000/(1+0.07/12)^3*12
			=	$6000/(1.005833)^36
			=	$6000/1.232911
			=	4866.53
Present Value of annuity			=	P[{1-(1+r)^-n}/r]
		where P	=	Monthly payment = $565
		t	=	3*12=36
		r	=	0.07/12=0.5833% or 0.005833
Present Value of annuity			=	565*[{1-(1+0.005833)^-36}/0.005833]
				565*[{1-(1.005833)^-36}/0.005833]
				565*[{1-1.005833^-36}/0.005833]
				565*[{1-0.811088633}/0.005833]
				565*[{0.18891136607}/0.005833]
				565*[0.18891136607/0.005833]
				565*[32.386656]
				18298.4606
Cost of buying a car in present value				=	$2100+$18298-$4866.53
				=	15531.47
Relative benfit of buying as opposed to lesing				=	PV of leasing-PV of buyig
				=	$13906.5-$15531.47
				=	$                       (1,624.97)
				I.e there is loss in buying.Leasing has benefit of $1624.97
		there amy be little difference due to decimal places
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