Question

"You receive a $40,000 car LEASE at 5% nominal annual terest for 4 years. Interest is comp ounded monthly and yo make month pay m s. ou RESID aTue at the end of your lease is $12,000. Assume LEASE payments are made at the END of the month, with the first payment due at the end of the 1st month. You can also get a LOAN for the same terms (although you will pay off the entire car in 4 years). Assume your MARR (or your investment interest rate) is 3% nominal annual, compounded monthly as wel. Do you prefer the LEASE or the LOAN/ inter the present value of the alternative that you should choose. Don't forget to include that you have a car worth $12,000 if you go with the LOAN alternative."

Answer 1

Solution: The working of the problem is as under:-

Lease Option for use of Car
CAR VALUE	$40,000.00	Lease Value or Present Value
Term	4 years	In months it will be 48 months
Nominal Annual Interest	5%	In month the rate will be =5% / 12 = 0.42%
Payment per month will be calculated using Present Value Annuity Formula i.e. PVIA = A*[1-(1+r)^-n] / r
N= Term i.e. 48 months; R= Monthly Rate i.e. 0.42% and PVIA = 40000, thus we have to calculate only for A i.e. monthly payment
40000	= A*[1 - (1+0.42%)^-48] / 0.42%
40000	=A*43.389
A=	=40000/43.389		921.893
As this is a monthly payment it is an outflow
Present Value of Annuity Leasing @ 3% annual compounded monthly i.e. @3% / 12 = 0.250%
PVIA =	A*[1-(1+r)^-n] / r
	=921.893*[1- (1+0.250%)^-48]/ 0.250%
Present Value of Leasing =	41649.9181
Borrowing Option for use of Car
CAR VALUE	$40,000.00	Borrowed Value or Present Value
Term	4 years	In months it will be 48 months
Nominal Annual Interest	5%	In month the rate will be =5% / 12 = 0.42%
Payment per month will be calculated using Present Value Annuity Formula i.e. PVIA = A*[1-(1+r)^-n] / r
N= Term i.e. 48 months; R= Monthly Rate i.e. 0.42% and PVIA = 40000, thus we have to calculate only for A i.e. monthly payment
40000	= A*[1 - (1+0.42%)^-48] / 0.42%
40000	=A*43.389
A=	=40000/43.389		921.893
As this is a monthly payment it is an outflow
Present Value of Borrowing @ 3% annual compounded monthly i.e. @3% / 12 = 0.250%
PVIA =	A*[1-(1+r)^-n] / r
	=921.893*[1- (1+0.250%)^-48]/ 0.250%
	41649.9181		Cash Outflow
In the Last 48th month we have Car Worth $12000.00 whose Present Value if calculated will be
Present Value of Car Residual Value = 12000*(1/(1+0.250%)^48					=12000*0.887
PV of Residual Car Value	10644.000		Cash Inflow
Thus Present Value of Borrowing Option is = PV of Cash Outflow = -41649.918 Adding Present Value of Residual Car Value = 10644
Net Present Value =	-31005.918
We conclude our Answer by choosing the Borrowing Option