Question

Your old car costs you $400 per month in gas and repairs. If you replace it, you could sell the old car immediately for $2,300. To buy a new car that would last seven years, you need to pay out $11,000 immediately. Gas and repairs would cost you only $170 per month on the new car. Interest is 6% compounded monthly. Should you buy a new car (alternative 1) or keep your old car (alternative 2)? Compute the net present value of each alternative and determine which alternative should be accepted or rejected according to the new present value criterion.

Answers:

The net present value for Alternative 1 is $-17,373

the net present value for Alternative 2 is $-20,408

The preferred alternative is Alternative 1

Answer 1

It is assumed that the payments take place at the end of each month.

Alternative 1: Buy a new car:

Net present value is the sum of Cost of new car less sale price of old one, and the present value of monthly expenses for 7 years.

Given, Cost of new car= $11,000

Sale price of old car= $2,300

Monthly expenses constitute an annuity for 84 months.

Present value of the monthly expenses= $11,637.02 calculated as follows:

C D Α. B 1 Present Value of Annuity Payments at the end of each period 2 3 Present value of annuity is calculated using the formula PVA=P*{[1-(1+r)^-n]/r]} 4 Where P= Periodical payments, 5 n= Number of payments. 6 r= Rate of interest per period in decimals All amounts in $ Given that 9 Periodical payments (P)= 170 No. of years (N)= 10 Rate of interest (% p.a)=R= 6 No. of payments a year (t)= 8 11 Value 0.005 84 12 Calculations: 13 Formula Cell reference 14 Interest rate per period in decimals (r )= R/(100*t) D10/(100*G10) 15 No. of periods (n)= N*t G9*G10 16 (1+r)= 1+G14 17 (1+r)^-n= G164-G15 18 [1-(1+r)^-n]= 1-G17 19 [1-(1+r)^-n]/r= G18/614 20 Present value of annuity PV= P*{[1-(1+r)^-n]/r]} D9*G19 21 Rounded to 1.005 0.65773479 0.34226521 68.4530424 11637.0172 $11,637.02

Net present value of alternative 1= -$11,000 + $2,300 - $11,637.02 = -$20,337.017

Alternative 2: Keep the old car:

It is assumed that the existing car will be in use for 7 years and that there will be no salvage value at the end of 7 years.

Monthly expenses of $400 constitutes an annuity for 84 months. Present value of these expenses is -$27,381.22   calculated as follows:

A B C D 1 Present Value of Annuity Payments at the end of each period 2 3 Present value of annuity is calculated using the formula PVA=P*{[1-(1+r)^-n]/r]} 4 Where P= Periodical payments, 5 n= Number of payments. 6 r= Rate of interest per period in decimals All amounts in $ Given that 9 Periodical payments (P)= 400 No. of years (N)= 10 Rate of interest (% p.a)= R= 6 No. of payments a year (t)= 12 11 12 Calculations: Formula Cell reference Value 14 Interest rate per period in decimals (r )= R/(100*t) D10/(100*G10) 0.005 15 No. of periods (n)= N*t G9*G10 84 16 (1+r)= 1+G14 1.005 (1+r)^-n= G164-615 0.65773479 [1-(1+r)^-n]= 1-617 0.34226521 [1-(1+r)^-n]/r= G18/614 68.4530424 20 Present value of annuity PV= P*{[1-(1+r)^-n]/r]} D9*G19 27381.217 21 Rounded to $27,381.22 13 17 19 27

Net present value (negative) of alternative 1 is llower and hence the same is preferred.