# Choosing between two car deals on the same car - NPV

This is basic question, but I have starred myself blind on it - I am stuck. I am not sure which formulae I should use. Here is the story.

I am looking to buy a car. I have found two dealerships willing to sell the car.

1. XCars can sell the car for 25,000 with a down payment of 2,000 and the balance (25,000 - 2,000 = 23,000) paid in equal monthly installments (annuity) over 3 years - i.e. 36 months of equal loan payments.

2. YCars are willing sell the same type of car for 22,500 if I pay cash, i.e. they give a 2,500 discount.

Effective annual discount rate: 7 %.

Which car dealer shoud I choose?

Net present value (NPV) calculation seem like the right way to solve the problem, but I am really unsure if I am on the right path.

Hoping for some help / some pointers.

• Go to your local credit union and get a loan for 22.5K as long as it is under 7% you will save money. The monthly payment will be higher but the initial down will be zero. Dec 13 '13 at 18:14
• Thanks for replying - that is not possible in this case - it was finance question where only two options should be considered.
– Sha
Dec 13 '13 at 21:45
• Is this a homework problem? It doesn't sound like you are debating between actually buying these two, but I could be wrong. Dec 13 '13 at 22:02
• Yes - homework problem.
– Sha
Dec 14 '13 at 1:41
• Buy a used car for \$3000, keep the \$19,500, and avoid the \$638/month payments for 3 years. Dec 19 '13 at 23:50

Here's the way I think you should analyze it:

The car costs \$22500. That's the cash price if you walk into the dealership and plunk down 225 \$100 bills, and that's what YCars is charging you.

XCars is selling the same car in exchange for \$2000 up front, and 36 equal monthly payments of \$638.88. In effect, XCars is lending you money at some interest rate, and folding the interest charge into an up-front increase in the so-called selling price. Excel. or a mortgage calculator can find the interest rate implied by these two options.

I make it 7.63% nominal interest rate, compounded monthly.

If you can borrow for less than that rate, then borrow the cash and go with YCars.

If you have the cash, and can invest it at better than this rate (after taxes on your earnings!) then go with XCars.

This is all math; it doesn't include the psychological benefit of going with XCars and having the cash cushion in your hands

Suppose then that you have available an investment vehicle that pays 7%, tax free, compounded monthly, and allows monthly withdrawals.

On the day you buy the car, you have two choices:

You can leave the house with \$22,500.00 go to YCars, pay for the car with the cash and leave having paid in full.

Or:

You can leave the house with \$22,691.06, put \$20,691.06 in your investment, and pay \$2,000 to XCars. That strange amount, \$20,691.06 is the amount that will fund 36 monthly payments of \$638.88 to XCars from the investment account. at the stated investment rate, leaving a zero balance after the 36th payment. So, again, you can leave XCars with the car, no cash, with all payments arranged for.

Now its easy: paying by installments cost \$191.06 extra as of the day of purchase.

Any change in the investment income available will change that amount to invest to cover the payments, and change the relative benefits of the two options.

• Thanks for replying - now I understand how to solve this problem - I just needed some pointers and you gave them. I have to find the effective monthly rate, r and forget about NPV. `r = 1.07^(1/12)-1 = 0.00565` I use this `r` for calculation of the annuity factor for 36 months (AF36), which becomes `32.4898`. If I multiply the AF36 with the installments of `-638.88`, I get `-20757.38`. In addition I have to put `-2000` upfront. The correct answer: `-2000 + -20757.38 = -22757.38 < -22500, and in terms |-22,757.38| > |-22500|` which means I should buy the car YCars today for 22500.
– Sha
Dec 13 '13 at 21:43
• I'm puzzled where the 7% figure comes from. Neither seller is quoting any rate; they give you two different payment schedules to buy the same item. And you can't pick one over the other without knowing what you would do if you took the time payments and had more investment income. Dec 14 '13 at 2:49
• The 7 % is an example, but it refers to the interest rate that you could get from investing the money risk free elsewhere. Normally this would be lower and have the rate of a 30 year state bond - depending on the country. In this example I could put the 25k in a bond an earn 7 % interest every year and don't buy a car, but instead buy a bicycle, taxis, train-ride etc. Opportunity costs.
– Sha
Dec 14 '13 at 17:56
• Your are using the wrong calculation in your recent edits to solve this problem. The 7 % is an effective interest rate, which is to be discounted monthly over the 36 months, at 12 times per year. That gives an r of = 0.00565. Remember this is simple solution - you have two options only to choose from. You cannot invest in other assets in this problem.
– Sha
Dec 16 '13 at 23:55
• As @Yamikuronue says above, why in the name of Warren Buffett would you borrow at 7% when you have the cash and it's not working for you at better than 7%? Dec 17 '13 at 2:20

To solve the problem, we find the effective monthly interest rate `r`, the annuity factor `AF` and then calculate the `NPV` and compare.

XCars will sell the car for 25,000 with a down payment of 2,000, which gives the following loan amount:

``````Car price (XCars) = -25000
Loan amount = -25000 + 2000 = -23000
``````

The effective monthly interest rate `r` is then calculated by taking the effective annual interest rate of `7%` to the power of `1/12` (for 12 months):

``````r = 1.07^(1/12)-1 = 0.00565 (Correct)
r = 0.07 / 12 = 0.00583 (Wrong!)
``````

Note: you cannot just do `0.07 / 12 = 0.00583` as this does not take into account compounding!

Next we find the annuity factor (AF) for 36 months `AF36` which becomes:

``````AF = (1 - (1 + r)^-n) / r
AF36 = (1 - 1.00565^-36) / 0.00565 = 32.4898
``````

Remember, we have 36 monthly equal payments, so we divide the entire loan amount by 36 so we get the annuity (equal payment per period of time):

`````` Monthly installment (annuity) = -23000/36 = -638.88
``````

Then multiply `AF36` with the `monthly installment` to get the present value `PV` and finally add the down-payment to get `NPV` for XCars.

`````` PV  = AF36 * monthly installment = 32.4898 * -638.88 = -20757.38
NPV = -2000 + -20757.38 = -22757.38
``````

Conclusion: The rule for choosing between investments using NPV is to select the investment with the highest NPV.

Since `-22757.38` (XCars) is lower than `-22500.00` (YCars) you should buy the car today at YCars, i.e. do not take a loan to buy the car at XCars!

On the other hand, if the interest rate `r` is e.g. `8%` it would make more sense to take a loan because the NPV of the loan would be larger (i.e. closer to 0).

`````` Interest rate r = 9% => NPV = -22194.66
Interest rate r = 8% => NPV = -22472.13
Interest rate r = 7% => NPV = -22757.38
Interest rate r = 6% => NPV = -23050.72
...
``````

So we can see the `interest rate` determines whether we should get a loan or not.

In this example, we find that, at an interest rate of exactly `7.9011%` the NPVs of the loan and the "buy now" option are equal and therefor - ceteris paribus - equally good. NOTE: For all options, we do not take into account tax perspectives like interest rate deductions, which may lead to an entirely different conclusion.

The formulas used here are valid for real life applications even though this is a constructed example.

For those interested, check out the annuity factor formula here:

https://en.wikipedia.org/wiki/Annuity_(finance_theory)

To learn about the mechanics behind finance, look at free courses from the various MOOCs out there.

• This boils down to: if the guaranteed rate of investment is greater than the interest rate of the loan take the loan. Otherwise minimizing the amount of the loan makes the most sense. Your question only applies to people who have enough cash on hand to pay for the car in one payment. Dec 20 '13 at 13:04
• No, homework doesn't, in of itself make it invalid for real life, but being limited to a few options does. I will leave judgement to the community. I think you pulled a better answer out than before. Good job on that! I think you have better clarified the math to evaluate such a position, but leaving out the points mhoran_psprep makes does a disservice to the answer. Dec 20 '13 at 13:25
• @mhoran_psprep - yes this question is as I have pointex limited
– Sha
Dec 21 '13 at 12:19