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
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
The rule for choosing between investments using NPV is to select the investment with the highest NPV.
-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:
To learn about the mechanics behind finance, look at free courses from the various MOOCs out there.