Should I lease or buy my first car?

In a year I will start university in a town with poor public transport, and my parents have said they will help with a car only if I can show them the cheapest sensible way to get one. Every car advert I have looked at shows a low monthly price, but the adverts compare different things — a lease, a loan, a deposit, a “balloon payment” — so I could not tell which option was actually cheapest.

The aim of this exploration is to find which of three ways of having the same model of car for four years — leasing it new, buying it new with a loan, or buying a three-year-old one with savings — costs least overall, and how sensitive that answer is to the interest rate and the car's future value.

I chose four years because that is the length of my course. I expected leasing to be the most expensive (because the car is never yours) and the used car to be the cheapest, but I did not know how big the differences would be, or whether they would survive realistic changes in interest rates.

I chose a popular small hatchback because it is the car my older cousin drives and it is common on used-car websites. The new price is €19,500. The three options are:

Lease (personal contract hire): an initial payment of €2,000 then 47 monthly payments of €255, for up to 10,000 km a year, after which the car is returned. These figures come from a leasing website's quote in October (Appendix B).

Buy new with a loan: a 10% deposit (€1,950) and a 48-month loan for the remaining €17,550 at a nominal annual interest rate of 6.9%, compounded monthly (the rate I was quoted by a bank's online calculator). At the end I own the car and could sell it.

Buy a three-year-old car with savings: pay for it outright, keep it for four years, and sell it when it is seven years old.

To compare the options fairly I measure each one's net cost over four years: everything paid, minus the value of the car at the end. I also include the interest my family's savings would have earned (3% per year, compounded annually) on any money taken out of savings, because that money would otherwise have grown. I assume the same mileage, insurance and fuel for all three options, so I leave those out; the difference in repair costs is considered in Section 8.

Two of the options depend on what the car will be worth in the future, so I needed a model for its value. I recorded the asking price and age of every listing of this model (same engine, same trim) on a large used-car website on one day, which gave 30 cars aged 1 to 9 years (Table 1, Appendix A). I define \(t\) = age of the car in years and \(P\) = asking price in euros.

The website listed 34 cars when I searched. I removed four before analysing the data: two were listed as “accident damaged”, one was a different engine size mislabelled as my model, and one had covered over 200,000 km, far more than any other car of its age. I decided to remove these because my question is about the value of a normal car of this model, and each of them would have pulled the model towards prices that do not apply to the car I would buy. I kept one 6-year-old car that looked cheap but had no stated reason, because removing data just because it does not fit would bias my model.

The scatter graph shows prices falling quickly at first and then more slowly, which suggested an exponential model: a car loses a similar percentage of its value each year, not a similar amount. I compared this with a linear model. The linear model has a strong correlation (\(r \approx −0.973\)), but it predicts a price of about €-1,299 for a 12-year-old car, which is impossible, and its residuals are positive at both ends and negative in the middle.

To fit the exponential model I used my GDC's exponential regression, which fits a straight line to \(\ln P\) against \(t\) and then converts back. This gave \(P \approx 20,308 \times 0.850^{t}\) with \(R^2 \approx 0.981\) for the transformed data. The value 0.850 means the car keeps about 85% of its value each year, i.e. it loses about 15% a year. The model predicts a new price (\(t = 0\)) of about €20,308, which is close to the actual €19,500; I think it is a little higher because asking prices are usually above the price people finally pay.

From the model: a 3-year-old car costs about €12,482, a 4-year-old one is worth about €10,613, and a 7-year-old one about €6,523.

The aim was to find the cheapest of three ways of having the same car for four years. With the figures I found, buying a three-year-old car with savings costs about €7,525 (about €9,925 with extra repairs), buying new with a loan about €11,715 and leasing about €13,985. Leasing is the most expensive option unless loan rates rise above about 12.5% or the car loses value much faster than the market suggests.

The most important thing I learned is that monthly payments are misleading: the lease has the lowest monthly payment of the two new-car options, but because the car is returned it is the most expensive overall. The exponential model was central to the answer, because both buying options depend on what the car is worth later.

My conclusion is limited by the data. The listings were asking prices on one day from one website, and there were only two 9-year-old cars; a model fitted to actual sale prices might show faster depreciation. I also ignored inflation: €1 in four years is worth less than €1 now, which slightly favours options where the money is paid later (the lease and the loan). A fuller model would discount future payments, which I would like to learn how to do. Finally, the cheapest option is not automatically the best one for me: a new car comes with a warranty and less risk, which may be worth paying for — but now I can tell my parents exactly how much.