The key factors on the leasing spreadsheet are the capital cost for the car, it's residual value at the end of the lease (which tells us the depreciation), and the cost of borrowing the money. This last item is called the "money factor".
I'm told the money factor is the APR / 2400. What does this mean? Who came up with this conversion rate? (The use of confusing terminology is one big problem with leasing.) But we can break down the formulaes on the lease worksheet arithmetically to figure out what's going on.
The pivotal formula is that for the "rent charge". Strangely, the Federal Trade Commission's model lease form does not explain how the "rent charge" is calculated. This page explains the most common method.
According to the worksheet, the resdiual value of the car is as follows. (Each formula is shown first in a verbose format, then in an abbreviated format. Further reductions are performed using the abbreviations.)
| residual value | = | adjusted capital cost - depreciation |
| rv | = | acc - dep |
This should be clear. However, now comes the wierd part. The standard theory on leasing says that the lease costs cover the carrying costs for the car and the depreciation. But next on the worksheet is the "rent charge". We are not helped by being told that "this is *not* calculated the same as interest on a loan". Instead, it is:
| rent charge | = | ( adjusted capital cost + residual value ) * money factor * nm |
| rc | = | ( acc + rv ) * mf * nm |
But the residual value depends on the capital cost of the car. So, really...
| rc | = | ( acc + rv ) * mf * nm |
| = | ( acc + (acc - dep ) ) * mf * nm | |
| = | ( 2*acc - dep ) * mf * nm | |
| = | 2*acc*mf*nm - dep*mf*nm |
This is where I start to wonder what is going on. Why does the rent charge depend on twice the accumulated value? And what does that money factor reduce to in terms of interest rate?
Meanwhile, the total cost of the lease from which the monthly costs are calculated...
| total | = | rent charge + depreciation |
| total | = | rc + dep |
We can now take the total cost of the lease, and plug in the rent charge:
| total | = | 2*acc*mf*nm - dep*mf*nm + dep |
| = | 2*acc*mf*nm + dep - dep*mf*nm | |
| = | 2*acc*mf*nm + dep * (1-mf*nm) |
Clear as mud. The rent charge is equal to some factor times twice the cost of the car plus some factor of the depreciation costs. Can we make some sense of those factors? Let's look at the "money factor * number of months" term. As I mentioned, the worksheet mentions that the money factor is the APR divided by 2400. This converts as follows:
| mf * nm | = | annual percentage rate / 2400 * nm |
| = | apr / 2400 * nm |
Note that the annual percentage rate (e.g. 8) is equal to the annual interest rate * 100 (e.g. .08 * 100)
| mf * nm | = | annual interest rate * 100 / 2400 * nm |
| = | air / 24 * nm |
And the number of months is the number of years * 12
| mf * nm | = | air / 24 * number of years * 12 |
| = | air * number of years / 2 | |
| = | air * y / 2 |
OK. This makes a bit more sense. The money factor times the number of months is another way to express the annual interest rate times the number of years, divided by 2. We can use this to further reduce the formula for the total charge from above:
| total | = | 2*acc*mf*nm + dep*(1-mf*nm) |
| = | 2*acc * (air*y/2) + dep * (1- (air*y/2)) | |
| = | acc*air*y + dep*(1-air*y/2) |
At least some of this is beginning to make sense. There's no more money factor. And you can see that the total cost of the lease is determined by adding together
| "acc*air*y" | - the interest cost of carrying the car loan |
| "dep*(1-air*y/2)" | - the present value of the depreciation cost |
Finally, this looks right. When you lease a car, a finance company buys the car and rents it to you. Their costs are the interest on the car loan and the depreciation of the car when they finally sell it back to a dealer when the term is up.
But wait...Normally, when you calculate the cost of something over several years, you exponentiate, not multiply. For an interest rate of 8% and a 3 year lease, .08*3 = 24%, whereas .08^3 = 26%. Why do we get the cheaper rate? Is "interest * num years" just a way to simplify the math?
And who thought up the second expression for the present value of the depreciation cost? For an 8% rate, (1-air*y/2) comes to 88%. But why do we divide by the annual interest rate by 2? And again, what about exponentiation?
Let's check reality. A hypothetical Ford Explorer costs $25,000 and after three years has a value of $15,000. At 8%, the original forumla gives us:
| total | = | dep + ( acc + rv ) * mf * nm |
| = | $10,000 + ( $25,000 + $15,000 ) * .00333 * 36 | |
| = | $14,795.20 |
The reduced formula comes to:
| = | acc*air*y + dep*(1-air*y/2) | |
| = | $25,000*.08*3 + $10,000*(1-.08*3/2) | |
| = | $6,000 + $8,800 | |
| = | $14,800 |
The monthly costs are $14,800 / 36 = $411.
Note that the formula reduces correctly in degenerate cases. If the car looses no value during the term of the lease, the depreciation is zero, and so the total cost is just the carrying cost of the loan:
| total | = | acc*air*y + dep*(1-air*y/2) |
| = | acc*air*y + 0*(1-air*y/2) | |
| = | acc*air*y |
If the residual value of the car is understood to be zero, the depreciation is therefore equal to the adjusted capital cost of the car. The total cost of the lease is:
| total | = | acc*air*y + dep*(1-air*y/2) |
| = | acc*air*y + acc*(1-air*y/2) | |
| = | acc*air*y + acc - acc*air*y/2) | |
| = | acc + acc*air*y/2 |
This is the same as if you had purchased the car with a car loan. The total
is the cost of the car plus the interest payments.
(ok, i'm not quite sure about the air*y/2 term)
Leasing is right if you don't care about owning a car, you have no need of a depreciating equity investment, and you won't put more than the mileage allowed on the car.
You negotiate the lease the same way as with a regular car purchase but with two additional items to discuss. First agree on a price for the car; as usual, total up the invoice prices for the car and any options, then add on a few per-cent maximum for reasonable profit. Don't allow any other items - advertising costs, lot costs, undercoating, etc.
If you want to, you can put some money down or trade in a car to take some money off the car cost. This results in the adjusted capital cost.
Then, negotiate the lowest money factor - use the conversion above to make sure it's a reasonable APR.
Then, negotiate the residual value of the car. Many times this is expressed as a percentage. Unfortunately, the residual value is not the same as the blue book value. There is an industry bible for this, the "ALG" - Auto Leasing Guide - but you'd have to buy the book or buy some software that incorporates the book.
Once you've negotiated your numbers, make sure that they are the same numbers used when the dealer figures your monthly costs. If you can, bring a laptop or Palm Pilot along to check the math.
Don't permit any additional costs. A Big-Three auto credit corp is under investigation for adding per-monthly fees. Have each line of the monthly costs explained before signing off.
| monthyly depreciation | = | = ( acc - rv ) / nm |
| monthyly finance | = | = ( acc + rv ) * mf |
| total monthly cost | = | = monthly depreciation + monthly finance |
| total cost | = | = ((acc - rv)/nm + (acc + rv)*mf) * nm |
| = | = (acc - rv) + (acc + rv)*mf*nm |