## Wednesday, November 25, 2009

### Understanding Car Lease Payments

Frugal Trader at Million Dollar Journey had an interesting post explaining how car lease payments are calculated. The formula is simple enough until it adds a lease fee that involves a mysterious “money factor”. It all seems like extra profit for the dealership, but the truth is less nefarious.  According to commenter Robert, this money factor is only used in the U.S.; Canadians use the exact calculation given at the end of this post.

An Example

I’ll use the same example that Frugal Trader used:

– Honda CRV: MSRP + freight + PDI: \$29,880
– Residual Value after 3 years: \$15,276.60
– Depreciation (price minus residual): \$14,603.40
– Depreciation per month: \$405.65

So, if the interest rate were 0%, then the payments should be just this \$405.65 per month. But interest is a fact of life and we need to figure that out too. The accurate way to calculate interest involves present value calculations. I’ll leave the details of this accurate method to the end of this post for those who are interested.

Estimating Interest

At the beginning of the lease the customer owes the entire value of the car. At the end of the 3-year lease, the customer will owe only the residual value. On average over the 3 years, the balance owing by the customer will be about halfway in between these two values:

– Approximate average balance owing:
(\$29,880 + \$15,276.60)/2 = \$22,578.30
– Annual lease rate: 4.9% (1/12 of this per month)
– Monthly interest = \$22,578.30 x 4.9%/12 = \$92.19
– Total Lease Payment:
\$405.65 + \$92.19 = \$497.84 per month plus sales tax

Typically, the interest part of the payment isn’t explained the way I did. The official method begins with a mysterious money factor which is the annual interest rate divided by 24 (0.049/24). This is then multiplied by (purchase price + residual value) to get the “lease fee”. This lease fee calculation amounts to exactly the same thing as the estimated interest calculation I did.

How good is this interest estimate?

Many readers are no doubt suspicious of this method of estimating the lease payment. Using the proper calculation for this example gives a lease payment of \$499.40. So, the estimation method actually gives slightly lower payments.

However, dividing the yearly interest rate by 12 to get the monthly rate isn’t really correct because the yearly rate becomes larger when the monthly rate is compounded 12 times. If we adjust the monthly rate to give a true yearly rate of 4.9%, then the payments should be \$497.34, which is 50 cents less than the estimated payment. So, it’s a matter of taste whether the estimated payment is too high or too low, but either way it is very close.

The truth is that there is nothing sinister going on with the “money factor” and “lease fee”. There are no extra profits hidden in the math. The real problem is the advertised price. Few cars are worth as much as their MSRP (manufacturer suggested retail price). An error of a dollar or two on the lease payment is small potatoes compared to thousands of dollars added to the MSRP.

Exact Lease Payment Calculation

Some definitions:

M – price of car (MSRP + freight + PDI)
r – interest rate
n – lease duration in months
V – residual value
P’ – estimated payment (without sales taxes)
P – exact payment (without sales taxes)

The estimated payment is

P’ = (M-v)/n + (M+V)r/24

To get the exact value of the payment, the price should be equal to the present value of the payments and residual value:

M = V/y + (P/(r/12))(1-1/y), where y = (1+r/12)^n.

With some algebra we get

P = (r/12)(M + (M-V)/(y-1)).

If we treat y as an unknown variable and set P = P’, we get

y = (24 + rn)/(24 – rn).

This turns out to be a reasonably close approximation to the true value y = (1+r/12)^n. This can be verified by examining the power series of each expression. Aren’t you sorry you asked?

1. Useful info, but at the end of it, you have nothing to show for your money (if you buy a car you have a worthless asset as well, however, at least you can keep driving it until it falls apart (and the fall apart factor differs from car to car)).

Is there a useful set of numbers to show now much more you pay for cars buying vs. leasing, say over a 6 year period?

2. Big Cajun Man: I'm no fan of leasing -- too many people use it to get a more expensive car for the same payments. Then as you say, they're left with nothing for their money at the end of the lease.

I'm not sure I fully understand your question, but let's try a 5-year lease vs. buying for a Honda CRV over 5 years assuming 4.9% interest and a residual value of 40% of purchase price. Lease payments work out to \$384+sales tax and loan payments work out to \$563+sales tax. Sadly, some car shoppers who can afford the higher payment would just move up to a lease on a more expensive vehicle.

3. Hey Michael, thanks for taking the time to explain the "money factor". I really had no idea where it came from!

Oh and I forgot to reply to you in the comments, my annual interest rate on the depreciation is way off as I did not account for the time factor.

Thanks again!

4. For 15 years I developed and sold the CarCalculator. Canada's first and only car lease/loan software for consumers. I also developed and sold the HomeCalculator for mortgages.
Here's how leasing math works:
Price of car is \$20,000.
Residual is \$8,000.
Interest is 5%. Term is 48 months.
Your payment on the part of the lease you are paying off is calculated just like a loan by taking \$20,000-\$8,000=\$12,000 over 48 months=\$276.35
The interest on your residual is \$8,000 x 5%/12 months=\$33.33
5. 