Proving Money Factor x 2400 = Interest Rate

[irwinr]

Apologies is this is explained elsewhere. I searched and could not find anything to answer this specific question:

So I’m already well aware of the rule of thumb to convert MF to APR you multiply MF by 2400. However: I’m trying to “prove” this and the math isn’t working out for me and I’m not sure what I’m missing. So I figured I would pop in here and see if someone can point out where I’m going wrong.

I’m exploring the idea of leasing a Kia EV9 because the only way to access the EV tax credit with Kia right now is to do a lease. (IRS loophole allows the tax credit for leases on Kia EVs but not on purchases)

So as an easy example: Take this lease:

You can see total purchase price is $70,120 after you subtract the $7500 credit.
RV = $42,691 (55% for a 15k mile 3 year lease)

So the depreciation paid over the lease term would be $27,429. Now suppose I want a “zero down” lease where taxes and fees are financed and the total amount “financed” through the lease would be:
$27429 + $743 (fees) + $4383 (taxes)
For a total amount “financed” (If you will) of $32,555

Assumption check: I’ve always thought of leasing as though it’s just financing the first 3 years of depreciation (Rather than the full lifetime depreciation when you purchase)

So essentially if that assumption is true: With this lease I am effectively “borrowing” $32,555 - The 3 years of depreciation plus the taxes and fees. Have I missed anything on this assumption?

The lease payment quoted here is $1295.19 – No money factor is mentioned and this seems a bit high to me so I went over to the Leasehackr calculator to see if I could reverse engineer the money factor:

You can see that here: CALCULATOR | LEASEHACKR

When I plug all the numbers into the Leasehackr calculator, I end up having to adjust the money factor to 0.00269 to end up with a payment of $1295. 0.00269 isn’t great but that works out to an APR of 6.46% which is fairly competitive with bank interest rates. (IE: If I was going to finance the vehicle, mid 6% is roughly what I would expect my interest rate to be).

However I noticed that something doesn’t add up. If we take that total amount “financed” of $32,555 (see my assumption above) and did a bog standard 3 year loan at 6.46%: The payment for a 3 year loan at 6.46% would be $997.19.

In Fact: in order to get a payment amount of $1,295 on a 3 year loan of $32,555 – I’d need to crank the interest rate up to about 25% (That’s credit card level interest)

And the numbers on the Leasehackr calculator seem to agree with me:

Notice the “Depreciation” of $33,673 (Very close to my $32,555) and the “Rent charge” of $10,988 – There’s no way a 6.46% APR can result in $11k in “interest” on a $33.5k paid over 3 years.

So can someone help me understand what I’ve missed here? I just cannot understand how a money factor of 0.00269 = 6.46% APR when it results in nearly $11k in finance charges from a $33,500 “loan” over 3 years.

Thanks!

[Sarafil]

The bank is loaning you a complete car for 3 years, not just 1/3 of a car.

Nice way of saying you are forgetting the interest expense on the residual - since the bank is basically loaning you that value for 3 years until they get their car back.

The monthly interest is effectively the interest on the balance of cap cost - residual.

[mllcb42]

@irwinr

I just cannot understand how a money factor of 0.00269 = 6.46% APR when it results in nearly $11k in finance charges from a $33,500 “loan” over 3 years.

Its nkt a $33500 loan for 3 years.

Its more like paying the interest on an 8 year loan of $70k, but only for the first 3 years.

Youre paying monthly interest on the adjusted lease balance as it lowers from the adjusted cap cost to the residual value.

[irwinr]

Yep, that’s what I missed. I knew I had done this before and the numbers always matched up pretty closely so I knew I had to be fudging something in the math.

Thanks for the quick (and super clear) answer.

[themachine]

The reason it’s 2400:

2400 = 100 * 12 * 2

100 gets you from a percentage to a decimal. 12 gets you from annual to monthly. 2 gets you to the average of adj cap cost and residual since you add them together when multiplying by the MF.

[delta737h]

@irwinr

So I’m already well aware of the rule of thumb to convert MF to APR you multiply MF by 2400. However: I’m trying to “prove” this and the math isn’t working out for me and I’m not sure what I’m missing.

Let’s look at the MF formula for computing the monthly base payment…

The above shows that interest is levied on the average depreciated lease balance and illustrates that…

It’s a bit more complicated than this and involves a Taylor series expansion encountered in calculus which is why the above only gives an estimate of the interest rate but it’s very close.

The amount financed in a lease is always the adjusted capitalized cost (Adj. Cap) and is determined as follows…

Gross Cap = Sell Price + Amounts Financed (e.g., capitalized fees such as the acquisition fee)
Adj. Cap = Gross Cap – Cap Reduction
Depreciation = Adj. Cap – RV Note: Depreciation is not Sell Price – RV

There are a few items missing in the dealer worksheet preventing me from computing the MF and that is the Adj. Cap or sufficient information that would enable me to compute the Adj. Cap. For instance, there is no mention of the acquisition fee or dealer doc fee. The amounts capitalized in the lease are not itemized. It appears to me that there are capped items. The reason I know this is because if I treat the sell price as the gross cap and subtract the cap reduction to get the adj. cap and use it to compute the MF, I get a funky MF = .0066889 which I know is not accurate. Therefore, there must be fees that are being capitalized. One of those fees is most likely the acquisition fee.

Questions? Please let me know. Glad to help.

EDIT: average depreciated lease balance

[irwinr]

This is the kind of math I was hoping to find when I couldn’t make my numbers come out as expected.

@delta737h

There are a few items missing in the dealer worksheet preventing me from computing the MF and that is the Adj. Cap or sufficient information that would enable me to compute the Adj. Cap. For instance, there is no mention of the acquisition fee or dealer doc fee.

Those details are in the fine print here:

Of course this isn’t an official offer from a dealer. Just their advertised lease deal on the website. I was just trying to get a rough idea if doing a lease to get the $7500 tax credit was worth it vs doing a purchase where I would not be able to get the tax credit.

[socal59]

Haven’t seen these formulas since my business finance classes. I don’t remember much of it

[delta737h]

@Sarafil

The monthly interest is effectively the interest on the balance of cap cost - residual.

No, it’s not. The interest is computed on the adjusted lease balance. Let’s say that the lease amortization rate (actuarial rate) is 4%. If the adjusted cap cost is 50,000 and the base monthly payment is 600, then the first period (first full month) interest charge is (4%/12) x (50,000 - 600) = 164.67 and the amount allocated to depreciation is 600 - 164.67 = 435.33. Interest is always levied on the unpaid lease balance and then, subtracted from the base payment to get the depreciation payment.

[delta737h]

@irwinr

Those details are in the fine print here:

Thanks, now it makes sense. Your MF = 0.00281 as your adj. Cap = 77,190.69. Your payment cranks out as …

I won’t bore you with a complete analysis at this point unless you would like me to do so.

[Sarafil]

Technically yes if it worked like a loan, but a lease has fixed ratio of depreciation and interest in each payment, it doesn’t amortize like a traditional loan (the interest charge is the same every month of the lease). Typo in my original reply - the interest is calculated on the average balance of the lease which is the mid point of the cap cost and residual, times the interest rate, and then you pay that amount through the life of the lease each month.

[mllcb42]

@Sarafil

but a lease has fixed ratio of depreciation and interest in each payment, it doesn’t amortize like a traditional loan (the interest charge is the same every month of the lease).

With most leases, this isn’t actually true. Your adjusted lease balance doesn’t decrease linearly.

[delta737h]

I’m a mathematician and have taught courses in the mathematics of finance at both the undergrad and grad levels. I’ve also created countless lease amortization schedules as well as loan amortization schedules and so, I know the difference. What you’re describing is inaccurate and reflects an estimated levelized interest charge based on the first part of the money factor formula … MF x (Adj Cap + RV). Apparently, you’re getting the idea of mid-point from…

@Sarafil

the interest is calculated on the average balance of the lease which is the mid point of the cap cost and residual, times the interest rate

This is leading you to the wrong conclusion as to how a lease is actually amortized. You do not pay this fixed interest charge every month throughout the term of the lease. You pay interest on the outstanding lease balance at the beginning of each month. The interest charge declines each month while the depreciation charge increases each month. The monthly depreciation charge is subtracted from the previous lease balance to determine the current lease balance and is analogous to the way a principal payment is subtracted from the previous loan balance to determine the current loan balance.

Below is an example of a lease amortization schedule which has an implicit lease interest rate of 1.2081% and an adjusted cap = 28,744.24, RV = 16,710.40, MF = 0.00050, Term = 36, Payment = 357.00

As you can see, the schedule begins with the adj cap = 28,744.27 and terminates with the RV = 16,710.40. Notice how the interest charge (lease charge) declines while the depreciation charge increases throughout the term (except at lease origination where the first payment is due upfront so that it reflects a depreciation payment). You may want to try creating a lease amortization schedule so you can see how it really works. I’m happy to help you if you like.

[djrabbi]

Is this Calc class all over again ?

[delta737h]

nope… not calculus… just multiplication and subtraction.

[delta737h]

One other thing… The dealer WS is screwed up… BIG SURPRIZE! It shows a cap reduction of 6,421. This is not true. The 6,421 is that portion of the 7,500-rebate used to cover all the upfront fees. The remaining portion of 1,079 is used as the cap cost reduction (CCR). Therefore, the adjusted cap cost is calculated as follows…

77,620 KIA Price + 625 acq fee - 1,079 CCR = 77,191

Apparently, some of the dealer’s numbers were rounded up and so I lost some accuracy (e.g., adj cap should be 77,190.69).

[Sarafil]

I follow your math and don’t disagree with it - just basing my answer on practical reality.

I have a copy of a several payoff docs from my last BMW lease that I pulled at different dates. BMW shows the remaining payment balance and then “unearned amount” which is interest portion of remaining payments. The unearned interest per month (unearned interest / number of months remaining) was exactly the same amount regardless of when during the lease I pulled the payoff.

Ultimately happy to let you “win” this one but not sure your mathematical theory follows reality.

[mllcb42]

@Sarafil

Ultimately happy to let you “win” this one but not sure your mathematical theory follows reality

The nice thing here is that lease contracts have specific language in them that describe the method by which the adjust lease balance is calculated. For most, that is as @delta737h is describing. One could look at the terms of the BMW contract to see if they specify something different than the rest of the industry.

[max_g]

Does the amortized method match up with lease payoff statements we typically see posted on LH?

Remaining payments + RV - unearned rent

[delta737h]

@Sarafil

just basing my answer on practical reality.

That’s part of the problem. Your answer isn’t based on practical reality or what actually happens. Sometimes when people see mathematical equations, they dismiss them as not being practical… HUGE MISTAKE. One way to determine your lease payoff is to create a lease amortization schedule like the one above or, you can use the following equations to determine the adjusted lease balance…

I’ve always used them to determine my lease buyouts and my calculations have always been spot on.

@Sarafil

Ultimately happy to let you “win” this one but not sure your mathematical theory follows reality.

It’s not a question of winning or losing. It’s a matter of accuracy and correctness that mirrors reality.

@Sarafil

BMW shows the remaining payment balance and then “unearned amount” which is interest portion of remaining payments. The unearned interest per month (unearned interest / number of months remaining) was exactly the same amount regardless of when during the lease I pulled the payoff.

This doesn’t sound right and am willing to bet that the BMW contract states something different than what you’re describing. I believe you may be misinterpreting the language. You’re not accounting for the time-value of money (e.g., present value of unearned interest charges) There are many ways to compute the lease balance. One way to do it is to calculate the present value of the remaining base payments and add that to the present value of the residual. Another way to do it is to add the remaining depreciation payments to the residual. I’ve computed BMW buyouts every which way to midnight and, with all due respect, what you’ve described doesn’t seem accurate to me. If you post an BMW lease contract with the description of how the adjusted lease balance is calculated, I’ll be happy to go through it with you step by step. The language may be different among fund providers, but they are all equivalent and result in the same adjusted lease balance based on my experience.