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We're buying a brand new car at $60K. We technically have the cash on hand, but have opted to get a loan instead and keep the cash in investments. For the purposes of this question & simplicity, let's assume a guaranteed 7% return on investing.

We're planning on renting the car out on Turo/using it with Uber/Lyft to subsidize the car payments. Let's assume (for simplicity) that we can make $800/month by doing this.

My available auto loan options are as follows:

  • 3 year, 2.49%
  • 4 year, 3.24%
  • 5 year, also 3.24%
  • 6 year, 3.49%
  • 7 year, 4.99%

If I can finance the car anywhere from 0-100% (paid fully in cash to 100% financed), how do I calculate the best loan term + downpayment to make? Additionally, if I don't want to pay more than, say, $200 out of pocket ($1,000/month loan payment after the $800 the car brings in), how does that factor into the calculation?

  • 5
    You'd let a stranger "abuse" your $60k car? And you'd really "work" for about minimum wage? – topshot May 7 at 12:26
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    I wasn't referring to external damage as in they got in an accident, but hard acceleration/braking, etc that will cause wear you cannot see. People drive rental cars harder than they do their own car. – topshot May 7 at 14:51
  • It's a Tesla Model 3, so there are no internal moving parts with the electric motor. Wear and tear is mostly limited to brakes and tires, and (very eventual) battery life (Tesla has a great warranty). Even if the car is shot within the loan term, but Turo paid for the loan, I'd still be happy to have gotten a nearly free Tesla for that amount of time. Also, my friend who does this said people have been treating his car better than their own. (Obviously not a guarantee.) – bobbyz May 7 at 14:55
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    It wouldn't take you 5 minutes to learn this doesn't make sense. If there is cash flow, car maker already leased their car to drive sharing drivers and corner the market. – mootmoot Sep 11 at 8:11
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If you have $60K on hand and just invest it at 7% effective p.a. for 7 years the interest i is

s = 60000
i = s ((1 + 7.0/100)^7 - 1) = 36346.89

Likewise if compounded monthly

m = (1 + 7.0/100)^(1/12) - 1 = 0.00565415
i = s ((1 + m)^(7*12) - 1) = 36346.89

If you buy the car on the seven year plan the monthly payments d are

r = 4.99/100/12 = 0.00415833
n = 12*7
d = r s (1 + 1/((1 + r)^n - 1)) = 847.75        (formula 1)

You make $800 per month on the car. If this is reinvested, at the end of 7 years your gain g is

x = 800
g = (d - x + (1 + m)^n (m s + x - d))/m - s = 31230.70

This is based on the investment principal changing from month-to-month by

p[n + 1] = p[n] (1 + m) - d + x    starting with    p[0] = s

∴ p[n] = (d - x + (1 + m)^n (m s + x - d))/m    (formula 2)

The difference in gains is

g - i = 31230.70 - 36346.89 = -5116.19

So with the car deal you lose $5116.19 but you gain a 7 year-old car.

Comparing other interest and term offers

years                 3          4          5          6          7
invest. interest  13502.58   18647.76   24153.10   30043.82   36346.89
car interest          2.49       3.24       3.24       3.49       4.99
car payment        1731.42    1334.43    1084.53     924.83     847.75
rental gain      -23569.23  -10728.88    3895.57   18988.63   31230.70
comparative loss  37071.81   29376.64   20257.53   11055.19    5116.19

If the car resale value exceeds the comparative loss that is a profit over simple investing.

Adding continued earnings (from formula 3) to the capital p[n] up to a total of 7 years.

car loan years        3          4          5          6          7
after car paid    36430.77   49271.12   63895.57   78988.63   91230.70
continued rent    43974.25   31841.13   20501.77    9904.24       0.00
total             80405.02   81112.25   84397.34   88892.87   91230.70

Considered over seven years, the longer car loan is more advantageous.

Returning to the seven year deal, if you buy two-thirds of the car, paying out $40K straight away leaving $20K owed and as capital

s = 20000
r = 4.99/100/12 = 0.00415833
n = 12*7
d = r s (1 + 1/((1 + r)^n - 1)) = 282.58

x = 800
p[n] = (d - x + (1 + m)^n (m s + x - d))/m = 87551.22

Paying out car payments and receiving rent grows the $20K capital to $87551.22

The gain is

g = 87551.22 - 60000 = 27551.22

and compared with simply investing the $60K

i = 60000 ((1 + 7.0/100)^7 - 1) = 36346.89

g - i = -8795.67

Now there is a loss of $8795.67 plus a 7 year-old car.

If you buy the car outright the income is equivalent to investing $800 at the end of every month for 7 years

(((1 + m)^n - 1) x)/m = 85711.48                (formula 3)

g = 85711.48 - 60000 = 25711.48

g - i = -10635.41

Now the loss is $10635.41 compared with investing, although you have the car at the end.

It is better to keep any capital invested rather than making any down-payment on the car. Obvious really, since the 7% rate is higher.

Notes

Formula 1 - loan payment formula

enter image description here

Formula 2 - inhomogeneous difference equation (Arne Jensen, Aalborg Uni.)

enter image description here

Formula 3 - future value of an ordinary annuity

enter image description here

  • This is awesome! Maybe clarify something for me - the $800/month wouldn’t be used to reinvest, but would go towards paying the loan. Does that change any of these calculations? Also, since it sounds like you know the math pretty well, is there any way you could put the official formula names in here? (Like NPV, e.g.) I’d like to learn more about what goes into these calculations as a personal interest. – bobbyz May 7 at 15:04
  • Also, how come the compounded monthly calculation comes out to the same amount before compounding? – bobbyz May 7 at 15:05
  • Hi, the $800 income nets against the monthly payment as - d + x in the derivation p[n + 1] = p[n] (1 + m) - d + x. That is to say, the capital at month n has interest added, then the payment is subtracted and the income added to produce the capital at month n + 1. I will add derivations for the other two formulae. The compounded monthly calculation comes out the same as the compounded annual calculation because the 7% is assumed to be an effective annual rate, normal for market performance, (APY rather than APR). – Chris Degnen May 7 at 15:34
  • Makes sense. Thanks again, this is above and beyond! – bobbyz May 7 at 15:39
  • One more question - what would the final recommendation be? I'm parsing through all this slowly, but if you had to make a financial decision with these parameters, what would it be? – bobbyz May 8 at 21:25
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Pay cash and the the cost of the car is $714.29 a month when on a seven year depreciation. The enterprise is not monthly profitable at $800 monthly revenue when including insurance and maintenance costs. However, there may be a value to the car after depreciation so possibly adjust the depreciation to make the enterprise break-even. Also, there is "off meter" use of the car.

Of course the three-year financing only adds about $2331 to the seven year cost. That's $714.29 in monthly depreciation and $27.75 in monthly interest cost as spread across seven years. Then four-year financing adds $48.25 monthly to the seven year cost, five-year financing adds $60.38 monthly to the seven year cost, six-year financing adds $78.43 monthly to the seven year cost, or seven-year financing adds $133.46 monthly to the seven year cost

Here is a Tesla taxi with 400,000 miles:

https://electrek.co/2018/07/17/tesla-model-s-holds-up-400000-miles-3-years/

Note the maintenance cost per mile of $0.05 .

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