How to calculate the optimal downpayment + loan term when considering investing cash on hand

Question

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?

Answer 1

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

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

Formula 3 - future value of an ordinary annuity

Answer 2

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 .