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I just read this article on Reddit yesterday, and I'm trying to figure out if it makes financial sense to invest in this thing.

Assuming the system costs \$16,790 and saves $2110/yr, at what interest rate does it become better to leave your money in the bank than to buy this thing?

If you purchased this setup, you would start generating a profit after nearly 8 years (16790/2110=7.96). We can graph our savings on a line:

$$ -$16,790+$2110x $$

However, if I could invest that same \$16790 for the same period of time at 5% interest, then I would have \$24,754.91 at the 8 year mark.

$$ $16,790*(1+.05)^{7.957345972} = $24,754.91 $$

Then on my 9th year, I will earn 24,754.91*.05=\$1,237.75 in interest, which is less than the $2110 I would have saved with the Powerwall.

What interest rate do I have to earn such that on my 9th year I make $2110? i.e., what interest do I have to earn such that at no point do I come out ahead by investing in a Powerwall?

I can't seem to figure out how to formulate this as an equation.

For the purposes of this question, please ignore maintenance costs of the Powerwall or appreciation of your property due to its installation, etc.

It will also be somewhere below 7.2% because at that rate, after 8 years, our investment will have grown to 28,848.35 just as the Powerwall reaches a net positive, our bank investment will be earning \$2019.38/yr, plus we're still \$12,058.35 ahead (which would take the Powerwall another 5.71 years to catch up, assuming we didn't earn any more).

At 5% interest, I'm not sure if it's a win or not. We're earning less per year at the 8 year mark, but we're still ahead.

I drew a graph to better explain the problem:

enter image description here

Edit: I think the black line should actually start at 0. Let's assume the lines represent your bank account balance and you have exactly \$16790 in your account. You can either buy a Tesla wall with that money (bringing you down to $0) or invest it in stocks, which are somewhat liquid so it doesn't reduce your balance.

I think the problem comes out to:

$$ y=16790+(1+r)^x, y=2110x, y>0, x>0, r>0 $$

Solve for $r$.

But Wolfram|Alpha times out.

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    The "at no point" question is simlpe: the 2110 USD saved each year must equal the yearly interest accumulated for 16790 USD, so that's about 12.5% (though this does not consider wear and amaintenance and stuff) – 2017-02-22
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    @HagenvonEitzen 12.5% of 16,790 ≈ 2110, sure, but by the time the Powerwall investment breaks even (crosses over 0), we're earning a lot *more* interest than just 2110 due to compounding. So the number is somewhere below 12.5% and above the 5% I gave in my example above. – 2017-02-22
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    Note: An important factor should be how much your home appreciates in value with this improvement. If this amount is, say, \$16790, that would dramatically change this calculation. – 2017-02-22
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    Your picture is misleading (even taking your edit into account). If you buy a Tesla wall, you can then take the $2110 you're saving every year and invest that in whatever interest-bearing thing you would have put your initial capital in. So actually both curves should be growing exponentially, not just the Tesla-free curve. – 2017-02-22
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    @Micah Is it solvable if we ignore that? And even if we factored that in, and we could earn the same interest rate on our \$2110 as we could on our stocks, which would be better? We'd have a much higher starting balance if we don't buy Powerwall, but with the Powerwall, we could add an extra \$2110 each year to our investment which might actually cause it to grow faster at some point. – 2017-02-23

2 Answers 2

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What interest rate do I have to earn such that on my 9th year I make $2110?

You want to find $r$ such that $$16790(1+r)^8r=2110$$

There's no nice exact solution, since this is a 9-th degree polynomial equation. However, Wolfram Alpha can approximate its solution, as $$r\approx0.0720367780964360645184$$

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    Thank you. I don't think I formulated my question quite properly though. At 7.2%, then after 8 years, I would be generating more in an interest than I would be saving with the Powerwall. However, in addition to saving more, I also socked away $12,058 extra over those 8 years, which means the Powerwall actually needed even more time than that to break even with my investment. I think there's actually two variables we need to solve for: the interest rate, *and* the number of years after which both would be even. – 2017-02-22
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    With two variables, there are many possible solutions. Perhaps you should reflect and ask a different question. Further, as the commenters have pointed out, the Tesla annual savings (a) can be invested, and (b) may actually increase, should the price of electricity rise. – 2017-02-23
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I figured out how to formulate this. Using "Wolfram Development Platform" we can plot the two functions as below. Since there's actually 3 variables, this is a 3D plot, and there are actually many solutions:

$$ Plot3D[{16790 + (1 + r)^t, y=2110*t}, {t, 7, 15}, {r, 0, 2}] $$

enter image description here

The blue plane represents the Powerwall investment, and the orange represents your stock investment. As you can see, for sufficiently high $r$, the blue plane never exceeds the orange plane.

It seems the numbers are a bit bigger than I expected. Will need to play around with the formula a bit more, but this is the gist of what I was getting at.