by Julian M Bucknall

…in which your blogger finds he has to take several deep breaths at the sheer vacuity and cluelessness of an anti-Tesla-solar-roof guy who tries to use a Net Present Value calculation to show that said solar roofing is not worth it. In essence, he’s saying, “I think it’s crap, ergo I’ll fudge something I don’t understand to show I’m right.”

<another deep breath>

Let’s see what the issue is. There’s an article on Inverse (not a site I’ve come across before) that discusses how much one of Tesla’s solar roofs would cost over time. Or, rather, how much you’d make from it after the initial outlay. Tesla has an online calculator to help you evaluate the costs and benefits. Here’s the screenshot from that article for 40% coverage on a 1800 sq ft house in California.

In other words, for an initial outlay of $59,100, you’d get a tax credit of $13,300, and you’d make $85,700 in energy (essentially, the savings from your power company) over 30 years.

Ah ha, says our doofus, quoting some site about Net Present Value (NPV) together with a mathematical expression to make him look like a Real Expert, natch, this means that if you ignored the tax credit, the NPV of the whole thing over 30 years comes to -$11,788 when using a discount rate of 2%. NOT WORTH IT!

Adding in the tax credit back in? A measly profit of $1,512.

The calculations and results just made me go WTF? I pulled out my trusty calculator (it used to be the HP 12C, but nowadays I use an iOS calculator app): the first scenario gave me an NPV of $4,879, the second with the tax credit $18,179. Why the big differences?

When I was in the banking biz writing trading software for interest rate swaps and options, the one thing I learned about calculations that took into account the Time Value of Money (TVM) was: draw a diagram. Understand the cash flows over time by looking at a picture. Only then can you do some calculation like find the internal rate of return, find the NPV, or whatever. Our friend essentially drew this diagram for the no tax break scenario, at least in a virtual sense:

In other words, he imagines that at the end of the 30-year period (having received *no payments* in the interim in the form of energy savings) Elon Musk is going to appear at the end of his driveway, with a big goofy smile, camera flashes popping, holding a check for $85,700. If you imagine that’s the way it works, then sure, the PV of the lump sum at the end is $47,312, so adding in the outlay of $59,100 makes an NPV of -$11,788. Yay! Except…

…that’s not what this savings calculation is all about. The calculator *expressly* states that the savings you get are over the 30-year period, not a lump sum at the end of 30 years. In other words, you would be receiving $85,700 / 30, or $2,857 at the end of every year:

The total PV of those 30 “receipts”? $63,979. So, if you subtract that initial outlay, you get an NPV of $4,879. Add in the tax break at the start and you get $18,179 as the overall NPV.

Of course, having said all that, we all pay our electricity bills *monthly*. So we should actually do these NPV calculations based on monthly savings *over 360 months*, at $238 per month. For that I get $5,306 NPV without the tax benefit, or $18,606 with it.

Of course, now we can play around. The cashflow diagram is like a mortgage: an initial outlay (for the bank) and then receiving monthly payments to settle it. The future value (FV) after 30 years is zero: the mortgager will have paid it all off. Let’s imagine that the initial outlay is a mortgage loan, and the loan is being paid monthly from the power savings, what is the interest rate? Without the tax advantage: 2.65% APR. (It’s at this point that the incoming savings, once present valued, equals the initial outlay.) With the tax advantage? 4.72% APR.

(If you have an HP 12C or equivalent: you specify four of N, i, PV, PMT, and FV, and the calculator will evaluate the fifth value. For the “pretend it’s a mortgage without the tax benefit” case, N is 360, PV is -59,100, PMT is 238, and FV is 0. The calculator makes the monthly interest rate 0.22%, which, when multiplied by 12 for the annual rate, makes 2.65% APR. For the tax benefit scenario, make the PV -45,800, then solve for i. I get 0.393% for the monthly interest rate, which is 4.72% APR.)

At the end of all that, what conclusions can I draw? With regard to the solar roof, well, if I had the house for which the Tesla calculator made this estimate, I’m afraid I don’t have $59,100 lying around to pay for the install and thereby reap the benefits. So I’d have to get an equity loan in all probability which would skew those NPVs I just calculated. Of course, such an improvement to the house (much as a remodeled kitchen or bathroom does) would increase the value of the house by some undetermined amount as well, offsetting the costs over time. In other words, if you want it, you’d possibly make the finances work and install it. If you don’t want it, this set of (corrected) calculations is not going to change your mind.

As for the article that raised my blood pressure in the first place (what’s up with that?), it’s from a climate-change denial site – no, I’m not linking to it – and it just makes me wonder about the accuracy of the other posts there. If this simple enough proposition cannot be argued *correctly* (this is friggin’ mathematics, people, as free from opinionated points of view as you can get), what hope does a reader have for accuracy for the remainder of the site? Or maybe the inaccuracies aren’t spotted because the posts reinforce a certain viewpoint? The article has at present over 200 comments and *not one commenter has seen the flaws*. In fact, oh no, the author has made a comment showing that he doesn’t understand treasury bonds either and how they work and oh god I’ve got to stop reading this shit and instead mow the lawn and water the plants and do anything to stop this madness welling up inside my brain…

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