# California Solar Panels and Understanding Net Present Value

(Bloomberg) -- In 1202, the Pisan customs official’s son who has come to be known as Fibonacci gave the world the mathematical tools to calculate the present value of a future stream of money. Now, 816 years later, net present value remains such a foreign concept to most people that it’s deemed too arcane to mention before the general public.

At least, that’s the impression I got after reading through the news coverage of the California Energy Commission’s decision last week to require solar panels on virtually all new houses and low-rise apartment and condominium buildings (those with roofs that are especially tiny or are in the shade most of the time are exempted). I’m not the hugest fan of the move, given that the state’s need for more housing seems to be greater than its need for more solar panels. But that’s an argument for another day. What I’m curious about is whether it’s a good deal for prospective homebuyers in California or a bad one.

The basic numbers on this, provided by the California Energy Commission and cited in multiple news articles, are that the new standard “will increase the cost of constructing a new home by about \$9,500 but will save \$19,000 in energy and maintenance costs over 30 years.”

OK, \$19,000 is twice as much as \$9,500. But you wouldn’t be getting all those savings at once, would you? Divide that \$19,000 over 30 years, and plug the resulting \$633.33 annual savings into Fibonacci’s formula, and it’s worth nowhere near twice as much as \$9,500. A key element in the formula is the interest rate by which future values are discounted. Use 4.3 percent — the lowest 30-year mortgage rate I could find online — and the net present value is \$10,563. That’s not a big savings!

Except … the next line of the energy commission’s FAQ on the new standards says that, if one buys a new solar-equipped house with a 30-year mortgage, the added mortgage cost will be \$40 a month and the energy savings \$80 a month. If you divide \$19,000 by 360 months (30 years times 12 months), you only get \$53. So maybe that \$19,000 is the net present value of the estimated future savings. Sure enough, after some digging, I found the “PreRulemaking” document for the new solar rule, which indeed makes clear that what they’re talking about are “present-value energy cost savings over the 30-year period of analysis.”

Why didn’t they say that in the documents intended for public and media consumption? I’m guessing it’s because they figured the words “present value” would be off-putting. Or maybe it’s that everybody already knows that solar cost savings estimates are expressed in present-value terms and I’m getting all worked up over nothing, but I don’t think so. I did a search on the phrase “present value” on Google News and all I came up with was a bunch of exercises in corporate valuation from Simply Wall Street, a CFO Magazine article on “The Renewed Importance of Purchase Price Allocations” and a syndicated newspaper column by “The Mortgage Professor.” That last one was the only one aimed at a general readership.

Present value is a concept essential to informed decision-making about the future — and thus to modern life — yet very few people outside the financial sector (and even there I have some doubts) seem to know what it entails. Certainly most journalists don’t. (I didn’t until I went and wrote a book about the history of academic financial theory.) But the calculation is really simple, and it makes intuitive sense.

First, the calculation: The present value of a sum of money to be received n years in the future is that sum of money divided by 1 plus the interest rate (which means 1.05 if the interest rate is 5 percent) to the power of n. So \$10 in two years at 5 percent interest is \$10 divided by 1.05 squared, which comes to \$9.07. If you’re getting \$10 a year for a number of years, all you need to do is add up the values for all the years, which is easy enough to do in Excel, Google Sheets, Apple’s Numbers or another spreadsheet of your choice, although you can also just use an online NPV calculator.

That \$10 a year from now is worth less than \$10 today just makes sense, right? You might die in the interim, or you might have some use you could put the money to over the next year (an investment, say). Plus, there’s the threat of inflation: You might not be able to buy as much with \$10 years from now as you can now. Figuring out exactly how much less that \$10 is worth is a bit less straightforward, of course. It’s all about the choice of discount rate, and while sometimes that’s obvious, in many real-world applications, it is not. “Discount-rate variation,” John Cochrane said in his presidential address to the American Finance Association in 2011, “is the central organizing question of current asset-pricing research.”

So present-value calculations don’t always give the right answer, in asset valuation, solar power savings estimates or other endeavors. But they’re often essential to framing the right questions. Such as: What discount rate does the California Energy Commission use in calculating the present value of future energy savings? According to the commission’s latest “Time Dependent Valuation of Energy for Developing Building Efficiency Standards” report, it’s 3 percent. That’s less than the 30-year mortgage rate, but the lower rate makes sense given that they’re estimating the future savings in current dollars, not in inflated future dollars, whereas mortgage rates factor in an estimate of future inflation.

The estimated average future savings add up to about \$30,000, the present value of that is \$19,000, and that’s twice the present cost. This sounds pretty advantageous for California homebuyers. It’s most advantageous if you’re buying in sunny and often scorchingly hot Palm Springs or points south and east (aka California Climate Zone 15), where the average estimated electricity cost savings per new house is \$33,911, and least advantageous if you’re buying just over the mountains in San Diego (Climate Zone 7), where the estimated cost savings is only \$15,900 because it gets foggy in the mornings and the mild climate keeps home energy use down. But it seems like a pretty good deal all around.

Another big variable in this particular present-value calculation, though, is the future path of electricity prices. The California Energy Commission makes periodic forecasts, and it seems pretty transparent about the assumptions involved. Energy economist Severin Borenstein of the University of California at Berkeley complained last week that “the savings calculated for the households are based on residential electricity rates that are far above the actual cost of providing incremental energy.” But if electricity prices go down — and requiring solar panels on virtually every new house in California may push them even further in that direction — then, well, energy costs will be lower. As a matter of public policy, it may be better, as Borenstein urges, to focus resources on large-scale solar generation instead of rooftop installations. But for people buying new houses in California, it seems like they’ll be paying a lot less for electricity in either case.

It is true that many, perhaps most of us, intuitively discount the future in ways that don’t comport with standard present-value models. This has come to be known in the behavioral economics literature as “hyperbolic discounting,” the idea being that the discount rate in your personal present-value calculation might start out low but rises sharply after a year or two before settling down again in the distant future. If your discount rate is 3 percent for the first two years and 15 percent after that, \$30,000 in energy savings evenly allotted over 30 years has a present value of just \$4,341. In that case, you would find the new California rule to be a travesty. You would also be, how shall we put it, a little bonkers. Sometimes it makes sense for the government to step in and wrest such decisions from individuals’ hands, as it does with Social Security. Sometimes it doesn’t.

Which side of that line does the California Energy Commission’s decision — which also involves considerations of air pollution, climate change and energy reliability — fall on? Really, I don’t know, and I could have arrived at that same unhelpful conclusion without a tutorial in present-value calculation. But wasn’t it more fun this way?

To contact the author of this story: Justin Fox at justinfox@bloomberg.net.

1. The classic treatment of all these factors is economist Irving Fisher's "The Theory of Interest, as Determined by impatience to Spend Income and Opportunity to Invest It" which is available online from the Library Fund BUT BE WARNED THE TEXT IS ALL IN CAPS.