Home PV as an investment

The purpose of this post is to illustrate some of the finance related issues involved in considering a rooftop photovoltaic (PV) system. If you set aside the environmental benefits, then the decision to install PV is an investment decision. To illustrate the decision making process I will consider three options for a homeowner considering a PV system. 1. Don't buy system and invest the money into some security. 2. Buy the system with cash and use the saving to invest in a security. 3. Obtain a loan to purchase the system and invest the cash not used to purchase the system into some security, along with investing the savings from the financed system into the same security. To focus only on the investment problem I will use Allan R. Hoffman's assumptions and calculations as the basic economics of the PV system found here and just consider the rooftop system with no storage. This is not a complete model, and we're going to leave some considerations (risk, volatility, panel degradation, etc..) out. If there is enough interest in this post I will write some follow-up posts including some of those factors, and construct more sophisticated models. Lastly, a warning: doing the math is unavoidable, but it requires nothing more than algebra, and the formulas are easy enough to be plugged into a calculator or excel spreadsheet.
The first we must consider is the "cost of money," which we can do by running scenario 1. Let's assume we have $12,250, let's call this "P₀", handed to us, to do with what we please. I don't buy the system and invest the money. Say there exist a fund that yields 7% APR (annual percentage rate as compounded monthly), and I place P₀ into that fund. Then after i months the money in that fund grows to "P_i", is given by the compound interest equation.

Where r is 0.07. If you consider 15 years, or 180 months, the compound interest equation tells you that the $12,250 has grown to $34,900. The other factor you must consider is the cost of keeping the money liquid as you save. Since I don't know anyone willing to hand me $12,250 I must save up money over time. It is likely the case that my fund is either not liquid (has some specified term in which the money cannot be withdrawn), or that it isn't wise to treat the money in the fund as liquid. Thus the fund cannot be used to save money for the PV system. Then to derive the formula I need to calculate the potential account balance in this case let's assume my only option is to stick the money into a basic bank account at 0% interest. Assuming I can save $1,020.8 (call this "s") a month so I can buy the system in a year, then if I don't plan on buying the system I could put that money into my fund each month. To make the calculation easier let me call  (1 + r / 12) = γ. Then the compound interest equation reads P_i = P₀ * γⁱ. Consider the money we have in our investment account for the 1st (A₁), 2nd (A₂), 3rd (A₃), months below.

As you can see every month the previous month's balance grows by a factor of γ, and then I deposit "s" dollars into the account. I can factor out the "s" and switch the order of terms to reveal a pattern. If you then continue on then for the i^th month you get.

 Where the ... represents the terms I didn't write to abbreviate the expression. The sum in the parenthesis is very famous, it's probably one of the first sums mankind ever solved. Which is probably because its solution would be very useful to ancient bankers. If you want to derive the solution to the sum yourself then just write the sum as equal to some variable K, multiply both sides by 1-γ, and then solve for K. In any event this gives us

This formula probably has name that I don't know, but let's just call it the basic investor's formula (BIF). It's useful any time you have a fixed recurring payment into a loan or investment at a fixed interest rate. Using this formula we get $12,250*(1.0328)=$12,651. Plugging this into the interest rate equation we get an investment value of $36,041. Not a huge effect in this case, but if it takes you longer to save or you're considering a more expensive system it can have a large effect on your opportunity cost. With the potential value of this investment nailed down we can now consider purchasing a PV system with cash.

If you need to save up the money for the system you can use the above formula with the interest rate of your money market account or other similar short term investment option. The highest rate I found for a money market account was ~0.85%, plugging this in gives $12,298 after 12 months. In this case I only get an extra $48 which I will put into my fund immediately. For generality I will keep this in my formula as "k₀." Note that if k₀ is greater than s, you can buy the system early. Now from Allan's calculations this system saves me $815 a year which I will label "k." Each month I will put in k/12 dollars into my fund which nicely cancels the 12 in the BIF formula. Then on the i^th month the return on my solar investment "S_i" is obtained by using the interest rate equation on k₀ and BIF on my utility bill savings and adding the two.

Let's name this one the solar investment formula (SIF). If you have the money immediately available just drop the first term, and use the first potential investment value to compare with. Plugging this in for 180 months gives me $137 + $21,527 = $21,664. Now there's a question about the value of the PV system. When installed it might be worth $12,250 in additional value to the house. It might even be worth more than you paid. But after 15 years you must also factor in some amount of deprecation. This is an open question, as bob wallace pointed out the panels may degrade between 0.1%-0.4% per year according to NREL, and they may last 50 year or more. However appraisers are likely to base their valuation on the warranted life of the panels since their performance is not guaranteed after then. On the other hand a potential homebuyer may be willing to pay a fairly large fraction of the systems original value because they're betting the panels will still produce significant amounts of energy after the warranted life. However if the buyer is getting mortgage the price he pays for the home can't deviate from the appraised value by too much because the bank will see the loan as too risky. In any event I will be generous and have the system retain 95% of its value after 15 years. In that case our total value is $21,527 + (0.95)*$12,250 = $33,165. This is less than scenario 1., but I haven't considered taxes because it depends on location, and whether you use something like a Roth IRA or 401K. You'll have to consider the taxes on all your options on a case by case basis, but I'll include one example. Most often in the US only the capital gains are taxable, and if it's a long term investment, the rate is 15% unless you're too wealthy. If you sell a primary residence you don't have to pay taxes on $250,000- or $500,000 if married- of capitals gains from the sell. So most likely the value of the PV system isn't taxed when realized. In this case after taxes the solar investment yields $21,527 - (0.15) * $(21,527 - 15*815) + (0.95) * $12,250 = $31,769 while the previous investment gives $32,472. So it looks like for 15 years investing the cash wins by ~ $700. However I only picked 15 years because it was the payback time suggested by Allen's calculations so I thought it was an interesting time horizon to investigate. There's also the matter of the degradation we ignored. If panels have only lost 1.5%-6% efficiency over 15 years I'm confident that our error is less than a few percent. Nonetheless you can just easily plug in 20 or 25 years into the formulas I've derived. However the value of the solar investment will never equal the value of the other investment because the rate return on investment (ROI) is 815/12250 ~ 6.65%. An investment with 7% ROI will always grow more quickly. If you read nothing else from this post consider these few lines. The best investment is typically the one with the better ROI if they have similar levels of risk. I say typically because there are sometimes other factors to consider. With all that out of the way let's consider a loan.

Let's assume I have the cash in hand, and that I don't need a down payment. Using a home equity loan will likely get you the lowest interest rate, and a few minutes on the internet shows an offer for 2.7% APR. I'll skip the calculation of the monthly payment "p" and just give you the result. If you're interested I'll make short post, but it works exactly like my derivation of SIF except with an added minus sign. If We pick a 30 year loan the payment ends up being $50/month = $600/year. This eats up most of my savings, but I still have the $12,250. What's our ROI? The 12,250 gets 7% and we get solar investment gives use (815-600)/12250 ~1.76%. A combined ROI gives us 8.78% which tells immediately that this is probably our best option. But the numbers are easy to calculate since we already did all the work. Just use the SIF formula k=$215 and k₀=$12,250. The result is $34,900 + $5,679 = $40,579. Of course if you don't have the cash on hand this option is even better. You can get the PV system now while putting all of the money into an investment account. This early start nets you $36,041 + $6,311 = $42,352. Now since it's a 30 year loan at this point you only own a bit less than half the value of PV system, but our investment account is already far ahead of the other options. And I haven't even included the tax benefit of deductible interest on the loan. So I think I've made a clear case for finance in this example.

Of course all of these figures are only plausible assumptions, and we didn't discuss risk. A higher yield investment (even if it's a general index which is extremely unlikely to lose money over a long term) is only appropriate if you're flexible about when you need to realize the gains by selling the investment. So your options should be chosen based upon your risk tolerance. Lastly the many issues we neglected will have to be factored into your decision. Some of these things can likely be done by hand to make them more transparent, while others will require a little bit of computer code. If there's interest I can write posts exploring some of these topics. If there's even more interest I can write a completely independent solar investment calculator.