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Should you buy LED bulbs? What about replacing an old vehicle with a more fuel-efficient vehicle? When exactly is an energy efficient upgrade worth the investment? While part of these questions has environmental and safety considerations - making the answer a definite “it depends” - the financial aspects are a little more straight-forward. Essentially the answer is “yes” if the item - LED bulbs, a car, or what have you - pays for itself in a certain time frame.

When something “pays for itself” it has cost less to run or use the same amount of money that it cost to purchase than if you had not replaced the item. In essence, this is the same as doubling your money. If it pays for itself in exactly one year, that means it’s the same as an equivalent investment that returns 100% APR.

I like math, so let’s do some math.

First, we look at the normal compound interest formula:

``````fv = pv (1 + r) ^ t
``````

Note that pv is present value, fv is final value, r is rate, and t is time in years.

Since we’re looking at when our purchase pays for itself, essentially doubling the investment income, we know:

``````fv = 2pv
``````

And can plug that into our equation:

``````2pv = pv (1 + r) ^ t
``````

Solve for r:

``````2pv = pv (1 + r) ^ t
2 = (1 + r) ^ t
2 ^ (1/t) = 1 + r
2 ^ (1/t) - 1 = r
r = 2 ^ (1/t) - 1
``````

Remember that taking a number to the power of a reciprocal is the same as taking the root of that number. So `x ^ (1/2)` is the square root of x, `x ^ (1/3)` is the cube root of x, and so on.

I put my equation into Excel and figured out the equivalent rates for years 1 through 20:

years rate
1 100%
2 41%
3 26%
4 19%
5 15%
6 12%
7 10%
8 9.1%
9 8.0%
10 7.2%
11 6.5%
12 5.9%
13 5.5%
14 5.1%
15 4.7%
16 4.4%
17 4.2%
18 3.9%
19 3.7%
20 3.5%

This means something that pays for itself in less than a year is definitely worth the investment. I’d even say something that pays for itself less than ten years is worth it. Your mileage may vary, of course. Note that I’m only compounding annually. An investment that compounds more often (e.g. monthly or daily) would get slightly better returns.

Now let’s look at a couple of examples. I’m going to write these as math word problems. Sorry if it gives anyone flashbacks to math class!

A 60-watt equivalent LED lightbulb costs \$3.25 and uses 6.5 watts of electricity. Assuming the bulb is on for eight hours a day and electricity costs 11¢ a kilowatt-hour, how soon will it pay for itself? What’s the approximate rate of return and is it worth it?

``````old daily cost: 60 watts * (1 kilowatt / 1000 watts) * 8 hours *  \$0.11 = \$0.0528
new daily cost: 6.5 watts * (1 kilowatt / 1000 watts) * 8 hours *  \$0.11 = \$0.00572
daily savings: \$0.0528 - \$0.00572 = \$0.04708
days to payoff:  \$3.25 / \$0.04708 = 69 days
approximate rate of return = 2 ^ (1/[69/365]) - 1 = 3,812%
``````

Is it worth it? Is over a 1000% return on your money worth it? Hell yeah!

Your current car gets 20 mpg. You drive an average amount, which is 3,000 miles a quarter. Gas costs \$2.00 a gallon. How much should you pay for a car that gets 50 mpg in order for it to pay for itself in ten years (7.2% interest)? Assume all other maintenance costs are the same, gas prices are constant, and your current car is worth zero.

``````old yearly cost: 3,000 miles * (4 quarters / year) * (1 gallon / 20 miles) * \$2.00 = \$1,200
new yearly cost: 3,000 miles * (4 quarters / year) *  (1 gallon / 50 miles) * \$2.00 = \$480
yearly savings: \$1,200 - \$480 = \$720
ten-year savings = \$720 * 10 years = \$7,200
``````

Oh, wow, I didn’t even use the compound interest equation here! I just decided, based on the table above, that ten years was a reasonable time since it was equivalent to 7.2% interest. Of course, the assumptions in this question were bad: maintenance costs for a 50 mpg car is probably going to be higher than maintenance costs for a 20 mpg car (since the 50 mpg car is probably a hybrid), gas is probably going to go up, and you could probably sell your current car for at least \$500 (if it’s a crappy car).

### 15 comments for When Is Energy Efficiency Worth the Investment?

• Ooohhh I like how the car example reverses it a bit to help you find out your budget. Because that’s one of the things I was going to say: for me, at least, we’re in such a crunch with the planet right now that I’m willing to pay a little more for anything that’s going to help reduce the harm.

• Hey Joe,

I really like the light bulb example, it makes it very clear, how energy efficiency can result in real money savings. It’s easy to wrongly assume that something more expensive is always more costly, even if it’s actually cheaper over the long run.

Cheers, Miguel

• Just over two months to pay off the light bulb!?! That’s pretty awesome. And yeah, our planet is in a bit of a crunch, so it’s worth making some changes just to consider future generations.

• I know! I even tried to make it worse in my example by only using it 8 hours a day instead of a full 24, and it still paid for itself fairly quickly.

• Thanks for the post–I’ve been trying to crunch the numbers on solar panels, but the different lease/buy options make it tough. This reminds me to take another look. And it’s hard to shell out so much money for a single LED lightbulb, so it’s gratifying to see how the math works.

• It felt like I was in math class all over again! Thanks for the lesson.

• Well done. I spent my career evaluating capital projects for the complex I worked at. We typically didn’t fund projects that took longer than three years to payout, with the exception of energy saving projects that we’d accept maybe a five year payout on. But that was because we had more projects than we had money to invest. For a family 8 to 10 year payback is probably about right.

• I’ve found LED’s at the dollar store and swapped out. They work great and a quick break even

• onfirebuilder

Steve Thanks for sharing,

I test and analyze existing and new home efficiency for a living, and I would love to add some of the nuances to the discussion. The problem is that when most people including many mechanical engineers are “running the numbers”, the assumptions being used don’t match the reality of the building. Question, should I add insulation or air seal or both? That’s easy every ignores air sealing because they don’t measure it or understand how it impacts the insulation ability to work. So they insulate but don’t get the return cause the insulation doesn’t work as expected. So in addition to running the numbers, validating the assumptions is important, otherwise all you end up with is deemed savings which aren’t as good as actual returns.

So with the LED example the important factors are to get a light has a high lumen per watt ratio at a minimum, and if adoption is important, get one the matches the quality of light being replaced. To get the answer for lighting you need frequency of use, so the kitchen lights will have a faster payback than the back closet, even with same bulb swap. Another factor that comes into play is that the reduction in watts from incandescent = reduction in BTU’s, 1Kwh=3412BTU/hr. One Ton of cooling capacity is 12000Btu/hr so replacing the lights can have a substantial internal gain reduction. If you don’t use ac or any types fans to cool, maybe you get to sweat a little less (priceless) If the lights are located in recess fixtures exposed to an attic it gets more complex because of the heat transfer through the attic.

There is one other factor to consider in payback that doesn’t occur in most other investments. What is the tax rate on gains from efficiency measures? if you have a simple payback of 10 years, and the life of the improvement is 30 years, you have no tax on the investment “returns” that you experience in year 20-30. Even better your tax rate may be negative due to government incentive. As the FI community well understands a dollar that you never have to spend is way better than a dollar that you have to generate by work or investment income.

Calculating the ROI is like creating a budget, looking back on your expenses after the fact helps you to calibrate the budget to reality.

• This is a nice explanation and discussion. I think LED bulbs are an interesting test case. The payback period was much quicker than I expected. I know with older CFLs I felt that they rarely lived long enough to justify the premium cost.

• RS

I’m not sure what you think of your car example. The question posed was How much should you pay for a car that gets 50 mpg? Your answer is \$7,200, which is inflated PV because you neglected the return rate. Because you are not going to find a new car for \$7,200, and extremely unlikely to find even a used car, the decision should be keep driving the current car until it’s dead. The largest cost of a car is the capital expense. If you have a paid off car- keep driving it. This also provides the opportunity to save for the next car. If you actually have the money in hand, invest it and earn more. When you are in the market for a car, you can compare the standard efficiency and high efficiency models this way too determine if the EXTRA cost is worth it.

• PV = present value? I guess my car example had one more assumption that I didn’t list: you’re paying for the car in cash (i.e. not financing). The other assumptions (all other maintenance costs are the same, gas prices are constant, and your current car is worth zero) are not realistic; I admit this in the explanation below the “answer.” This problem of bad assumptions is simply to make the math easier in the case of examples - you see it all the time in physics problems with frictionless planes and spherical cows.

And you’re absolutely right - a car that gets 50 mpg probably will cost a lot more than \$7,200, which means, money-wise at least, it’s best to stick with your current car.

• RS

Yes, I was trying to indicate present value with PV. I agree, you have to make some assumptions but only time will tell how good some assumptions are. A couple years ago we would have assumed gas prices would continue to be more than \$4/gallon. I think we agree, even with assumptions and simplifications it helps to run the numbers.