Back of envelope financial calculations: Warranties, fine; insurance, no! (statistics)
created: 10 Mar 2009; modified: 06 Dec 2016; status: finished; belief: highly likely

Consider the poor consumer. Insurance is offered for all sorts of things, and often the consumer buys it - even when he shouldn’t. One of the problems in an inefficient marketplace - like the ones we often must purchase in - is that there’s a no-trade theorem of sorts in play: if the insurance was fair, the insurer would make no profit, so why would they offer it at all? They’ll only offer one which makes them a profit. Therefore, all the insurances on offer are unfair (you’ll get less out of it than you paid) and you shouldn’t buy any!

Of course, we know why one would purchase insurance: because the risks one is insuring against are too large to be borne at any given time (even though one can pay for them eventually). A house burning down, chemotherapy, a car totaled, etc. One buys insurance as a way to trade many small doable payments for a single large impossible payment. This is a valuable service to you, so you don’t mind buying unfair insurance; your lower expected value is traded off against a smaller variance of your future expenses. (People are well known to be risk averse; the rich are less so than the poor, which is sad1.)

But not all insurance is of the catastrophic variety. I’ve seen insurance offered on travel trips, airplane flights, TVs, and even video game consoles! And that insurance is expensive, dozens or hundreds of dollars. It’s strange that people apparently think they can afford the steep insurance fees but not bear the cost of just buying a new console or whatever. The irony is especially rich when one considers that people grossly overestimate how unhappy they would become after major traumas, and also smaller losses2; how much less so if their iPod or Xbox broke? This is probably due to the endowment effect; especially ironic is that insurance - the option to change one’s mind - may sabotage one’s enjoyment of the purchase3.

These purchases make no sense from the original rationale for buying insurance. Nor are these insurances trivial side-lines companies run to humor their consumers: they are popular services (~31%4 of purchases in one sample), and they are highly profitable5.

Worst Buy

Let’s at a real example. The smaller the good, the more obviously a bad deal these kinds of insurances are. I recently saw offers of console insurance at Best Buy (sandwiched between $100 HDMI cables), and was struck at how exploitative the offer must be. (Best Buy is notorious among geeks for its extraordinarily high prices and poor customer service6.)

What a deal

The price varies per item, but let’s take a presumably common one. As of 9 March 2009, offered an Xbox 360 for $399.99. The basic insurance plan on that Xbox 360 will run $59.99. That is, you will pay ~16% extra for insurance (). And it’s good only for 2 years or so. What does this considerable premium buy you?

Our obligations under this Plan will be fulfilled in their entirety if we replace your product, issue you a voucher or gift card or reimburse you for replacement of your product pursuant to these terms and conditions.

It buys you one replacement; or:

After three qualified (3) service repairs have been completed on an individual product and that individual product requires a fourth qualified (4th) repair, as determined by us, we will replace it with a product of comparable performance of like kind and quality not to exceed the original purchase price.

3 repairs and a replacement. And that’s it. But of course, it’s not unconditional. You can’t simply walk into Best Buy after a few days and announce you want a new Xbox 360. Here’s invalid reasons to seek repair:

This Plan does not cover:

  1. damage to your product caused by accident (unless you have purchased the optional ADH Coverage), abuse, neglect, intentional physical damage, misuse (including faulty installation, repair, or maintenance by anyone other than an authorized service provider), unauthorized modification, viruses, extreme environment (including extreme temperature or humidity), external condensation, lightning, static electricity, fire, flood, insect infestation, rodents, war, terrorism, computer software related failures (unless you have the Vi-Spy Coverage) Acts of God or other external causes;
  2. products that have been lost or stolen. This Plan only covers products that are returned to us in their entirety;
  3. cosmetic damage to your product including but not limited to scratches, dents and broken plastic on parts, that does not otherwise affect its functionality or materially impair your use;
  4. products with a serial number that has been altered, defaced or removed;
  5. problems caused by a device that is not your product, including equipment purchased at the same time as your product;
  6. consumable parts, such as batteries, unless expressly provided for herein;
  7. damage to, or loss of any software or data residing or recorded in your product…
  8. damage to your xBox 360 due to Microsoft’s Red Ring of Death

Well, shucks. That seems to cover just about everything that would damage my new Xbox 360! If it ships broken - as is extremely likely and seems to cover almost all of the reported breakage rates7, then it’s the original manufacturer’s problem (Microsoft); if I break it by accident, it’s my problem; if it gets scratched up, it’s my problem (and god help me if the scratches are anywhere near the serial number! Then I can’t use the plan at all!); if my house damages it, it’s my problem; if one of the many mechanical and electronic problems with the Xbox manifests, it’s either my or Microsoft’s problem; etc. This doesn’t seem to cover very much, although I suppose it covers if my Xbox stops working for no reason whatsoever one day. (Which isn’t a terribly common problem with video game consoles.)

We’re interested not in what happens if we get lucky and immediately the Xbox 360 breaks and the manufacturer’s warranty doesn’t cover it and Best Buy will give us a new one8 - after all, if we only go by best case scenario, the lottery is the greatest deal around - but rather in what we can expect on average; we have no a priori reason to expect to be luckier or unluckier than average. So we use the usual expected value formula:

payoff×probabilityOfPayoff=expectedValue\text{payoff} \times \text{probabilityOfPayoff} = \text{expectedValue}

Generally, one only wants to engage in deals or actions which have a positive expected value. Does the Xbox insurance have a positive value? Let’s say I have no idea of the true rates9. Certainly neither you nor I have access to good information about how many of Best Buy’s customers actually use the insurance successfully (although Best Buy surely does). But we can still put some bounds on it: we can ask the question, how likely would Xbox-breaking have to be before the insurance has a positive payoff? That is, we know the payoff - it’s 340. And we know the expected value we want by definition is for it to be greater than 0; but we want to get more out than we put in, so we need to get back at least $60 if we want to just break-even. With 2 out of the 3 variables solved, basic algebra can now figure out the final variable, the probability:

  1. 340×p=60340 \times p = 60
  2. 340×p60=6060\frac{340 \times p}{60} = \frac{60}{60}
  3. 34060×p=1\frac{340}{60} \times p = 1
  4. 5.7×p=15.7 \times p = 1
  5. p=15.7p = \frac{1}{5.7}

More details

So we need a 1-in-6 chance of redeeming the insurance. But wait! The 16\frac{1}{6} doesn’t apply directly to the chance of breaking an Xbox. Rather, it applies to the conjunction I expressed before: breaking the Xbox and Microsoft not covering it and Best Buy making good10 and us actually using it. As we all know, a conjunction of a bunch of probabilities is going to be less likely than any of the probabilities in themselves. So the first probability, breaking the Xbox is going to be more likely than just 16\frac{1}{6}.

How much more likely? Well, we can play with some more numbers here. Let’s see what happens if we are very generous & assign all the other probabilities something like a 90% chance. We know the total is 16\frac{1}{6}, and we know the other 3 we’ve defined to 0.90, and we leave one variable to solve for, breaking the Xbox (pxpx):

  1. 16=0.90×0.90×0.90×px\frac{1}{6} = 0.90 \times 0.90 \times 0.90 \times px
  2. 16=0.729×px\frac{1}{6} = 0.729 \times px
  3. 160.729=px\frac{\frac{1}{6}}{0.729} = px
  4. px=160.729px = \frac{\frac{1}{6}}{0.729}
  5. px=0.170.729px = \frac{0.17}{0.729}
  6. px=0.229px = 0.229

So just by being a little more realistic, we see that for the insurance to make sense, we need an awfully high rate of Xbox breakage. And the necessary breakage rate for us to break-even becomes still worse, still more incredible, if we were to be less generous in the conjunction.

I don’t know about you, but I don’t believe I have a 1 in 4 chance of breaking my new Xbox within the 2 years the insurance covers. I believe the chances are much lower11. And that makes this insurance one lousy deal.

Deeper down the (rabbit) hole

Nor is console insurance the worst offer Best Buy has. It recently (February 2011) began a curious buy/lease program12. In this scheme, the customer pays 10% up front, and in exchange, if the customer returns the item for any reasons within 6 months, Best Buy will give them a 50% discount on a new/replacement item. It’s hard to see who this trade-in offer is for. One’s time is not free and one doesn’t want to switch products too frequently; and if we assume generously that a compelling new product comes out randomly every 2 or 3 years, the numbers are even worse.

  1. Chen et al 2009:

    The results from the heterogeneity analysis are intriguing. In contrast to prior empirical findings (Padmanabhan 1995; Padmanabhan and Rao 1993), we find that, as compared to high-income consumers, low-income consumers are more likely to purchase ESCs. The analysis reveals that they are likely to do so because they are more sensitive to the replacement costs in the event of product failure. Unlike in the case of automobiles, where high-income consumers buy ESCs to avoid maintenance because they have higher time costs, in the electronics product category, low-income consumers buy insurance to hedge against the out-of-pocket costs of replacing the product. Additionally, lower-income consumers are also more predisposed to exercise the savings obtained from promotions to purchase ESCs.

    If the ESCs do, in fact, offer little value, the results imply a perverse impact on consumer welfare. The lack of financial ability of low-income consumers to replace products induces them to pay a potentially unnecessary and overpriced insurance premium. High-income consumers, for whom product replacement is not a cause for anxiety, incur a lower total cost of product acquisition. These findings are somewhat ironic in light of observations that poor sick patients who are unable to afford health insurance pay the highest prices for drugs (e.g., Frank 2001). In the health domain, poor patients are unable to afford insurance in a category where it is salubrious to buy it, but they opt for insurance in an area where the investment is inexpedient.

  2. If money doesn’t make you happy, then you probably aren’t spending it right, Dunn et al 2011, Journal of Consumer Psychology:

    Research on how well people cope with a wide variety of traumas and tragedies-from heart attacks to terrorist attacks-suggests that people are not the emotionally fragile creatures they often imagine themselves to be (Bonanno, 2004; Ubel, 2006).

    …research suggests that people don’t know much about their own psychological immune systems (Gilbert, Pinel, Wilson, Blumberg, & Wheatley, 1998), and as a result they overestimate their vulnerability to negative affect…research shows that this expectation is wrong. Kermer et al. (2006) gave participants $5, and then flipped a coin. Participants were told that if the coin came up one way they would get an additional $5, and if it came up the other way they would lose $3 of their initial endowment. Although participants expected to be more emotionally affected by the loss of $3 than by the gain of $5, they were not.

    …When passengers on a train were asked to estimate how much regret they would feel have felt if they had missed the train by 5 minutes or 1 minute, they estimated that they would have felt more regret in the latter case than the former. And yet, passengers who had actually missed their trains by 1 and 5 min reported remarkably little regret, and equally little regret regardless of whether they had missed the train by 5 min or by 1 (Gilbert, Morewedge, Risen, and Wilson 2004).

    For another good example of people failing to understand themselves, see the next footnote on how return policies (and thus warranties/insurance?) can sabotage one’s satisfaction with a purchase.

  3. Dunn et al 2011:

    Unfortunately, this handy mental mechanism may actually be short-circuited by generous return policies. Gilbert and Ebert (2002) offered participants the choice between prints of paintings by artists ranging from Van Gogh to El Greco. After participants made their selection, half of them were presented with the equivalent of a generous store return policy: they were told, If you change your mind about which poster you want to take home before you leave today or even any time in the next month, you can just let me know and we will exchange it for you. The remaining participants were informed that no such exchange would be possible and that their choice was final. Participants who knew they were stuck with the poster they had chosen responded by inflating their appreciation of it, seeing the poster in a more positive light than they had initially. In contrast, participants who knew they could exchange their poster anytime were deprived of this emotional benefit of commitment and found the poster no more attractive than they had before selecting it (see also Frey, 1981; Frey, Kumpf, Irle, & Gniech, 1984; Girard, 1968; Jecker, 1964). Interestingly, however, participants failed to predict this difference and thought they would be equally happy whether they could exchange their poster or not.

  4. From A Chance to Keep Up With New Technology (for a Price), Stross 2011, New York Times:

    Professor Kalra was co-author of a 2009 article published in the Journal of Consumer Research that examined purchase records from the electronics department of an unidentified retailer. [Tao Chen, Ajay Kalra and Baohong Sun (2009), Why do Consumers Buy Extended Service Contracts, Journal of Consumer Research Volume 36 (December), 611-623] An extended service contract was bought in about 3 of every 10 transactions.

  5. From Stross 2011:

    Mr. Fassler says Best Buy has not disclosed for many years how much extended warranties contribute to its profits. Consumer electronics retailing is historically a low-margin business that is dependent on extended warranties for profitability, he says. Perhaps extended warranties have become even more valuable to Best Buy recently, as its content businesses - like movies and music - have shrunk.

    Warranty Week provides aggregate and broken-down figures in Extended Warranty Administrators: While auto and PC manufacturers have the top spots, insurance companies and third party administrators grab the bulk of the pie:

    Just as in previous editions we’ve sized the product warranty industry to be in the neighborhood of $25 billion per year, in this edition we’re suggesting that extended warranties generate in the vicinity of $15 billion per year in premiums paid. Not all that money goes to the actual administrators and underwriters of the policies, however. In fact, we’re estimating that roughly half is kept by the actual sellers – the retailers and dealers in the world of warranty. Only $7.5 billion passes through to the administrators, we estimate.

    The Warranty Windfall by R. Berner Business Week (2004), offers more details on household names:

    Warranties cost virtually nothing to market, and the products they insure rarely need repairs. Says FTN Midwest Securities Corp. analyst Daryl Boehringer: It’s just pure profit flowing down to the bottom line.

    Last year, profits from warranties accounted for all of Circuit City’s operating income and almost half of Best Buy’s, say analysts. They figure that profit margins on contracts are between 50% and 60%. That’s nearly 18 times the margin on the goods themselves. For example, a four-year contract on a $3,000 flat-panel TV costs about $400. Best Buy gives its insurers $160 and keeps $240 for itself.

    …As service contracts become more critical to its bottom line, Best Buy has actually cut back on disclosure. The Richfield (Minn.)-based chain doesn’t report its warranty profits separately, though it used to give the percentage of sales that the contracts comprised. It stopped doing that after fiscal 2001 and buried the number in a revenue category labeled other. Then for fiscal 2004 it stopped reporting the other category altogether.

    Circuit City is more forthcoming. The Richmond (Va.)-based outfit reports how much revenue the contracts generate, along with the percentage of sales they make up – but not the profit they produce. For the year ended Feb. 29, it said its warranty revenue totaled $326 million, or 3.3% of sales. Clearly, says SAFE’s Sebastian, the retailers don’t want to disclose to J.Q. Public how much money they are making on these contracts.

    Using details gleaned from industry sources, though, analyst Boehringer put together estimates of just how lucrative these contracts are. For the year ended Feb. 28, he estimates contract profits accounted for 45%, or $600 million, of Best Buy’s $1.3 billion operating profit. He figures that without contract profits, Circuit City would have posted an operating loss from continuing operations of $195 million last year instead of a $564,000 profit.

  6. See eg. Wikipedia’s Best Buy#Criticism and controversy section, Google for best buy sucks, the break-down by category of complaints to the Better Business Bureau; or you can just read Fortune Magazine’s Best Buy’s giant gamble article for yourself. (If you can read through this article without realizing that Best Buy admits that its corporate strategy is charge suckers high prices, then you need to work on your critical thinking skills.)

  7. Reportedly, Game Informer found a 54.2% breakage rate for Xbox 360s by surveying readers; Examiner claims just <16.5% based on how many Xbox 360s are active online. Neither figure seems very reliable.

  8. A situation in which the insurance is obviously profitable, as what would have been an $800 expenditure becomes a $460 expenditure - which saves a healthy $340.

  9. Chen et al 2009 note that Consumer Reports listed the overall failure rate of video game equipment at 9%.

  10. not a trivial consideration, given their customer service and the lengthy list of excluded issues

  11. From Stross 2011 (NYT):

    Mark Kotkin, director of survey research at Consumer Reports, says, The salesperson often tells you, This will give you peace of mind in case you need an expensive repair. I would get it. But the odds of a product breaking down during a typical extended warranty period are low. Even if the product does break, the cost of repair is not much more, on average, than the cost of the warranty, he says. He concludes that extended warranties are a bad bet.

    Ajay Kalra, a marketing professor at Rice University, agrees. All the statistics are compelling: in almost all cases, you shouldn’t buy the extended warranty, he says.

  12. Stross 2011.