In very rough terms the world of insurance is divided into life and non-life insurance. Non-life insurance is for things like your car, house, travel, company, and other non-life things. We all know how it works. You pay £500 to insure your car against a number of things, including for example theft. Let's say it is a £10,000 value car. In simple terms, the probability of making a claim against the full value of the car in any one year has to be 5%. Without necessarily doing it in those terms, most buyers of insurance probably consider that about right and therefore worth it.
The reason I would not buy the £500 insurance on my £10,000 car (other than the 3rd party insurance which is required by law) has to do with my knowledge of the insurance company's combined ratio. The combined ratio is the sum of the claims and expense ratio. The claims ratio is exactly that - what the company pays out in claims to people whose cars were stolen or damaged. And the expense ratio is all the other costs of the insurance company; marketing, administration, overhead, etc. Insurance companies can have combined ratios over 100%; if claims don't come due for a while the insurers earn an interest on the premiums they collected until the claim falls due. But since car insurance is typically a one year policy the combined ratio for this policy should be below 100% to be profitable.
For car insurance the risks are somewhat predictable and the insurance company are likely to have a good idea of the number of claims and expenses they will face (insurers can reinsure risks they don't wish to hold fully themselves). Using very rough numbers the insurance company might have a combined ratio of 95% for these policies made up of a 70% claims ratio and 25% expense ratio (my friends in insurance will bemoan this simplification). So essentially if you are an average risk customer, for every time you pay £100 in premium on your car insurance you get £70 back in claims and it costs £25 for the insurance company to make it all happen, and they take a £5 profit. Put in other words, you are paying £30 for the peace of mind of having the insurance. You obviously don't get £70 back. Most of the time you get nothing back as you didn't make a claim on the insurance company, and then when misfortune strikes you get your £10,000 back; on average you get £70 back.
So the reason I don't buy insurance is that I don't want to pay the 30% in cases where I can afford the loss (25% expenses plus 5% profit to the insurance company). Obviously it would really stink to have my car stolen or damaged to the tune of the full £10,000, but I see this as a risk I can afford to bear and don't need to pay to protect against. Importantly I don't think that I save the full £500 in annual car insurance. I think that I save the 30% difference between what I paid and the average claims. In my view the insurance company knows as much about my risk as buyer of insurance as I do, and if they set the average pay-out for me at 70% of a £500 policy then that is probably about right. So using this case of car insurance to extrapolate how I think about insurance in general, on average over all the insurance policies I don't buy I would expect to have a loss of £350 (70% of £500) on my car in any one year, and have saved £150 by not buying insurance (30% of £500).
Not buying insurance against things we can afford to replace or have happen does not mean that those things don't happen. It just means that instead of having the small bleed of constantly paying small premiums for lots of small things we will once in a while be paying out larger replacements amounts for things we did not insure against. Personally I also think the whole hassle of keeping track of insurance policies is a pain I would rather avoid and I also seem to constantly hear stories about insurance companies that either fought claims or made claiming on a policy a huge headache.
Without being scientific about it including all insurance forms that I don't buy (including life insurance) I think I save about £500 per year in expense ratio and insurance company profit. Assuming that I took this money every year for the next 30 years and invested it in the broader equity markets and was able to return 5% on that money, my savings from not buying insurance over the period would amount to around £35,000 in present money. This is money I have instead of it being in the insurance company's pockets in 30 years. Importantly this saving does not assume that I do not have accidents or have my car stolen. In fact it assumes that I am at risk of those things exactly with the same probability that the insurance companies assume.
Investment advice typically have an "always seek expert advice" or "don't try this at home" disclaimer, but here it really applies. You should not save on insurance premium payments in instances where you can't afford the loss; and everyone are different in terms of what they can afford to lose. Almost everyone couldn't afford to lose their house in a fire so they should insure against this possibility (you probably couldn't get a mortgage if you didn't). Most people in countries without national health services couldn't afford bad health reverses and should get health insurance. Many can't afford to have bad things happen to their car or their homes broken in to, so they should insure against that. But most people can afford to lose their mobile phone, having to cancel a flight or vacation, or an increase in the price of their electricity bill, and they should not insure against those things. And even if there are things you need to buy insurance for you should always get a high deductible which in turn will lower the cost of the insurance policy. Over time having no insurance or a high deductible when you do will save you quite a bit of money, and that should make you sleep better at night. And perhaps you will look after that mobile phone just a little bit more because it is not insured which in turn will lower the risk that you inadvertently lose it.
Similarly there are many instances where life insurance makes sense. As with the case of annuities many life products have an investment component to them, but obviously also a life component. If you are in a situation where your death or disability will cause unbearable financial stress on your descendants then the premium you pay on these policies make sense. As with the example of car insurance, you should do so when you or your descendants can't afford the loss. Whether they can or not is obviously a highly individual thing, but bear in mind that as with all insurance products there is a tangible financial cost to the intangible peace of mind many people cherish in insurance. Make sure it is worth it.