Insurance of any kind is always going to have a lower expected value than self-insuring. If there's a 10% chance of a loss of ₹100 occurring, insurance companies will charge more than ₹10, to cover their overheads and earn a profit. If you self-insure, you keep the profit, and eliminate the overheads.
If you can't handle the worst-case scenario, if it will cost too much relative to your assets, then you should buy insurance. End of story. This question refers only to situations in which you can comfortably handle the worst-case scenario.
In my case, instead of paying the insurance company, I invest the money. I switched from comprehensive insurance for my car to third-party. I don't buy insurance for my phone or computer when I'm offered one. And so on.
I instead invest the money, most in equity mutual funds. I'm taking a bit of risk to make a profit, and I'm comfortable with that.
But it recently struck me that I could instead take the risk in a different way. Buy the insurance, and adjust my investment portfolio towards higher risk/return than I'd otherwise choose.
In other words:
BASELINE SCENARIO: Third-party car insurance, investment portfolio consists of 80% equity and 20% debt
ALTERNATIVE SCENARIO: Comprehensive car insurance, investment portfolio consists of >80% equity and <20% debt.
In effect, I'm substituting the risk of loss from my car being destroyed for the risk of a stock market decline.
Which of these scenarios provides more returns for the same level of risk? Or, alternatively, provides the same returns with less risk? How would you quantify and analyse this?
If someone came to you and presented their portfolio of investments and insurance policies, and asked you to reduce risk in their financial life, would you do so by increasing the percentage of debt in their investments, or by buying insurance for things that they haven't insured? Would you be able to justify mathematically why your choice is better?
Here's a concrete example:
One of my family members has medical insurance for a low amount which may not even be sufficient to cover routine hospitalisation without surgery. It's also low relative to my expenses — it's just 1-2 months of expenses.
I considered discontinuing this insurance, but my financial advisor suggested I keep it, since the sum insured is 17 times the annual premium. As he put it, it's a good deal to spend ₹1 to get back ₹17 if things go wrong .
Is he correct? Is 17x a good deal? 40x? 7x? How do I analyse this?
 Let's rule out the possibility of getting insurance for a higher amount, for reasons I prefer not to go into here. The question is whether the given insurance policy is worth it relative to self-insuring.