Insurance: When to get insured, and when not to

Personal Finance covers a wide range of topics, from savings, budgeting, investments, loans, tax, insurance and housing, for instance. However, while there are many sources for us to learn, cross check and refer to on most personal finance topics, insurance is the one where the discussion is largely driven by the people who sell them for a living. In particular, we need to understand why we may want to get insured, when to get it, and when not to. Going beyond just knowing insurance product specifications may help us better understand why and how insurance works best for us.

Why do we need insurance?

Let’s start at the starting point – why do we need insurance? If all events in our life are certain, then there is no need for insurance. But life is not so simple. Life is uncertain. For our personal finances, this uncertainty is financial gains or losses, which may be out of our control.

For example, we may have a winning lucky draw ticket which we are going to claim $5,000 with. But on the way to the remote location to claim it, there is a chance that we may meet with an accident, which not only deprives us of the chance to be $5,000 richer, but also leaves us $3,000 poorer due to the medical costs involved. Hence we may want to insure against the possibility being $3,000 poorer. Perhaps by giving up some of that possible $5,000 gain? How do we evaluate this need for insurance?

The explanation to this is by using expected utility. But first, let’s talk about utility theory. Utility theory is a hypothesis that we don’t value money for itself, but for the utility or happiness it brings. In addition, the Law of Diminishing Marginal Utility tells us that the first $1,000 we gain is a big deal, and the second $1,000 is still a big deal, but less so than the first $1,000. And so on for the next $1,000, and the next $1,000 after that. So we can picture the relationship between wealth and utility as something like this:

Wealth and Utility under Diminishing Marginal Utility

So, if we have wealth of $A, we have utility of a. And we can see that utility keeps going up with wealth, but at a slower and slower rate. At some point, the wealth-utility line may flatten out, like what some studies finding that happiness does not increase after a certain level of income. All we can be sure of is that it will take more and more wealth to increase your utility.

Now, let’s talk about expected utility. Suppose, like in the example earlier, there is a 50% chance we can be richer with wealth of $A, as well as a 50% chance that we will be poorer with wealth of only $B. How will we feel about that? The answer is that we don’t feel like we have $C = $(A+B)/2. Instead, our utility will be c’ = (a+b)/2, as shown below:

Expected Wealth and Utility

So here is the problem. Whenever there is uncertainty about our wealth, our utility (or happiness) is going to be less than what it would be if we had the expected amount of wealth for sure. That is, we will have a utility level of c’ instead of c. And this is where insurance comes in to help!

How does insurance help? Well, in the simplest case, we know that we will have a higher level of utility or happiness if we had $C, instead of a 50:50 chance between $A and $B. So, how about we pay $(A-C) to an insurer (ending up with $C), who will pay us $(C–B) if we end up with an accident? Regardless of the outcome, we will have $C for sure, and so our level of utility or happiness will now be at c instead!

Using Insurance to Provide Certainty to Uncertain Wealth Outcomes

In the diagram above, the insurance premium is at the actuarially fair level, which is the chance of the adverse event happening, multiplied by the amount of loss. But it is evident from the diagram that even if you paid more than this actuarially fair premium, you might be still be better off in terms of utility than if you did not insure. After all, the Wealth-Utility line is still above the level of utility c’ even when wealth is below $C.

What does this mean? It means that a commercial insurer will attempt to charge more than the actuarially fair premium (to cover their costs and profits) whenever they can. And this has implications for when to get insured and when not to. Because if you did not get insured, you can still self insure by saving the amount of the actuarially fair premium (or more) for a rainy day. In some situations and for some risks, you may end up better off!

1. Commercial insurers will always try to charge you a premium which is higher than the actuarially fair premium. In some situations, this is okay. In other situations, you are better off saving the premiums in a rainy day fund and self insuring.

A second point of interest jumps out of the diagram above. Which is that you need to have a downside scenario, such as ending up with amount $B, in order to insure yourself against this outcome. A lot of us get the message to buy life insurance when we first enter the workforce “because it is cheaper”.

But firstly, as we show in How much does it cost to insure my life, it is not cheaper to insure yourself the younger you are. In fact, the sweet spot is when you reach you mid 30’s. Secondly, at that point in life, most of us actually have no dependents and hence no liabilities to meet upon death. In banking and finance, if you use derivatives or options when you do not have an underlying risk to hedge, you are actually speculating. So it is too, when you use insurance in such a way before you have dependents and liabilities to meet. Hence, in terms of when to get insured and when not to:

2. Only get insured when you have a downside risk to insure against, to ensure you are not wasting your money on premiums for nothing.

When to get Insured

Let’s look at a few more realistic examples instead of the artificial one above. For most of the large risks we face in life, they do not occur with 50:50 likelihood (the fate of the human race will be in doubt if that is so!). More often than not, these risks are rare, but very severe in terms of the financial consequences. For example, a middle aged breadwinner may have 3 or more dependents needing financial support upon death. How does this fit into the framework we have looked at?

Insurance when the chance of loss is small but size of loss is large

In the diagram above, the chance of loss is small, at around 20%, but the size of the loss is large. This is similar to the risk of death (less than 0.5% per year for middle aged people) which we usually get term life insurance for. In the diagram, the expected wealth, C, is calculated using the chance of nothing happening, p, and the chance of death occurring (1-p).

In the same way as the earlier, artificial examples, paying an insurance premium will lead to a gain in the level of utility or happiness. While the gain in utility is smaller than before, the premium payable is smaller as well. This way, even if the insurer charges a premium above the actuarially fair premium, the impact on our wallets is still small. Moreover, and more importantly, paying a small premium will help us mitigate a potentially very large loss, which is way beyond our ability to save up and self insure. Hence:

3. When to get insured: The best types of insurance to get are for risks that have a small chance of occurring, but a very large impact in terms of financial loss when they do occur.

Life insurance and disability insurance will fall into this category, as will hospitalisation and surgery insurance (not the riders though). Another type of risk which will fall into this category will be critical illness, especially when it covers late stage illnesses (although Early Critical Illness insurance coverage and Multiple Pay insurance may not).

When not to get Insured

Are there instances when we should not get insurance? Using the framework, we can look into the case where the chance of loss is low, but the size of the loss is small. We show this below, where the chance of loss is (1-p) = 50%:

Insurance when the chance of loss is small and size of loss is small

In the diagram above, when the loss occurs, wealth is at a level $B, which is close to $A. Depending on the level of wealth $A represents, the Wealth-Utility line gets flatter and flatter, and gets closer to being a straight line. Note that the difference between the expected utility c’ and certain utility c become very close. If the chance of loss is smaller, the gap between c and c’ will be even smaller. However, the actuarially fair insurance premium payable still remains substantial, and is very large relative to the gain (if any) in utility or happiness. Therefore:

4. When not to get insured: When the size of the possible loss is small, and/or the chance of the loss is large, it is better not to get insured and to self insure instead.

Examples of these types of risks and insurance products are the Integrated Plan riders which cover the deductible and copayments for hospitalisation and surgery. Another example is travel insurance. In most cases, the amounts covered by such insurance plans are small, in the range of a few thousands (compared to the tens of thousands for hospitalisation and surgery). While the sales pitch may sound seductive e.g. “You do not have a pay a single cent (or only a low fixed amount now) for hospitalisation and surgery!”, the truth is that such insurance rarely leads to a higher level of utility or happiness, as all you are doing is prepaying your costs with the premiums.

And because the size of the losses are small, to cover the operational costs, the insurers will have to charge premiums well above the actuarially fair level for these policies. Meaning that the sum of the premiums you pay will be more than enough to fully cover the medical or travel costs and losses covered by the insurance policies! This is where it is far better to self insure by saving the premiums payable, in anticipation of incurring those costs in the future.

But that is not all! Pricing of insurance policies also serve to differentiate between customers. For example, Integrated Plan riders which cover most of the medical costs are targetted at people who know that they are more likely to be hospitalised, and hence have a higher risk than the general populace. Therefore, these insurance riders will be priced higher, to match the risks. But alongside these higher risk people, there will be low risk people who buy these policies which appeal to them “for peace of mind”. These low risk people will be relatively indifferent to the pricing, and so the insurers can price these policies well above the actuarially fair price to profit even more.

When you cannot get insured

The last case that we can look at is when you cannot get insurance. This happens when the chance of the loss happening is high. This is even worse when the size of the loss is large as well. See below, where the chance of loss is (1-p) = 75%:

No insurance possible when the chance of loss is high

In theory, insurance is still possible here, as there is a gain to the level of utility when insured. But in practice, insurers will not offer such products, because most of the premium ends up for paying claims, and there will be very little ability to diversify away the risks among a large pool of people. What is worse for such risks is that the premiums will be so high that only the high risk people will be willing to pay for it. And this means that there will be no diversification of risk possible for the insurer between the high risk and the low risk people. Which leads to higher premiums, and then to only even higher risk people getting insurance, which leads to even higher premiums and so on. In short, the insurance model will break down completely!

An example of such risks are life insurance for people who have become disabled through illness. Or those who have been diagnosed with critical illnesses. This may be a reason why term life insurance for older people (as discussed in Investing for Life, Investing in Life Mortality Risk) is cheap, because the high risk people are uninsurable by then. In this instance, the only avenue available is self insurance.

Conclusions

Among all the topics in Personal Finance, insurance is the one where discussions are conducted almost without any rigorous basis, unlike investments for example. And that is not because there is no rigorous basis for insurance, but probably because the discussions are sales-focused. And also because the discussions tend to be focused on specific risks, rather than the general principles of insurance. But when we look at the principles behind insurance, we can find a few rules to keep in mind:

Commercial insurers will always try to charge you a premium which is higher than the actuarially fair premium. In some situations, this is okay. In other situations, you are better off saving the premiums in a rainy day fund and self insuring.
Only get insured when you have a downside risk to insure against. This ensures you are not wasting your money on premiums for nothing.
The best types of insurance to get are for risks that have a small chance of occurring, but a very large impact in terms of financial loss when they do occur.
When the size of the possible loss is small, and/or the chance of the loss is large, it is better not to get insured and to self insure instead.

The idea of self insurance comes up quite frequently when we talk about insurance. How exactly do we self insure? The answer is the emergency fund that most personal finance sources talk about, especially for the young. This emergency fund ties in with the entire insurance framework. But for now, that will have to be a topic to cover another time (see Insurance: Emergency Funds as Self-Insurance).