Do Insurance and Investments Mix?

An impressive trait of the current generation of financial advisors is how they can confidently advise us on most financial matters, like insurance to protect ourselves, and like investments to profit from it. But do insurance and investments mix? Aren’t they all about the risks we have, on the one hand using insurance to reduce it, and on the other, using investments to add to it? Yet while the financial advisor is usually at pains to assess our tolerance for risk when it comes to investments, trying to gauge how much loss we are able and willing to bear, why don’t they do the same for insurance matters? Instead, when it comes to insurance, we are often advised to cover ourselves to the hilt, regardless of whether we are willing, or able to bear some of that risk ourselves.

If we try to delve into the details, we may told that this is because the risks we insure against and the risks we bear in investing are different risks. But at the end of the day, it is all about how these different risks affect the amount of wealth we have, when we die, or when we fall ill, or when the market crashes. So is it possible to reconcile these two ends of the spectrum of risk? And is it possible to justify advice for us to do both at the same time, or products like Whole Life insurance or Investment Linked Policies (ILPs) which are both insurance and investments simultaneously?

Do Insurance and Investments Mix At All?

Why Do We Insure or Invest?

Let’s start off by thinking about why we insure or invest in the first place. The framework for thinking about this is expected utility, and we covered a lot about it in Insurance: When to get insured, and when not to. A quick recap of this will start from utility theory.

Utility theory is a hypothesis that we don’t value money or wealth 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.

Now, let’s talk about expected utility. Suppose, we start off with wealth of $C. And we invest it such that 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, we don’t feel like we have wealth of $C = $(A+B)/2, with utility of c. Instead, our expected utility will be c’ = (a+b)/2, i.e. the probability-weighted utilities of wealth of $A and $B, as shown below:

Expected Wealth and Utility: Investment

So this is the investment problem. Investment results in uncertainty about our wealth. When this happens, 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. So why do we even bother to invest?

Now, utility levels cannot be compared to each other directly, except to say that one level, c, is higher than another, c’. But we can transform these utility levels back to the equivalent level of wealth in each case (that is, along the thick dark curved line in the chart above). So, while utility level of c corresponds to a wealth of $C, utility level c’ corresponds to a lower level of wealth of $C’ (not shown). We can assume then that if the expected returns to an investment is greater than the difference $C – $C’, then we should be happy to invest. The expected return more than makes up for the lower expected utility of investing. If the expected return is less, then we should not bother with the investment.

What about insurance? It is in fact driven by the same framework of expected utility as well!

In the diagram below, suppose we start off with a wealth of $A. But now, there is a small chance, say 1%, of this wealth dropping sharply to $B. This is like the case when the breadwinner of the family passes away, leaving the family with very little income. Now wealth and income are not quite the same, but let’s think of wealth in a period as the income earned during that period. In the diagram, the expected wealth, $C, is calculated using the chance of nothing happening, p, and the chance of death occurring (1-p).

Expected Wealth and Utility: Insurance

This is the classic case of life insurance. Although the fall in wealth and utility from the starting level of $A and a may be small, there is considerable dread of actually ending up at wealth level $B and utility b. So, what do we do? We insure ourselves by paying a small premium to ensure that no matter what happens, our wealth (dead or alive) will remain at $C.

Should we always insure ourselves fully? Well, using the expected utility framework again, we can work out for ourselves whether the gain in wealth corresponding to the gain in expected utility from insuring ourselves (i.e. c – c’) is higher or lower than the cost of insurance (which corresponds to $A – $C, or in utility terms, a – c).

So the same expected utility framework should determine how we look at insurance and investments. But there is one more piece of the puzzle to put together – our risk tolerance or aversion.

How Does Our Risk Tolerance or Risk Aversion Affect Our Insurance and Investment Decisions?

Now that we know a bit about utility and expected utility, we can ask why do some people invest or insure, while others don’t? It turns out that while everyone logically should have a similar curved utility function with respect to wealth, the degree of curvature is not quite the same. A person with a utility function which curves less is more risk tolerant, or less risk averse, while person who has a more curved utility function is less risk tolerant, or more risk averse. Using a Constant Relative Risk Aversion (CRRA) isoelastic utility function, we can see how this curvature changes as a person gets more risk averse:

CRRA Isoelastic Utility Functions

*Note that CRRA isoelastic utility functions imply that our allocation between risky and risk-free assets will not change as we get wealthier. A person who takes less and less risk as he/she becomes wealthier, or conversely starts gambling more and more, is uncommon, to say the least.

it is quite evident that the utility functions curve more and more as a person gets more risk averse. But who do these utility functions represent? Professional traders and/or investors can be thought of as people with a risk aversion of 1. Perhaps normal investors in the stock markets can be thought of as people with risk aversion of 2. At the end of the spectrum, people who only leave their money in banks deposits are likely to have a risk aversion of 4. So when a financial advisor tries to work out our risk tolerance or risk aversion for the purposes of recommending an investment, this is essentially what they are looking for.

Investments

How then does our risk aversion affect our insurance and investment choices? Following what we discussed earlier, we can use the various isoelastic utility functions to compute the corresponding loss in wealth when an investor puts his/her money in an investment which can go up or down by 20%. This corresponds to a volatility of around 14%, which is like a well diversified global stock portfolio.

Certainty Equivalent Loss in Wealth from Investing

Risk Aversion Level	Certainty Equivalent Loss in Wealth
Risk Aversion = 1 (least risk averse)	-2.0%
Risk Aversion = 2	-4.0%
Risk Aversion = 3	-5.9%
Risk Aversion = 4 (most risk averse)	-7.6%

So, for the least risk averse investors, as long as the investment has an expected return of at least 2%, it is still worthwhile investing. For a normal investors, the required level of expected return is 4%. But for the more risk averse people, only expected returns of more than 6% or 8% (before any adjustment for inflation) I’ll make them feel comfortable investing! Given that most stock investments have a volatility higher than 14%, and have nominal expected returns in the range of 6% – 8%, it should not come as a surprise that only people with low risk aversion (i.e. 1 or 2) will invest in stocks.

Insurance

What about our insurance decisions? We can use the same computation as above, but apply it to an insurance setting, where we have a 99% chance of remaining at the same level of wealth, and a 1% chance of having only 20% of our wealth (i.e. a fall of 80% versus a gain of 0%). What do the certainty equivalent loss in wealth be if we do not insure ourselves?

Certainty Equivalent Loss in Wealth from Not Insuring

Risk Aversion Level	Certainty Equivalent Loss in Wealth
Risk Aversion = 1 (least risk averse)	-3.2%
Risk Aversion = 2	-3.8%
Risk Aversion = 3	-10.2%
Risk Aversion = 4 (most risk averse)	-23.6%

Here, the least risk averse people may still want to get insured, but they will not want to spend a lot of their income on it, may in the range of 2% – 3% of the income they are protecting. Hence, “Buy term and invest the rest” may really be all the financial advice these people need. Going a step further, only partially insuring themselves, getting decreasing balance insurance etc., may all be viable approaches to insurance for these people.

On the other hand, for the more risk averse people, “Insure and don’t invest” is probably the more appropriate advice to give them From the calculations above, they may be willing to put 10% or even more of their income towards pure insurance to protect themselves and their loved ones. Fortunately, term insurance doesn’t need to cost them so much, and what they should be doing is to put whatever else they have left into bank deposits or other low and zero risk products.

Are We Getting The Right Sort of Financial Advice To Mix Insurance and Investment?

At the start of this post, we noted how financial advisors can advise us on most financial matters, like insurance to protect ourselves, and like investments to profit from it. But is this advice really sound? Do insurance and investments mix? Aren’t they all about the risks we have, on the one hand using insurance to reduce it, and on the other, using investments to add to it?

If we have truly consistent views and tolerance to risk, the assessment of our tolerance for risk when it comes to investments, trying to gauge how much loss we are able and willing to bear, should be broader, and extend to insurance matters as well. Instead, when it comes to insurance, we often get the advice to cover ourselves to the hilt, regardless of whether we are willing, or able to bear some of that risk ourselves. And regardless of the advice given with respect to investments.

But insurance and investments are linked, by our tolerance for risk, or risk aversion. And in some cases, insurance and investments do not really mix.

You can read some of our other blogposts on insurance here:

• Term Life Insurance (2): How much does it cost to insure my life?
• Insurance: How Much Life Insurance Do I Really Need?
• Insurance: When to get insured, and when not to
• Insurance: Emergency Funds As Self-Insurance
• Insurance: How Much Should I Be Spending on Insurance?
• But Why Do We Still Mix Insurance With Investments?