Investment and Insurance From a Risk Tolerance Perspective

lifefinance
February 2, 2025
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Insurance, Investing, Personal Finance, Saving

At first sight, investment and insurance look like the diametric opposite of each other. Investment is for growing our wealth, while insurance is for protecting our wealth. How can we grow our wealth while protecting it? But on second thought, investment and asset allocation is all about how we manage our tolerance for risk versus return. While insurance is about managing our tolerance for rare events which may have a large financial impact on us. So, both investment and insurance can be viewed or analysed through the lens of our risk tolerance.

Previously, we have used the expected utility theory to model our risk tolerance. This is then applied, firstly to insurance:

And then, to investment and asset allocation:

Since both investment and insurance can be based on the same risk tolerance framework, can we use the same approach to work out how we can allocate between investments of stocks, bonds and cash, with insurance at the same time? How then, does investment and insurance look from a risk tolerance perspective?

Will Insurance Hinder or Help with our Investments?

Insurance, investment and risk tolerance

Why Do We Insure or Invest?

Let’s start off with a quick recap about why we insure or invest in the first place. Expected utility theory states that we value money or wealth for the utility or happiness it brings. 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. Hence, the relationship between wealth and utility is something like this:

Wealth and Utility under Diminishing Marginal Utility

So, if we have wealth of $A, we have utility of a. We can see that utility keeps going up with wealth, but at a slower and slower rate. The curve of the utility function, which is how fast utility goes up with wealth, depends on our risk tolerance. A higher risk tolerance (or lower risk aversion) means that the curve is straighter, while a lower risk tolerance (or higher risk aversion) means that the line is more curved.

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, and a 50% chance that we will be poorer with wealth of $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. That is, the probability-weighted utilities of wealth of $A and $B, as shown below:

Expected Wealth and Utility: Investment

Investment results in uncertainty about our wealth. When this happens, our utility (or happiness) is going to be less than what it would have been if we had the expected amount of wealth for sure. That is, we will have a utility level of c’ instead of c. If the difference between c’ and c is larger, it means that the utility function is more curved, and we have lower risk tolerance, and will invest more conservatively. If the difference is smaller, the utility function is straighter and we have a higher risk tolerance, and will take more risks in 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 2%, of this wealth dropping sharply to $B. This is the case where the breadwinner of the family passes away, leaving very little savings or assets. Therefore, wealth falls sharply. 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

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

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.

Just like for investment, how much we insure depends on our risk tolerance. If the difference between c’ and c is larger, it means that the utility function is more curved, and we have lower risk tolerance, and will insure more. If the difference is smaller, the utility function is straighter and we have a higher risk tolerance, and will insure less.

How Do Extreme Losses Affect our Investment Decisions?

Up to this point, we have covered the same risk tolerance framework we have used elsewhere previously. We can now add more flesh to it by using the isoelastic or Constant Relative Risk Aversion (CRRA) utility function to put this framework into numbers and examples:

With this utility function, it is straightforward to convert risk-free wealth (i.e. cash deposits earnings) and uncertain wealth (i.e. investment) to expected utility numbers to compare between different asset allocations, and also to covert these utility numbers back into a certainty equivalent wealth figure, to see how much the risk of uncertainty in investment returns means to us.

A World With No Insurance

Let us first consider a world where we do not have insurance.

On the investment side, we assume that risk-free interest rates are 4%, and the assets we can invest in are

Equities: the World All Countries (VT) ETF.
Bonds: the International (ex US) Total Bonds (BNDX) ETF, and the US total Bonds (BND) ETF

The returns and volatility of the various portfolios are computed with the historical returns on LazyPortfolio ETF portfolio Backtest and Simulation Tool for the period 2004 to 2024.

Previously, we show that given these assumptions, the asset allocation with the highest Sharpe Ratio is the one which has 70% allocated to equities and 30% to bonds. Based on the 20 years of returns, this portfolio has:

Expected return = 7.18%
Annual Volatility = 11.29%
Max Drawdown = -40.09%

What this means is that regardless of risk tolerance, all asset allocations chosen when interest rates are 4% will be composed of this 70:30 portfolio, plus cash or borrowing.

At the same time, we assume that a large non-investment related loss occurs with 2% probability. This loss is uncorrelated with investment returns, and can be due to death (which occurs with roughly 1% probability per year) or critical illness (which also has a likelihood of 1% a year).

Using the CRRA utility function shown above, we can work out both the expected values of the optimal asset allocation chosen, as well as the certainty equivalent wealth, when there are no extreme losses, as follows:

Certainty Equivalent Wealth (initial wealth of 100) when Interest Rates = 4%

Risk Tolerance	Risk-free Wealth if not Invested	Investment Asset Allocation	Expected Wealth of Asset Allocation	Certainty Equivalent Invested Wealth
Normal (Risk Aversion = 2)	104.00	90% in 70:30 10% in Cash	106.86	105.06
Moderate (Risk Aversion = 3)	104.00	50% in 70:30 50% in Cash	105.59	104.51
Low (Risk Aversion = 4)	104.00	30% in 70:30 70% in Cash	104.95	104.25

If we look at the top row of the table above, we see that for a person of normal risk tolerance (risk aversion = 2), the optimal asset allocation is to put 90% of wealth in the 70:30 portfolio, and the remaining 10% in cash earning 4% interest. While the expected returns from such an asset allocation is 6.86%, due to the risk involved in investing, this is only equivalent to the 5.06% risk free return! However, as this is higher than the 4% risk-free return of putting everything in cash and not investing, the preferred outcome for such a person would be to invest.

Now, suppose in addition to investment risk, there is also the risk of an extreme loss, with a likelihood of 2%. Let’s also assume that this loss will result in wealth decreasing by 90%. This would possibly be the case of a young household where the main breadwinner passes away, leaving little in savings. How would the table above change?

Certainty Equivalent Wealth when there is a 2% Risk of an Extreme Loss (-90%)

Risk Tolerance	Expected Return on Cash	Certainty Equivalent Wealth if not Invested	Expected Return of Optimal Portfolio	Certainty Equivalent Wealth if Invested
Normal (Risk Aversion = 2)	2.12%	87.54	4.92%	89.23
Moderate (Risk Aversion = 3)	2.12%	58.66	4.48%	58.87
Low (Risk Aversion = 4)	2.12%	36.32	3.85%	36.33

Once we assume that there may be a small chance of a large loss (unrelated to investment risk), the table above shows us some very interesting results.

Firstly, the 2% chance of a low of 90% reduces the expected return by almost 2%, whether you invest or not. This means that in such a situation, it is always better to invest than not too, since the upside is higher when invested, while the downside (-90% return) is the same
Secondly, even if the chance of a large loss is small, it weights very heavily on our minds. For someone with a low risk tolerance (risk aversion = 4), it is almost like losing more than 60% of your wealth straightaway! No wonder in countries where there is no social safety net, most people are fearful and tend to save as much as they can, forgoing consumption. However, people with nornakl risk tolerance fare much better under these assumptions, they only feel a pinch of between 10-12% of their wealth, if they cannot insure against a large loss.
Thirdly, it shows us how important insurance is. To insure against an uncorrelated small chance of a large loss, insurance such as a term life policy shock not cost more than 2-5% of the amount of loss we are trying to insure against. Even though the table above shows that a person may be willing to exchange anything between 10-60% of their wealth to protect against such a risk of a large loss, with efficient insurance markets, they can actually do so for 5% of their wealth! Hence, insurance is absolutely essential for personal finance, as long as we do not overpay for it.

But what if the amount of the possible large loss is somewhat smaller, say at 50% of our wealth instead of 90%? This would correspond to the situation when we’ve had a few years to build up our savings and investments, and hence may be in a better position to self insure. The table below shows what it looks like:

Certainty Equivalent Wealth when there is a 2% Risk of a Large Loss (-50%)

Risk Tolerance	Expected Return on Cash	Certainty Equivalent Wealth if not Invested	Expected Return of Optimal Portfolio	Certainty Equivalent Wealth if Invested
Normal (Risk Aversion = 2)	2.92%	101.80	5.72%	104.90
Moderate (Risk Aversion = 3)	2.92%	100.70	4.48%	101.76
Low (Risk Aversion = 4)	2.92%	98.98	3.05%	99.48

Looking at this latest table, we see an interesting change in the optimal behaviour across all levels of risk tolerance when the size of the large, non-investment loss decreases as a proportion of existing wealth:

Firstly, the decrease in certainty equivalent wealth compared to the case where there is no large non-investment loss is much smaller than before. Previously, for someone with low risk tolerance (risk aversion = 4), the possibility of losing 90% of wealth reduced the certainty equivalent wealth to 36.32 when the wealth is not invested. Now, given the possibility of a 50% loss, the certainty equivalent wealth is 98.98, which is much, much closer to the original 104 when there is no non-investment loss.
What this implies is that the need for insurance will fall as wealth increase, which is the basis for self insurance. Granted, for those with low risk tolerance, they would still be willing to pay up to 6% of their wealth to insure against a 2% possibility of a 50% loss, but those with higher risk tolerances may not be willing to do so, given the actuarial cost of such insurance is 1% of wealth. At the very least, those with higher risk tolerance may opt for partial; insurance coverage only.

So, insurance for large losses which occur very infrequently, such as life insurance, or critical illness insurance, become less and less relevant once our wealth starts increasing. For this with normal and moderate risk tolerance, partial or even zero coverage of these risks would be optimal once wealth reaches twice or more of the required level of protection. But what about smaller non-investment losses which happen more frequently? These are things like hospitalisation insurance coverage, or the riders which cover the copayment and deductible portfolios of medical bills. Let’s look at the final case, where there is a 5% chance of a loss of 33% of wealth:

Certainty Equivalent Wealth when there is a 5% Risk of a Moderate Loss (-33%)

Risk Tolerance	Expected Return on Cash	Certainty Equivalent Wealth if not Invested	Expected Return of Optimal Portfolio	Certainty Equivalent Wealth if Invested
Normal (Risk Aversion = 2)	2.81%	101.21	5.53%	104.50
Moderate (Risk Aversion = 3)	2.81%	100.52	4.32%	102.10
Low (Risk Aversion = 4)	2.81%	99.64	3.71%	100.48

Looking at this final table, we see that when it comes to smaller non-investment losses which happen more frequently (at a 5% probability), the case for insurance is actually quite poor. In the case above, insuring against a 5% chance for a 33% loss costs 1.67% of the amount covered in actuarial terms. In commercial terms, it is likely to be higher, at 3.5% of so. Those with normal risk tolerance should not bother with this sort of insurance, as it will make them worst off compared with not having insurance cover for these risks. Those with moderate or low risk tolerance may consider partial or zero coverage as well, depending on the cost of insurance.

A World With Insurance

Having looked at how large non-investment losses can affect our expected utility and (largely) mental well-being in spite of the low probability (2% or less in each year) of their occurrence, living in a world where insurance exists for such risks makes our financial lives much, much better off. To be more specific, this applies to insurance for the risk of loss of 90% or more of our current wealth, and term life insurance, and perhaps critical illness insurance, especially for the young and those without much savings and investments.

For those who have the time and opportunity to accumulate more savings and investments, the need for insurance diminishes, regardless the level of risk tolerance. For protection against infrequent smaller losses of less than 50% of wealth, insurance is probably not needed by those with normal risk tolerance, and partial insurance may be more suited for those with moderate of low risk tolerance.

As the non-investment loss sizes get smaller, and perhaps occur more frequently, those with normal risk tolerance can forgo insurance altogether. For those of moderate or low risk tolerance, the costs of this sort of insurance can become fairly prohibitive. Hence, they too, can consider forgoing such insurance protection as well.

Conclusions

The risk tolerance framework, when applied to both investment asset allocation and insurance questions can lead to results and answers which are both practical and interesting. In particular, a common question we often ask ourselves is when we should self-insure, versus when we buy insurance.

Two main results stand out here:

When these non-investment losses are large relative to our wealth, e.g. more than 50% of our wealth, insurance is critical. The improvement to our well being of being insured in such a case far outweighs the cost of insurance, and it is best to make insurance decisions independently and separately from investment and asset allocation decisions.
When these non-investment losses are smaller relative to our wealth, e.g. less than 50% of our wealth, then it is more useful to integrate our insurance decision making with our investment and asset allocation decision making. More often than not, we may conclude that we’d be better off with just partial insurance coverage.