How Risk Tolerance and Asset Allocation Go Together

One of the first things which anyone who wants to get started in investing needs to tackle, is trying to figure out what their asset allocation should be. And if you get an investment professional, such as a bank relationship or product manager, or a financial advisor to help you with this question, the next thing you have to figure out is your risk tolerance. But how does risk tolerance and asset allocation go together? And what other factors may be relevant to helping us figure out our asset allocation?

Fortunately, there is plenty of received wisdom to help you on this front. For example, Investopedia defines the issue as such:

Asset allocation is how investors divide their portfolios among different assets that might include equities, fixed-income assets, and cash and its equivalents. Investors ordinarily aim to balance risks and rewards based on financial goals, risk tolerance, and the investment horizon

Wikipedia has this take on asset allocation:

Asset allocation is the implementation of an investment strategy that attempts to balance risk versus reward by adjusting the percentage of each asset in an investment portfolio according to the investor’s risk tolerance, goals and investment time frame.The focus is on the characteristics of the overall portfolio. Such a strategy contrasts with an approach that focuses on individual assets

So, we see that risk tolerance, investment goals, and investment horizon are the key factors for approaching asset allocation decisions. And the basic building blocks of asset allocation are equities, bonds/fixed-income, and cash. Obviously, these definitions are not enough for defining your asset allocation. There is no model here to operationalise it! And there is always the questions at the back of our heads: Is there more to this? Let’s look at how risk tolerance helps to determine asset allocation, between equities and bonds.

Of course, there are many more assets we can consider, such as cash, or real estate, or commodities. But all these other assets only clutter our thoughts without really adding anything to the determination of asset allocation. Also, cash is held for various reasons other than investment, for example, for daily needs, or as self insurance. So we shall look at adding cash as part of the asset allocation in a future article instead.

What Determines our Asset Allocation for Investment?

What is Risk Tolerance?

Let’s talk about risk tolerance. To understand risk tolerance, we need to go back to the utility theory. This, we discussed in Do insurance and investments mix?, and also in Insurance: When to get insured, and when not to. Utility theory says 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:

CRRA Isoelastic Utility Function for Risk Aversion = 2

Due to diminishing marginal utility, our joy at getting more and more wealth does not go up in a straight line. Instead, the line is curved, and this curvature is our risk tolerance or risk aversion. When our risk tolerance is high (or risk aversion is low), the line is less curved. When our risk tolerance is lower (or risk aversion is higher), the line is more curved (because we don’t like to move away from the point we are at). So, what the curvature of the utility function and risk aversion tells us is how much risk we are willing to take (investment) and how much risk we do not want (insurance). For example, working through the implications of risk aversion for investments with an isoelastic or Constant Relative Risk Aversion (CRRA) utility function:

Real Returns Required to Invest in a Well Diversified Portfolio of Equities (Annualised volatility of 14%)

Risk Tolerance	Real Returns Required
High: Risk Aversion = 1	2.0%
Normal: Risk Aversion = 2	4.0%
Moderate: Risk Aversion = 3	5.9%
Low: Risk Aversion = 4	7.6%

So as far as investments are concerned, the theory makes sense. The people with high or normal risk tolerance (or low risk aversion) usually invest in equities. Why? Because as long as the expected return is higher than 4% in real terms (or 6% in nominal terms i.e. with inflation factored in), they can tolerate the risk. But for the people with lower risk tolerance (or higher risk aversion), they will only invest in bonds, or even just cash and deposits. Why? Again, they demand at least 5.9% in real terms (7.9%-8.9% in nominal terms) or more to invest in equities, and no honest portfolio manager is going to promise consistent returns that high!

How Does Risk Tolerance Affect Asset Allocation?

Most of the time, when we go to a bank or to a financial advisor to make investments, we are put through a questionnaire to assess our risk tolerance. This risk tolerance is then compared with the riskiness of various investments to determine which assets we can invest in. For example, someone with a high risk tolerance (e.g. being able to tolerate an investment loss of 40% or more in a year) will be allowed to invest in higher risk technology funds. Conversely, someone with a moderate risk tolerance (e.g. being able to tolerate an investment loss of not more than 20% a year) will be allowed to invest in broadly diversified equity index funds or balanced funds, and not the higher risk sectoral funds. Asset allocation then consists of selecting a range of assets which you are qualified to invest in based on your risk tolerance.

But is this approach to asset allocation correct? An investor with high risk tolerance can select a whole range of equity funds, with no bond funds. Sure, his/her risk tolerance may justify this asset allocation, but is it correct? Basically, this approach can end up investing in a range of high risk and high return equity funds with little consideration of how the risk or volatility interacts, or is compensated for by the return in the portfolio! Is there another way to go about it?

Using Expected Utility and Risk Tolerance to Determine Asset Allocation Between Equities and Bonds

Let’s tackle this by going back to the isoelastic or CRRA utility function we discuss earlier. The great thing about this function is that the risk tolerance is constant no matter how rich you are. That is, suppose you have a risk tolerance to invest 80% of your wealth in equities and 20% in bonds. This proportion allocated to each asset class will not change whether you have $1 million to invest, or $10 million. Which is a reasonable assumption to make. And it is also consistent with how institutional investors, such as an investment fund, invests. They do not increase or decrease the level of risk they take in their investments when the fund gets larger.

On the other side of this analysis, we have a range of asset allocation options available to the investor:

• For the equities part, we use the 20 year returns and volatility of the World All Countries (VT) ETF.
• For the bonds part, we use the 20 year returns and risk of two ETFs:
  • half in the International (ex US) Total Bonds (BNDX) ETF
  • the other half in 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.

Historical Risk and Returns of Equities and Bonds ETF portfolios (2004 – 2024)

Asset Allocation	Risk (Annual)	Return (Annual)	Max Drawdown	Risk Level
100% Equities 0% Bonds	15.78%	8.23%	-55.18%	Very High
90% Equities 10% Bonds	14.25%	7.92%	-50.37%	Very High
80% Equities 20% Bonds	12.75%	7.57%	-45.34%	Very High
70% Equities 30% Bonds	11.29%	7.18%	-40.09%	High
60% Equities 40% Bonds	9.88%	6.75%	-34.62%	High
50% Equities 50% Bonds	8.52%	6.29%	-28.93%	Medium
40% Equities 60% Bonds	7.24%	5.80%	-23.00%	Medium
30% Equities 70% Bonds	6.07%	5.28%	-17.34%	Medium
20% Equities 80% Bonds	5.08%	4.73%	-16.43%	Low
10% Equities 90% Bonds	4.36%	4.15%	-15.56%	Low
0% Equities 100% Bonds	4.06%	3.55%	-15.75%	Low

With the returns, risk/volatility and maximum annual drawdown figures, we can work out the entire distribution of returns for each asset allocation. With this distribution of returns, we then calculate the utility of the investment returns over the entire range of possible outcomes (from the maximum drawdown return, to the mean return, to the highest). The probability-weighted average utility of investing in each of the asset portfolios (i.e. the expected utility) is then compared with the utility of simply putting everything in a term deposit at the current interest rate. This comparison gives us two conclusions:

1. If the expected utility from investing is higher than the term deposits, it is better to invest
2. Amongst the portfolios worth investing in, the one with the highest expected utility is the optimal asset allocation

We run these calculations across a range of different risk tolerances (risk aversion from 1 to 4) to work out which asset allocation works best for different levels of risk tolerances. We show this in the table below:

Certainty Equivalent Returns (Based on Risk Tolerance) Across Asset Allocations (Interest Rate = 5%)

Asset Allocation	High Risk Tolerance Risk Aversion = 1	Normal Risk Tolerance Risk Aversion = 2	Moderate Risk Tolerance Risk Aversion = 3	Low Risk Tolerance Risk Aversion = 4
100% Equities 0% Bonds	6.06%	4.10%	1.45%	-2.16%
90% Equities 10% Bonds	6.09%	4.55%	2.57%	-0.04%
80% Equities 20% Bonds	6.05%	4.86%	3.40%	1.55%
70% Equities 30% Bonds	5.93%	5.04%	3.98%	2.70%
60% Equities 40% Bonds	5.76%	5.10%	4.35%	3.48%
50% Equities 50% Bonds	5.52%	5.06%	4.55%	3.97%
40% Equities 60% Bonds	5.24%	4.92%	4.59%	4.22%
30% Equities 70% Bonds	4.89%	4.68%	4.47%	4.24%
20% Equities 80% Bonds	4.38%	4.22%	4.06%	3.88%
10% Equities 90% Bonds	3.83%	3.71%	3.58%	3.44%
0% Equities 100% Bonds	3.23%	3.12%	3.01%	2.88%

The table showing the risk tolerance and expected utility analysis above is a bit complicated and needs some explaining. Utility by itself is a dimensionless figure, and only tells us whether one asset allocation has higher utility than another. It does not actually tell us how much better one portfolio is compared to another. But what we can do is to convert the utility number back into a wealth figure, and this wealth figure is used to give a certainty equivalent return.

In the example above, when the interest rate is 5%, an investor can get a 5% return for sure by placing everything in a 1 year term deposit. Or, he/she can take some risk and put it in a 60:40 portfolio to try to earn an expected return of 6.75%. We cannot compare a risky 6.75% return directly with a risk-free 5% return, as the presence of risk in the 60:40 portfolio must make the investor feel worse-off. Hence the certainty equivalent return is a sort of “risk adjusted” return based on how he/she feels, which in turn is based on his/her risk tolerance.

Normal Risk Tolerance

So, for an investor with normal risk tolerance (i.e. risk aversion = 2), investing in a 60:40 portfolio has a certainty equivalent return of 5.10% (lower than an expected return of 6.75% due to the risk). This is higher than the risk-free return of 5% from investing in a term deposit. Hence, for this investor, given a choice between everything in term deposits, or everything in a 60:40 portfolio, the choice based on his/her risk tolerance should be to invest in the 60:40 portfolio. In fact, across all the asset allocations, a 60:40 portfolio is the one which suits his/her risk tolerance the best. However, a 70:30, and a 50:50 portfolio are also viable choices

High Risk Tolerance

By way of comparison, look at the person with a high risk tolerance (i.e. risk aversion = 1). He/She will choose to invest in any portfolio which has at least 40% in equities. But amongst all these eligible asset allocations, the one which suits his/her risk tolerance best, is a 90:10 portfolio. So, having a higher risk tolerance does not automatically mean that we should put everything in equities, because there are still limits to how much risk we can tolerate!

Moderate and Low Risk Tolerance

Now, let us look at the cases where the investor has a lower risk tolerance (i.e. risk aversion = 3 or 4). Every single one of their certainty equivalent returns is below the risk-free case of 5%. Which means that they should not invest at all, not even in bonds. Instead, they should put everything into term deposits. Yet the normal financial advisory practice is to recommend bond portfolios for these people. Is there any situation where they can invest, based on their risk tolerance?

And the answer is YES! The table below shows the certainty equivalent returns when interest rates drop to 4%. This means that the risk-free term deposit return is now 4%:

Certainty Equivalent Returns (Based on Risk Tolerance) Across Asset Allocations (Interest Rate = 4%)

Asset Allocation	High Risk Tolerance Risk Aversion = 1	Normal Risk Tolerance Risk Aversion = 2	Moderate Risk Tolerance Risk Aversion = 3	Low Risk Tolerance Risk Aversion = 4
100% Equities 0% Bonds	6.06%	4.10%	1.45%	-2.16%
90% Equities 10% Bonds	6.09%	4.55%	2.57%	-0.04%
80% Equities 20% Bonds	6.05%	4.86%	3.40%	1.55%
70% Equities 30% Bonds	5.93%	5.04%	3.98%	2.70%
60% Equities 40% Bonds	5.76%	5.10%	4.35%	3.48%
50% Equities 50% Bonds	5.52%	5.06%	4.55%	3.97%
40% Equities 60% Bonds	5.24%	4.92%	4.59%	4.22%
30% Equities 70% Bonds	4.89%	4.68%	4.47%	4.24%
20% Equities 80% Bonds	4.38%	4.22%	4.06%	3.88%
10% Equities 90% Bonds	3.83%	3.71%	3.58%	3.44%
0% Equities 100% Bonds	3.23%	3.12%	3.01%	2.88%

If you think you are seeing exactly the same table as before, well, your eyes are not deceiving you! As we assume that the long term expected investment returns from equities and bonds do not shift with changes in interest rates, the certainty equivalent returns are exactly the same. But for the investors of different risk tolerances, anything which gives a certainty equivalent return higher than 4% now is worth investing in. And we see the following:

High Risk Tolerance

Now, every portfolio with at least 20% in equities is worth investing in. However, the optimal portfolio is still the 90:10 one.

Normal Risk Tolerance

The optimal portfolio is still the 60:40 one. But any portfolio with at least 20% in equities is also worth investing in.

Moderate and Low Risk Tolerance

When term deposit interest rates drop to 4%, portfolios with between 30% to 40% in equities become optimal to invest in, even for investors with low risk tolerance. Again, this is at odds with received wisdom that investors with low risk tolerance should only invest in bond portfolios. This model in fact does make a nice prediction that as interest rates go lower and lower, even these investors with low risk tolerance will start looking around for better alternatives to term deposits.

A model of asset allocation based on expected utility and risk tolerance predicts that even investors with low risk tolerance will start looking for better alternatives to term deposits when interest rates fall

Some Preliminary Conclusions from Using Risk Tolerance to Determine Investment Asset Allocation

At this stage, any conclusions we can make of this model if using risk tolerance to determine investment asset allocation are very preliminary. One aspect of real-world asset allocation that has not been modelled yet is including cash/term deposits/T-bills as part of the asset allocation instead of just having a choice of “everything in term deposits” versus “everything in equities and bonds”. This is not trivial, as it could very well become the investment version of the 3-body problem!

But what we can deduce is that the practice of assessing an investor’s risk tolerance is just not put to good use in the investment advisory process:

• There is no structured process for recommending asset allocation between equities and bonds
• No recognition that even investors with low risk tolerance may want to take on risks of equities and bonds when interest rate fall, and vice versa when rates rise

In fact, by using risk tolerance of investors solely as a means to restricting or allowing access to different investment products, and thereafter not incorporating the risks of different investments, the investment advisory process runs the risk of ending up as a pursuit of higher returns without paying any attention to the risk incurred!