This has many different applications. It can be used to determine the correct mixture of stocks and bonds for your personal risk tolerance, and to calculate the optimal glidepath for accumulation. It can also be used to calculate the correct proportion of stocks and bonds in a market timing strategy where the return assumptions change continuously.

In this post I will be writing about the CRRA utility, some math details, the relation between risk tolerance and the kelly criterion, assumptions of target volatility, multi-asset optimization, effect of time horizon on asset allocation, and uninformed market timing.

Introduction to utility functions

A fundamental concept in asset allocation is that of risk aversion. Risk aversion models the behavior of humans with respect to uncertainty. Most investors are risk averse, that means that they would rather accept a certain payment instead of taking a gamble with the same expected return. Your risk aversion is encoded mathematically in what's called an utility function. If you lose money, you're unhappy and the utility function responds by outputting a small value. If you make money, you're supposed to be happy and the utility function gives a large value. Most investors are risk averse, that means that losing $10 hurts more than twice as much as losing $5.

The goal of every investor is to maximize their utility function. Obviously it's extremely difficult to determine your exact utility function, but there are a few functions that are common in literature. One of those is the constant relative risk aversion (CRRA). Under this form of risk aversion, your decisions depend only on the bet size relative to your total net worth, not on the absolute dollar amount. That means that a billionaire that loses 5% of his net worth is equally unhappy as a middle class worker that loses 5% of his net worth, if both have the same risk aversion.

Another form of risk aversion is constant absolute risk aversion (CARA). This is the opposite of CRRA, the decision depends only on absolute dollar amounts. I will not be discussing this type of risk aversion further.

Math stuff

The CRRA utility is shown below, c stands for consumption.

Throughout the remainder of this post I will use the symbol γ (gamma) to indicate the coefficient of relative risk aversion. A γ=0 indicates a risk neutral investor. The value of the parameter γ depends on your personal risk tolerance.

This utility function is quite abstract. But if we make the assumption that returns are normally distributed, we fund two extremely useful and simple results. The first result is the expression for the investor's utility, which can be expressed as:

U(X) = E(X) - 1/2 * γ * Var(X)

X indicates the random variable for your investment. E indicates the expectancy, or the average return (not the geometric return) of the investment. Var indicates the variance (standard deviation squared). The investor's utility decreases linearly with the variance (not the standard deviation).

The second result is the expression we get when maximizing the investor's utility. The following expression gives the optimal proportion allocated to the risky asset (stocks).

proportion of risky asset = (1 / γ) * E(X - Rf) / Var(X - Rf)

Rf indicates the risk free rate. It is important to realize that the optimal proportion does not depend on the absolute return, but on the risk

*premium*compared to a risk free benchmark.

We can plug in some numbers to get a feel for the correct value of γ. I will use a risk premium of 5% and standard deviation of 16%.

γ=5 corresponds to an asset allocation of 40% stocks, 60% bonds

γ=4 corresponds to an asset allocation of 50% stocks, 50% bonds

γ=3 corresponds to an asset allocation of 65% stocks, 35% bonds

γ=2 corresponds to an asset allocation of 100% stocks

γ=1 corresponds to an asset allocation of 200% stocks

γ=0 corresponds to an asset allocation of infinitely many stocks.

I think a value of 3 is a good starting point for most investors. I personally think I'm a bit more risk tolerant than the average investor, and a lot less likely to panic sell, my risk aversion is between 2 and 3.

Certainty equivalent return

Also related to the concept of utility is certainty equivalent return (CER). The CER is the inverse of the utility function. Conveniently, the inverse utility of a normal random variable simplifies to U

^{-1}(u) = N(u, 0) = an investment with a constant return of u

Suppose that we have an investor with γ=2 and a 100% stock allocation. The utility of this investor is E(X) - 1/2 * γ * Var(X) = .0244. The CER is 2.44%. This means that if the investor can choose between a 100% stock allocation or an investment with a guaranteed return of 2.44% + risk free rate, both investments are equally good.

The certainty equivalent return is extremely useful for comparing investment decisions. If we have two strategies A and B, and A has a higher return and more risk than B, we don't have enough information to decide which is better. The better investment depends on the personal utility function of the investor. But if we say that A has a higher certainty equivalent return than B for an investor with γ=3, we can immediately conclude that it is better.

Arithmetic, geometric returns, kelly criterion, and sharpe ratio

How does utility relate to commonly used metrics, such as geometric returns and sharpe ratio?

If we use γ=0, we find that the investor's expected utility is simply E(X), the investment's expected return.

If we use γ=1, we find that the investor's expected utility is E(X) - 1/2 * Var(X), which is equal to the geometric return. (under the assumption of normally distributed returns).

How about the kelly criterion? The kelly criterion is an optimization method that attempts to maximize the geometric growth rate. This is equivalent to maximizing the log wealth. Which is equivalent to maximizing the CRRA utility function with γ=1. We can say that the kelly criterion is useful

*if and only if γ=1*. Have you ever wondered why you should use the kelly criterion? Now you know.

If you're a frequent visitor on this forum, you may have heard that there is no reason to go past the kelly criterion. But that is false. For investors that have γ<1 (very risk tolerant), it is rational to take more risk than the kelly criterion.

What about the sharpe ratio? The max sharpe ratio portfolio does not correspond with any specific γ. Therefore, for the investor with a CRRA utility, optimizing for the maximum sharpe ratio is completely pointless.

The assumptions of target volatility strategies

Target volatility is a strategy that attempts to forecast future volatility. The strategy then adjusts the asset allocation such that the volatility is constant throughout time. If we have an investor with a CRRA, which assumptions do we need to make to rationalize this strategy? We denote the target volatility investor's allocation to stocks with the following formula:

C / σ

Where C is a constant related to the investor's personal risk tolerance. The question we are interested in is: when does the target volatility investor agree with an investor with CCRA utility? Recall that the CRRA investor's optimal stock allocation is:

(1 / γ) * r / σ

^{2}

With r being the return, and σ

^{2}the variance. We assume that return is linear to volatility, r = D * σ, with D being a constant greater than zero. We now have:

= (1 / γ) * (D * σ) / σ

^{2}

= (1 / γ) * D / σ

Therefore C = (1 / γ) * D, with D being equal to r / σ, which is the sharpe ratio (a constant).

What have we learned here? If you are using target volatility and have a CRRA utility, you must believe that the sharpe ratio is a constant, and that return is proportional to volatility. This is at odds with the research literature, return is not at all or weakly

*negatively*correlated with volatility. for example see this blog.

Mean-variance optimization.

The equations above are capable of finding the optimal allocation to a single asset. It is possible to use mean-variance optimization to find the optimal proportion of multiple assets. Two such examples are shown below (UPRO is a 3x leveraged S&P500 fund, and TMF is a 3x leveraged 20 year treasury fund):

The top panel shows the efficient frontier and utility. The bottom panel shows the proportion of different funds.

I advice against using these specific images to determine your asset allocation, as these asset allocations have not been peer reviewed. Nevertheless, this should give some indication how your asset allocation changes as your risk tolerance increases or decreases. It also shows that specific proportions of funds are only optimal for one specific risk tolerance. This is especially concerning for portfolio's such as Dalio's all-seasons portfolio, or HEDGEFUNDIE's 55/45 strategy. I always find myself wondering: for which investor, which assumptions and which risk tolerance is this strategy appropriate? It turns out nobody knows.

A limitation of mean variance optimization is that returns must be normally distributed and independent throughout time (no mean reversion). It is possible to use other optimization techniques to calculate the optimal allocations for non-normally distributed asset returns, but this is a significantly more complicated. Usually, a normal distribution results in an asset allocation that is good enough. See also https://www.aacalc.com/docs/ef_non_normal.

By tweaking the inputs to mean variance optimization, it's possible to create many different asset allocations strategies. For example, if you assume all assets have the same return and no correlation, it becomes equivalent with risk parity. The advantage of this method is that it forces you to make your assumptions explicit. Rather than running a backtest with some specific portfolio (e.g. all-seasons) we can say: all-seasons is optimal under assumptions X Y and Z. This (potentially) results in much more robust asset allocations because the individual assumptions can be tested. My favorite example is that the statistical significance of the value factor is greater than that of gold. Therefore, it would appear to be irrational to have a portfolio with gold that does not contains the value factor. See also viewtopic.php?f=10&t=277661 for some inspiration for different assumptions.

The effect of time horizon on investment strategy.

What happens if we try to optimize the utility over time, instead of on a fixed time horizon? This problem is known as Merton's portfolio problem. This is essentially a continuous time version of modern portfolio theory. Conveniently, if consumption is optimally chosen the asset allocation does not depend on the time horizon!

Various improvements to Merton's portfolio problem have been proposed. If human capital is taken into account, then the strategy turns into lifecycle investing. In the case of lifecycle investing, a constant amount of lifetime wealth is invested into stocks instead of a proportion of current wealth. If a mean reverting market is considered, or stochastic lifespan is taken into account, the solution also takes slightly different forms. Nevertheless, a simple application with lifecycle investing is good enough for most investors.

The utility loss of uninformed market timing.

The optimal proportion of stocks for an investor with γ=4 is approximately 50%. This investor has a CER (certainty equivalent return) of 1.22%.

Now suppose an investor that is market timing. 50% of the time, the asset allocation is 100% stocks, and 50% of the time the asset allocation is 100% bonds. We assume that the equity risk premium is constant at 5%. You would think that the utility of the market timing investor is similar to that of the buy and hold investor because the average allocation is the same. The CER for the market timing investor is:

= P(asset allocation stocks) * U(100% stocks) + P(asset allocation bonds) * U(100% bonds)

= .5 * U(100% stocks) + .5 * U(100% bonds)

= .5 * -0.12% + .5 * 0

= -.06%

Which is worse than just investing everything in a savings account. If the stocks outperform by 2.56% in periods where the market timer holds stocks, then both strategies are equivalent, that's quite a high bar to reach. This shows that you should never try to time the market by going 100% in and out of stocks.

We can also reverse-engineer the market timer's decisions. If the market timer with γ=4 is in bonds, he must believe that the equity risk premium is zero or negative. And if the market timer is in 100% stocks, he must believe that the equity risk premium is 10% or greater.

Can the market timer do better? Instead of going 100% in and out of stocks, suppose that the market timer estimates that the equity risk premium is 4% in bad times and 6% in good time. This corresponds to an asset allocation of approximately 40% stocks and 60% stocks, respectively. If the market timer is completely wrong and the underlying equity risk premium is actually a constant 5%, the utility is:

= P(erp forecast = 4%) * U(40% stocks) + P(erp forecast = 6%) * U(60% stocks)

= .5 * U(40% stocks) + .5 * U(60% stocks)

= .5 * 1.18% + .5 * 1.16%

= 1.17%

Which is very close to the buy & hold investor's CER of 1.22%. Even if the market timer is completely wrong, the damage is limited. If the marker timer forecasts the equity risk premium perfectly, the CER increases to 1.27%.