Abstract

Does the pain of a dollar lost outweigh the pleasure of a dollar gained? Many real-world investors believe it does. This article explores the utility of loss aversion and introduces the concept of hedonically adjusted return, a metric that expresses the utility of a risky investment in terms of the risk-free rate of return for which an investor would be willing to exchange the risky investment with indifference, given the investor's degree of loss aversion and target return. Hedonically adjusted return can be generally described by the following equation:

Hedonically adjusted return (i.e. utility)

=   function

(investment return, target return, loss aversion)

By hedonically adjusting the returns of portfolios that lie on a mean-variance efficient frontier, the frontier can be recast as a hedonically adjusted efficient frontier. Depending on the investor's degree of loss aversion, the hedonically adjusted frontier may be significantly flatter than the traditional efficient frontier. Thus, calculation of the hedonically adjusted efficient frontier may serve investors who wish to maximize utility per unit risk (rather than return per unit risk) in their portfolio selection decision.

Introduction

Modern portfolio theory (MPT) is built on two main assumptions:

• an investor will seek maximum expected return
• an investor will seek maximum consistency of return (i.e. lowest risk), where consistency or risk is measured as standard deviation of return, a statistical metric whose calculation assigns equal importance to downside and upside risk.

Unfortunately, while these two assumptions offer computational simplicity, they do not appear to serve the interests of some real-world investors.

MPT’s first assumption, that investors seek to maximize return, while simple, may not always reflect an investor’s objectives. In general, a real-world investor seeks to maximize utility (i.e. quality of life, or well-being), not simply return. While return is a dominant factor in determining utility, other factors often come into play.

For example, when seeking to choose an investment or portfolio with maximum utility, an investor will consider utility factors such as:

• Will this investment complicate my tax calculation?
• Can I gain any lifestyle benefits from this investment (i.e. art, real estate)?
• Will this investment help my family, local community, or other social objective?
• Would I be better off consuming this wealth instead?
• If this investment fails, will my lifestyle or reputation be damaged?

Loss Aversion

While modeling every utility factor would be impossible, there is one particularly important factor that stands out as worthy of further attention: the investor’s loss aversion, or “pain-to-pleasure” ratio that an investor experiences from losses or gains on a portfolio.

By adjusting investment return for loss aversion, we model a portfolio's utility to the investor. Portfolio utility, expressed as a perceived rate of return, is termed hedonically adjusted return. (Note 1)

To better understand how loss aversion can influence a portfolio's utility, consider, for example, a physically infirm retiree of limited means. Such an investor, while not averse to upside risk, may be extremely averse to downside risk, even if such risk could lead to higher long-term returns. This loss-averse disposition may be based on a concern that an investment shortfall could lead to severe hardships (e.g. loss of food, shelter, and health care) that could far outweigh any quality-of-life benefits that might be obtained by a corresponding investment surplus. The infirm retiree has a high "pain to pleasure" ratio, which we define as loss aversion, at a given target level of return:

 
Thus, a loss-averse investor has a high loss aversion ratio.

Target return, also known as minimum acceptable return, will vary from investor to investor. For a pension fund, the target return may represent the return required to fund the pension liability. For a foundation, it might be equal to the required spending rate plus inflation. Given that an investor's loss aversion ratio is specific to his target level of return, it follows that for the hedonic adjustment process to work properly, the investor must set a reasonable target return.

Loss aversion is not unique to private investors. Institutional investors face similar challenges, and are often required to meet inflexible spending obligations. Depending on circumstances, the failure of investments to meet the required target could jeopardize an institution's ability to fulfill its mission, potentially damaging the institution’s operations, organization, and reputation for years to come.

A money market fund, therefore, is extremely loss-averse. Aware that its utility as a cash substitute depends on maintaining a $1.00 share value, the fund can not risk any loss of principal. Such an event would not only run contrary to the fund’s capital preservation objective, but it would also almost certainly trigger investor withdrawals that would threaten the fund's existence.

While some investors derive utility from loss aversion, others don't. The following table identifies factors that distinguish between the two investor archetypes: the classical MPT investor and the loss-averse investor. Most real-world investors will fall somewhere between these two extremes.

Table 1 Investor Types
Characteristic	Classical MPT investor	Loss-averse investor
Loss Aversion	May be as low as 1.0	May be 2.0 or higher
Perception of Risk	Pain and pleasure of downside and upside risk are experienced roughly equally.	Experiences more pain from a loss than pleasure from a gain.
Spending Flexibility	Flexible.	Inflexible.
Funding Level	Well-funded; holds sufficient assets to allow for a high degree of certainty that spending goals will be achieved.	Marginally funded; holds barely enough assets to assure that ongoing spending goals are met.
Investment Horizon	Long-term.	Inflexible.
Liquidity Demands	Low.	High. May need funds to access funds at a moment's notice.
Taxes	Symmetrical; gains and losses receive equal and offsetting tax treatment.	Asymmetrical; gains are taxed, but losses receive limited tax credits.
Lifestyle Penalties	The investor is not subject to any personal lifestyle penalties if investments fail to perform.	Underperformance may trigger an avalanche of hardships that would be difficult or impossible to reverse.
Access to Capital	Has access to low-cost capital, even if portfolio underperforms.	Unreliable; may involve unfavorable rates and terms.
Other Sources of Income	None. Relies exclusively on investments for survival.	None. Relies exclusively on investments for survival.
Portfolio Size	Small; is a component of a larger portfolio.	Large; represents investor's entire pool of assets.

 
Estimating Loss Aversion Ratio

Determining an investor's appropriate level of loss aversion begins with an objective examination of the investor’s circumstances, focusing particularly those factors identified in Table 1. Is the investor's situation similar to that of the archetypical MPT investor? If so, the investor's loss aversion will likely be low, perhaps as low as 1.0, meaning a dollar lost is as painful as a dollar gained is pleasurable.

If, when evaluated versus the factors in Table 1, the investor appears to have a reasonable basis for being risk averse, she may have a loss aversion ratio that is much higher.

The following quiz can help estimate an investor's loss aversion. It should be noted that this quiz merely measures an investor's perceptions; it does not confirm whether those perceptions are reasonable. Therefore, the results of the quiz should only be considered in conjunction with an independent and objective review of the investor’s circumstances; it would be inappropriate to determine an investor's loss-aversion on the basis of the quiz alone.

Loss Aversion Quiz

This quiz consists of a single question. For the best results, you should try to answer the question in a way that is as relevant as possible to your real-world investment situation.

Suppose that you have evaluated your present investment situation, and have determined that you seek a target return of 10% during the next year.

You are given a choice:

You may invest in a risk-free Investment A that will provide you with a 10% return, with 100% certainty, thus meeting your target.

Or, you may choose Investment B, which has a 50% chance of returning 15%, and a 50% chance of returning some level X% that is less than or equal to 10%.

What is the lowest level of X at which you would be willing to choose Investment B over Investment A?

Your Response: __________ (Note: your response must be between 5% and 10%)

Scoring the Results

Your loss aversion ratio can be estimated as 5/(10-X). As a measure of "pain-to-pleasure", your loss aversion ratio measures how much more sensitive you are to losses than gains. For example, a loss aversion ratio of 2.0 means that you perceive losses as twice as painful as gains are pleasurable. Some scoring examples:

Table 2 Loss Aversion Quiz Scoring
Quiz Response	Scoring Calculation	Loss aversion
X=5	5/(10-5) =	1.0
X=6	5/(10-6) =	1.25
X=7	5/(10-7) =	1.66
X=7.5	5/(10-7.5) =	2.0
X=8	5/(10-8) =	2.5
X=9	5/(10-9) =	5.0

 
Hedonically Adjusted Return

Once an investor’s degree of loss aversion has been determined, his or her performance can be recast on a hedonically adjusted basis that provides an estimate of how the performance felt to the investor.

For the loss-averse investor, hedonically adjusted returns (or more briefly, hedonic returns) are simply portfolio returns, where any underperformance relative to the target return is adjusted by the loss aversion ratio, to capture the "pain penalty" experienced by the investor.
 

		investment return	...if investment return >= target return
Hedonically adjusted return	=	or
		target return - (target return - investment return) * loss aversion ratio	...if investment return < target return

For example, suppose an investor has a loss aversion ratio of 2.0 and a target return of 5.5%, but receives an investment return of 3.0%, underperforming the target by 2.5%. On a hedonically adjusted basis, the investment’s return would be 5.5% - (5.5%-3.0%) * 2 = 0.5% (see Figure 3).

Portfolio Construction

Figure 1 presents an example of traditional mean-variance optimization, where portfolios along the efficient frontier are constructed from asset blends with the goal of maximizing return and minimizing standard deviation of return.

 
Traditional mean-variance optimization's use of standard deviation as a measure of risk is best suited for the classical rational investor, for whom a dollar foregone is no more painful than a dollar gained.

For the loss-averse investor, a different approach is needed, one that takes the investor's target return and level of loss aversion into account.

In recent years, downside risk optimizers (also known as semi-variance optimizers) have been developed as a solution to assist the loss-averse investor. However, downside risk optimizers, like their mean-variance progenitors, are generally not built to accommodate the loss aversion parameter and hedonically adjusted efficient frontier concepts that are described in this text.

Constructing the Hedonically Adjusted Efficient Frontier

By applying a simple hedonic performance adjustment, we can convert a standard mean-variance efficient frontier into a hedonically adjusted efficient frontier.

To accomplish this, we deconstruct the traditional efficient frontier into a "cloud" of randomly generated performance outcomes, where for each point on the efficient frontier (per Figure 1), a sampling of normally distributed outcomes are discretely plotted (see Figure 2). Note that the cloud is visually vertically symmetrical around the efficient frontier. 

To each point on the outcome cloud, a hedonic adjustment is applied, whereby each instance of performance below the investor’s target is penalized by the investor's loss aversion ratio. The adjustment of a single outcome point is presented in Figure 3. This process is repeated for all the points in the outcome cloud.

 

Given that the hedonic adjustment process penalizes the underperforming outcomes, the downside outcomes, when adjusted, "feather out" in proportion to the degree of loss aversion (see Figure 4). The resulting performance distribution represents the investment results as they are perceived by the loss-averse investor. Note the loss-averse investor’s hedonically adjusted outcome cloud is negatively skewed.

The hedonically adjusted cloud can now be collapsed into a hedonically adjusted efficient frontier by averaging the hedonically adjusted outcome points. By repeating this process for a variety of levels of loss aversion, we can create multiple hedonically adjusted efficient frontiers (see Figure 5).

When the loss aversion ratio is 1.0, the hedonically adjusted frontier is identical to the traditional mean-variance efficient frontier. However, as the loss aversion ratio increases, the hedonically adjusted frontier moves downward, and its slope flattens because the hedonic adjustment process penalizes the substantial underperformance that can occur in a higher-volatility portfolio.

Interpreting the Hedonically Adjusted Efficient Frontier

The hedonically adjusted efficient frontier enables the loss-averse investor to compare the hedonically adjusted return of various portfolios. The strongly loss-averse investor will discover that as riskier portfolios are contemplated, the hedonically adjusted return may actually decline, a phenomenon that is intuitively satisfying to the loss-averse investor.

For example, in Figure 5, we find that for an investor with a loss aversion ratio of 2.5, a portfolio with approximately 7% standard deviation has the highest hedonic return. In this example, further risk-seeking only serves to reduce hedonic returns. However, for the investor with loss aversion ratio of 1.0, the highest hedonic return is expected at 22% standard deviation level.

Thus interpreted, the hedonically adjusted efficient frontier may potentially serve as a useful tool for portfolio selection, as the peak in each curve represents the point of greatest modeled utility. (Note 2)

Limitations of the Analysis

The hedonically adjusted efficient frontier described in this article is not the result of a hedonic optimization process. Rather, it is an adjustment that is built onto a pre-existing mean-variance optimization. The piggy-back approach used in this article, while portable, is limited to looking at the portfolio as a whole. It is incapable of looking at individual investments within the portfolio, should that ever be necessary. A more sophisticated approach would be to incorporate the hedonic adjustment process within the portfolio optimization process.

The hedonic adjustment process, as described in this article, is also limited by being a simple linear function where the loss aversion ratio is multiplied by the level of underperformance. While this simple linear function provides a reasonable hedonic adjustment at return levels near the investor’s target return, investors’ perceived utility functions are not truly linear. Better nonlinear approximations could be developed for return levels further from the investor’s target return.

Finally, hedonically adjusted returns have the potential to be misunderstood and misused. Hedonically adjusted returns do not represent actual portfolio returns, and should not be reported as such. Given that hedonic returns are always less than or equal to actual returns, any misunderstanding or misstatement would lead to a performance understatement.

Notes

1. The term hedonically adjusted return expresses the utility of a risky investment in terms of the risk-free rate of return for which an investor would be willing to exchange the risky investment with indifference, given the investor's degree of loss aversion and target return. Thus, hedonically adjusted return is a unitized metric of utility.

2. While hedonically adjusted return attempts to represent utility by incorporating loss aversion relative to a level of target return, it must be recognized that future models of utility may incorporate additional factors (e.g. tax aversion).

References

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Peter L. Bernstein. "Most Nobel Minds." CFA Magazine, November/December 2005, Vol. 16, No. 6: 36-43.

Amos Tversky and Daniel Kahneman. "The Framing of Decisions and the Psychology of Choice." Science, 1981.

Daniel Kahneman and Amos Tversky. "Prospect Theory: An Analysis of Decision Under Risk." Econometrica, 1979.

Harry Markowitz. "Efficient Diversification of Investments." 1991.