Are investors essentially rational beings who look to maximize their portfolio values?

Or are they ruled by an assortment of motivations that affect the way they structure their investments?

If you belong to the former camp, as would Spock from Star Trek, you may find value in Modern Portfolio Theory, the Capital Asset Pricing Model, and the Arbitrage Pricing Theory.

However, if you think that fear and hope are the kings of Wall Street, consider Behavioral Portfolio Theory (BPT).

## 1. Assumptions

Many portfolio theories make certain assumptions about markets and traders, including:

- Investors want to maximize return and minimize risk, i.e., investors are rational.
- Investors desire
*mean-variance efficiency*, meaning the maximum return for any given level of risk or the minimum risk for any given level of return. - The
*risk aversion coefficient*measures the amount of utility we gain as we add to our wealth or lose when we subtract from our wealth. - Investment returns are normally distributed around a mean.
- Investors all have access to the same information (i.e., no inside information) and share the same views on expected returns. This is highly disputed.
- Ignore trading costs and taxes.
- Individual investors do not influence market prices.
- You can borrow as much money as you want at the risk-free rate.

Obviously, some of these assumptions are questionable. Although these assumptions are adopted by Modern Portfolio Theory, they are relaxed or expanded by other theories.

## 2. Risk Measures

Most theories use *variance* or *standard deviation* to measure risk. Variance is the average variation of each point in a data set from the set’s mean. Variance is expressed in squared units relative to the mean.

Standard deviation is the square root of variance, so that it is expressed in the same units as the mean. It is a measure of how much a set of data is dispersed relative to the average value. The greater the dispersion, the greater the risk and the higher the standard deviation.

*Beta* is another risk measure. It is a portfolio’s sensitivity to the market’s systematic risk (beta = 1.0), which is risk you can’t reduce through diversification within the market. A beta below 1.0 is less risky than the market, whereas betas greater than 1.0 are riskier than the market portfolio.

## 3. The Modern Portfolio Theory

Modern Portfolio Theory (MPT) describes a way for investors to construct portfolios such that return is maximized for any level of risk.

MPT assumes that investors are averse to risk, only willing to assume more risk if they get a higher return.

MPT introduces the *efficient frontier*, which is a region on a graph of return (Y-axis) vs risk (X-axis) known as the risk-expected-return space.

The space maps out a parabola, and any point on the upper surface is part of the efficient frontier.

Any portfolio with risk and reward equal to a point on the efficient frontier is optimized.

Conversely, any portfolio below the efficient frontier is suboptimal and thus undesirable. MPT states that you can reduce the risk of your portfolio through diversification.

If you add a risk-free asset to the risk-expected-return-space, the efficient frontier becomes a straight line from the Y-axis.

If you draw the tangent to the upper boundary of the parabola, you get the *capital allocation line*, which shows the amount of return to expect from a given level of risk.

You can increase the riskiness of a portfolio through leverage. That is, you can borrow money to buy more shares.

## 4. The Capital Asset Pricing Model (CAPM)

The CAPM derives from MPT. It describes the relationship between expected returns and systematic risk.

CAPM is used to price assets based on asset risk and the cost of capital.

The CAPM equates the expected return of an investment with the sum of the risk-free rate and the relationship of the market risk premium (i.e., the expected return of the market minus the risk-free rate) to the investment. That relationship is called *beta* and is a measure the additional riskiness an investment will add to the market portfolio.

Adding in a low-beta stock (i.e., beta less than 1.0, which is the market beta) reduces the risk of a portfolio. The reverse is true for high-beta stocks.

You use CAPM to evaluate whether a stock is priced rationally compared to its riskiness, interest rates, and expected return.

## 5. What is Behavioral Portfolio Theory?

BPT is not one theory.

It is an umbrella term for progressive refinements of the idea that describes how investors try to protect against losses while seeking high rewards.

BPT is sometimes called the “government bond and lottery ticket theory” because it addresses widely divergent objectives.

### Safety-First Portfolio Theory

Safety-First Portfolio Theory was introduced in 1952. The thrust was that investors care first about avoiding ruin, defined as wealth falling below subsistence levels.

It introduced the SFRatio, which defines the required return an investor demands for different risk levels.

The SFRatio is applied to a portfolio. It is the difference between the expected and minimum required returns divided by the portfolio’s standard deviation.

The SFRatio indicates the probability that a portfolio will achieve an investor’s minimum required return. The trick is to find the highest available SFRatio.

Admittedly, this is a hit-and-miss endeavor in which investors evaluate different investments and different asset allocations.

The behavioral outcome of the Safety-First theory is restful sleep. Knowing you will achieve your required minimum return lets you get a good night’s sleep. It’s a bonus if you achieve a higher-than-minimum return.

Safety First Theory does not assume unlimited access to borrowed money at the risk-free rate.

Subsequent improvements to Safety First theory have been developed over the last 60 years.

In sum, Safety First Theory is risk-averse by nature, an assumption that is perhaps a little too tame for more adventurous investors. SFPT is a behavior theory of choice when conditions are uncertain. Risk-taking hangs in the balance between hope and fear.

### Behavioral Portfolio Theory Multiple Accounts (BPT-MA)

BPT-MA[1] imagines that investors maintain “mental accounts,” each with its own purpose and risk/reward structure.

These need not be actual subaccounts — rather, it’s a way of mentally picturing how one thinks about different investments.

For example, the base account might be oriented toward preventing losses and contains low-risk, low-reward investments such as Treasury debt. Investors then construct other sub-portfolios with specific goals.

Other mental accounts have different degrees of risks and rewards. For example, one might imagine a lottery account where the investor is willing to take overly aggressive positions in the hope of making a big score.

Mental accounts might exist for different goals, target wealths, risk attitudes, and target dates.

In BPT-MA, investors ignore the covariance between different mental accounts. It leads to quite different asset allocations then you would expect from Modern Portfolio Theory.

### SP/A Theory

SP/A Theory deals with security, potential, and aspiration. It was developed in 1987 by Lola Lopes. The name of the theory stems from these three attributes:

- S = security, a general worry about being poor.
- P = potential, the desire to be rich.
- A = aspiration, the need to reach a specific goal, such as achieving a certain minimum amount of wealth.

Fear is the dominant factor because fearful people overestimate the chances of the worst outcomes and underestimate the chances for the best outcomes. This causes investors to minimize the chances of obtaining the highest amount of expected wealth.

In other words, fearful individuals are pessimistic.

Hope represents the optimistic side of investors. Optimism causes hopeful investors to overstate the probability of achieving the highest level of expected wealth.

Lopes uses a sophisticated symbolism to allow us to gauge the probability that the value of a portfolio will exceed the investor’s subsistence level (i.e., Probability of wealth > subsistence level).

Pessimistic investors set the probability of success too low. Optimistic investors set the probability too high.

The theory assumes that all individuals have fear and hope, and that the relative strength of each emotion serves to modify the wealth attained by the investor’s portfolio.

SP/A theory teaches that you should control your emotions when constructing your portfolio so that neither hope nor fear predominates. Unfortunately, it doesn’t provide the recipe for accomplishing this goal.

## 6. Advancements in BPT Since 2000

Work has continued on the theoretical, empirical, and experimental aspects of BPT in the 21^{st} century.

### Theoretical Advancements

BPT theoretical advancements mostly deal with human psychological factors, including beliefs, choices, preferences, and biases, such as:

- Changes in consumer confidence influence individual investor outlooks. Growing confidence equates to bullishness.
- Indexes of socially responsible stocks outperform standard indexes like the S&P 500.
- Investors sometimes make separate decisions that should be combined. For instance, some investors choose different functions for different asset groups. This can skew consumption decisions and other behaviors toward irrational, negative outcomes.
- Investor biases introduce cognitive errors when constructing a portfolio. This is especially evident when making decisions under uncertain and difficult circumstances.
- Biases include judgements, beliefs, rules of thumb, emotions, and preferences.
- Successful traders tend towards overconfidence as evidenced by higher trading volumes.
- Our emotional states and subsequent behaviors (or
*affects*[2]) impact the willingness to purchase individual stocks, which in turn influences stock prices. For example, if we don’t like Facebook the company, we may refrain from buying Facebook stock, thereby decreasing demand and its share price.

### Empirical Advancements

Empirical (or observable) BPT advancements have touched upon diversification strategy, individual aspirations, risk aversion, and the use of certain pricing models. These include:

- Many investors are not diversified enough. The average investor holds three or four stocks, whereas Modern Portfolio Theory suggests a portfolio size in excess of 300 stocks. The mental accounts of BPT help explain this phenomenon, in which the base account correlates to risk aversion or risk seeking.
- Stock price studies have been able to reconcile MPT and BPT, but the two are not interchangeable. MPT portfolios are more mean-variance efficient than are BPT portfolios.
- BPT portfolios produce higher risks, higher returns, and positively skewed returns.
- BPT portfolios have a risk aversion coefficient up to 10X lower than do MPT portfolios.
- A 2010 study[3] found that residual risk (i.e., risks associated with a single company rather than the overall market) has no impact on the expected returns of MPT portfolios.

### Experimental Advancements

Recent experimental studies have covered risk aversion attitudes and how they affect individual aspirations, risk aversion, diversification strategy, and the use of models. They include:

- Investors ignore covariances (i.e., directional relationships) when putting portfolios together. That means that investors tend to hold suboptimal portfolios.
- The risk perceptions, risk attitudes, and investment time horizons of investors strongly impact the riskiness of their portfolios.
- Investors use two mental accounts for risky and risk-free investments, and risk attitude impacts the risky asset’s weight in the portfolio.
- Option traders use puts and calls based on their risk tolerances. They use puts for downside-protection mental accounts and calls for upside-potential mental accounts.
- When investors set a targeted goal, they will tend to use the least-risky asset to obtain that goal, even if a riskier asset could provide a greater return. They will use the riskier asset if the less-risky one would fail to provide the desired return.

## 7. Conclusion

Research has been able to connect the robotic Modern Portfolio Theory with the oh-so-human Behavioral Portfolio Theory, but the two are not interchangeable and come from different perspectives.

With BPT, one hopes to make models that better match the way investors and traders actually operate. This is a robust area for research, and acts as a counterbalance to the algorithms of high-volume trading that are at a distant remove from hope and fear.