Unit Overview

Unit 2 focusses on financial wealth. It examines how households allocate their financial wealth between a risk-free asset and risky assets. The unit considers why many households do not hold risky financial assets (stocks), and what household characteristics explain the probability of holding stocks. The unit also investigates how households select stocks and other risky securities, and whether households follow the basic principle of portfolio diversification. Finally, the unit examines whether households respond to market movements by rebalancing their portfolios, and discusses the role of risk aversion in portfolio rebalancing.

Learning outcomes

When you have completed your study of this unit and its readings, you will be able to:

• analyse the optimal allocation of financial wealth between risk-free and risky assets
• explain the so-called ‘stock market participation puzzle’ (SMPP)
• examine the factors affecting stock market participation by households
• discuss potential reasons for under-diversification in household portfolios
• analyse portfolio rebalancing by households.

2.1Introduction

In this unit you will examine the potential reasons for the limited participation by households in the stock market and in other financial instruments. In theory, the risk premium on stocks, represented by a higher expected return relative to risk-free bonds, should encourage households to invest in the stock market. However, many households do not hold stocks in their financial portfolios. This issue is known in the academic literature as the ‘stock market participation puzzle’ (SMPP).

In Section 2.2 you will study what characteristics differentiate households that buy stocks from households that do not buy stocks. You will also discuss the potential reasons for limited participation of households in other types of financial instruments such as insurance products. You will focus on the role of three main factors: (1) participation costs; (2) non-standard preferences; (3) beliefs about the distribution of future stock returns. In addition, household participation in stock markets is influenced by perceptions relating to the overall reliability of the financial market (that is, whether or not households ‘trust’ that the financial system is fair).

In Section 2.3 you will examine a central topic in household portfolio choice: how do households select the securities in their financial portfolios? This is another area where households do not seem to follow strictly what is suggested by academic theory. In particular, some households tend to under-diversify their portfolios. This section explains why diversification is important and how we can measure diversification in household portfolios. It then discusses which household features tend to be associated with under-diversification. Another important feature of household portfolio choice is trading frequency and its relationship with portfolio performance. You will also consider the role of financial advisers in household portfolio choice.

In Section 2.4 you will take the analysis one step further and examine when households decide to trade. In particular you will examine how households choose a target for the proportion of their financial wealth invested in risky assets, dependent on their risk aversion, the expected market excess return, and the variance of returns on risky assets. This target is assumed to stay relatively constant over time. However, the actual proportion of financial wealth invested in risky assets can deviate from the target due to changes in asset prices. This is known as passive rebalancing. If households buy or sell risky assets to return the proportion of financial wealth invested in risky assets to the target proportion, this is known as active rebalancing.

One key assumption made by studies in the asset-pricing literature (among others, Campbell, 1993, and Campbell and Vuolteenaho, 2004) is that households have a constant degree of risk aversion over time. If this is not true, and risk aversion alters over the lifetime of a household, this will affect the allocation of financial wealth between risk-free and risky assets. You will examine the consequences of this in Section 2.4.

In Unit 1 you studied how to estimate the value of human capital as the present value of expected future labour income. You also saw that the value of human capital changes over time as a household ages. In Section 2.4 you will analyse the effect of including the value of human capital in household wealth. You will see that if the target proportion of financial wealth invested in risky assets depends on the ratio of human capital to financial wealth, then the target proportion will also change over time.

Much of the analysis in this unit involves examining the predictions from theoretical models of household portfolio allocation, comparing these predictions with actual household behaviour, and discussing the reasons why household behaviour does not always match the predictions from the models. This allows us to examine the household characteristics that affect portfolio choices. You will also see the potential exists for households to make a variety of investment mistakes. This is of relevance to households, obviously, and for policymakers. It also highlights opportunities for banks to develop markets for financial products, services and advice, which will generate revenues for the banks but also improve the financial well-being of households.

2.2Household Participation in the Stock Market

In this section you will use the capital asset pricing model (CAPM) to develop your understanding of how households allocate their financial wealth between a risk-free asset and a portfolio of risky assets. The section first considers the optimal allocation of wealth between a risk-free asset and a portfolio of risky assets, and how this allocation is affected by an investor’s degree of risk aversion. The section then examines the stock market participation puzzle – why do households not behave according to the predictions of the model?

You may already be familiar with the CAPM from your work or your previous studies. The section starts with a brief statement of the main elements of the CAPM and the fundamental predictions from the model. The CAPM is studied in more detail in the modules Risk Management: Principles and Applications and Corporate Finance.

2.2.1Expected return and risk

You will see that the allocation of financial wealth between risk-free and risky assets is determined by household preferences towards risk, the return on risk-free assets, the expected returns on risky assets, and the variance of returns on those risky assets.

Risky assets

As you saw in Unit 1, we differentiate between a risk-free asset paying a risk-free return, and risky assets. Each risky asset is characterised by its expected return (the return we expect to receive by holding the asset) and its risk, which we measure by the standard deviation of the asset return (reflecting how much the return varies above, and below, the expected return).

As you also saw in Unit 1, risk-averse investors require a higher expected return for holding a risky asset. So the expected return on risky assets will, in general, be higher than the risk-free return. If we consider two risky assets that have the same level of risk, investors would prefer the asset paying the higher expected return. Likewise, if two risky assets pay the same expected return, we would prefer the asset with the lower risk.

Portfolios of risky assets

A risky portfolio consists of combinations of risky assets. It is possible to form portfolios of risky assets such that the overall risk of the portfolio is less than the risk of the individual assets. This is called diversification. We take advantage of the fact that the returns on assets are not perfectly related. If the return on one asset is lower than normal, this can be offset by a better performance on other assets in the portfolio.

The statements above about a pair of assets also apply to portfolios of assets. If we consider two portfolios that have the same risk, we would prefer the portfolio with the higher expected return. And if two portfolios have the same expected return, we would prefer the portfolio with the lower risk. If we consider all of the risky assets that are available, and all possible combinations of those risky assets, then it is possible to plot out those portfolios that, for a given expected return, have the lowest portfolio standard deviation. Or, equivalently, we could map out the portfolios that, for a given standard deviation, have the highest possible expected return. This produces what is known as the minimum variance set of portfolios (MVS). The MVS represents portfolios yielding the minimum level of risk we can obtain for a particular level of expected return from all possible portfolios. This is shown in Figure 2.1 where the expected return of the overall portfolio, including risk-free and risky assets, E(RP)E(RP), is measured on the vertical axis, and portfolio risk, σ(RP)σ(RP), is measured on the horizontal axis. The minimum variance set is the curve passing through points MVP and M. The minimum variance portfolio (MVP) is the portfolio on the MVS associated with the lowest attainable level of risk. The upper section of the MVS (that is, the section above the MVP) is the efficient frontier of risky assets. This is the set of risky portfolios offering the highest level of expected return for a particular level of risk.

Figure 2.1Utility maximisation for a risk-averse investor

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Review Question 2.1

Please take a moment to consider the efficient frontier in Figure 2.1. For example, would you choose a portfolio that was inside and not on the frontier?

If a portfolio is inside the efficient frontier, then it is possible to choose a different portfolio that has the same risk but higher expected return. Or you could find a different portfolio that has the same expected return but a lower risk.

Now consider point M on the efficient frontier in Figure 2.1. This is the best portfolio out of the portfolios on the efficient frontier. Why best? The portfolio represented by point M has the highest ratio of expected return to risk. In addition, if everyone has the same information about expected returns and asset risk, then this portfolio M will be the same for all investors. If investors hold any risky assets at all, then they will hold this portfolio M, known as the market portfolio.

Suppose an investor was completely averse to risk and invests only in the risk-free asset. This is represented by point RFRF on the vertical axis in Figure 2.1. The investor earns the risk-free rate and the risk of their position is zero.

If an investor decides to hold some of their wealth in the risk-free asset and some in the market portfolio, then the expected return on their financial wealth will increase, and the riskiness of their financial wealth will increase. This is represented in Figure 2.1 by the straight line linking point RFRF on the vertical axis and point M on the efficient frontier. This is known as the capital market line (CML), and is the set of efficient portfolios when a risk-free asset exists in the economy.

2.2.2Optimal allocation between risk-free asset and market portfolio

According to the CAPM, investors, regardless of their level of risk aversion, should invest their wealth either in a risk-free asset or in the so-called market portfolio, or a combination. If they decide to invest in risky assets, they should not hold risky portfolios different from the market portfolio. Investors with different levels of risk aversion may invest different weights of their wealth in the risk-free asset. The higher the level of risk aversion, the larger the portion of their wealth they will invest in the risk-free asset.

Figures 2.1 and 2.2 describe graphically how a risk-averse investor may choose what percentage of their financial wealth to invest in the risk-free asset relative to the market portfolio. The key idea of these graphs is to show how the degree of risk aversion of an individual investor affects the choice between the risk-free asset and the market portfolio.

How are risk preferences represented? We represent an investor’s preferences concerning expected return and risk with indifference curves. An indifference curve represents the combinations of expected return and risk generating the same level of expected utility. An investor is indifferent to any combination of portfolios lying on the same indifference curve.

Consider the indifference curve labelled U in Figure 2.1. The indifference curve slopes upwards. If the investor were to take on more risk, they could maintain the same level of utility if they received a higher expected return. Risk aversion influences the shape of the indifference curves for the individual investor. If the investor is more risk averse, then they would need to receive a larger increase in expected return in order to take on increased risk and maintain the same level of utility.

As stated above, each indifference curve is drawn for a particular level of utility. An indifference curve for an increased level of utility would be drawn higher in Figure 2.1. To see this, consider a point above indifference curve U. For a given level of risk (a point on curve U), a point about curve U represents higher expected return for the same risk, and therefore higher utility. Please also note that indifference curves cannot cross each other.

When allocating their financial wealth between the risk-free asset and the market portfolio, investors would try to be on the highest indifference curve possible (corresponding to their highest level of utility). This optimal allocation of wealth is achieved at the point where the indifference curve is tangential (just touching) the capital market line. The slope of the capital market line represents the trade-off between expected return and risk: this is the extent to which the expected return on the investor’s financial wealth can be increased by holding more of the market portfolio (at the cost of increased risk). The slope of the indifference curve represents the rate at which the investor is prepared to substitute higher expected return for increased risk, and for their utility to remain unchanged. At the tangent point, these slopes are the same.

In Figure 2.1, an investor with indifference curve U would buy both the risk-free asset and the market portfolio and would choose a position on the capital market line corresponding to point J. Therefore, in this case, the investor would participate in the stock market. If an investor, on the other hand, decided to buy only the risk-free asset, but not stocks, then they would choose a position on the CML corresponding to the intercept (point R_F in the graph). In this case, they would not buy any stocks.

Review Question 2.2

Consider the optimal allocation between the risk-free asset and the market portfolio, represented by point J in Figure 2.1. Suppose there is an increase in volatility of returns on all risky assets. What would happen to the capital market line? (Assume the risk-free return and the expected returns on risky assets are unchanged.)

Increased volatility of returns on risky assets would shift the minimum variance set to the right: The expected return corresponding to each portfolio is now associated with more risk. The market portfolio would also shift to the right. The trade-off between expected return and increased risk, represented by the capital market line, would become flatter. This would have implications for the investor’s optimal allocation of financial wealth between the risk-free asset and the market portfolio, and for their level of utility. Let us now examine the indifference curves in more detail.

In Figure 2.2 you can see more clearly how different degrees of risk aversion affect where the indifference curves are tangential to the CML. The degree of risk aversion affects the curvature of the indifference curve: the higher the degree of risk aversion, the more convex is the indifference curve. This is illustrated graphically in Figure 2.2, which shows the indifference curves for three investors U₁, U₂, and U₃ related to a particular value for utility (0.099625). The three investors have different coefficients of absolute risk aversion, A: for investor 1, A = 1; for investor 2, A = 2; and for investor 3, A = 3.

Study Note 2.1

As you saw in Unit 1, the coefficient of absolute risk aversion is equal to

A=−u′′(w)u′(w)A=−u″(w)u′(w)(2.1)

where:

w is the level of wealth

u(w)u(w) is the utility function of wealth

u′(w)u′(w) is the first derivative of the utility function with respect to wealth

u′′(w)u″(w) is the second derivative of the utility function with respect to wealth.

In Figure 2.2 investor 1 has A = 1; investor 2 has A = 2 and investor 3 has A = 3. The indifference curves in Figure 2.1, U1, U2, and U3, are based on the following equation

U=E(RP)−0.05σ(RP)A=0.099625U=E(RP)−0.05σ(RP)A=0.099625(2.2)

The indifference curves are drawn for a particular level of utility. Utility is usually referred to as an ordinal concept: this means it is possible to rank various outcomes according to which outcome you prefer, but you cannot measure a particular level of utility. The value for total utility in equation (2.2), 0.099625, is a theoretical idea. The indifference curves are drawn with a particular level of utility in mind, but recognising that utility cannot be measured. It means we can also examine what an indifference curve would look like for a lower level of utility. So the indifference curve labelled U3′ in Figure 2.2 is based on the equation

U=E(RP)−0.05σ(RP)A=0.064625U=E(RP)−0.05σ(RP)A=0.064625(2.3)

Please now consider Figure 2.2. In Figure 2.2, the investor with a coefficient of absolute risk aversion equal to 1 (curve U₁) has an indifference curve which is tangential to the capital market line (at point J₁). For investors 2 and 3, the indifference curves U₂ and U₃ do not touch the CML. How can these investors choose their position on the CML? They need to accept a lower level of utility. Put another way, given their degree of risk aversion, investors 2 and 3 are unable to achieve the same level of utility as investor 1 (who is less risk averse). Given the current market conditions, they must accept a lower expected return, and/or higher risk.

Indifference curves for the same investor all have the same shape (in terms of slope and curvature), but indifference curves corresponding to higher levels of utility are located in a higher section of the graph, because a higher expected return E(RP)E(RP) for the same level of risk, σ(RP)σ(RP), increases the utility of a risk-averse investor.

For example, let us examine the optimal allocation for investor 3. Figure 2.2 also shows the indifference curve U’3U’3, which corresponds to a level of utility equal to 0.064625 (which is lower than 0.099625). This indifference curve is tangential to the CML at point J’3J’3. Note that the indifference curves U₃ and U’3U’3 have exactly the same shape, because the coefficient of absolute risk aversion is 3 for both of them.

Review Question 2.3

Comment on the optimal allocation chosen by investor 3 in Figure 2.2.

Investor 3 chooses a portfolio on the CML closer to the risk-free asset, compared to investor 1, because investor 3 is more risk averse than investor 1.

Figure 2.2Indifference curves for investors with different degrees of risk aversion

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2.2.3Stock market participation puzzle (SMPP)

So far in this section you have examined the optimal allocation of financial wealth between a risk-free asset and the risky market portfolio, and how the degree of risk aversion for an investor affects this allocation. Do you agree that the model makes some reasonable predictions? The analysis suggests risk-adverse investors will choose a combination of the risk-free asset and the market portfolio; and investors with a higher degree of risk aversion will allocate more of their financial wealth in the risk-free asset, compared to investors with a lower degree of risk aversion.

Contrary to what the theory predicts, many households do not invest in risky assets at all. The term ‘limited participation’ used in the literature refers to the fact that the actual participation rate in the stock market (the proportion of households holding stocks) is lower than the participation rate predicted by theory. Moreover, when households hold stocks, in many cases they do not hold the market portfolio.

The fact that a substantial fraction of households does not hold risky assets is a major puzzle in the household finance literature: why should so many households forego the opportunity to benefit from the additional expected return associated with investing in stocks?

The stock market participation puzzle (SMPP) is not limited to a particular country. For example, the Survey of Health, Ageing and Retirement in Europe (SHARE), which contains data on wealth and portfolio composition of households aged over 50 for 11 European countries, provides evidence consistent with limited stock market participation. In addition, countries with similar levels of GDP per capita exhibit substantial differences in terms of rates of participation in the stock market (Christelis, Jappelli, and Padula, 2010)¹. What are the determinants of such differences? You will consider this topic in the next section.

2.2.4Determinants of stock market participation

In this section you will consider three potential explanations for the SMPP:

• costs of participation in the stock market
• non-standard preferences (loss aversion and ambiguity aversion)
• beliefs concerning future returns.

Participation costs

The costs to participate in stock market investments may affect the decision to buy stocks. If such costs are fixed, or if fixed costs are an important part of the cost structure of activities involved in stock trading, this could lead to non-participation in the stock market for a substantial number of households.

For example, two individuals are deciding whether they should buy stocks or not: Charles has £1 million in his bank account (which is his total financial wealth); Marika has £1,000. The cost to set up a stock trading account at the bank is £100 (for simplicity, we assume this is the only cost associated with stock trading), and there are no variable costs associated with trading. Therefore, for Charles the cost of participation is equal to 0.01% of his financial wealth; while for Marika the cost is equal to 10%. Clearly, all other things being equal, Charles is much more likely to buy stocks than Marika. Charles is likely to buy stocks even if the expected return is very low, (let’s say 5%); on the other hand, for Marika an expected return on the stock market of less than 10% would make any investment in stocks undesirable.

Note that we have not said anything about the level of risk aversion of the investors. We have focused on expected returns, net of participation costs. Thus, even if Charles and Marika were risk neutral, fixed participation costs would make a difference to their decision.

Now, let’s consider another example. In this case, we consider Marika (the same investor as before), and Lynne. Both of them have £1,000 in their bank account, but Marika is less risk averse than Lynne: Marika is willing to invest £1,000 in stocks, provided that the expected return is higher than 10%. On the other hand, Lynne is willing to invest only half of her money in stocks (£500), which means that the set-up cost of the stock trading account is equal to 20% of the amount Lynne is prepared to invest in stocks (£100/£500 = 20%).

Thus, stock market participation is related to both the overall level of financial wealth and the degree of risk aversion of an individual: less risk-averse households are more likely to buy stocks than more risk-averse households.

Unfortunately, participation costs cannot explain very well the cross-country differences in participation rates, especially for countries with similar GDP per capita. Moreover, there is evidence that even rich households do not buy stocks (see Table 6 in Guiso and Sodini, 2013).

Non-standard preferences

Non-standard preferences refer to the possibility that households evaluate risk according to principles that are inconsistent with the expected utility framework. In particular, households may be loss averse, rather than risk averse. Loss-averse households dislike losses more than they like gains of an equal amount. For example, if you are loss averse, losing £5 by accident leads to a decrease in satisfaction larger than the increase in satisfaction resulting from finding £5 by accident. This type of behaviour is inconsistent with standard risk aversion, where losses and gains are treated symmetrically. You will study the concept of loss aversion in more detail in Unit 3.

However, loss aversion may not be enough to explain the SMPP. Behavioural economists have suggested that households may display other types of behaviour that are inconsistent with a traditional economics framework. For example, households may also evaluate risky investments separately from other risks they face, a condition known as narrow framing. If households engage in narrow framing, they fail to consider the impact of a potential new investment on the overall riskiness of their portfolio. There is evidence that a combination of loss aversion and narrow framing can provide an explanation for the SMPP. Finally, households may also be ambiguity averse: they prefer known risks to unknown risks. In the context of portfolio decisions, households prefer an investment where the probability distribution of the potential outcomes is known, compared to an investment where the probability distribution of the outcomes is unknown. If stock returns are perceived to be ‘ambiguous’ (that is, the probability distribution of stock returns is unknown), then ambiguity-averse households are unlikely to participate in the stock market.

Beliefs about future stock market returns

Beliefs about future stock market returns can also lead to a very low participation rate in the stock market. For example, Hurd, Van Rooij and Winter (2011) provide evidence that more optimistic households (with respect to future stock market returns) are more likely to invest in the stock market than pessimistic households. To elicit households’ expectations, Hurd, Van Rooij and Winter (2011) survey Dutch households to collect information on their expectations regarding the Dutch stock market. The respondents are asked for their subjective probability that a particular investment fund linked to the stock market will generate gains or losses of different amounts in a one-year horizon. In order to estimate the mean and variance of the distribution of the expected stock market returns for each respondent, Hurd, Van Rooij and Winter (2011) develop a model of stock market expectations. The key features of the model are as follows.

Given the stock market price (the value of the index) at time t, the log returns (between period t and t + 1) follow a random walk with drift process:

ln(st+1st)=α+vtln⁡(st+1st)=α+vt(2.4)

where:

α is the drift component

vtvt is a normally, independently and identically distributed (NIID) variable, with mean 0 and variance σ2σ2.

If we re-arrange equation (2.4) it provides a more intuitive interpretation:

lnst+1=lnst+α+vtln⁡st+1=ln⁡st+α+vt(2.5)

These equations suggest that the (log of) the stock market index in the next period is the same as the (log of) the stock index in this period, plus a constant and a random element. The random element is normally distributed, its mean and variance do not change over time (it is identically distributed) and the value in one period is not correlated with the value in other periods (it is independently distributed). Suppose the realised value of the random element was zero in a period. The log of the stock market index would increase by an amount equal to α. If α is positive then the stock index would have a tendency to increase, and if α is negative, the stock index has a tendency to decrease, with random fluctuations around these trends. If the drift component is zero, then the stock market index increases or decreases by a random element from period to period.

Review Question 2.4

What is the stock market index two periods ahead, according to equation (2.5)?

Two periods ahead, the stock index is given by

lnst+2=lnst+1+α+vt+1=(lnst+α+vt)+α+vt+1=lnst+2α+vt+vt+1ln⁡st+2=ln⁡st+1+α+vt+1=(ln⁡st+α+vt)+α+vt+1=ln⁡st+2α+vt+vt+1(2.6)

If we look further ahead, equation (2.5) can be generalised for any future investment horizon, τ, as follows:

ln(st+τst)=ατ+τ−1∑j=tvjln⁡(st+τst)=ατ+∑j=tτ−1vj(2.7)

or

ln st+τ=lnst+ατ+τ−1∑j=tvjln st+τ=ln⁡st+ατ+∑j=tτ−1vj(2.8)

In words, equation (2.8) says that if the stock market index follows a random walk with drift, the (log of) the stock index for an investment horizon is equal to the (log of) the stock index now, plus the drift component multiplied by the number of years in the investment horizon, and the sum of the random disturbance terms over that investment horizon.

What can we say about the distribution of the logged stock market return over the investment horizon shown in equation (2.7)? The mean of the random disturbance term vtvt is zero, so the mean for the logged returns at a particular horizon τ is ατ. The variance of vtvt is σ2σ2, so the variance of the logged return over the investment horizon τ is τσ2τσ2.

Beliefs about ατ and τσ² may vary between individuals (known as heterogeneous beliefs). Suppose that an individual forms expectations about future stock market returns based on the formulas above. If they believe the value of the drift term is positive, that is α > 0, the individual will report that the probability of observing gains is larger than the probability of losses. Vice versa, α < 0 indicates that the individual is pessimistic about future stock market returns.

Review Question 2.5

Do you think the random walk with drift model provides a useful and realistic description of stock price movements? What is your belief concerning the drift component, α?

To identify investors’ beliefs in relation to α and σ2σ2 the researchers asked respondents to answer eight questions relating to the probability of gains from holding stocks over the investment horizon, and the probability of losses.

Four questions concerned the probability of a gain (a return greater than 0%, 10%, 20%, and 30%):

P(st+τst>δ) where δ=1.0, 1.1, 1.2, and 1.3P(st+τst>δ) where δ=1.0, 1.1, 1.2, and 1.3(2.9)

And four questions concerned the probability of a loss (a return less than 0%, −10%, −20%, and −30%):

 P(st+τst<δ) where δ=1.0, 0.9, 0.8, and 0.7 P(st+τst<δ) where δ=1.0, 0.9, 0.8, and 0.7(2.10)

Hurd, Van Rooij and Winter (2011) estimate α and σ² based on the responses from 1,251 individuals in 2004 and 1,273 individuals in 2006. For both 2004 and 2006 the median value for α is positive, but there is a strong degree of heterogeneity in the responses. For example, the 5th percentile of α in 2004 was −0.169 and the 95th percentile was 0.105. The estimates for σ² also suggest strong heterogeneity in beliefs.

Review Question 2.6

Please take a moment to think about these estimated values for the drift component. Recall that this component is the predicted movement in stock prices from one period to the next, (the expected value of the random element is zero, so the forecast for the random element is zero). What are the implications for the optimal allocation of financial wealth by households if households have such diverse expectations concerning the expected return on risky assets?

You have seen that perceptions concerning expected returns are important for determining if and how investors participate in risky assets. In addition, the perceived riskiness of the stock market is also important. Investors who perceive the stock market to be less risky (in terms of volatility of stock returns) are more likely to buy stocks. Empirical research on this topic suggest that beliefs may play an important role in explaining cross-country differences in stock market participation. Participation tends to be lower in countries with higher stock market volatility, but higher in countries where the excess return on stocks is high compared to the risk, as measured by the Sharpe ratio (Dimson, Marsh and Staunton, 2002).

The Sharpe ratio for a particular portfolio of securities is equal to the average portfolio return in excess of the risk-free rate, divided by the standard deviation of the portfolio returns

Sharperatio=¯RP−RFσ(RP)Sharperatio=R¯P−RFσ(RP)(2.11)

The examples above consider beliefs in terms of subjective probabilities about future stock returns. By ‘subjective’ we mean that the probability assigned to a particular outcome depends on the individual. Another important component in the decision-making process concerning household portfolios is trust.

When households are in the process of choosing whether or not to invest in the stock market, they are likely to gather information on stock market performance from a variety of sources, such as family members, financial analysts, portfolio managers, or financial advisers. The degree to which households trust the information about the financial market is likely to affect their decision to participate in the stock market. If households think that the financial system is ‘rigged’, they are less likely to invest in stocks. Clearly, investment scandals and instances of fraud (for example, the Bernie Madoff Ponzi scheme, uncovered in 2008) are likely to decrease the level of trust in the stock market (Gurun, Stoffman and Yonker, 2016).

The degree of trust in the stock market may be related to underlying historical and institutional characteristics of the country where the household is located. For example, in countries where the quality of minority shareholder protection is higher, households may be more likely to trust the stock market.

2.2.5Limited participation in other financial instruments

You have seen that for some households, participation in stock market investments is unlikely, and you have examined the reasons why this might be so. We can also ask whether there is limited participation in relation to other financial instruments, including debt instruments (eg mortgages), and other financial instruments such as insurance. It is also the case that there is limited participation with respect to these types of financial instruments.

Households might not borrow if they are particularly risk averse, and are especially concerned about bankruptcy risk. In the UK the application cost for filing for bankruptcy is £680. In the US, after the implementation of the Bankruptcy Abuse Prevention and Consumer Protection Act in 2005, the average cost of filing for bankruptcy for households, including credit counselling and other legal and administrative fees, increased from $921 to $1,377 (Gross, Notowidigdo, Wang, 2014). In addition to these monetary costs, the process of filing for personal bankruptcy can be very stressful and can affect a person’s life considerably². In addition, the tax system in most countries does not provide an incentive for households to take on debt. In many countries, companies are able to deduct interest expenses before taxes are calculated. This creates a tax shield for companies which reduces the cost of debt finance (relative to equity finance). However, in most countries, households are not able to deduct interest expense before tax is calculated on their income, and they do not benefit from this tax shield. For particularly risk averse households, the absence of any tax benefits from holding a mortgage may increase their sense of bankruptcy risk and may discourage them from holding any type of debt instruments.

Many households do not hold insurance products. In some cases insurance is compulsory by law. For example, third party motor insurance is a legal requirement in the UK. However, in other cases households can decide whether or not to buy an insurance product. For example, travel insurance is often not a legal requirement. In the UK, third party house insurance is a legal requirement for the owners of a house, but households do not have to insure the contents of their house. In the situations where insurance is not a legal requirement, a lower degree of risk aversion may lead to limited participation: more risk-tolerant households are likely not to insure themselves against certain types of risk.

Trust may also play a role in participation in insurance markets. Some households might not trust insurance companies, perhaps because they believe that insurance pricing is not fair, or that the insurance companies will refuse a claim. For these households, the likelihood of buying insurance will be lower than for households that trust insurance companies.

The next reading examines these topics in more detail. Please remember that in the readings from Guiso and Sodini (2013) the authors use the term ‘risky share’ to mean the proportion of a household’s financial wealth (or of their overall financial portfolio) that is invested in risky assets. In Guiso and Sodini (2013) ‘risky share’ does not mean the share or stock of a particular company.

Reading 2.1

Guiso L and P Sodini (2013) Section 4.1 ‘Stock Market Participation’ of ‘Household finance: An emerging field’. In: GM Constantinides, M Harris and RM Stulz (Eds.) Handbook of the Economics of Finance. Volume 2B. Oxford and Amsterdam: North Holland. pp. 1452–59.

Reading 2.1

To consolidate your understanding, please now read the start of Section 4, and Section 4.1 ‘Stock Market Participation’, pages 1452–59 of Guiso and Sodini (2013).

If you are especially interested in this area, further reading for this section is R La Porta, F Lopez-de-Silanes and A Shleifer (2006) ‘What Works in Securities Laws?’. However, this is an optional reading and you will not be examined on it.

Review Questions 2.7

1. Explain, in your own words, the difference between ambiguity aversion, loss aversion, and risk aversion.
2. Discuss the potential impact of the following financial, institutional and macroeconomic features on stock market participation:
   1. a low Sharpe ratio
   2. a strong level of protection for minority shareholders
   3. GDP per capita
   4. an efficient judiciary
   5. strict disclosure requirements for companies listed on the stock market.

2.3Household Portfolio Selection

Earlier in this unit you saw how the optimal allocation of financial wealth between risk-free and risky assets depends, in part, on the ability of investors to construct well-diversified portfolios. The efficient frontier and the market portfolio are based on this assumption. In this section you will examine diversification, the extent to which households do not diversify their portfolios sufficiently, the frequency with which households trade in stocks, and the role of financial advisers.

2.3.1Diversification

Earlier in this unit you saw that it is possible to combine securities into a portfolio, so that the risk of the portfolio is less than the risks of the individual securities. This was the logic behind the minimum variance set in Figure 2.1. You will now consider the method of diversification in more detail.

To see the benefits of diversification it will be useful to break down the risk associated with a security into systematic and unsystematic risk. The return on a stock can be written as (Elton et al., 2014)

Ri=ai+βiRmRi=ai+βiRm(2.12)

where:

RiRi is the return on stock i

aiai is that part of the return on the security that is company specific and independent of the market

RmRm is the return on the market index

βiβi is the company beta, showing how the return on stock i varies with the market return.

We can further break down the company-specific part of the stock return into its expected value and a random component

ai=αi+eiai=αi+ei(2.13)

where:

αiαi is the expected value of the part of the stock return that is independent of the market

eiei is a random, uncertain part of the return on the security that is independent of the market.

The return on the stock is then given by

Ri=αi+βiRm+eiRi=αi+βiRm+ei(2.14)

The variance of the return on stock i is

σ2i=β2iσ2m+σ2eiσi2=βi2σm2+σei2(2.15)

where:

σ2mσm2 is the variance of the return on the market index, and

σ2eiσei2 is the variance of the company-specific element of the return.

The variance of the company-specific element of the stock return is known as non-market risk, or unsystematic risk, or idiosyncratic risk. Unsystematic risk refers to firm-specific events that are uncorrelated between each company. If a portfolio is formed of many different stocks, for firms that are not related, then it is possible to reduce the contribution of unsystematic risk to the overall portfolio risk. In fact, the effect of unsystematic risk can be eliminated in a portfolio if the portfolio contains a sufficient number of different stocks. Therefore unsystematic risk is also known as diversifiable risk.

The term

β2iσ2mβi2σm2

is referred to as systematic risk or market risk. It depends on the company’s beta and also the risk of the market. This element of risk cannot be reduced by diversification, because events that affect the economy as a whole influence the returns on the stocks of all companies. Therefore this part of company risk is known as non-diversifiable risk.

In summary, households can reduce the risk of their financial wealth for a particular expected return if they invest in a sufficiently diversified portfolio. Or equivalently, they can earn a higher expected return, adjusted for risk, if they engage in diversification.

Of course, some events may affect many securities even if they do not affect the whole economy. For example, new regulation on patents in the pharmaceutical industry would affect the stock price of pharmaceutical companies without necessarily influencing the market as a whole.

As a result of possible correlations among stock returns of companies in the same industry, diversification can improve if households invest in stocks belonging to companies in different industries. International diversification can also lead to better risk-adjusted portfolio returns, because foreign stocks are less likely to be influenced by domestic events. However, many investors do not invest in international securities, or do not exploit fully the benefits of international diversification, a phenomenon known as ‘home bias’.

Unfortunately, estimating the degree of diversification in household portfolios is difficult because of the lack of available data with sufficient detail on the securities held by households.

There are several types of information sources for household portfolios, including:

• surveys
• ownership registers
• tax accounts (for example, in the US, the 401 (k) pension account is a useful source of information)
• data provided by financial institutions (for example, brokerage houses).

Different databases may provide different kinds of information, and usually there is a trade-off in terms of the level of detail in the database and the representativeness of the sample. For example, data from brokerage houses are usually very accurate, but may suffer from two drawbacks. Firstly, the clients of the brokerage house may not be representative of the population of households (there is sample selection bias), because many households do not hold accounts at brokerage houses. Secondly, the data may not contain information on the total wealth of the household.

Surveys can improve the representativeness of the sample, but may suffer from misreporting, which can be either accidental or deliberate. For example, wealthy investors may not want to reveal detailed information on their financial wealth, and there may be under-reporting of income.

In many cases, tax registers and registers of securities ownership provide information on much larger samples of households, which can reduce sample selection bias. However, they may not be as detailed as data from brokerage houses. For example, the Vardepappererscentralen, the Swedish Security Register Center, has detailed information on the stocks directly held by households, but does not contain information on stocks held indirectly via mutual funds.

Why is it important to have information on stocks indirectly held via mutual funds?

Because this information is important to estimate the degree of diversification: if data on mutual funds is missing, we may under-estimate the degree of diversification of household portfolios.

Calvet, Campbell and Sodini (CCS, 2007) use a database that is able to allow for mutual fund holdings, as well as other demographic features of households. This paper is based on Swedish households, and has the advantage of providing detailed data on the overall wealth of the households included in the sample. An additional advantage is that the database is not based on surveys, and therefore the probability of measurement error due to accidental or deliberate misreporting is low. The findings from this research are considered in the next section.

2.3.2Under-diversification

There has been much research on portfolio diversification. The research suggests that many households are under-diversified. For example, many households tend to hold directly very few stocks. This is the finding from Goetzmann and Kumar (2008), who study US individual investors, and CCS (2007) who study Swedish investors. Households are also found to hold local stocks (Ivković and Weisbenner, 2005), and such a low level of diversification is usually not compensated by mutual fund holdings. The advantage of a mutual fund is that the investments are more likely to be in a broad, diversified portfolio. Ivković and Weisbenner (2005) provide evidence for the US that investing in local stocks may be related to superior knowledge about local firms. This is consistent with what is reported by Goetzmann and Kumar (2008), who also find that a minority of individual investors profit from concentrating their equity investments on stocks for which they have superior information.

In fact, there is evidence that households do tend to use mutual funds to obtain good levels of diversification (CCS, 2007). Recall that idiosyncratic risk is the risk associated with particular companies. In a well-diversified portfolio the overall effect of this risk can be reduced. In a portfolio that is concentrated on a few stocks, the benefits of diversification will not be obtained. CCS (2007) find that households with high idiosyncratic risk have concentrated portfolios of individual stocks, while households with low idiosyncratic risk have portfolios that are diversified because of investments in mutual funds. These findings support the view that households tend to attain good levels of diversification via mutual funds.

Under-diversification may be a problem for households, and especially so for those households with a large share of their wealth invested in risky assets. However, CCS (2007) provide evidence consistent with relatively low losses due to idiosyncratic risk. This happens because most of the households that are willing to bear high levels of idiosyncratic risk actually invest a relatively small share of their wealth in risky assets. In other words, households tend to offset high levels of idiosyncratic risk by investing only a small portion of their wealth in risky assets. However, a small percentage of Swedish households lose a substantial amount of money because of under-diversification.

What drives under-diversification?

Three factors play an important role:

• Level of education: better educated households tend to earn higher risk-adjusted returns.
• Wealth: richer households tend to earn higher risk-adjusted returns.
• Financial literacy: households with a better understanding of basic financial concepts (such as interest rates and inflation) tend to have better-diversified portfolios.

Therefore, households which we might describe as being more financially ‘sophisticated’, defined as being able to avoid making investment mistakes, tend to exploit the benefits from diversification more than households which might be considered to be financially ‘unsophisticated’ (less able to avoid making financial mistakes). However, the term ‘sophistication’ is quite broad and general. You will examine the role of financial literacy in more detail in Unit 3.

2.3.3Trading frequency

Earlier in this unit you considered the optimal allocation of household financial wealth between a risk-free asset and the market portfolio of risky assets. According to financial economics theory, the relationship between the share of financial wealth invested in risky assets for household i, defined as ω_i, is negatively related to the household’s degree of risk aversion, γ_i, and the variance of returns on risky assets, σ2iσi2, and positively related with the expected excess return on risky assets, EreiErie. If households have a constant degree of risk aversion, then any change in the riskiness of returns, σ2iσi2, or a change in the expected excess return on risky assets, EreiErie, should lead investors to adjust their allocation, leading to a change in the proportion of financial wealth held in the risky market portfolio, ω_i. The frequency with which households trade in stocks should coincide with the frequency of news affecting the stock market (Guiso and Sodini, 2013). New information on financial assets occurs very frequently, so we should expect very frequent trading by households.

However, the trading frequency of households is not very high. Guiso and Sodini (2013) report that, on average, households trade 4.5 times a year (this is based on the Unicredit Survey, UCS). Moreover, the frequency at which they observe their portfolio is also quite low (12 times a year for the median investor). This suggests that perhaps households incur substantial costs for gathering information on their investments and keeping track of possible changes in the value of their portfolio. The research also finds that investors tend to check the value of their investments before trading, which is a result we would expect to see.

Does frequency of trading actually improve portfolio performance? Perhaps surprisingly, the answer is No. Individuals that trade more frequently earn on average lower returns, net of transaction costs (Barber et al., 2009). Men are more inclined to trade more frequently than women, and more likely to make losses (Barber and Odean, 2001). However, there is also evidence that individual investors learn from their past mistakes, and if they experience losses repeatedly they stop trading in risky assets (Barber et al., 2014).

What factors might explain infrequent trading by households? For individual households transaction costs can be substantial. In particular, the costs of trading on the stock market are high.

Exercise 2.1

Find information on the costs of trading shares, for a bank or trading platform with which you are familiar.

For example, at the time of writing, HSBC Plc charges more than £10 for each transaction in shares, with frequent traders getting a small discount (HSBC, n.d.).

Suppose you plan to invest £1,000 in the shares of a company via HSBC. The cost of a round-trip transaction (purchase and later sale) will be more than 2% of the investment (£20/£1,000). For wealthier investors, who can afford to make larger transactions in shares, the impact of transaction costs is of course lower.

We might also ask if households make wise trading decisions. Another kind of investment mistake, relating to the frequency and timing of trading, is to sell shares too late when the share price is falling and the investor is losing money, and to sell shares too soon when the share price is rising and they are earning money. This is called the disposition effect, which describes how investors tend to sell winning stocks and to hold losing stocks.

Review Questions 2.8

Do you think these research findings are reasonable? If you actively participate in stock market investments in a personal capacity, are these research findings consistent with your own behaviour? For example, what is the frequency with which you monitor your investments, and how often do you buy and sell stocks? Do you suffer from the disposition effect, or under-diversification?

2.3.4The role of financial advisers

If a household wants to trade in stocks, they can do so via a broker, or a bank. It is possible to trade in stocks without financial advice, but many households tend to delegate the management of their portfolio to a portfolio manager, or to seek advice from a finance professional. For example, you can ask at your bank to meet a financial adviser to discuss potential investment in equities.

As reported in Guiso and Sodini (2013), 60% of the Italian investors considered in the UCS survey make use of financial advisers, and 73% of investors in the US use professional advisers. The evidence suggests that a majority of households tend to delegate portfolio management to mutual funds, a phenomenon called ‘portfolio delegation’, or to seek financial advice from a financial intermediary like a bank.

Does financial advice lead to higher portfolio returns, net of fees?

Professionals are likely to engage in many more transactions than individual investors, which means they are able to benefit from economies of scale in relation to information, and economies of scale in relation to other types of transaction costs. By operating on a much larger scale, professionals should be able to trade with a lower average unit cost, relative to individual investors. In which case, taking financial advice from professionals should be worthwhile, and the information should be obtained at a lower cost than if the household did the research themselves.

You have seen that some households choose to delegate the management of their portfolios to the managers of mutual funds. We can examine the performance of fund managers by comparing actively managed funds and passively managed funds. In an actively managed fund, the manager attempts to pick stocks that will provide the best performance, and which match the investor’s preferences regarding risk. In contrast, passively managed funds are funds whose composition does not change over time. These funds attempt to replicate a benchmark stock market index, like the Dow Jones Industrial Average. This means that the components of the fund are the same as the index being tracked, and the relative weights of each stock within the fund are the same as those in the index. The fund manager does not try to buy or sell stocks to improve performance, so the management fees should be lower than for actively managed funds. Such passive funds are usually named index funds or index trackers, and they can be either mutual funds or exchange traded funds (ETF). There is evidence, reported in Guiso and Sodini (2013), that actively managed portfolios, once management fees are considered, do not provide investors with better returns compared to passively managed funds. This suggests that the performance of the managers of actively managed funds is not sufficient to outweigh the additional costs involved in active management.

What are the factors driving inflows and outflows of money in mutual funds? In other words, when do investors put money in a mutual fund, and when do they decide to take the money out of a mutual fund?

Research using information on US individual brokerage accounts finds that investors tend to invest in mutual funds that are performing better than the average for the sector in that particular period. The household decision to invest in a fund is driven by relative performance. However, when individual investors sell their share in a mutual fund, they do so because of bad past performance of that particular mutual fund (they sell funds with low past returns). So the household decision to withdraw is driven by absolute performance of the mutual fund, and the performance is not judged relative to the performance of other funds.

Does consulting a financial adviser improve the performance of individual investors’ portfolios?

Empirical evidence does not support the view that households that receive financial advice earn higher returns in their portfolios. However, financial advice leads to better diversification in clients’ portfolios. The evidence on the impact of financial advice on other investment mistakes, such as the disposition effect, is inconclusive.

Reading 2.2

Guiso L and P Sodini (2013) sections 4.2.2, 4.2.3 and 4.2.4,of ‘Household finance: An emerging field’. In: GM Constantinides, M Harris and RM Stulz (Eds.) Handbook of the Economics of Finance. Volume 2B. Oxford and Amsterdam: North Holland. pp. 1464–75.

Reading 2.2

Please now read sections 4.2.2, 4.2.3 and 4.2.4 of pages 1464–75 of Guiso and Sodini (2013). This reading mentions prospect theory as an explanation for under-diversification. You will study prospect theory in Unit 3. Equation (4.2) on page 1470 is the equivalent of equation (3.1) on page 1424. Equation (4.2) refers to a vector of individual assets, and expresses the relation between the proportion of a household’s portfolio invested in various risky assets, and the expected excess returns for these assets. The variance-covariance matrix in equation (4.2) represents the variances and covariances of the returns on the various assets. In equation (3.1) the equivalent term was the variance of the returns of the market portfolio of risky assets.

Make a note of the main points of the reading, focussing on:

• potential reasons for which households choose to hold idiosyncratic risk
• the relation between under-diversification and portfolio performance (ie is it always bad to hold directly only a few stocks)
• how ‘familiarity’ can affect portfolio choice.

The reading from Guiso and Sodini suggests that hedging is another reason why investors might choose to hold portfolios of risky assets that are under-diversified relative to the market portfolio: Investors use their investments in risky assets to hedge the risks associated with their professions, or their location, or other aspects of their lives. For example, someone working for Apple might choose to hold investments in Microsoft or Samsung to hedge the risks associated with their employment income.

2.4Household Portfolio Rebalancing

In this section you will examine how households adjust, or rebalance, their financial portfolios. Some of this rebalancing occurs passively. For example, when asset prices change, the composition of household financial wealth also changes. Some of the rebalancing is active, and is the result of household decisions. You will examine rebalancing in response to changes in stock markets, rebalancing that takes place over the life of the household, and the influence of risk preferences on portfolio rebalancing.

2.4.1Response to market movements

As you have seen, financial theory suggests a household’s optimal allocation of financial wealth between risk-free assets and a portfolio of risky assets depends on the household’s degree of risk aversion, and the trade-off between the excess return on the market portfolio, and the volatility of the returns on risky assts. This analysis suggests that households have a target for the proportion of their wealth invested in risky assets, given by

ω∗i=Erei γiσ2i=[E(RM)−RF] γiσ2iωi∗=Erie γiσi2=[E(RM)−RF] γiσi2(2.16)

where:

ω∗iωi∗ is the target proportion of financial wealth invested in risky assets

Erei=[E(RM)−RF]Erie=[E(RM)−RF] is the expected excess return on the market

E(RM)E(RM) is the expected market return

RFRF is the risk-free rate of return

γiγi is the degree of relative risk aversion

σ2iσi2 is the variance of returns on the portfolio of risky assets.

It is assumed that households will attempt to keep the actual proportion of wealth invested in risky assets close to the target value. Households are predicted to ‘rebalance’ their financial wealth so that the current proportion is consistent with the target. Some portfolio balancing will be active, in the sense that the household actively trades to restore the proportion invested in risky assets to its target value. But some balancing will be passive, and occurs because asset prices change, which changes the value of the investment in risky assets. As Calvet, Campbell and Sodini (2009) observe, ‘the change in a household’s risky share is partly determined by the household’s active trades and partly by the returns on its risky securities. For instance, the risky share tends to fall mechanically in a severe bear market’.

Review Question 2.9

Consider a household with a coefficient of relative risk aversion equal to 3.5. The expected return on the market is 10%; the risk-free rate is 2%; and the variance of risky returns is 3%. What is the target proportion of wealth invested in risky assets?

Using equation (2.16), the target proportion invested in risky assets is 76%:

ω∗i=[E(RM)−RF] γiσ2i=0.10−0.023.5×0.03=0.76ωi∗=[E(RM)−RF] γiσi2=0.10−0.023.5×0.03=0.76

Review Question 2.10

Suppose the same household has 100 units of financial wealth, of which 24 units are invested in the risk-free asset and 76 units are invested in a portfolio of risky assets. If asset prices decrease such that the value of the investment in risky assets falls to 45, what is the effect on the proportion of wealth invested in risky assets?

Initially the household’s proportion of financial wealth invested in risky assets equals its target value of 76%. If the value of risky assets falls to 45, the proportion of financial wealth invested in risky assets falls to 65%:

4525+45=0.654525+45=0.65

To return the proportion of financial wealth invested in risky assets to the household’s target, the household would have to purchase risky assets.

Conversely, large increases in the prices of risky assets cause the household’s proportion of financial wealth invested in risky assets to increase above the target. In that situation, the household would have to sell risky assets to return the proportion of financial wealth invested in risky assets to the target value. This active rebalancing in response to share price movements leads to what is called a ‘contrarian’ strategy: the household sells risky assets when prices are increasing, and buys risky assets when the prices are falling.

Review Question 2.11

Consider a household with a coefficient of relative risk aversion equal to 3.5. The risk-free rate is 2%; and the variance of risky returns is 3%. What is the effect on the target proportion of wealth invested in risky assets if the expected market return falls from 10% to 8%?

The expected market return is a determinant of the target proportion of financial wealth invested in risky assets, which would fall from 76% to 57%:

ω∗i=[E(RM)−RF] γiσ2i=0.08−0.023.5×0.03=0.57ωi∗=[E(RM)−RF] γiσi2=0.08−0.023.5×0.03=0.57

If the household begins with an actual proportion of financial wealth invested in risky assets of 76%, then they would need to sell risky assets until the actual risky share is equal to the new, lower target proportion. If you look back to Figure 2.1, a fall in the expected market return is presented by a flatter capital market line. The trade-off between market excess return and risk worsens. The risk-free rate is still less than the expected market return, but the risk-free asset now appears relatively more attractive, and the household would invest a little more in the risk-free asset and a little less in the portfolio of risky assets. This is also active rebalancing, as the household responds to changes in the fundamental determinants of the target proportion of financial wealth invested in risky assets.

When do households actively rebalance their portfolios?

Empirical studies have shown that households tend to engage in the contrarian rebalancing noted above: they buy risky assets when prices are falling and sell risky assets when prices are increasing. Grinblatt and Keloharju (2000) observe this for Finland, and Goetzmann and Massa, (2002) observe this in the US. This finding is particularly strong for short-term and medium-term investment horizons (that is, up to one year). Financial sophistication, defined earlier in the unit as the ability to avoid investment mistakes, tends to decrease this type of investment mistake. This is true for both individual stocks and the overall financial portfolio of households.

There is evidence that households that have been ‘lucky’, meaning that the value of the risky assets in the portfolio has increased, tend to sell risky assets. This is consistent with ‘active’ rebalancing of the type described above. They can sell risky assets in two ways: by selling stocks they hold in their portfolio or by reducing the weight of wealth invested in mutual funds. ‘Unlucky’ households, on the other hand, who have experienced a fall in the value of the risky assets in their portfolio, buy more stocks, both directly and indirectly (that is, via mutual funds).

Researchers have also attempted to measure the speed at which rebalancing takes place. As you have seen earlier in the unit, households do not trade very frequently, and rebalancing is unlikely to happen on a daily basis. Empirical evidence shows that it takes nine months for households to adjust their portfolio by active rebalancing. Enhanced financial sophistication increases the speed of adjustment.

The research on rebalancing has also provided evidence of diminishing relative risk aversion: when households become richer, they tend to increase the proportion of their overall financial portfolio invested in risky assets. This is consistent with the view that as wealth increases, risk aversion decreases (you studied DRRA preferences in Unit 1).

2.4.2Rebalancing over the life-cycle

Classical financial economics theory suggests that households should participate in risky asset markets, regardless of their age. This is one prediction from the portfolio models suggested by Merton (1969, 1971), Mossin (1968) and Samuelson (1969) (MMS). In addition, these models also suggest that the proportion of financial wealth invested in risky assets will not vary over time. In contrast, financial advisers tend to suggest that younger people should invest more in stocks than older people. According to the ‘100 minus age’ rule of thumb, a 25 year old individual should invest 75% of their portfolio in stocks, but a 50 year old should invest 50% of their portfolio in stocks. Is there any way to reconcile the predictions from the theoretical models with this professional advice?

The MMS model is based on some assumptions, some of which we might consider to be unrealistic. In particular, these models do not consider the role of human capital in portfolio decisions. As you saw in Unit 1, the value of human capital is the sum of discounted expected future income from employment. You also saw that human capital changes with age. For this reason, the early MMS model may require an adjustment to allow for human capital. Human capital is higher for young people (they have more years of labour income ahead of them) than for older people. Normally, as time goes by, the portion of household total wealth represented by human capital decreases, because the present value of future labour income decreases. However, as time passes, household financial wealth increases, because the household saves.

Earlier in this section you saw that the target proportion of financial wealth invested in risky assets can be represented as

ω∗i=Erei γiσ2i=[E(RM)−RF] γiσ2iωi∗=Erie γiσi2=[E(RM)−RF] γiσi2(2.16)

We can amend equation (2.16) to allow for human capital at age a and an investment horizon of T years, as follows, using equation (4.1) in Guiso and Sodini (2013):

ω∗i=Erei γiσ2i[1+Ha,TWi,a]ωi∗=Erie γiσi2[1+Ha,TWi,a](2.17)

where:

Wi,aWi,a is financial wealth for household i at age a

Ha,THa,T is human capital for the household at age a and a remaining lifetime horizon of T years.

As time goes by, T decreases, and the value of human capital also decreases because there are fewer years of employment left before retirement. As you can see from equation (2.17) the target proportion of financial wealth invested in risky assets should decrease as the ratio of human capital to financial wealth decreases.

Review Question 2.12

Consider the same household from the earlier example, with coefficient of relative risk aversion of 3.5. The expected market return is 10% and the risk-free rate is 2%. The household has a ratio of human capital to financial wealth of 30%.

• What is the household’s target proportion of financial wealth to be invested in risky assets?
• Over time, the household’s ratio of human capital to financial wealth decreases to 10%. What is the effect on the target proportion of financial wealth invested in risky assets? (Assume the value of other variables is unchanged.)

From equation (2.17)

ω∗i=Erei γiσ2i[1+Ha,TWi,a]=0.76(1+0.30)=0.988ωi∗=Erie γiσi2[1+Ha,TWi,a]=0.76(1+0.30)=0.988

Over time, the ratio of human capital to financial wealth has decreased to 10%. Then

ω∗i=Erei γiσ2i[1+Ha,TWi,a]=0.76(1+0.10)=0.836ωi∗=Erie γiσi2[1+Ha,TWi,a]=0.76(1+0.10)=0.836

As you can see, the target for the proportion of financial wealth invested in risky assets has decreased from 98.8% to 83.6% as the ratio of human capital to financial wealth has decreased.

Review Question 2.13

Guiso and Sodini (2013, pp. 1480–81) note that equation (2.17) above, their equation (4.1), is obtained as the solution from a model in which human capital is tradable and without risk. How realistic are these assumptions?

You saw in Unit 1 that human capital is an element of intangible wealth and cannot be traded. Employment income in future years is uncertain, so human capital is risky. These restrictive assumptions were made by researchers to obtain a solution to these models. If the assumptions are relaxed, the models have no solution. To address this, researchers attempt to calibrate the models (which involves suggesting reasonable values for some of the parameters). The models can then be used to predict the patterns of investments in risky assets over a household’s lifetime, and these can be checked against the patterns we observe.

By including human capital in the analysis of portfolio choice, we have reconciled the predictions from the theoretical models with the advice from financial advisers. Younger people have a higher ratio of human capital to financial wealth, so it makes sense for them to invest in risky assets to a greater extent than older people, for whom the ratio of human capital to financial wealth is lower.

2.4.3Time-varying risk aversion

So far we have assumed that risk aversion remains constant over time. Therefore, for a particular household, i, the coefficient of risk aversion γ_i remains constant throughout their lifetime. However, this may not be the case, and risk aversion might change as the household ages. Time-varying risk aversion can be a possible cause of portfolio rebalancing.

Review Question 2.14

Consider again the household from the earlier example. The expected market return is 10% and the risk-free rate is 2%. The variance of returns on risky assets is 3%. Ignore the role of human capital. What happens to the household’s target for the proportion of financial wealth invested in risky assets if the coefficient of risk aversion changes from 3.5 to 4.5?

The target proportion of financial wealth invested in risky assets changes from 76% to 59%:

ω∗i=[E(RM)−RF] γiσ2i=0.10−0.024.5×0.03=0.59ωi∗=[E(RM)−RF] γiσi2=0.10−0.024.5×0.03=0.59

As expected, when people become more risk averse the target for the proportion of financial wealth invested in risky assets decreases.

To allow for time-varying risk aversion, researchers have introduced habit-formation models. These models consider the possibility that households have a preferred level of wealth and deviations from this level result in a change in the degree of risk aversion. In particular, if current wealth is lower than the habit³, then risk aversion increases. And vice versa, if current wealth is higher than the habit, risk aversion decreases.

Therefore, the proportion of financial wealth invested in risky assets is predicted to decrease with variables that are correlated with habit. Habits are not generally observable, so researchers use proxies for habit. This analysis also suggests that the proportion of financial wealth invested in risky assets will increase with the level of financial wealth of the household. For example, there is empirical evidence suggesting that households with higher average income (a proxy for habit) are less prone to taking on financial risk.

However, risk aversion can vary over time because of factors other than habit. For example, sudden changes in macroeconomic variables can affect the degree of risk aversion of households: risk aversion could increase in periods of low or negative economic growth. The global financial crisis of 2007–09 caused recessions in many countries, and may have influenced the level of risk aversion of households.

You saw in Unit 1 how researchers can estimate elicited measures of risk aversion. Empirical research has used elicited measures of risk aversion to test the impact of the 2007–09 crisis on risk aversion. This research shows that the recession did indeed cause an increase in the average degree of risk aversion of Italian households. In particular, the percentage of households unwilling to take on any financial risk increased from 18% at the beginning of 2007 (before the crisis) to 42% in 2009 (Guiso, Sapienza and Zingales, 2011).

Therefore risk aversion could be affected over time by a ‘fear’ factor. For example, in an experimental setting, Kuhnen and Knutson (2011) find that showing pictures that are related to negative feelings (for example, pictures of rotten food) results in lower participation in risky assets.

Finally, risk aversion could also be affected by age: older people tend to be more risk averse than younger people. An increase in the degree of risk aversion with age, taken together with a decrease in the ratio of human capital to financial wealth, suggest that young households will have a larger proportion of their financial wealth invested in risky assets compared to older households. However, another channel through which risk aversion can affect the relationship between age and portfolio choice is background risk. For young households, uncertainty regarding future income, household size, and human capital is higher than it is for older households. Therefore background risk may reduce the proportion of financial wealth invested in risky assets for young households to a level that is lower than the proportion that would be predicted by a simple life-cycle model of the kind you saw earlier in the unit in section 2.4.2.

Reading 2.3

Guiso L and P Sodini (2013) Section 4.4.3 of ‘Household finance: An emerging field’. In: GM Constantinides, M Harris and RM Stulz (Eds.) Handbook of the Economics of Finance. Volume 2B. Oxford and Amsterdam: North Holland. pp. 1483–85.

Reading 2.3

Please now read Section 4.4.3, pages 1483–85 of Guiso and Sodini (2013), and Section 4.4.6, pages 1489–95. Ensure you make a note of the key points of these readings, especially with respect to:

• the relationship between age and the proportion of the household portfolio invested in risky-assets
• potential reasons why this proportion tends to be smaller for young households compared to theoretical predictions
• potential reasons for the patterns of the participation rates and the proportion of financial wealth invested in risky assets during the life cycle as shown in Figure 29, on page 1493 of Guiso and Sodini (2013).

The next reading, from Kick, Onali, Ruprecht and Schaeck (2016), examines how households (and non-financial firms) adjusted their portfolios in response to the Eurozone crisis, beginning in 2009. It is a useful example of how households rebalance their portfolios in response to an exogenous shock. The database used by the researchers enables them to examine precisely how and to what extent portfolio rebalancing takes place. The asset classes considered include shares, bonds, and mutual funds. The data also provides information on the issuers of the securities, including governments, non-financial firms, and credit institutions. The research examines portfolio rebalancing by considering portfolio concentration levels, represented by Herfindahl-Hirschman indices, across different asset classes and issuers, given by

HHI=N∑i=1s2iHHI=∑i=1Nsi2(2.18)

where:

sisi is the proportion or weight of the security category i = 1, 2, … , N in the portfolio.

The difference-in-difference approach used in the paper involves establishing a treatment group and a control group, identifying the differences between the groups, and examining how these differences evolve in response to the shock.

Reading 2.4

Kick T, E Onali, B Ruprecht, and K Schaeck (2016) How Does the Eurozone Crisis Affect Securities Portfolios? CEB working paper No. 16/022.

Reading 2.4

Please now read the introduction of the paper by Kick, Onali, Ruprecht and Schaeck (2016). Make sure you can:

• describe the event that causes portfolio rebalancing in German household portfolios
• identify and explain the main findings of the paper and the possible interpretations for these findings
• discuss the extent to which you agree with the authors’ interpretations.

2.5Conclusion

This unit started by analysing the optimal allocation of household financial wealth between a risk-free asset and a portfolio of risky assets. You saw how this optimal allocation depended on the risk-free return, the expected return on the market portfolio, the standard deviation of returns on risky assets, and household preferences towards risk. This is a useful starting point, because it allows us to examine how, to what extent, and why households might not behave exactly according to the predictions of the model. You then used this framework to consider the stock market participation puzzle, under-diversification, and rebalancing. The framework also allows us to examine how and why households make investment mistakes.

In relation to the stock market participation puzzle, you identified possible explanations for why so many households do not buy stocks, including participation costs, non-standard preferences (including loss aversion and ambiguity aversion), and beliefs in relation to expected returns and also the possibility of fraud.

You also investigated why households tend to hold under-diversified portfolios. Under-diversification leads to lower risk-adjusted returns and should be avoided. However, under-diversification rarely results in substantial losses, because usually households that hold under-diversified portfolios do not invest much in risky assets. The evidence on the role of financial advice suggests that households that ask for advice tend to hold better diversified portfolios.

In this unit you investigated if and when households rebalance their portfolio. In particular, you focussed on passive changes in the proportion of financial wealth invested in risky assets that occurs, for example, when asset prices change, and if households engage in active rebalancing to keep the proportion of financial wealth invested in risky assets consistent with their target proportion. In addition, you considered rebalancing over a household’s lifetime, and discussed the role of age and time-varying risk aversion.

You have seen that the probability of households making mistakes in their portfolio choices is high, especially for households with low levels of financial sophistication. This obviously has implications for the households themselves, but it is also very relevant for policymakers and financial intermediaries. It means there is an opportunity for retail banks, and the financial advisers working for them, to identify market opportunities in terms of financial advice, products and services, that would generate revenues for the bank, and reduce the adverse consequences for households of their behavioural biases and under-diversification. In relation to this, Unit 8 in particular will focus on private banking.

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