## Important insights from portfolio and capital market theory

The portfolio and capital market theory topics rank among the most important finance topics; after all, the scientific foundations for these topics won Nobel Prizes for Markowitz (portfolio theory) and Sharpe (capital market theory) in 1990. Here’s a succinct outline of these topics (as covered in Finance 4335):

• Portfolio Theory
1. Mean-variance efficiency
2. Portfolio Mean-Variance calculations
3. Minimum variance portfolio (n = 2 case)
4. Efficient frontier (n = 2 case under various correlation assumptions)
• Capital Market Theory
1. Efficient frontiers with many (large “n”) risky assets (also known as the “general” case)
2. Portfolio allocation under the general case
• degree of risk aversion/risk tolerance determines how steeply sloped indifference curves are
• indifference curves for investors with high (low) degrees of risk tolerance (aversion) are less steeply sloped than indifference curves for investors with low (high) degrees of risk tolerance (aversion)).
• Optimal portfolios (i.e., portfolios that maximize expected utility) occur at points of tangency between indifference curves and the efficient frontier.
3. The introduction of a risk-free asset simplifies the portfolio selection problem since the efficient frontier becomes a straight line rather than an ellipse in $E({r_p}), {\sigma _p}$ space. The same selection principle holds as in the previous point (point 2); i.e., investors determine optimal portfolios by identifying the point of tangency between their indifference curves and the efficient frontier. This occurs on the capital market line (CML) where the Sharpe ratio is maximized; everyone chooses some combination of the risk-free asset and the market portfolio, and risk tolerance determines whether the point of tangency involves either a lending (low risk tolerance) or borrowing (high risk tolerance) allocation strategy.
4. The security market line (SML), aka the CAPM, is deduced by arbitrage arguments. Specifically, it must be the case that all risk-return trade-offs (as measured by the ratio of “excess” return ($E({r_j}) - {r_f}$) from investing in a risky rather than risk-free asset, divided by the risk taken on by the investor (${\sigma _{j,M}}$) are the same. If not, then there will be excess demand for investments with more favorable risk-return trade-offs and excess supply for investments with less favorable risk-return trade-offs). “Equilibrium” occurs when markets clear; i.e., when there is neither excess demand nor supply, which is characterized by risk-return ratios being the same for all possible investments. When this occurs, then the CAPM obtains: $E({r_j}) = {r_f} + {\beta _j}(E({r_M}) - {r_f})$.

## A (non-technical) Summary of Portfolio Theory and Capital Market Theory

One of the better non-technical summaries of portfolio theory and capital market theory that I am aware of appears as part of a press release put out by The Royal Swedish Academy of Sciences in commemoration of the Nobel prizes won in 1990 by Harry Markowitz on the topic of portfolio theory, and by William F. Sharpe on the topic of capital market theory (see http://www.nobelprize.org/nobel_prizes/economic-sciences/laureates/1990/press.html). I have included an appropriately edited version of that press release below for your further consideration below:

==========================

Financial markets serve a key purpose in a modern market economy by allocating productive resources among various areas of production. It is to a large extent through financial markets that saving in different sectors of the economy is transferred to firms for investments in buildings and machines. Financial markets also reflect firms’ expected prospects and risks, which implies that risks can be spread and that savers and investors can acquire valuable information for their investment decisions.

The first pioneering contribution in the field of financial economics was made in the 1950s by Harry Markowitz who developed a theory for households’ and firms’ allocation of financial assets under uncertainty, the so-called theory of portfolio choice. This theory analyzes how wealth can be optimally invested in assets which differ in regard to their expected return and risk, and thereby also how risks can be reduced.

A second significant contribution to the theory of financial economics occurred during the 1960s when a number of researchers, among whom William Sharpe was the leading figure, used Markowitz’s portfolio theory as a basis for developing a theory of price formation for financial assets, the so-called Capital Asset Pricing Model, or CAPM.

Harrv M. Markowitz
The contribution for which Harry Markowitz now receives his award was first published in an essay entitled “Portfolio Selection” (1952), and later, more extensively, in his book, Portfolio Selection: Efficient Diversification (1959). The so-called theory of portfolio selection that was developed in this early work was originally a normative theory for investment managers, i.e., a theory for optimal investment of wealth in assets which differ in regard to their expected return and risk. On a general level, of course, investment managers and academic economists have long been aware of the necessity of taking returns as well as risk into account: “all the eggs should not be placed in the same basket”. Markowitz’s primary contribution consisted of developing a rigorously formulated, operational theory for portfolio selection under uncertainty – a theory which evolved into a foundation for further research in financial economics.

Markowitz showed that under certain given conditions, an investor’s portfolio choice can be reduced to balancing two dimensions, i.e., the expected return on the portfolio and its variance. Due to the possibility of reducing risk through diversification, the risk of the portfolio, measured as its variance, will depend not only on the individual variances of the return on different assets, but also on the pairwise covariances of all assets.

Hence, the essential aspect pertaining to the risk of an asset is not the risk of each asset in isolation, but the contribution of each asset to the risk of the aggregate portfolio. However, the “law of large numbers” is not wholly applicable to the diversification of risks in portfolio choice because the returns on different assets are correlated in practice. Thus, in general, risk cannot be totally eliminated, regardless of how many types of securities are represented in a portfolio.

In this way, the complicated and multidimensional problem of portfolio choice with respect to a large number of different assets, each with varying properties, is reduced to a conceptually simple two-dimensional problem – known as mean-variance analysis. In an essay in 1956, Markowitz also showed how the problem of actually calculating the optimal portfolio could be solved. (In technical terms, this means that the analysis is formulated as a quadratic programming problem; the building blocks are a quadratic utility function, expected returns on the different assets, the variance and covariance of the assets and the investor’s budget restrictions.) The model has won wide acclaim due to its algebraic simplicity and suitability for empirical applications.

Generally speaking, Markowitz’s work on portfolio theory may be regarded as having established financial micro analysis as a respectable research area in economic analysis.

William F. Sharpe

With the formulation of the so-called Capital Asset Pricing Model, or CAPM, which used Markowitz’s model as a “positive” (explanatory) theory, the step was taken from micro analysis to market analysis of price formation for financial assets. In the mid-1960s, several researchers – independently of one another – contributed to this development. William Sharpe’s pioneering achievement in this field was contained in his essay entitled, Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk (1964).

The basis of the CAPM is that an individual investor can choose exposure to risk through a combination of lending-borrowing and a suitably composed (optimal) portfolio of risky securities. According to the CAPM, the composition of this optimal risk portfolio depends on the investor’s assessment of the future prospects of different securities, and not on the investors’ own attitudes towards risk. The latter is reflected solely in the choice of a combination of a risk portfolio and risk-free investment (for instance treasury bills) or borrowing. In the case of an investor who does not have any special information, i.e., better information than other investors, there is no reason to hold a different portfolio of shares than other investors, i.e., a so-called market portfolio of shares.

What is known as the “beta value” of a specific share indicates its marginal contribution to the risk of the entire market portfolio of risky securities. Shares with a beta coefficient greater than 1 have an above-average effect on the risk of the aggregate portfolio, whereas shares with a beta coefficient of less than 1 have a lower than average effect on the risk of the aggregate portfolio. According to the CAPM, in an efficient capital market, the risk premium and thus also the expected return on an asset, will vary in direct proportion to the beta value. These relations are generated by equilibrium price formation on efficient capital markets.

An important result is that the expected return on an asset is determined by the beta coefficient on the asset, which also measures the covariance between the return on the asset and the return on the market portfolio. The CAPM shows that risks can be shifted to the capital market, where risks can be bought, sold and evaluated. In this way, the prices of risky assets are adjusted so that portfolio decisions become consistent.

The CAPM is considered the backbone of modern price theory for financial markets. It is also widely used in empirical analysis, so that the abundance of financial statistical data can be utilized systematically and efficiently. Moreover, the model is applied extensively in practical research and has thus become an important basis for decision-making in different areas. This is related to the fact that such studies require information about firms’ costs of capital, where the risk premium is an essential component. Risk premiums which are specific to an industry can thus be determined using information on the beta value of the industry in question.

Important examples of areas where the CAPM and its beta coefficients are used routinely, include calculations of costs of capital associated with investment and takeover decisions (in order to arrive at a discount factor); estimates of costs of capital as a basis for pricing in regulated public utilities; and judicial inquiries related to court decisions regarding compensation to expropriated firms whose shares are not listed on the stock market. The CAPM is also applied in comparative analyses of the success of different investors.

Along with Markowitz’ portfolio model, the CAPM has also become the framework in textbooks on financial economics throughout the world.

## Rothschild-Stiglitz model (numerical and graphical illustration)

According to the Rothschild-Stiglitz model (referenced in the Adverse Selection – a definition, some examples, and some solutions blog posting, and on page 23-25 of the Asymmetric Information: Moral Hazard and Adverse Selection lecture note), insurers will limit contract choices such that there is no adverse selection. Rothschild and Stiglitz refer to such an outcome as a “separating equilibrium”.

To see this, assume there are equal numbers of high-risk and low-risk insureds, all of whom have an initial wealth of $125 and square root utility. There are two states of the world – loss and no loss, and the probabilities of loss are 75% for high-risk types and 25% for low-risk types. By offering high-risk types full coverage at their actually fair price of$75 and offering low-risk types partial (10%) coverage at their actuarially fair price of \$2.50, both types of risks buy insurance and there is no adverse selection.

This is illustrated in the figure below and in the spreadsheet located at http://fin4335.garven.com/fall2023/rothschild-stiglitz-model.xls.  Clearly, neither the B (full coverage for low-risk insureds) nor C (based on the average cost of the actuarially fair prices for the low-risk and high-risk) contracts would ever be offered because both of these contracts incentivize high-risk types to adversely select against the insurer.

Rothschild-Stiglitz model (numerical and graphical illustration)

## Adverse Selection – a definition, some examples, and some solutions

The adverse selection problem (also known as the “hidden information” problem) is especially easy to grasp in an insurance market setting; if you are an insurer, you need to be concerned that the worst potential risks may be the ones who wish to purchase insurance. However, it is important to note that adverse selection also occurs in many other market settings. Adverse selection occurs whenever one party has superior information compared with his or her counterparty. When this occurs, there is a risk that the more informed party may take advantage of the other, less informed party.

In an insurance setting, adverse selection is an issue whenever insurers know less about the actual risk characteristics of a potential client than the client herself.  In lending markets, banks have limited information about their potential clients’ willingness and ability to pay back their loan commitments. In the used car market, the seller of a used car has more information about the car that is for sale than potential buyers. In the labor market, employers typically know less than the worker does about his or her abilities. In product markets, the product’s manufacturer often knows more about product failure rates than the consumer, and so forth…

Several strategies exist for mitigating adverse selection. In financial services markets, risk classification represents an important strategy. Insurers and banks want to know your credit score because consumers with bad credit not only often lack the willingness and ability to pay their debts, but they also have more accidents on average than consumers with good credit. Signaling is used in various settings; for example, one solution to the “lemons” problem in the market for used cars is for the seller to “signal” by providing credible third-party certification; e.g., by paying for Carfax reports or relying on vehicle inspections by an independent third-party intermediary such as Carmax. Students “signal” their quality by selecting a high-quality university (e.g., Baylor! :-)). Here, the university provides potential employers with credible third-party certification concerning the quality of human capital. In product markets, if a manufacturer provides a long-term warranty, this may signal that quality is better than average.

Sometimes it’s not possible to mitigate fully adverse selection via the methods described above. Thus, insurers commonly employ pricing and contract design strategies that financially reward policyholders for revealing their true risk characteristics according to the contract choices they make; i.e., they voluntarily reveal their preferences. Thus, we get what’s commonly referred to as a “separating” (Rothschild-Stiglitz) equilibrium in which high-risk insureds select full coverage “high-risk” contracts whereas low-risk insureds select partial coverage “low-risk” contracts:

The Rothschild-Stiglitz equilibrium cleverly restricts the menu of choices in such a way that the insurer induces self-selection; i.e., low-risk insureds select the (low-risk and profitable) partial coverage contract designed with them in mind, and the high-risk insureds select the (high-risk and profitable) full coverage contract designed for them.  Here, the insurer offers contract L, which involves partial coverage at an actuarially fair price (based on the loss probability of the low-risk insured), and contract H, which provides full coverage at an actuarially fair price (based on the loss probability of the high risk insured). The indifference curve slopes are steeper for the low-risk insureds than they are for the high-risk insureds.  Here, the high-risk policyholder optimally chooses contract H and the low-risk policyholder optimally chooses contract L. The high-risk policyholder prefers H to L because L would represent a point of intersection with a marginally lower indifference curve (here, the Ih curve lies slightly above contract L, which implies that contract H provides the high-risk policyholder with higher expected utility than contract L).  The low-risk policyholder prefers L to H, even though only provides partial coverage.   Thus, one inefficiency related to adverse selection is that insurance opportunities available to low-risk policyholders are limited compared with the world where there is no adverse selection.

There is a very practical implication of this model. If you are a good risk, then you owe it to yourself to select high-deductible insurance, since insurers price low-deductible insurance expecting high-risk policyholders will be the primary purchasers of such coverage (and therefore, low-deductible policies will be more costly per dollar of coverage than high-deductible policies).

## Synopsis of the Moral Hazard topic…

In finance, the moral hazard problem is commonly referred to as the “agency” problem. Many, if not most real-world contracts involve two parties – a “principal” and an “agent”. Contracts formed by principals and agents also usually have two key features: 1) the principal delegates some decision-making authority to the agent, and 2) the principal and agent decide upon the extent to which they share risk.

The principal has good reason to be concerned that the agent may take actions that are not in her best interests. Consequently, the principal has strong incentives to monitor the agent’s actions. However, since it is costly to closely monitor and enforce contracts, some actions can be “hidden” from the principal in the sense that she is not willing to expend the resources necessary to discover them since the costs of discovery may exceed the benefits of obtaining this information. Thus, moral hazard is often described as a problem of “hidden action”.

Since it is not economically feasible to perfectly monitor all the agent’s actions, the principal needs to be concerned about whether the agent’s incentives line up, or are compatible with the principal’s objectives. This concern quickly becomes reflected in the contract terms defining the formal relationship between the principal and the agent. A contract is said to be incentive-compatible if it causes principal and agent incentives to coincide. In other words, actions taken by the agent usually also benefit the principal. In practice, contracts typically scale agent compensation to the benefit received by the principal. Thus, in insurance markets, insurers are not willing to offer full coverage contracts; instead, they offer partial insurance coverage which exposes policyholders to some of the risks that they wish to transfer. In turn, partial coverage reinforces incentives for policyholders to prevent/mitigate loss.

Similarly, in a completely different setting, consider the principal/agent relationship which exists between the owner and manager of a business. If the manager’s effort level is high, then the owner may earn higher profits compared with when the manager’s effort level is low. However, if managerial pay consists of a fixed salary and lacks any form of incentive compensation (e.g., bonuses based upon meeting or beating specific earnings targets), then the manager may be inclined to not exert extra effort, which results in less corporate profit. Thus, compensation contracts can be made more incentive-compatible by including performance-based pay in addition to a fixed salary. This way, the owner and manager are both better off because incentives are better aligned.

## On the Determinants of Risk Aversion

Several years ago, The Economist published a particularly interesting article about various behavioral determinants of risk aversion, entitled “Risk off: Why some people are more cautious with their finances than others”. Here are some key takeaways from this (somewhat dated, but still quite timely) article:

1. Economists have long known that people are risk averse, yet the willingness to run risks varies enormously among individuals and over time.
2. Genetics explains a third of the difference in risk-taking; e.g., a Swedish study of twins finds that identical twins had “… a closer propensity to invest in shares” than fraternal ones.
3. Upbringing, environment, and experience also matter; e.g., “… the educated and the rich are more daring financially. So are men, but apparently not for genetic reasons.”
4. People’s financial history has a strong impact on their taste for risk; e.g., “… people who experienced high (low) returns on the stock market earlier in life were, years later, likelier to report a higher (lower) tolerance for risk, to own (not own) shares and to invest a bigger (smaller) slice of their assets in shares.”
5. “Exposure to economic turmoil appears to dampen people’s appetite for risk irrespective of their personal financial losses.” Furthermore, low tolerance for risk is linked to past emotional trauma.

## Some important intuitions from today’s class meeting of Finance 4335

1. The most important concept covered in class today is that people vary in terms of their preferences for bearing risk. Although we focused most of our attention on modeling risk-averse behavior, we also briefly considered examples of risk neutrality (where you only care about expected wealth and are indifferent about the riskiness of wealth) and risk loving (where you actually prefer to bear risk and are willing to pay money for the opportunity to do so).
2. Related to point 1: irrespective of whether you are risk averse, risk neutral, or risk loving, the foundation for decision-making under conditions of risk and uncertainty is expected utility. Given a choice among various risky alternatives, one selects the choice that has the highest utility ranking.
3. If you are risk averse, then $E(W) > {W_{CE}}$ and the difference between $E(W)$ and ${W_{CE}}$ is equal to the risk premium $\lambda$. Some practical implications — if you are risk averse, then you are okay with buying “expensive” insurance at a price that exceeds the expected value of payment provided by the insurer, since (other things equal) you’d prefer to transfer risk to someone else if it’s not too expensive to do so. On the other hand, you are not willing to pay more than the certainty equivalent for a bet on a sporting event or a game of chance.
4. If you are risk neutral, then $E(W) = {W_{CE}}$ and $\lambda = 0$; risk is inconsequential and all you care about is maximizing the expected value of wealth.
5. If you are risk loving, then $E(W) < {W_{CE}}$ and $\lambda < 0$; you are quite willing to pay for the opportunity to (on average) lose money.

## On the ancient origin of the word “algorithm”

The August 29th assigned reading entitled “The New Religion of Risk Management” (by Peter Bernstein, March-April 1996 issue of Harvard Business Review) offers a concise overview of the same author’s 1996 book entitled “Against the Gods: The Remarkable Story of Risk“. An intriguing excerpt from page 33 of “Against the Gods” elucidates the historical roots of the term “algorithm.” An intriguing excerpt from page 33 of “Against the Gods” elucidates the historical roots of the word “algorithm.”

“The earliest known work in Arabic arithmetic was written by al­Khowarizmi, a mathematician who lived around 825, some four hun­dred years before Fibonacci. Although few beneficiaries of his work are likely to have heard of him, most of us know of him indirectly. Try saying “al­Khowarizmi” fast. That’s where we get the word “algo­rithm,” which means rules for computing.”

Note: The book cover shown above is a copy of a 1633 oil-on-canvas painting by the Dutch Golden Age painter Rembrandt van Rijn.