Synopsis of the Capital Market theory topic that we’ll cover during tomorrow’s Finance 4335 class meeting

The Capital Market Theory lecture note upon which last tomorrow’s Finance 4335 class discussion will be based provides the following important insights:

  1. Borrowing and lending at the riskless rate of interest in combination with investing in (mean-variance efficient) risky portfolios makes it possible for investors to obtain superior risk-return trade-offs compared with investing only in mean-variance efficient risky portfolios. In the figure below (taken from page 13 of the Capital Market Theory lecture note), investors select portfolios along the Capital Market Line, which is given by the following equation: E({r_p}) = {r_f} + \left[ {\displaystyle\frac{{E({r_m}) - {r_f}}}{{{\sigma _m}}}} \right]{\sigma _p}.Capital Market LineIn the above figure, \alpha corresponds to the optimal level of exposure to the market index which is labled as point M. When \alpha = 0, the investor is fully invested in the riskless asset. When 0 < \alpha < 1, the investor is partially invested in the riskless asset and in the market index; such portfolios are referred to as “lending” portfolios. When \alpha = 1, the investor is fully invested in the market index. Finally, when \alpha> 1, the investor funds her investment in the market index with her initial wealth plus borrowed money; such portfolios are referred to as “borrowing” portfolios.
  2. Given that investors select (based upon their level of tolerance for risk) portfolios that lie on the Capital Market Line, this behavior has implications for the pricing of risk for individual securities. Specifically, the Capital Market Line implies that for individual securities, the Security Market Line must hold. The equation for the Security Market Line (which is commonly referred to as the Capital Asset Pricing Model, or CAPM) is given by the following equation:E({r_i}) = {r_f} + \left[ {E({r_m}) - {r_f}} \right]{\beta _i},where {\beta _i} = {\sigma _{i,m}}/\sigma _m^2.
  3. According to the CAPM, the appropriate measure of risk for an individual stock is its beta, which indicates how much systematic risk the stock has compared with an average risk investment such as the market portfolio. Beta for security i ({\beta _i}) is measured by dividing the covariance between i and the market ({\sigma _{i,m}}) by market variance (\sigma _m^2). If the investor purchases an average risk security, then its beta is 1 and the expected return on such a security is the same as the expected return on the market. On the other hand, if the security is riskier (safer) than an average risk security, then it’s expected return is higher (lower) than the same as the expected return on the market.
  4. If the expected return on a security is higher (lower) than the expected return indicated by the CAPM equation, this means that the security is under-priced (over-priced). Investors will recognize this mispricing and bid up (down) the under-priced (over-priced) security until its expected return conforms to the CAPM equation.
  5. According to the CAPM, only systematic (i.e., non-diversifiable) risk is priced. Systematic risks are risks which are common to all firms (e.g., return fluctuations caused by macroeconomic factors which affect all risky assets). On the other hand, unsystematic (i.e., diversifiable) risk is not priced since its impact on a diversified asset portfolio is negligible. Diversifiable risks comprise risks that are firm-specific (e.g., the risk that a particular company will lose market share or go bankrupt).

A Summary of Portfolio and Capital Market Theory (source: The Royal Swedish Academy of Sciences)

During tomorrow’s Finance 4335 class meeting, we will complete our study of portfolio and capital market theory. The portfolio theory topic won Professor Harry Markowitz the Nobel Prize in Economics in 1990, and Professor William F. Sharpe shared the 1990 Nobel Prize with Markowitz for his work on capital market theory.

The very best summary of portfolio theory and capital market theory that I am aware of appears as part of an October 16, 1990 press release put out  by The Royal Swedish Academy of Sciences in commemoration of the prizes won by Markowitz and Sharpe (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 (it is important to also note that University of Chicago Finance Professor Merton Miller was cited that same year along with Markowitz and Sharpe for his work on the theory of corporate finance; I include below only the sections of the Royal Swedish Academy press release pertaining to the work by Messrs. Markowitz and Sharpe on the topics of portfolio and capital market theory):

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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.

Correction of typo in the Moral Hazard Extra Credit assignment due today by 5 p.m.

In my extra credit blog posting located at http://risk.garven.com/2017/10/11/moral-hazard-extra-credit-problem-due-via-email-sent-to-riskgarven-com-by-5-p-m-on-monday-october-16/, there was a typo in the assignment that I gave. Specifically, in the table provided, the estimated firm’s profit for an "Average Economy" when the CEO works hard is $12,000,000, not $1,200,000.

Insurance death spiral in the news…

It turns out that the so-called “insurance death spiral” which we modeled in class this past Tuesday (see page 21 of the Moral Hazard and Adverse Selection lecture note for a verbal description and the Dynamic Adverse Selection Spreadsheet for a numerical illustration of the problem) is very much in the news these days; e.g., see the editorial entitled Salvaging Private Health Insurance in today’s Wall Street Journal and yesterday’s page 1 WSJ article entitled In Start to Unwinding the Health Law, Trump to Ease Insurance Rules.  Quoting from today’s WSJ editorial,

“ObamaCare’s defenders are calling all of this “sabotage” and warning about “adverse selection,” in which a more robust individual market will siphon off the healthy customers that prop up ObamaCare’s exchanges. They predict a death spiral of higher premiums for the sick or elderly left on the exchanges.”

Sound familiar?  As we discussed in class last Tuesday, since the implementation of combined premium schemes effectively force good risks to pay too much whereas bad risks pay too little, the good risks opt out.  When this occurs,  expected cost of claims (and correspondingly higher premiums) are in store for those who remain in the risk pool.

minimum variance portfolio spreadsheet from today’s class

Linked below, please find the minimum variance portfolio spreadsheet from today’s class. The learning objective is to showcase how differences in correlation give rise to different portfolio weightings for determining the minimum risk combination of two risky assets. The spreadsheet also decomposes portfolio variance into component parts – risk that originates from the variances of the assets which comprise the portfolio, as well as risk that originates from covariance.

4335 mvp.xlsx

Moral Hazard Extra Credit Problem due (via email sent to risk@garven.com) by 5 p.m. on Monday, October 16

I have written an extra credit problem set on the topic of moral hazard that will be due (via email sent to risk@garven.com) by 5 p.m. on Monday, October 16 (see http://fin4335.garven.com/fall2017/moralhazardEC.pdf). The grade received on this assignment will replace your lowest problem set grade, so long as it is higher than your lowest problem set grade.

Here are some helpful suggestions concerning how to go about working this problem set. The key learning objective for this problem set is to understand how compensation contract design affects managerial incentives to look after the interests of the firm’s owners. For simplicity, I assume here that investors are risk neutral; thus, they expect the manager to maximize the expected value of profit. Put yourself in the shoes of a compensation consultant and show the relationship between contract designs and expected profits under the various compensation scenarios that are provided in the problem set.

At your option, you may solve this problem set by hand or by creating a spreadsheet model. If you rely upon a spreadsheet model, you’ll need to include the spreadsheet as a file attachment when you send it to the risk@garven.com email address. Although a spreadsheet model is not required, you’ll find that Solver will come in very handy, particularly for answering the questions posed in parts 3-5 of the problem set.

Some observations concerning the Rothschild-Stiglitz numerical example from today’s class…

Today, we considered the following problem in class:

  • Assume that consumers are identical in all respects expect for their loss probabilities; some are high risk, and others are low risk.
    • Members of the high-risk group have loss probability pH = 65%, whereas members of the low risk group have loss probability pL = 35%.
  • Each consumer has initial wealth of $100 and utility U(W)=W.5.
  • There are only two possible states of the world, loss and no loss.  If a loss occurs, then consumers lose their initial wealth of $100.
  • Insurance contract offerings include the following:
    • Policy A provides full coverage for a price of $65.
    • Policy B provides full coverage for a price of $45.50.
    • Policy C provides 60% coverage for a price of $39.
    • Policy D provides 30% coverage for a price of $13.65.

The objective here is to identify the set of contract offerings that would prevent adverse selection.  If you consider the pricing of these 4 insurance contracts, Policy A involves full insurance that is actuarially fair for high-risk consumers.  We know from the Bernoulli principle that these consumers would like to purchase this contract.  The challenge is to identify contracts that are favorable for the low-risk consumers but not for the high-risk consumers.  Clearly we would not want to offer contract B, since everyone would select this contract and we would lose $19.50 on every high-risk consumer who purchased it (while breaking even on every low-risk consumer).  High-risk consumers won’t want Policy C because it offers actuarially fair partial coverage, which provides lower expected utility than actuarially fair full coverage.  However, low-risk consumers would be willing to purchase Policy C, so if A and C were offered, the insurer would break even on A and make $18 in profit from low-risk consumers who purchase Policy C.  Given a choice between being uninsured, buying Policy A, or buying Policy C, low-risk consumers would purchase Policy C since it would offer higher expected utility than the other alternatives.  Policy D would also be an acceptable alternative; if high-risk consumers purchased this contract, the insurer would lose $5.85 per high-risk consumer.  However, if Policy A was also offered, none of the high-risk consumers would purchase Policy D.  But low-risk consumers would prefer Policy D since it would offer higher expected utility than the other alternatives.

Here’s a spreadsheet consisting of expected utility calculations:

Rs_spreadsheet

Problem Set 6 Hints and Spreadsheet

In question 1, part C of Problem Set 6, I ask you to “Find the maximum price which the insurer can charge for the coinsurance contract such that profit can still be earned while at the same time providing the typical Florida homeowner with higher expected utility from insuring and retrofitting. How much profit will the insurer earn on a per policy basis?” Here are some hints which you’ll hopefully find helpful.

For starters, keep in mind that in part A, insurance is compulsory (i.e., required by law), whereas, in parts B and C, insurance is not compulsory. In part B of question 1, you are asked to show that the homeowner retrofits while choosing not to purchase insurance. However, in part C, the homeowner might retrofit and purchase coverage if the coinsurance contract is priced more affordably. Therefore, the insurer’s problem is to figure out how much it must reduce the price of the coinsurance contract so that the typical Florida homeowner will have higher expected utility from insuring and retrofitting compared with the part B’s alternative of retrofitting only. It turns out that this is a fairly easy problem to solve using Solver. If you download the spreadsheet located at http://fin4335.garven.com/fall2017/ps6.xls, you can find the price at which the consumer would be indifferent between buying coinsurance and retrofitting compared with self-insurance and retrofitting. You can determine this “breakeven” price by having Solver set the D28 target cell (which is expected utility for coinsurance with retrofitting) equal to a value of 703.8730 (which is the expected utility of self-insurance and retrofitting) by changing cell D12, which is the premium charged for the coinsurance. Once you have the “breakeven” price, all you have to do to get the homeowner to buy the coinsurance contract is cut its price slightly below that level so that the alternative of coinsurance plus retrofitting is greater than the expected utility of retrofitting only. The insurer’s profit then is simply the difference between the price that motivates the purchase of insurance less the expected claims cost to the insurer (which is $1,250).

Finance 4335