… are available at http://fin4335.garven.com/spring2018/ps7solutions.pdf.
This week, we introduce the topic of financial derivatives in Finance 4335. Regarding futures and forward contracts (see pp. 5-13 of the Derivatives Theory, part 1 lecture note), it’s hard to beat the following video tutorial on this topic:
When I posted problem set 7 to the course website earlier today, I inadvertently mistyped the problem sets webpage link so it pointed to problem set 6 (which y’all turned in today). Anyway, the link on the webpage is now fixed. The direct link to the problem set is http://fin4335.garven.com/spring2018/ps7.pdf.
The Capital Market Theory lecture note upon which today’s Finance 4335 class discussion was based provides the following important insights:
- 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: .In the above figure, corresponds to the optimal level of exposure to the market index which is labled as point M. When , the investor is fully invested in the riskless asset. When , the investor is partially invested in the riskless asset and in the market index; such portfolios are referred to as “lending” portfolios. When , the investor is fully invested in the market index. Finally, when , the investor funds her investment in the market index with her initial wealth plus borrowed money; such portfolios are referred to as “borrowing” portfolios.
- 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:,where
- 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 () is measured by dividing the covariance between i and the market () by market variance (). 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.
- 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.
- 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).
… are available at http://fin4335.garven.com/spring2018/ps6solutions.pdf.
Problem Set 7 is due at the beginning of class on Tuesday, March 20. Here are some helpful hints:
- The least risky combination of Security A and Security B in Problem 1 is found by calculating and .
- It will always be the case for 2 security portfolios that by following the minimum variance portfolio weighting scheme in the previous bullet point, such a portfolio must have zero variance if or -1.
- In part B of Problem 2, the Sharpe Ratio for security j is .
The topics covered during the course of the last couple of Finance 4335 class meetings (portfolio and capital market theory) 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). The following outline pretty much summarizes what we covered in class on Tuesday, March 13 and Thursday, March 15:
- Portfolio Theory (covered on Tuesday, March 13)
- Mean-variance efficiency
- Portfolio Mean-Variance calculations
- Minimum variance portfolio (n = 2 case)
- Efficient frontier (n = 2 case under various correlation assumptions)
- Capital Market Theory (covered on Thursday, March 15)
- Efficient frontiers with multiple number (“large” n) of risky assets (aka the “general” case)
- 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 than 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 efficient frontier.
- Introduction of a risk-free asset simplifies the portfolio selection problem since the efficient frontier is now a straight line rather than an ellipse in space. The same selection principle holds as in the previous point (point 2); i.e., investors determine optimal portfolios by identifying the tangency between their indifference curves and the efficient frontier. The point of tangency 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.
- 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 () from investing in a risky rather than risk-free asset, divided by the risk taken on by the investor () 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 or supply, which is characterized by risk-return ratios being the same for all possible investments. When this occurs, then the CAPM obtains: .
Here’s the “prospectus” for the Fall 2018 Finance 4335 risk management course; if you click on the image, this will bring up a full-page PDF version. If you have enjoyed Finance 4335 this semester, I hope you’ll tell your friends about the course and encourage them to enroll in it (and/or contact me if they have any further questions)!
Linked below, please find a spreadsheet which showcases how differences in correlation between two risky assets result in different minimum variance portfolio weightings. While we manually performed a similar analysis on the whiteboard in class yesterday, it is also helpful to see this analysis modeled in Excel. Besides determining minimum variance portfolio weightings under different correlation assumptions, the spreadsheet also calculates expected portfolio returns and risks. Furthermore, it decomposes portfolio risks into their component parts – risks related to the variances of the assets which comprise the portfolio, as well as risks which originate from covariance.
During tomorrow’s Finance 4335 class meeting, I expect to complete our discussion of portfolio and capital market theory. The portfolio theory topic (introduced during yesterday’s class meeting) 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 (tomorrow’s topic).
One of the better non-technical summaries 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):
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.