Category Archives: Risk

Source for tomorrow’s coverage of the underinvestment problem

Tomorrow’s coverage of the so-called underinvestment problem (where, in the absence of risk management, it may sometimes be “rational” to reject a positive net present value project)  is based on the How Insurance Solves the Underinvestment Problem assigned reading.  This reading is an unpublished appendix based on the following published (peer-reviewed) journal article:

Garven, James R. and Richard D. MacMinn, 1993, “The Underinvestment Problem, Bond Covenants and Insurance,” Journal of Risk and Insurance, Vol. 60, No. 4 (December), pp. 635-646.

The above-referenced article was recognized as the best (shorter) article published in the Journal of Risk and Insurance in the year that it was published, and according to Google Scholar, it has been cited 87 times.

Options and credit risk in the news…

See “Bridgewater Makes $1.5 Billion Options Bet on Falling Market” – this article details how Bridgewater has purchased $1.5 billion worth of put options written on the S&P 500 and Euro Stoxx 50 indices which would pay off if either the S&P 500 or the Euro Stoxx 50—or both—fall in value by this coming March.  However, the notional value of this options position is small relative to the firm’s net long positions in equities, so apparently Bridgewater has for all intents and purposes merely purchased some insurance against downside risk.

Also see “Lenders Brace for Private-Equity Loan Defaults” on how financial institutions have been raising their default probability estimates for loans to private-equity-owned companies; article notes a 10% deterioration in credit risk in these so-called leveraged loans.

Merton: Applications of Option-Pricing Theory (shameless self-promotion alert)…

During class today, we showed how the multi-timestep binomial option pricing formula (also known as the Cox-Ross-Rubinstein, or CRR model) converges in the limit (as the number of timesteps n become arbitrarily large and the length of each timestep \delta t becomes arbitrarily small) to the famous (and Nobel-prize winning) Black-Scholes-Merton (BSM) option pricing formula. Speaking of BSM, allow me to shamelessly plug a journal article that I published early in my academic career which Professor Robert C. Merton cites in his Nobel Prize lecture (Merton shared the Nobel Prize in economics in 1997 with Myron Scholes “for a new method to determine the value of derivatives”).

Here’s the citation (and link) to Merton’s lecture:

Merton, Robert C., 1998, Applications of Option-Pricing Theory: Twenty-Five Years Later, The American Economic Review, Vol. 88, No. 3 (Jun. 1998), pp. 323-349.

See page 337, footnote 11 of Merton’s paper for the reference to Neil A. Doherty and James R. Garven (1986)… (Doherty and I “pioneered” the application of a somewhat modified version of the BSM model to the pricing of insurance; thus Merton’s reference to our Journal of Finance paper in his Nobel Prize lecture)…

How Do Energy Companies Measure the Temperature? Not in Fahrenheit or Celsius

Instead of Fahrenheit or Celsius, a metric called “degree days” is used to capture variability in temperature. The risk management lesson here is that this metric makes it possible to create risk indices which companies can rely upon for pricing and hedging weather-related risks with weather derivatives.

On the role of replicating portfolios in the pricing of financial derivatives

Replicating portfolios play a central role in terms of pricing financial derivatives. Here is a succinct summary from today’s class meeting:

  1. Buying forward is equivalent to buying the underlying on margin, and selling forward is equivalent to shorting the underlying and lending money. Like options, forwards and futures are priced by pricing the replicating portfolio and invoking the “no-arbitrage” condition. If the forward/futures price it too low, then one can earn positive returns with zero risk and zero net investment by buying forward, shorting the underlying and lending money. Similarly, if the forward futures price is too high, one can earn positive returns with zero risk and zero net investment by selling forward and buying the underlying with borrowed money. This is commonly referred to as “riskless arbitrage”; it’s riskless because you’re perfectly hedged, and it’s arbitrage because you are buying low and selling high.
  2. The replicating portfolio for a call option is a margined investment in the underlying. During today’s class meeting, we priced a one timestep call option where the price of the underlying asset is $100, the exercise price is also $100, u = 1.05, d = .95, the interest rate r = 5%, and the timestep \delta t = 1/12. Given these parameters, the payoff on the call is $5 at the up (u) node and $0 at the down (d) node. The replicating value consists of half a share that is financed by a margin balance of $47.30; thus the “arbitrage-free” price of the call option is (.5(100) – 47.30) = $2.70.
  3. Since the replicating portfolio for a call option is a margined investment in the underlying, it should come as no surprise that the replicating portfolio for a put option consists of a short position in the underlying combined with lending. Thus, in order to price the put, we need to determine and price the components of the replicating portfolio. We will begin class next Tuesday by reviewing our analysis of the replicating portfolio approach to pricing calls and puts, and move on to other pricing methods such as delta hedging and risk neutral valuation.

Synopsis of today’s Capital Market theory topic

Our coverage of the Capital Market Theory topic has provided the following important insights:

  1. Borrowing and lending at the riskless rate of interest in combination with investing in (mean-variance efficient) risky portfolios enable 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 2 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).

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). 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 multiple number (“large” n) of risky assets (aka 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 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.
    3. 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 or 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

Yesterday’s Finance 4335 class meeting on the topic of portfolio theory positions us well for tomorrow’s discussion of capital market theory. Work in these particularly important areas of finance subsequently won Nobel Prizes for their proponents; specifically, the portfolio theory topic won Harry Markowitz the Nobel Prize in Economics in 1990, and William F. Sharpe shared the 1990 Nobel Prize with Markowitz for his work on 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 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 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.

How insurance premiums and deductibles work

CNBC presents Health insurance 101 – premiums, deductibles, coinsurance, and stop loss (AKA maximum out-of-pocket cost):

Health-care costs can add up fast. Understanding how medical costs and insurance are related and structured may help you anticipate expenses and budget better for your health. Watch this video to find out how premiums and deductibles affect your coverage – and what you can do to lower your medical bills.

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

During today’s Finance 4335 class meeting, we will delve deeper into the topic of adverse selection. Adverse selection is often referred to as the “hidden information” problem. This concept is particularly easy to understand in an insurance market setting; if you are an insurer, you have to be concerned that the worst possible risks are the ones that want to purchase insurance. However, it is important to note that adverse selection occurs in many market settings other than insurance markets. Adverse selection occurs whenever one party to a contract has superior information compared with his or her counter-party. When this occurs, often the party with the information advantage is tempted to take advantage of the uninformed party.

In an insurance setting, adverse selection is an issue whenever insurers know less about the actual risk characteristics of their policyholders than the policyholders themselves. In lending markets, banks have limited information about their clients’ willingness and ability to pay back on 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…

The problem with adverse selection is that if left unchecked, it can undermine the ability of firms and consumers to enter into contractual relationships, and in extreme cases, may even give rise to so-called market failures. For example, in the used car market, since the seller has more information than the buyer about the condition of the vehicle, the buyer cannot help but be naturally suspicious concerning product quality. Consequently, he or she may not be willing to pay as much for the car as it is worth (assuming that it is not a “lemon”). Similarly, insurers may be reticent about selling policies to bad risks, banks may be worried about loaning money to poor credit risks, employers may be concerned about hiring poor quality workers, consumers may be worried about buying poor quality products, and so on…

A number of different strategies exist for mitigating adverse selection. In financial services markets, risk classification represents an important strategy. The reason insurers and banks want to know your credit score is because consumers with bad credit not only often lack the willingness and ability to pay their debts, but they also tend to have more accidents 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 vehicle inspections by an independent third party. Students “signal” their quality by selecting a high-quality university (e.g., like 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 indicate that quality is better than average.

Sometimes it’s not possible to fully mitigate adverse selection via the methods described above. Thus, insurers commonly employ pricing and contract design strategies which incentivize policyholders to reveal their actual risk characteristics according to their contract choices. Thus, we obtain 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:

Rothschild-Stiglitz

The Rothschild-Stiglitz equilibrium cleverly restricts the menu of available choices in such a way that the insurer induces self-selection. Here, the insurer offers contract L, which involves partial coverage at an actuarially fair price (based upon the loss probability of the low risk insured), and contract H, which provides full coverage at an actuarially fair price (based upon the loss probability of the high risk insured). The differences in the shapes of the indifference curves are due to the different accident probabilities, with a lower accident probability resulting in a more steeply sloped indifference curve. 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 will prefer L, but would prefer a full coverage contract at the point of intersection of APl line with the full insurance (45 degrees) line. However, such a contract is not offered since both the low and high-risk policyholders would choose it, and this would cause the insurer to lose money. Thus, one of the inefficiencies 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 with the expectation that 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.