The crux of MM’s position is that the effect of dividend payments on shareholders wealth is offset exactly by other means of financing. Where the firm has made its investment decision, it must decide whether to retain earnings or to pay dividends and sell the new stock in order to finance the investments. But the issue of an additional stock of shares will cause a decline in the terminal value of shares. What is gained by the investors as a result of payment of dividends will be neutralized completely by the reduction in the terminal value of shares. MM suggest that the sum of discounted value per share after financing and dividends paid is equal to the market value per share before the payment of dividends. In other words, the stock’s decline in market price because of external financing offsets exactly the payment of dividends. Therefore the investors according to MM will have no preference between getting the increase in wealth in the form of dividends now or capital appreciation later. The dividend pay out is irrelevant.
Modigliani and Miller also showed the irrelevance of capital structure for investment decisions in perfect markets. The market value of a firm is independent of its capital structure. The sum of the parts must equal the whole; so regardless of financing mix, the total value of the firm stays the same, according to MM. The basic premise of MM approach is that, the total value of a firm must be constant irrespective of the degree of leverage
If the market values of any two firms differ then the process of arbitrage operates to equalize the values of the two companies. The central proposition of MM is that the weighted average cost of capital ( WACC) is independent of the debt-equity ratio and equal to the cost of capital which the firm would have with no gearing in its capital structure. MM argue that company value and the overall required return, Ko, are invariant to capital structure.
Markokwitz and Sharpe are widely known for their path breaking contributions to portfolio theory. According to Markowitz’s mean-variance maxim, an investor should seek a portfolio of securities that lies on the efficient frontier. A portfolio is not efficient if there is another portfolio with a higher expected value of return and with the same or a lower standard deviation or the same expected value with lesser risk. If inefficient portfolios are deleted, we get a set of efficient portfolios or efficient frontier. It is possible to draw a Risk-Return indifference map such that the investor is indifferent between any combination of risk and return on any indifference curve. The slope of the indifference curve is the investor’s marginal rate of substitution between risk and earnings. The point of tangency between the efficient frontier and the indifference curve is the optimal portfolio combination.
Markowitz devised an algorithm, using quadratic programming to calculate a set of efficient portfolios. His model is extremely demanding in its data needs and computational requirements.
Sharpe views that relationship between securities occurs mostly through their individual relationships with some index. This is known as Sharpe’s Index model. Sharpe’s model reduces the data requirement considerably.
In an article titled “A simplified model for portfolio analysis,” published in 1961, Sharpe relates each stocks return to the market as a whole rather than to every other stock. One way to capture this relationship is the market model. This can be expressed as
rs = ά + β rI + e
where rs = return on security
ά = intercept term
β = slope
rI = return on market Index and
e = error
This market model specifies that every risky security in a portfolio is related to the return on the market index such as SENSEX. The market model assumes that the return on a security is sensitive to the movements of the market Index (factor). Hence, the market model is also called Index model or Factor model.
Capital Asset Pricing Model (CAPM):
The CAPM shows the relation between risk and expected return for efficient portfolios. In the CAPM graph, we represent returns on the vertical axis and risk of the Portfolio on the horizontal axis. Efficient portfolios, plot along the line going from risk free return through the Market Portfolio. Efficient Portfolios consist of alternative combinations of risk and expected returns obtainable by combining market portfolio with risk free borrowing and lending. The lenear efficient set of the CAPM is known as the Capital Market Line (CML).
Because all investors face the same efficient set, the only reason they will choose different portfolios is that they have different preferences towards risk and return resulting in distinct indifference curves. Although the chosen portfolios will be different each investor will choose the same combination of risky securities. As a result each investor will spread his or her funds among risky securities in the same relative proportions, adding risk free borrowing or lending in order to achieve a personally preferred combination of risk and return. The tangency portfolio is referred to as market portfolio and it is same for all investors. Only there will be a certain amount of either risk free borrowing or lending that depends on that person’s indifference curves. The optimal combination of risky assets for an investor can be determined without any knowledge of the investor’s preferences toward risk and return. This feature of the CAPM is often referred to as the separation theorem.
In CAPM, the market will ultimately achieve equilibrium. In equilibrium the proportions of the tangency portfolio will correspond to the proportions of the market portfolio. The tangency portfolio is commonly referred to as market portfolio.
The vertical intercept of the Capital Market Line (CML) is the risk free rate of return which is often referred to as the reward for waiting. The slope of the CML is equal to the difference between the expected return of the market portfolio and the risk free security divided by the difference in their risk. The slope of the CML is often referred to as the reward per unit of risk borne. The intercept and slope of CML can be thought of as the price of time and the price of risk. In essence, security markets provide a place where time and risk can be traded, with the prices determined by supply and demand.
We have seen that the market model (Index model) uses market Index, whereas the CAPM involves the market portfolio. In practice the composition of the market portfolio is not precisely known; so a market Index must be used. As such beta determined by using market Index is used as an estimate of beta determined by market portfolio.
The Capital Market Line (CML) represents the equilibrium relationship between the expected return and standard deviation for efficient portfolios. The relation between covariance of security with the market and expected return is known as Security Market Line (SML). The securities with larger covariance with the market will be priced so as to have higher expected returns. Suppose the beta of an individual security is 1.5, the required rate on the market is 15% and risk free rate is 6% per annum. Then the required rate of return for the security is
0.06 + 1.5 (0.15 – 0.06) = 0.195 or 19.5%
The expected return for a security is the product of beta and the market risk premium plus the Risk free rate of return.
MYRON SCHOLES AND ROBERT MERTON:
In the 1970’s, Fischer Black, Myron Scholes and Robert Merton made a major contribution to the pricing of stock options. Before their work is recognised by the World, Black died. The remaining two received the Nobel Prize in 1997. Of their work, the most popular model is the Black-Scholes model. The Black-Scholes formula (BS formula) for pricing European Call Option on a non-dividend paying stock is given below. The buyer of Call Option gets the right but not the obligation to buy the Stock at a certain price. European call options can be exercised only on the expiration date only. The BSO formula for call options is given below.
C= S0.N(d1) – K. e–r.t . N(d2)
Where C is the value of the stock option
S0 is the current stock price at time zero.
K is the exercise price of the option
N (d) is the value of the cumulative Normal density function
e is an exponential, equal to 2.718
r is the short-term annual interest rate continuously compounded
t is the length of time to the expiration of the option, usually expressed as a proportion of an year
For the computation of d1 and d2, the formulas are
d1 = ln (S/K) + (r + σ2/2)t
------------------------
σ . √ t
d2 = d1 – σ . √t
Where ln = The natural Logarithm
σ = The standard deviation of the annual rate of return on the stock
The BSO formula is taken from the well known text book by Hull. The formulas appear differently in other text books like Redhead book. But they are one and the same.
For Stocks providing a dividend yield at rate Q, the BSO formula for European Call options is modified on the basis of results derived by Merton. In the revised formula, Stock price is reduced from S0 to S0.e-QT where Q is dividend and d1 computation also changed accordingly.
The revised BSO (Merton) formula can be used for calculating European Call option for Stock Index. S0 is the value of the index and K is the exercise price of the index and Q is equal to the annualized dividend yield on the index.
For currency options also we use the same BSO (Merton) formula. We define S0 as the spot exchange rate and replace Q (dividend rate) with rf , foreign risk free interest rate.
In the case of American Call options on Stocks, the right to buy the stock can be exercised at any time upto the expiration date of the option. When there are no dividends on Stocks, the American and European Call option prices are equal and the BSO formula can also be applied to determine the price of American Call options on Stocks. When there are dividends on stocks, Black suggested an approximate procedure to determine the price of American Call option.
While the Call options give the right, but not the obligation to buy stock (underlying asset) at a specified strike price on a future expiry date, the Put options give the right but not the obligation to sell the stock at a specified price on a future date. So Put options are exact opposites of Call options. So BSO formula of Call options can be used for Put options also, by changing signs of formula C and rewriting the formula for Put option. The calculation of d1 and d2 remain the same.
The Global Financial crisis of 2008 has led to blaming the Financial models and question their underlying assumptions, that markets function efficiently. Actually, the financial industry consisting of Banks, Investment Funds, Hedge funds and such others are to be blamed for causing the Financial crisis. Fiscal stimulus policies and liquidity injection policies have averted the Financial crisis deepening into a Depression.
Black and Scholes model pdf
touch : NYU, Stern, Chapter 5
Eugene Fama, Lars Peter Hansen, Robert J. Shiller won the Nobel Price for 2013 for their work on predictions in Financial Market and also for spotting trends in Financial Markets.
Chapter 15
INFORMATION ECONOMICS
(Mirrlees , William Vickrey, Akerloff, Spence, Stiglitz, Hurwitz.Myerson, Maskin & Alvin Roath)
In the 1950’s George Stigler explained that price differences of products are due to expensiveness of search and information. Since then, information economics has grown steadly.
Stiglitz observes that information is costly and further it is asymmetric. Because of asymmetics of information between buyers and sellers, markets fail and misallocate resources. As such Stiglitz argues for government intervention.
Three main themes arise in situation in which asymmetric information exists in a contractual relationship, that is to say, in which one participant knows something that another does not. They are: Moral Hazard, Adverse, Selection and Signaling. In the Contractual relationship in which the participants could be individuals, institutions or firm, let us refer to the participants as Principal and Agent. The Principal is responsible for designing and proposing the Contract, while the Agent, who is contracted to carryout some task decides, if he is interested, in signing or not.
In the context of Information asymmetry, Moral Hazard problems have to do with the behaviour of the Agent during the contractual relationship. The Agent’s behaviour is not observable by the Principal, it is not verifiable (for a Court of Law). The first formal papers on Moral Hazard are those of Mirrlees. A classic example here is Fire insurance where the insure (agent) may or may not take adequate care while storing the flammable materials. It is possible for the Fire insurance company to send inspectors to see that the insurer takes proper care. But perfect monitoring is not possible.
Agents frequently do not act in the best interests of the Principals. Against the interest of the employer (Principal), the workers may not be working very hard, preferring to idle away time by doing work at a slow pace. Similarly, in a company, the shareholders are owners and the managers are the agents. The managers (agents) may pursue goals other than that profit maximization. The resulting inefficiency due to such conflicting goals is termed ‘X’ inefficiency.
Such problems can be tackled by proper monitoring of the performance of the agents by the principals. Further, there must be incentives for the agents to behave in the principals interests. Thus, managers’ salaries could be linked to firms’ profitability.
William Vickery won the Noble Prize jointly with Mirrless in 1996. Vickry deals with the problems of design mechanism in economics, especially with the writing of Contracts among parties who will come to have private information. Vickery suggested a design mechanism known as ‘Second-Price Sealed Bid Auction’., termed as ‘Vickery ‘s Auction’
In this type of auction system suggested by Prof. Vickrey the person who is willing to pay the highest price gets the chance to buy the good and he pays the social opportunity cost, which is the second highest bid. Hence, this auction design is socially efficient. The bidders are induced to reveal their true valuations of the good.
The ‘Adverse Selection’ problem is present, when before signing the Contract, the party that establishes the conditions of the Contract (the Principal) has less information than the Agent on some important characteristics affecting the value of the Contract.
Early important contributions to ‘Adverse Selection’ problems came from Akerloff. Akerloff gives the illustration of the used cars, which are in good condition (Peaches) and in bad Condition (Lemons). In a secondhand car market, there is a tendency of ‘Bad Used cars’ driving out the good cars.
If the good and bad cars are sold at the same price, owners are more likely to offer a bad car for sale than a good one. Potential buyers of used cards suspect that the cars on the market are bad. Accordingly, they reduce the price they are willing to pay. At the reduced prices, the sellers will have no incentive to sell good cars. In such a vicious circle, the market for used cars may even collapse.
While information is imperfect and asymmetric, persons can take steps to provide others with the signals or proxies for the relevant variable. Michael Spence in a path-breaking article points out that education of candidates for a job serves as a signal to the employer. Employer has no prior information about the ability of candidates. They initially believe that persons having education, say degree are more able than others., who are believed to be less able.
On this basis, the employers offer higher wages or salary to more able candidates and low wages to less able persons. The candidates in turn fulfill the employers expectations. It is assumed that the cost of education (acquiring a degree) is higher for less able persons as they take more time to get it than the less able persons. In such a context, only the more able persons find it worthwhile to acquire the needed degree as a signal about their ability. And For the Employer taking the degree as a signal of their ability offers higher wages to all those candidates having a degree.
In the Moral hazard example, we referred to Employer and Employee relationship. The Employer (Principal)designs the contract to induce high effort on the part of the Agent. The agents expected pay-off for high effort must be at least as great as his pay-0ff from low effort. This kind of inequality is called the incentive compatibility constraint.
In the car example above it was assumed that the buyer had no information about the quality of the car. However, the buyer (the Principal) while designing the contract can motivate the seller (the agent) to reveal his private information about the quality of the car. The Contract specifies a guarantee for the car’. The seller would accept the contract only if the car is a good quality.
The 2007 Nobel Prize is awarded to Hurwitz, Myerson and Maskin for their Contributions to mechanism design theory. An important feature of collective decision making is that it takes into account individual preferences. But the individual preferences are not publicly observable. How the information can be elicited, and the extent to which the information revealation problem constrains the ways in which collective decisions can respond to individual preferences, is known as mechanical design problem. As Engineers design bridges and machines, analogously economists design exchange mechanisms such as telephone exchanges and auction markets. According to Hurwitz, the theory attempts to achieve incentive compatibility among the agents. Meyerson Revealation Principle induces people to reveal their private information truthfully. Maskins Implementation theory clarifies when mechanisms can be devised that only produce Nash equilibrium that is incentive efficient.
The work of these Nobel Economists of 2007 is closely related to that of non-cooperative game theories of Hasranyi, Nash and Selten and to the theories of Vickrey and Mirrlees referred earlier in this chapter.
As design theories form a constituent of game theory, books and chapters on game theory may be consulted for more details.
The Nobel Prize 2012 is awarded jointly to Professors Shapely & Alvin Roath. While Shapely used Game Theory to analyze different matching methods in the 1950’s & 960’s, Roath applied matching methods for allocations and the practice of market design. Those interested in knowing more about Roath’s contributions they may read his book (Co-edited with J.Kagel), Handbook of Experimental Economics, Princeton University press, 1995.
Design theories, Game theories and Experimental Economics are interrelated.
Chapter 16
W. Arthur Lewis (1915 – 1991)Arthur Lewis was born in 1915 at St. Lucia, an island in the Caribbian and other America’s Region. He had faced many difficulties during his childhood days. His father died when he was seven years old and his mother brought him up. He was a Black and from a Colonial country. Naturally he faced initial difficulties. He made best use of what opportunities came in his way.
He graduated from London School of Economics and worked as Professor in the Universities of Manchestor, U.K. and Princeton, U.S.A. He spent several years in administration also. He became a citizen of U.K. |