...multiplying the cost of each source of finance by the relevant weight and summing the products up. Formula For a company which has two sources of finance, namely equity and debt, WACC is calculated using the following formula: Cost of equity is calculated using different models for example dividend growth model and capital asset pricing model. Cost of debt is based on the yield to maturity of the relevant instruments. If no yield to maturity can be calculated we can base the estimate on the instrument's current yield, etc. The weights are based on the target market values of the relevant components. But if no market values are available we base the weights on book values. Cost of Preferred Stock Cost of preferred stock is the rate of return required by the holders of a company's preferred stock. It is calculated by dividing the annual dividend payment on the preferred stock by the preferred stock's current market price. In finance, the value of any asset equals the present value of its future net cash flows. In most cases, the cash flows stream of a preferred stock is a perpetuity because preferred stock has unlimited life and it pays a fixed amount of dividend each period. Value of a preferred stock is essentially the present value of the perpetuity. PV of Perpetuity = | Periodic Payment | | Discount Rate | We can modify this relationship to come up with the formula for cost of preferred stock. The discount rate in the above relationship is the rate of return required...
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...Copyright c 2005 by Karl Sigman 1 Capital Asset Pricing Model (CAPM) We now assume an idealized framework for an open market place, where all the risky assets refer to (say) all the tradeable stocks available to all. In addition we have a risk-free asset (for borrowing and/or lending in unlimited quantities) with interest rate rf . We assume that all information is available to all such as covariances, variances, mean rates of return of stocks and so on. We also assume that everyone is a risk-averse rational investor who uses the same financial engineering mean-variance portfolio theory from Markowitz. A little thought leads us to conclude that since everyone has the same assets to choose from, the same information about them, and the same decision methods, everyone has a portfolio on the same efficient frontier, and hence has a portfolio that is a mixture of the risk-free asset and a unique efficient fund F (of risky assets). In other words, everyone sets up the same optimization problem, does the same calculation, gets the same answer and chooses a portfolio accordingly. This efficient fund used by all is called the market portfolio and is denoted by M . The fact that it is the same for all leads us to conclude that it should be computable without using all the optimization methods from Markowitz: The market has already reached an equilibrium so that the weight for any asset in the market portfolio is given by its capital value (total worth of its shares) divided...
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...Copyright c 2005 by Karl Sigman 1 Capital Asset Pricing Model (CAPM) We now assume an idealized framework for an open market place, where all the risky assets refer to (say) all the tradeable stocks available to all. In addition we have a risk-free asset (for borrowing and/or lending in unlimited quantities) with interest rate rf . We assume that all information is available to all such as covariances, variances, mean rates of return of stocks and so on. We also assume that everyone is a risk-averse rational investor who uses the same financial engineering mean-variance portfolio theory from Markowitz. A little thought leads us to conclude that since everyone has the same assets to choose from, the same information about them, and the same decision methods, everyone has a portfolio on the same efficient frontier, and hence has a portfolio that is a mixture of the risk-free asset and a unique efficient fund F (of risky assets). In other words, everyone sets up the same optimization problem, does the same calculation, gets the same answer and chooses a portfolio accordingly. This efficient fund used by all is called the market portfolio and is denoted by M . The fact that it is the same for all leads us to conclude that it should be computable without using all the optimization methods from Markowitz: The market has already reached an equilibrium so that the weight for any asset in the market portfolio is given by its capital value (total worth of its shares) divided by the total capital...
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...Black–Scholes formula, which gives a theoretical estimate of the price of European-style options. The formula led to a boom in options trading and legitimised scientifically the activities of the Chicago Board Options Exchange and other options markets around the world.[2] lt is widely used, although often with adjustments and corrections, by options market participants.[3]:751 Many empirical tests have shown that the Black–Scholes price is "fairly close" to the observed prices, although there are well-known discrepancies such as the "option smile".[3]:770–771 The Black–Scholes was first published by Fischer Black and Myron Scholes in their 1973 paper, "The Pricing of Options and Corporate Liabilities", published in the Journal of Political Economy. They derived a stochastic partial differential equation, now called the Black–Scholes equation, which estimates the price of the option over time. The key idea behind the model is to hedge the option by buying and selling the underlying asset in just the right way, and consequently "eliminate risk". This hedge is called delta hedging and is the basis of more complicated hedging strategies such as those engaged in by investment banks and hedge funds. The hedge implies that there is a unique price for the option and this is given by the Black–Scholes formula. Robert C. Merton was the first to publish a paper expanding the mathematical understanding of the options pricing model, and coined the term Black–Scholes options pricing model. Merton...
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...Capital Asset Pricing Model Capital Asset Pricing Model (CAPM) is used to calculate the projected return on the equity of a single company. CAPM is based on risk free rate, the expected return rate on the market and beta coefficient of a single portfolio and security. Re = Rf + β [E(Rm) - (Rf)] According to the formula, Re represents the Return on Equity, Rf is for the risk-free rate, E(Rm) denoted to expected rate of return on the market, and β is the beta coefficient and E(Rm) - Rf is the difference among the expected market rate of return and the risk-free rate, is known as the market premium (Phillips, 2007). The total risk of stock is divided in two parts: the systematic risk, which is risk linked with the market and is impossible to...
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...Description of CAPM. The Capital Asset Pricing Model CAPM was introduced by Treynor ('61), Sharpe ('64) and Lintner ('65). By introducing the notions of systematic and specific risk, it extended the portfolio theory. In 1990, William Sharpe was Nobel price winner for Economics. "For his contributions to the theory of price formation for financial assets, the so-called Capital Asset Pricing Model (CAPM)." The CAPM model says that the expected return that the investors will demand, is equal to: the rate on a risk-free security plus a risk premium. If the expected return is not equal to or higher than the required return, the investors will refuse to invest and the investment should not be undertaken. CAPM decomposes a portfolio's risk into systematic risk and specific risk. Systematic risk is the risk of holding the market portfolio. When the market moves, each individual asset is more or less affected. To the extent that any asset participates in such general market moves, that asset entails systematic risk. Specific risk is the risk which is unique for an individual asset. It represents the component of an asset's return which is not correlated with general market moves. According to CAPM, the marketplace compensates investors for taking systematic risk but not for taking specific risk. This is because specific risk can be diversified away. When an investor holds the market portfolio, each individual asset in that portfolio entails specific risk. But through diversification...
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... Table of Contents Table of Contents 1 1.0 Multifactor model 2 2.0 Arbitrage pricing theory (APT) 2 3.0 Multifactor Models (APT) and Testing 4 Reference 7 Multifactor model Estimation of returns on security and APT on International level demonstrating Factors Those are statistically significant 1.0 Multifactor model Pardalos (1997) defines multifactor model as a financial model which uses multiple factors during computation to explain a given market phenomena or at a given equilibrium market prices. The model is also useful in explaining both the individual and portfolio market securities. This is capable through comparison of two or more factors which are being analyzed to determine the relationship between the securities performance and the variables. Formula can be used to express the relationship Return on equity (Ri), Market return (Rm), factor search (F 1, 2…) 2.0 Arbitrage pricing theory (APT) The relationship between literature theories and the stock market behavior is the Asset Pricing model (Levy and Thierry 2005). Consigli and Wallace (2000) in their study, indicates that both are used in whenever securities are being given price and the individual assets risk are also being priced and can also be used in between portfolio to give a more insights of business activities and behavior hence helps in calculating...
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...Compare and contrast CAPM and APT? Capital asset pricing model (CAPM) and arbitrage pricing theory (APT) are both methods of assessing an investment's risk in relation to its potential reward and whether the potential investment yield is worthwhile. CAPM developed by Sharpe 1964. The basic theory behind this model is that investor needs to be compensated for Time Value of Money and the risk that they are taking. The time value of money is represented by the risk-free (rf) rate in the formula and compensates the investors for placing money in any investment over a period of time. The other half of the formula represents risk. This is calculated by taking a risk measure of the market (beta) that compares the returns of the asset to the market over a period of time and to the market premium (Rm-rf). APT developed by Ross 1978. The basic theory of arbitrage pricing theory is the idea that the price of a security is driven by a number of factors such as macro factors, and company specific factors. Formula: r = rf + β1f1 + β2f2 + β3f3 + ⋅⋅ Where r is the expected return on the security, rf is the risk free rate, Each f is a separate factor and each β is a measure of the relationship between the security price and that factor. The CAPM bases the price of stock on the time value of money (risk-free rate of interest (rf)) and the stock's risk, or beta (b) and (rm) which is the overall stock market risk. APT does not regard market performance when it is calculated...
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...Jon M. Huntsman School of Business Master of Science in Financial Economics August 2013 Pricing and Hedging Asian Options By Vineet B. Lakhlani Pricing and Hedging Asian Options Table of Contents Table of Contents 1. Introduction to Derivatives 2. Exotic Options 2.1. Introduction to Asian Options 3.1. Binomial Option Pricing Model 3.2. Black-Scholes Model 3.2.1. Black-Scholes PDE Derivation 3.2.2. Black-Scholes Formula 1 2 3 4 4 5 6 7 3 3. Option Pricing Methodologies 4. Asian Option Pricing 4.1. 4.2. 4.3. 4.4. Closed Form Solution (Black-Scholes Formula) QuantLib/Boost Monte Carlo Simulations Price Characteristics 8 8 10 11 14 5. Hedging 5.1. Option Greeks 5.2. Characteristics of Option Delta (Δ) 5.3. Delta Hedging 5.3.1. Delta-Hedging for 1 Day 5.4. Hedging Asian Option 5.5. Other Strategies 6. Conclusion 16 17 17 19 20 22 25 26 27 32 34 Appendix i. ii. iii. Tables References Code: Black-Scholes Formula For European & Asian (Geometric) Option 1 Pricing and Hedging Asian Options 1. Introduction to Derivatives: Financial derivatives have been in existence as long as the invention of writing. The first derivative contracts—forward contracts—were written in cuneiform script on clay tablets. The evidence of the first written contract was dates back to in nineteenth century BC in Mesopotamia...
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...LECTURE 7: BLACK–SCHOLES THEORY 1. Introduction: The Black–Scholes Model In 1973 Fisher Black and Myron Scholes ushered in the modern era of derivative securities with a seminal paper1 on the pricing and hedging of (European) call and put options. In this paper the famous Black-Scholes formula made its debut, and the Itˆ calculus was unleashed upon the world o 2 of finance. In this lecture we shall explain the Black-Scholes argument in its original setting, the pricing and hedging of European contingent claims. In subsequent lectures, we will see how to use the Black–Scholes model in conjunction with the Itˆ calculus to price and hedge all manner of o exotic derivative securities. In its simplest form, the Black–Scholes(–Merton) model involves only two underlying assets, a riskless asset Cash Bond and a risky asset Stock.3 The asset Cash Bond appreciates at the short rate, or riskless rate of return rt , which (at least for now) is assumed to be nonrandom, although possibly time–varying. Thus, the price Bt of the Cash Bond at time t is assumed to satisfy the differential equation dBt (1) = rt Bt , dt whose unique solution for the value B0 = 1 is (as the reader will now check) t (2) rs ds . Bt = exp 0 The share price St of the risky asset Stock at time t is assumed to follow a stochastic differential equation (SDE) of the form (3) dSt = µt St dt + σSt dWt , where {Wt }t≥0 is a standard Brownian motion, µt is a nonrandom (but not necessarily...
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...Overview and Major Findings: This paper examines the problem of pricing a European call on an asset (Stock) that has a stochastic or variable volatility. Addressing this problem was done by investigating two cases: the first case is to determine the option price when the stochastic volatility is independent of stock price. The second case is to determine the option price when the stochastic volatility is correlated with the stock price. This paper provides a solution in series form for the stochastic volatility option, in addition to a discussion about the numerical methods that are used to examine pricing biases, and an investigation about the occurrence of the biases in the case of stochastic volatility. As for the results obtained, this paper presents interesting results for each of the two cases. When the stochastic volatility is independent of stock price, the results show that the price calculated using Black-Scholes equation is overestimated for at-the-money options and underestimated for deep in-and out-of-the-money options. This overpricing takes place for stock prices within about ten percent of the exercise price. Moreover, it is shown that the degree of the pricing bias can be up to five percent of the Black-Scholes price. For the second case when the stock price is positively correlated with the volatility, the results show that the Black-Scholes formula overprices in-the-money options and underprices out-of-the-money options. On the other hand, when the stock...
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...kij1 Master Thesis Supervisor: PETER LØCHTE JØRGENSEN Author: QIAN Zhang (402847) Pricing of principle protected notes embedded with Asian options in Denmark ---- Using a Monte Carlo Method with stochastic volatility (the Heston Model) Aarhus School of Business and Social Science 2011 2 Acknowledgements My gratitude and appreciation goes to my supervisor Peter Lø chte Jø rgensen, for his kind and insightful discussion and guide through my process of writing. I was always impressed by his wisdom, openness and patience whenever I wrote an email or came by to his office with some confusion and difficulty. Especially on access to the information on certain Danish structured products, I have gained great help and support from him. 3 Abstract My interest came after the reading of the thesis proposal on strucured products written by Henrik, as is pointed out and suggested at the last part of this proposal, one of the main limitations of this thesis may be the choice of model. This intrigues my curiosity on pricing Asian options under assumption of stochstic volatility. At first, after the general introduction of strucutred products, the Black Scholes Model and risk neutral pricing has been explained. The following comes the disadvanges of BS model and then moves to the stochastic volatility model, among which the Heston model is highlighted and elaborated. The next part of this thesis is an emricical studying of two structured products embbeded with Asian...
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...The Capital Assets Pricing Model Name: Course: Professor/ Tutor’s Name: University: City/State: Date: The Capital Assets Pricing Model Introduction The Capital Assets Pricing Model (CAPM) , is a method of pricing assets of capital nature. This model applies Beta (non-diversifiable risk) to link risks and returns of investments. According to Stahl (2016), Beta is a standard for measuring the systematic risk or the non-diversifiable risk. The uncertainty in the economy of a particular country causes the systematic risk. Systematic risk is that risk sharing or risk diversification cannot reduce. Economic downturns, war, natural calamities and a change of government policy are some of the activities that cause systematic risk. Both CAPM and Beta are measures of risk (Anon 2014). The capital assets pricing model defines the required rate of return of security. CAPM can be a mathematical equation, or a graphical representation is known as the security market line (SML) (Stahl 2015). An analysis of CAPM indicates that there are several critiques of this model. Nevertheless, there are multivariate models used to overcome these critiques. A).Formulas to Calculate CAPM and Beta 1). Capital Assets Pricing Model CAPM= [pic]= [pic]+ [pic] (RM-RF) Where; [pic] is the cutoff rate or even minimum required rate of return RM- RF is the risk premium and is above free rate RM is the market returns [pic] is the...
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...technical the capital asset pricing model relevant to ACCA Qualification Paper F9 the cost of equity Section F of the Study Guide for Paper F9 contains several references to the Capital Asset Pricing Model (CAPM). This article introduces the CAPM and its components, shows how it can be used to estimate the cost of equity, and introduces the asset beta formula. Two further articles will look at applying the CAPM in calculating a project-specific discount rate, and will review the theory, and the advantages and disadvantages of the CAPM. Whenever an investment is made, for example in the shares of a company listed on a stock market, there is a risk that the actual return on the investment will be different from the expected return. Investors take the risk of an investment into account when deciding on the return they wish to receive for making the investment. The CAPM is a method of calculating the return required on an investment, based on an assessment of its risk. SYSTEMATIC AND UNSYSTEMATIC RISK If an investor has a portfolio of investments in the shares of a number of different companies, it might be thought that the risk of the portfolio would be the average of the risks of the individual investments. In fact, it has been found that the risk of the portfolio is less than the average of the risks of the individual investments. By diversifying investments in a portfolio, therefore, an investor can reduce the overall level of risk faced. There is a limit to this risk...
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...the efficient ratios to be used when analyzing a company and its stock. Unlike other ratios, ROE is a profitability ratio from point of view of investor, not a company. It expresses the amount of net income returned as a percentage of average shareholders equity. The typical formula can be described as follows: ROE = Net Income ÷ Average stockholder equity. This is generally calculated over a year and expressed as a percentage. For example, the company A earned a net income of 400,000 in 2012. Average stockholder equity is 2,000,000. So the company’s ROE in 2012 equals to 20%. It means that for every $1 that has been invested by the shareholders, the company used it to earn $0.2 in its net income. ROE is often said to be the ultimate ratio that can be obtained from a company’s financial statement. Basically, the average ROE has been around at least 10% to be determined that a company does well on the market. Investors can compare this ratio to market benchmarks. The more increasing ROE is, the better company is doing business and employing investor’s capital to generate the profit. So the requirement for its management is how to grow ROE of next year faster than current year. Basing on the single formula above, a company can either increase its net income or decrease average stockholder equity. In order to...
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