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**Unformatted text preview: **UGBA 103 &Introduction to Finance Dmitry Livdan Walter A. Haas School Fall 2011 Dmitry Livdan (Haas) UGBA 103 Fall 2011 1 / 339 I.4 The Valuation of Risky Cash Flows Dmitry Livdan (Haas) UGBA 103 Fall 2011 2 / 339 Motivation We have by now developed a fairly complete theory of valuation and capital budgeting under certainty. In order to extend this theory to situations involving uncertainty (risk), we need to understand how risk is de&ned and what is the link between the risk of an investment and its expected rate of return. We therefore start by discussing what determines the expected rate of returns on risky assets (stocks) and then apply these results to capital budgeting. Before doing this, we need to review some basic concepts from statistics. Dmitry Livdan (Haas) UGBA 103 Fall 2011 3 / 339 I.4.1 Statistics Reminder Dmitry Livdan (Haas) UGBA 103 Fall 2011 4 / 339 Random Variables A random variable is a variable that can take several possible values. It is denoted by the superscript (e.g. X ). & A discrete random variable may take on only a nite number of values, while a continuous random variable may take on an innite number of values. Examples of random variables include the outcome of the next Super Bowl, tomorrows temperature, and the return on a stock. & Which of these are discrete and which are continuous random variables? The return on a stock at time t is dened as r t + 1 = D t + 1 + ( P t + 1 & P t ) P t = D t + 1 + P t + 1 P t & 1 where P denotes the price and D the dividend. The rate of return of a bond is dened similarly by replacing the dividend D with the coupon C . & Is this also random? Dmitry Livdan (Haas) UGBA 103 Fall 2011 5 / 339 Discrete Probability Distributions Associated to each random variable is a probability distribution , a list of all possible outcomes and the probability that each will occur. & Discrete RV: prob. distribution is specied by a probability function p : Prob ( X = x ) = p ( x ) . For the roll of a die, the associated probability function is: p ( x ) = & 1 6 if x in f 1 , 2 , 3 , 4 , 5 , 6 g otherwise For the number of heads in 10 coin tosses, the probability function is: Dmitry Livdan (Haas) UGBA 103 Fall 2011 6 / 339 Continuous Probability Distributions As an example of a continuous RV, consider the return on a stock. Empirically, the rate of return on any stock over short intervals (e.g. a day) conforms closely to a normal distribution : See Additional Materials for further technical details on continuous probability distributions. Dmitry Livdan (Haas) UGBA 103 Fall 2011 7 / 339 Joint Probability Distributions The joint probability distribution of two random variables X and Y speci&es the probabilities of all possible outcomes on both X and Y The following table shows Pr f X = x , Y = y g , which we often write as p ( x , y ) ....

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