Lecture06 Risk Return

Lecture06 Risk Return - Portfolio Theory: Risk, Return, and...

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Portfolio Theory: Risk, Return, and Asset Allocation Tyler R. Henry 1 FINA 4310 Outline Contents 1 Probability Distributions 1 2 Historical Returns 3 3 Portfolio Risk and Return 5 3.1 Risk w/ 2 Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Risk w/ 3 Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.3 Risk w/ many Assets . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4 Asset Allocation 11 4.1 Risky and Risk-Free Asset . . . . . . . . . . . . . . . . . . . . . . . 11 4.2 Multiple Risky Assets . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.3 Two Risky Assets and Risk-Free Asset . . . . . . . . . . . . . . . . . 17 1 Probability Distributions The Big Picture We invest in assets so that we can realize their future payoffs. Because the pay- offs to these assets are uncertain, they involve risk. Ultimately, we need measures to quantify the rewards from these payoffs and the risks that result from their uncertainty. Probability Distribution : The set of all possible values of a random variable and the probability associated with each possible outcome. Each outcome is assigned a probability. Each probability is non-negative. The probabilities must sum to 1. 1
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A Probability Distribution Example : Consider an asset, X, with a current price of $100. One year from now, this asset will have a payoff. There are three states of the world, and the asset has a different payoff in each state. Each state is assigned a probability of occurring. The probability distribution of these asset payoffs is given as: Outcome Probability Payoff Return Boom 0.2 $120 20% Normal 0.5 $110 10% Bust 0.3 $100 0% Expected Return Let S be the total number of possible states, and s denotes each possible state where s = 1 , 2 , 3 ,...,S . Each state has a probability of occurrence, denoted by p s . Only one state can occur, and all possible states are included in S . A risky asset has a rate of return in each of the possible states, denoted as: r 1 ,r 2 3 ,...,r S . Expected return : The expected return E ( r ) of an asset is the probability weighted av- erage return of all possible outcomes: E ( r ) = S X s =1 p s r s Standard Deviation: A measure of Risk The risk of an investment is the uncertainty of which payoff will be realized. The expected return is just an estimate, and the actual, or realized return , will likely differ from the expected return. This difference is a deviation from the expected return. The probability weighted average of these deviations is our measure of risk. Variance : the expected value of the squared deviation from the expected return, or mean. V ar ( r ) = σ 2 = S X s =1 p s [ r s - E ( r )] 2 Questions: Why do we need to square the deviations? What are the units of variance? Standard Deviation : the square-root of the variance. SD ( r ) = σ = p V ar ( r ) 2
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Calculating Expected Return Example : Calculate the expected return, variance, and standard deviation of asset X using its probability distribution. Outcome p s r s Boom 0.2 20% Normal 0.5 10% Bust 0.3 0% The Normal Distribution One specific probability distribution is the normal distribution . This simple probability dis- tribution is completely described by two statistics, its mean and standard deviation. The normal distribution is frequently used as an approximation for the probability distribution of stock re- turns.
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Lecture06 Risk Return - Portfolio Theory: Risk, Return, and...

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