Quantitative Analysis

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1. Introduction: A probability distribution specify the probabilities of the possible outcomes of a random variable. Here, well study the use of the uniform, binomial, normal, and lognormal probability distributions in investment analysis. Some examples of its use are: The Black-Scholes-Merton option pricing model The Binomial option pricing model The Capital asset pricing model Monte Carlo Simulation: Is a computer- based tool for solving complex investment problems. To use it the analyst need to identify the risk factors associated with the problem and the corresponding probability distributions. 2. Discrete Random Variables: Has a countable number of possible values. It can be a limited or unlimited number of outcomes. The key is that they are countable. At the other end Continuous Random Variables are not countable; we cant even define the possible outcomes because outcomes not included in the list will always be possible. Ex. Consider a random variable Z with a list of items, z1, z2, zn; an outcome of the form (z1+z2)/2 is possible even though it is not in the list. In most practical cases the probability distribution is just a mathematical ideal, an aproximate model of the relative frecuency of the possible outcomes of the random variable. Most of the time we have the choice between which probability distribution is more efficient for the task we are facing. Ex. Stocks traded in NASDAQ and NYSE quote in ticks of $0.01, since we know all possible outcomes it is a discrete random variable; however we can model (aproximate) the stock price using a continuous probability function (LOGNORMAL), as well see later in the chapter. Example 5.1 Pending Every random variable is associated with a probability distribution that completely describes it. Probability function: Specifies the probability that a random value X takes on a specific value x: P(X=x). If the random variable is discrete the notation used is p(x). For a continuous variable the probability function is called probability density function (pdf), and the notation used is f(x). Key properties of probability functions: 0p(x)1 The sum of the probabilities of all possible distinct outcomes must equal 1 Cumulative distribution function: Gives the probability of a range of values less than or equal to a particular value x, P(X x). The notation is F(x). To get F(x) we sum up the values of the probability function p(x) or f(x) for all outcomes less than or equal to x. Note that the cdf is related to the cumulative relative frecuency discussed in the chapter on statistical concepts. 2.1. The Discrete Uniform Distribution: Is the simplest of all. In this context, uniform mean that the probability for any value is the same. So, if the possible outcomes are integers from 1 to 8, then p(x) = 1/8 for all value of X. Table 5-1 summarizes the distribution P(X7) : From third column of table 5-1 the value is 0.875 = 87.5% P(4X6): F(6)-F(3) = p(4) + p(5) +p(6) = 0.125*3 = 0.375 = 37.5% P(40.36) = 1- 0.6406 = 0.3594 36% Solution 2: Z12= (12-12)/22 = 0 Z20= (20-12)/22 = 0.36 P(Z12) = P(0) = 0.50% P(Z20) = P(0.36) = 0.6406 P(12%X20%) = 0.6406 0.50 = 0.1406 = 14.06% Solution 3: P(X5.5) ; Z5.5 = (5.5-12)/22 = -0.2955 = -0.30 P(Z -0.30 ) = 0.3821 38%

if normality assumption is accurate.

3.3. Applications of The Normal Distribution According to modern portfolio theory (MPT) the value of investment opportunities can be adequately measured in terms of the mean return and the variance of the return. The proposition that returns are at least approximately normally distributed plays a key role in MPT. In economic theory mean-variance analysis2 (MVA - discussed Theory of combining risky assets so as to minimize the variance of return (i.e., risk) at any desired mean return. The locus of optimal mean-variance combinations is called the efficient frontier, on which all rational investors desire to be positioned. It Was developed by Harry W. Markowitz in the 1950 s.2

Quantitative Analysis

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in chapter on portfolios) works pretty well for risk-averse investors3. MVA considers risk symmetrically because captures variability in both sides of the mean. One approach that considers only shortfall risk is safety-first rules. It considers only the risk that a portfolio value fall below some minimum acceptable level over x time horizon. Roys Safety-First Criterion: Stablish that the optimal portfolio is the one that minimizes the probability of the return RP falling below a treshold level RL, minimizes P(RP < RL). If we let SFRATIO = [ E(RP)-RL ] / P. and assume that portfolio returns are normally distributed then the SFRATIO is the number of s that RL lies below the expected portfolio return E(RP). Note that E(RP)-RL is the distance from the mean return to the shortfall level; when divided by P we are expressing this distance in terms of standard deviations. The result is an standarized value that can use with the standard normal distribution to find the P(RP < RL). To choose portfolios using Roys criterion (assuming normality):4 1. Calculate each portfolio SFRatio 2. Choose the portfolio with the highest SFRatio For a portfolio with a given SFRatio, the probability that its return will be less than RL is N(-SFRatio). For example : Portfolio A: 12% return, = 15% Portfolio B: 14%, = 16% RL = 2% SFRatioA = (12-2)/15 = 0.667 SFRatioB = (14-2)/16 = 0.750 We choose the protfolio with largest SFRatio, in this case portfolio B. N(-0.75) = 0.227 22.7% SFRatio vs Sharpe Ratio: Sharpe Ratio = (Rp Risk-Free-Rate) / SFRatio = (Rp RL) / Sharpe ratio use Risk Free Rate while SFRatio uses RL Example 5-10 The Safety-First Optimal Portfolio for a Client. You are researching asset allocations for a client with an $800,000 portfolio. Although her investment objective is long-term growth, at the end of a year she may want to liquidate $30,000 of the portfolio to fund educational expenses. If that need arises, she would like to be able to take out the $30,000 without invading the initial capital o $800,000. Table 5-6 shows three alternative alloations. Assume normality for parts 2 and 3. Table 5-6 Mean and Standard Deviation for three alloation (in %)A Expected annual return Standard deviation of return 25 27 B 11 8 C 14 20

1. Given the clients desire not to invade the $800k principal, what is the shortfall level, RL? Use this shortfall level to answer part 2. 2. According to the safety-first criterion, which of the three allocations is the best? 3. What is the probability that the return on the safety-first optimal portfolio will be less than the short fall level? Solution 1: RL = 30,000 / 800,000 = 3.75%

Investors that choose investments to maximize satisfaction and when either (1)returns are normally distributed, or (2) investors have quadratic utility functions (Mathematical representations of attitudes toward risk and return). Mean-variance can still be usefull when either assumption (1) or (2) is violated. 4 If we can find an asset offering a risk-free return> RL over the time horizon being considered, then it is optimal to be fully invested in this risk-free asset because given the nature of the risk-free rate asset we have 100% assurance that RP > RL

3

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Solution 2: SFRatio1 = (25-3.75)/27 = 0.787 SFRatio2 = (11-3.75)/8 = 0.9062 SFRatio3 = (14-3.75)/20 = 0.512

Best option

Solution 3: P(RP