Reading 4: Common Probability Distribution
An asset's return is normally distributed with a mean of 7% and a standard deviation of 3%. The probability of a return less than 13% is closest to: A 0.475 B 0.950 C 0.975
C In a standard normal distribution, a z-value describes the number of standard deviations a given outcome is from the mean, where the mean equals 0 (µ = 0) and the standard deviation equals 1 (σ = 1). The z-value (z-score) in this scenario is 2: z = (13-7)/3 = 2 Since 13% is exactly two standard deviations above the mean, approximately 97.5% of the observations fall below a 13% return.
F distribution
Defined as a ratio of two chi-square random variables. It has two values of degrees of freedom: the numerator degree of freedom and the denominator degree of freedom. F follows a F-distribution with m numerator and n denominator degrees of freedom. Similar to the chi-square distribution, the F-distribution cannot be negative. Its shape approaches a bell shape when the numerator and denominator degrees of freedom increase. [LOS 4.o]
continuous uniform distribution
Defined over a range that spans between some lower limit, a, and some upper limit b, which serves as the parameters of the distribution. Formally: -For all a<=x1<=x2<=b -P(X<a or X<b) = 0 -P(x1<=X<=x2) = (x1-x2)/(b-a) [LOS 4.d]
Calculate Holding Period Return given annual continuously compounded rate
[LOS 4.m]
Assume that the annual earnings per share (EPS= fpr a population of firms are normally distrbuted with a mean of $6 and a SD of $2. What are the z-values for EPS of $2 and $8?
x = $8, z1 = (8-6)/2 = 1 x = $2, z2 = (2-6)/2 0 -2 Hence, z1 = 1 indicates that EPS of $8 is 1 SD above the mean, z2 = -2 indicates that EPS of $2 is 2 SD below the mean. [LOS 4.i]
Chi Square Distribution
Chi-square distribution: Defined as the sum of the squares of k independent normal random variables. Since it is the sum of squared values, the distribution is bounded from below by zero. As the number of degrees of freedom increases, the chi-square distribution becomes more symmetrical. [LOS 4.o]
Explain the role of correlation in the multivariate normal distribution
Correlation indicates the strength of lineat relationship between a pair of random variables. When building a portfolio of assets, all things being equal, it is desirable to combine assets having low returns correlation because it will result in a portfolio with a lower variance. [LOS 4.g]
X is uniformly distributed between 2 and 12. Calculate the probability that X will be between 4 and 8
P(4<=X<=8) = (8-4)/(12-2) = 40% [LOS 4.d]
Explain how to standardize a variable.
To standardise a variable from a given normal distribution, the z-value must be calculated. The z-value is the number of standard deviations the variable is from the population mean. It is calculated as the difference between the observation and the population mean divided by the standard devation and has a mean of 0 and SD of 1. [LOS 4.i]
Describe the properties of discrete uniform random variable
Variable where all possible outcomes for a discrete random variable are equal [LOS 4.c]
Confidence intervals for normal distribution
A specified percentage of confidence interval, e.g. 95%, is a range that the random variable is expected to be in for that specified percentage of time. For any normally distributed random variable, 68% of outcomes are within 1 standard deviation of the mean, 95% are within 2 sd of the mean and 99% are within 3 sd of the mean. [LOS 4.h]
Continuously compounded rate of return: A stock was purchased for $100 and sold one year later for $120. Calculate the investor's annual rate of return on a conitnuously compounded basis
ln(120/100) = 18.232% If we had been given the return of 20% instead, the annual rate of return is: ln(1+0.20) = 18.232% [LOS 4.m]
Assuming a binomial distribution, compute the probability of drawing three black beans from a bowl of black and white beans if the probability of selecting a blackbean in any given attempt is 0.6. You will draw 5 black beans from the bowl
n = 5, p = 0.6, x = 3 P(X=3) = p(3) = 5!/(3!2!) x (0.6)^3 x (0.4)^2 = 34.56% [LOS 4.e]
Determine p(6), F(6), and P(2<=X<=8) for the discrete uniform distribution defined as: X = {2, 4, 6, 8, 10}
p(6) = 0.2 since p(x)=0.2 for all x. F(6)=P(X<=6) nxp(x) = 3x0.2 = 0.6. Note that n=3 since 6 is the third outcome in the range of possible outcomes. P(2<=X<=8)=kp(x)=4x0.2=0.8. Note that k = 4, since there are 4 outcomes in the range 2<=X<=8. [LOS 4.c]
The average return of a mutual fund is 10.5% per year and the standard deviation of annual returns is 18%. If returns are approximately normal, what is the 95% confidence interval for the mutual fund return next year?
95% C.I.: 10.5 + 1.96x18 to 10.5 - 1.96x18 95% C.I.: -24.78% to 45.78% P (-24.78 < R < 45.78) = 95% [LOS 4.h]
Define cumulative density function and give example
A cumulative density function (CDF) gives the probability that a random variable will take on a value less than or equal to a specific value, i.e. the probability that the value will be between minus infinity and the specified value. e.g. For the function P(x) = x/15 for x=1,2,3,4,5 the CDF is xsum(x=1) x/15, so that F(3) or P(x<=3) is 1/15 + 2/15 + 3/15 = 6/15 = 40% [LOS 4.b]
You have invested in a portfolio with a mean return of 8% and a standard deviation of 17% per year. Assume that returns are normally distributed. Calculate: the probability the return will be positive for a given year. the probability the return will be between 10% and 20%.
A cumulative distribution function table must be used for this question. To calculate the probability that the return will be positive: Z=(X−μ)/σ=(0−8)/17=−0.47 From z table, F(0.47) = 0.68082 P(Z≥−0.47)=P(Z≤0.47)=68.08% To calculate the probability that the return will be between 10% and 20%: Z1=(X−μ)/σ=(10−8)/17=0.12 Z2=(20−8)/17=0.71 P(0.12≤Z≤0.71)=P(Z≤0.71)−P(Z≤0.12)=0.7611−0.5478=21.33% [LOS 4.j]
Multivariate normal distribution
A probability distribution for a group of random variables that is completely defined by the means and variances of the variables plus all the correlations between pairs of the variables. Correlations must be defined for multivariate normal distributions. There are n(n−1)/2 distinct correlations. [LOS 4.g]
Bernoulli Trial
A random process or experiment results in one of only two mutually exclusive outcomes. A binomial random variable X is the number of successes in n Bernoulli trials. The binomial distribution assumes p is constant for all trials and the trials are independent. The binomial random variable can be described by two parameters, n and p X∼B(n,p) [LOS 4.e]
Contrast univariate and multivariate distribution
A univariate distribution describes a single random variable. A multivariate distribution specifies the probabilities for a group of related random variables. It is only meaningful when the behaviour of each random variable in the group is somewhat dependent on that of the others. Multivariate distibutions between two discrete random variables are described using joint probability tables. For continuous random variables, a multivariate normal distribution may be used if all individual variables follow a normal distribution. [LOS 4.g]
A company generated positive annual earnings once over the past four years. An analyst uses this historical frequency to model the probability of future positive annual earnings. The probability that earnings will be positive in exactly 2 out of the next 5 years is closest to: A 0.26 B 0.53 C 0.63
B In this question, the company producing positive earnings is defined as success and the company not producing positive earnings is defined as failure. The analyst designates 0.25 (or 1/4) years as the probability of positive earnings each year (p), so the probability of nonpositive earnings each year (1-p) is 0.75. A binominal random variable X equals x number of successes in n Bernoulli trials. In this question, x equals 2 annual occurrences of positive earnings, and n equals 5 years. The combination formula calculates the number of different ways that exactly 2 successes (positive annual earnings) can occur in 5 years. The two occurrences can happen in Year 1 and Year 2; Year 1 and Year 3; or another other combination out of 10 possible sequences. The probability of any one sequence equals 0.0264. Multiplying the total number of sequences (10) by the probability of each sequence (0.0264) approximately equals 0.26.
The probability that a certain exchange-traded fund (ETF) outperforms its peer-group average is 28% in any given year. Assuming each year is an independent trial, the probability that the ETF will outperform its peer-group average more than once over the next 8 years is closest to: A 30.6% B 70.3% C 77.5%
B In this scenario, the exchange-traded fund (ETF) exceeding its peer-group average is designated a success. The number of successes (X) is a binomial random variable that can take on any integer value x from 0 to 8. Since these are the only possible values of x, it follows that: P(0) + P(1) + P(2) + P(3) + P(4) + P(5) + P(6) + P(7) + P(8) = 100% = 1 Thus: P(x > 1) = 1 − P(x ≤ 1) = 1 − [P(0) + P(1)] Calculate P(X=0): P(X=0) = 8!/(0!8!)x0.28^0x0.72^8 = 0.0722 P(X=1) = 8!(1!8!)x0.28^1x0.72^7 = 0.2247 P(x > 1) = 1 - 0.0722 - 0.2247 = 0.7031 (Choice A) 30.6% is the result of incorrectly finding the probability that the binomial random variable X takes on a value of 2 (in other words, that the ETF exceeds its peer-group average exactly 2 out of 8 years). (Choice C) 77.5% is the result of failing to account for the probability that the binomial random variable X can take on a value of zero, and therefore incorrectly assuming that P(x > 1) = 1 − P(x = 1).
An analyst predicts that a stock will have a 6% return this year. The standard deviation is estimated to be 2%. Assuming a normal distribution, the probability that the stock's return will be greater than 8.2% or less than 5% is closest to: A 13.6% B 44.4% C 55.6%
B z(8.2) = (8.2-6)/2 = 1.1 z(5) = (5-6)/2 = -0.5 P(Z≥8.2)=1-P(Z≤8.2)= 1-0.8643 = 0.1357 P(Z≤5) = 1-P(Z≥5) = 1-P(Z≤-5) = 1 - 0.6915 = 0.3085 P(Z≤5 and Z≥8.2) = 0.1357 + 0.3085 = 44.42%
A random variable Y has a continuous uniform distribution over the interval [0, 1]. The probability of Y having a value between 0.1 and 0.5 is closest to: A. 30%. B. 40%. C. 50%.
B. With a continuous uniform distribution, all outcomes are equally likely. In this example, the probability that the value of Y is between 0.1 and 0.5 is: F(0.5)−F(0.1)=0.5−0.1=0.4 Put differently, the range between 0.1 and 0.5 covers 40% of possible outcomes.
Which of the following is a discrete random variable? A The amount of time it takes to execute a trade order B The rate of return on a stock index over the past month C An investor's amount of wealth measured in whole dollars
C A discrete random variable has a countable number of possible values. This means that the unit of measurement for the variable is typically specified in fixed incremental units. In this question, the investor's wealth in dollars may be $1 million, $2 million, or any other countable amount of whole dollars. A continuous random variable has an uncountable number of possible values. This means that the unit of measurement for the variable is unspecified and may take on an unlimited range of measurement units. For example, the time that it takes to execute a trade can be measured in minutes, seconds, milliseconds, nanoseconds, or any other unit of measurement from an unlimited number of possibilities. (Choice A) The time it takes to execute a trade order can take on any unit of measurement; therefore, the time variable has an uncountable number of possible values. (Choice B) The rate of return on a stock index can take on any unit of measurement (eg, the nearest whole number, the nearest hundredth); therefore, the return variable has an uncountable number of possible values and is considered a continuous random variable.
An investor invests €1,000,000 today and plans to withdraw €50,000 one year from today. None of the withdrawal can be from principal. An advisor recommends three different portfolio allocations: E(R) SD 1 7% 9.5% 2 9% 11% 3 11% 12.5% The optimal allocation, according to Roy's safety-first criterion, has a safety-first ratio closest to: A 0.21 B 0.36 C 0.48
C Roy's safety-first criterion determines the optimal allocation for minimizing the probability of shortfall risk. Shortfall risk is the risk of a return falling below a minimum acceptable level (ie, threshold level). The threshold level (RL) here is 5%, or €50,000/€1,000,000. If the portfolio returns less than 5% this year, then the investor must "dip into" the original €1,000,000 to withdraw €50,000 at year-end, which the question specifies is unacceptable. The optimal allocation is indicated by the highest safety-first ratio (SF ratio). The SF ratio effectively converts the 5% RL value into a Z-value that measures the absolute number of standard deviations between the RL and the expected return (mean). The farther the RL is from the expected return, the lower the probability of shortfall risk. Allocation 3 has the highest SFR of 0.48
An analyst is pricing a stock using a five-step binomial model with annual nodes. The probability of the stock price moving upwards in any given year is 60%. The probability that the stock price will move downwards at most 1 time over the next 5 years is closest to: A. 0.207. B. 0.259. C. 0.337.
C.
t the end of the month, a stock will be worth between $0 and $128. There is an equally likely chance that the stock will take on any value in that range. The probability that the stock's month-end will fall outside a range of $40 to $65 is closest to: A. 20%. B. 50%. C. 80%.
C.
Explain the differences between discrete and continuous random variable and their probability functions
Discrete random variables have a countable number of possible values, which could still be unlimited (e.g., all integers). Continuous random variables do not have a countable number of outcomes and an infinite number of possible outcomes. For a discrete distributions, p(x) = 0 when x cannot occur, and p(x) > 0 if it can. e.g. probability of rain occurring 33 days in June is 0, but probability of it raining 25 days in June has some positive value For a continuous distribution, p(x) = 0 even though x can occur. We can only consider P(x1 <= X <=x2) where x1 and x2 are actual numbers. e.g. For example, the probability of receiving 2 inches of rain in June is zero because 2 inches is a single point in an infinite range of possible values. On the other hand, the probability of the amount of rain being between 1.99999999 and 2.00000001 inches has some positive value. [LOS 4.a]
A Binomial Model of Stock Price Movement
If the stock has an initial price of S, it can move to uS (with a probability of p) or dS (with a probability of 1−p) over one period. The stock moves up with constant probability p (the up transition probability); if it moves up, u is 1 plus the rate of return for an up move. The stock moves down with constant probability 1 − p (the down transition probability); if it moves down, d is 1 plus the rate of return for a down move. The binomial tree is shown, where we now associate each of the n = 3 stock price moves with time indexed by t; the shape of the graph suggests why it is a called a binomial tree. Each boxed value from which successive moves or outcomes branch out in the tree is called a node. The initial node, at t = 0, shows the beginning stock price, S. Each subsequent node represents a potential value for the stock price at the specified future time. The stock price at t = 3 has 4 possible values: uuuS, uudS, uddS, and dddS. The probability that the stock price equals any one of these 4values is given by the binomial distribution. For example, 3 sequences of moves result in a final stock price of uudS: These are uud, udu, and duu. These sequences have 2 up moves out of 3 moves in total; the combination formula confirms that the number of ways to get 2 up moves (successes) in 3 periods (trials) is 3!/(3 − 2)!2! = 3. Next, note that each of these sequences—uud, udu, and duu—has probability p^2(1 − p)^1, which equals 0.125 (= 0.50^2 × 0.50^1). So, P(S3 = uudS) = 3[p^2(1 − p)], or 0.375, where S3 indicates the stock's price after 3 moves. [LOS 4.e]
Binomial Distribution
In a binomial setting, suppose we let X = the number of successes. The probability distribution of X is a binomial distribution with parameters n and p, where n is the number of trials of the chance process and p is the probability of a success on any one trial. The possible values of X are the whole numbers from 0 to n. [LOS 4.e]
Monte Carlo Simulation
Monte Carlo simulation uses computers to generate many random samples from a specified probability distribution. It can be used for many purposes, such as estimating Value-at-Risk (VaR) or valuing securities with embedded options. The following steps provide a general overview of the Monte Carlo simulation: 1. Specify the quantities of interest in terms of underlying variables. 2. Split the time horizon into subperiods. 3. Specify the distributional assumptions for the underlying risk factors. 4. Use a computer program to draw K random values of each risk factor. 5. Calculate the underlying variables based on the random values drawn. 6. Compute the quantities of interest. 7. Repeat steps 4 to 6 for N trials. The Monte Carlo estimate is the mean quantity of interest over the N trials. [LOS 4.p]
Describe how to choose optimal portfolios based on Roy's Safety First Criterion
Roy's safety-first criterion states that the optimal portfolio minimizes the probability that portfolio return, RP, will fall below the threshold level, RL. In symbols, the investor's objective is to choose a portfolio that minimizes P(RP < RL). When portfolio returns are normally distributed, we can calculate P(RP < RL) using the number of standard deviations that RL lies below the expected portfolio return, E(RP). The portfolio for which E(RP) − RL is largest relative to standard deviation minimizes P(RP < RL). Therefore, if returns are normally distributed, the safety-first optimal portfolio maximizes the safety-first ratio. In summary, when choosing among portfolios with normally distributed returns using Roy's safety-first criterion: 1. Calculate each portfolio's SFRatio. 2. Choose the portfolio with the highest SFRatio [LOS 4.k]
Define shortfall risk and safety first ratio.
Safety-first rules focus on shortfall risk, the risk that portfolio value (or portfolio return) will fall below some minimum acceptable level over some time horizon. The risk that the assets in a defined benefit plan will fall below plan liabilities is an example of a shortfall risk. [LOS 4.k]
The Student's t-distribution, chi-square distribution, and F-distribution are appropriate for different types of hypothesis tests depending on what is being tested. For what purposes is each distribution most appropriate to be used for?
Student's t: - Test of a single population mean - Test of differences between two population means - Test of mean difference between paired populations - Test of population correlation coefficient Chi square: - Test of variance of a normally distributed population F distribution: - Test of equality of variances of normally distributed populations from two independent random samples [LOS 4.n], [LOS 4.o]
Suppose an investor's threshold return, RL, is 2%. He is presented with two portfolios. Portfolio 1 has an expected return of 12%, with a standard deviation of 15%. Portfolio 2 has an expected return of 14%, with a standard deviation of 16%. What are the SFRatios of the two portdolios and what is the probability of portfolio return being less than the threshold?
The SFRatios are 0.667 = (12 − 2)/15 and 0.75 = (14 − 2)/16 for Portfolios 1 and 2, respectively. For the superior Portfolio 2, the probability that portfolio return will be less than 2% is N(−0.75) = 1 − N(0.75) = 1 − 0.7734 = 0.227, or about 23%, assuming that portfolio returns are normally distributed. [LOS 4.k]
Student T-Distribution Properties
The Student's t-distribution is similar to the normal distribution because they both form symmetric bell curves. However, the t-distribution has longer, fatter tails, which means extreme values are more likely than the normal distribution, producing a more conservative downside risk estimate. The t-distribution only has one parameter, known as the degrees of freedom (df). The degrees of freedom can be interpreted as the number of independent variables used in defining sample statistics. The larger the sample size, the larger the degrees of freedom because there are more independent variations. Assume X¯and s represent the sample mean and sample standard deviation, respectively. The ratio below follows a t-distribution with a mean of 0 and n−1 degrees of freedom. [LOS 4.n]
Describe the lognormal Distribution and its useful ness for modelling asset prices
The function e^x where x is normally distributed; Positively skewed; Bound to the left by 0 hence it is useful for modeling asset prices, which never take negative values Using lognormal distribution to model price relatives avoids the problem of asset prices being less than 0. Price relative is the ending price divided by the starting price (S1/S0), equal to (1 + holding period return). [LOS 4.l]
Describe properties of normal distributions
The normal distribution can be completely described by two parameters - mean (μ) and variance (σ2). A normally distributed variable X can be defined as X∼N(μ,σ2). It is symmetric (skewness = 0) and has a kurtosis of 3. The mean, median, and mode are all the same. Also, a linear combination of two or more normal random variables is normally distributed. [LOS 4.f]
Return on equity for a firm is defined as a continuous distribution over the range from -20% to +30%and has a cumulative distribution function of F(x) = (x + 20) / 50. Calculate the probability that ROE will be between 0% and 15%.
To determine the probability that ROE will be between 0% and 15%, we can first calculate the probability that ROE will be less than or equal to 15%, or F(15), and then subtract the probability that ROE will be less than zero, or F(0): P(0 ≤ x ≤ 15) = F(15) - F(0) F(15) = (15 + 20) / 50 = 0.70 F(0) = (0 + 20) / 50 = 0.40 F(15) - F(0) = 0.70 - 0.40 = 0.30 = 30% [LOS 4.b]
What is Monte Carlo simluation's advantage over historical simulation and what is its drawbacks compared to other anlytical method such as Black-Scholes-Merton model?
Unlike the Monte Carlo method, historical simulation does not let itself to "what if" analyses because it only reflects the tendencies that appear in the data. Analytical methods, such as the Black-Scholes-Merton model provide more cause-and-effect analysis than the Monte Carlo method, which provides probability-based estimates rather than exact results. [LOS 4.p]
Binomial Random Variable
Variable may be defined as the number of successes in a given number of trials where the outcome can be either a success or failure; Expected value = (probability of success) * (number of trials); Variance = (expected value) * (1 - probability of success) [LOS 4.e]