Reading 9 LOS's Flashcards
LOS 9a: Define a probability distribution and distinguish between discrete and continuous random variables and their probability functions.
LOS 9b: Describe the set of possible outcomes of a specified discrete random variable.
LOS 9c: Interpret a cumulative distribution function
LOS 9d: Calculate and interpret probabilities for a random variable, given its cumulative distribution function
A discrete random variable is one that can take on a countable number of values. Each outcome has a specific probability of occurring, which can be measured.
The probability distribution of a random variable indentifies the probability of each of the possible outcomes of a random variable
All probability distributions have the following two properties:
- 0 <= p(x) <= 1.
- Σp(x) = 1
For a discrete random variable the probability of each outcome can be listed in the form of a probability function. p(x), which expressed that the probability that”X”, the random variable, takes on a specific value of “x”.
A continuous random variable is one for which the number of possible outcomes cannot be counted and therefore, probabilities cannot be attached to specific outcomes.
For continuous random variable, the probability of a specific outcome within a range of infinite outcomes is essentially zero. Therefore we use a probability density function (pdf), which is denoted by f(x) to interpret their probability structure. A pdf can be used to determine the probability that the outcome lies within a specified range of possible values
A cumulative distribution function (cdf) also known as a distribution function, expresses the probability that a random variable, X, takes on a value less than or equal to a specific value, x. It represents the sum of probabilities of all outcomes that are less than or equal to a specified value of x.
LOS 9e: Define a discrete uniform random variable, a Bernoulli random variable, and a binomial random variable
LOS 9f: calculate and interpret probabilities given the discrete uniform and the binomial distribution functions.
LOS 9g: Construct a binomial tree to describe stock price movement
A discrete uniform distribution is one in which the probability of each of the possible outcomes is the same. Best example is that of a die, all outcomes are 1/6.
The Binomial distribution
Suppose an experiment has only 2 possible outcomes which are labeled success and failure, and they are mutually exclusive and exhaustive
Such an experiment is called a Bernoulli trial. If this experiment is carried out n times, the number of successes, X, is called a Bernoulli random variable and the distribution that X follows is known as the binomial distribution.
Under these assumptions a random variable that follows the Binomial distribution is defined by n and p as:
- X~ B(n,p) which is read as:
- the randome variable X follows a binomial distribution with the number of trials, n, and a probability of success p
We use the combination formula in binomial experiments because the number of labels equals 2 and we are not concerned about the order in which these successess occur.
The expected value of a binomial random variable, X, is the product of p and n.
The variance of a binomial random variable is given by:
- σx2 = n x p x (1-p)
Binomial trees
These can be drawn to illustrate stock price movements. Probability of gains and losses in period one, leads to more branches in period 2, and then more in period 3
Skewness and Binomial Distribution
- If the probability of success is .5, binomial distribution is symmetric
- If it is less than .5, the distribution is skewed to the right
- More than .5 is skewed to the left
LOS 9h: Calculate and interpret tracking error
tracking Error is a measure of how closely a portfolio’s returns match the returns of the index which it is benchmarked.
Tracking error = Gross return on portfolio - Total return on benchmark index
LOS 9h: Define the continuous uniform distribution and calculate and interpret probabilities, given a continuous uniform distribution
A continuous uniform distribution is described by a lower limit, a, and an upper limit, b. These limits serve as the parameters of the distribution.
- The probability of the random variable taking on any set of values outside the parameters a and b equals 0
- The probability that the random variable will take a value that falls between x1 and x2 that both lie within the range, a to b, is the proportion of the total are taken up by the range, x1 to x2
- P (x1<= X <= x2) = (x2-x1) / (b-a)
LOS 9j: Explain the key properties of the normal distribution
LOS 9k: Distinguish between a univariate and a multivariate distribution, and explain the role of correlation in the multivariate normal distribution
The normal distribution is a continuous distribution that plays a central role in quantitative analysis and has the following properties
- It is completely described by its mean µ, and its variance, σ2
- The distribution has a skewness of 0, which means its symmetric around its mean
- Kurtosis equals 3, and excess kurtosis is 0
- A linear combination of normally distributed random variable is also normally distributed
- The probability of the random variable lying in ranges further away from the mean gets smaller and smaller but never goes to zero. The tail on either side extends to infinity
Univariate distributions describe the distribution of a single random variable.
Multivariate distributions specify probabilities associated with a group of random variables taking into account the interrelationships that may exist between them. AS mentioned, linear combinations of normally distributed random variables is also normally distributed. Therefore, the return on a portfolio of assets whose individual returns are normally distributed, the portfolio will also be normally distributed.
A multivariate normal distribution is defined by the following parameters:
- the mean returns on all n individual stocks
- the variances of returns of all n individual stocks
- The return correlations between each possible pair of stocks. There will be n(n-1) / 2 pairwise correlations in total. The need for correlations is what differentiates the multivariate from the univariate
LOS 9l: Determine the probability that a normally distributed random variable lies inside a given interval
A confidence interval represents the range of values within a certain population parameter is expected to lie in a specified percentage of time.
In practice, we do not usually know the values for population parameters, so we estimate them with samples. Confidence intervals use the mean and then +/- certain amount of standard deviations. This makes upper and lower bounds for our confidence interval.
For a random variable X that follows the normal distribution:
- 90% CI is sample mean (Xbar) +/- 1.65 standard deviations (s)
- 95% CI is Xbar +/- 1.96s
- 99% CI is Xbar +/- 2.58s
LOS 9m: Define the standard normal distribution, explain how to standardize a random variable, and calculate and interpret probabilities using the standard normal distribution.
The standard normal distribution curve is a normal distribution that has been stadardized so that it has a mean of zero and a standard deviation of 1. To standardize a given observation of a normally distributed random variable, its z-score must be calculated:
- z = (observed value - population mean) / standard deviation
Essentially the z-value represents the number of standard deviations away from the population mean, a given observation lies.
Calculating Probabilites using Z-scores
Once you find the z-score, find the associated probability on the z-table.
Remember if you are finding the probability of less than, use the probability given. If you are finding the probability of more than, subtract the probability given from 1.
LOS 9n; Define shortfall risk, calculate the safety-first ratio, and select an optimal portfolio using Roy’s safety-first criterion
Shortfal risk refers to the probability that a portfolio’s value or return, E (Rp), will fall below a particular target value or return (RT) over a given period
Roy’s safety-first criterion states that an optimal portfolio minimizes the probability that the actual portfolio return, Rp, will fall below the target return, RT
The minimum target level is also called the “threshold” level. If the portfolio’s returns are normal, we can calculate the probability that returns will be lower than the threshold by:
- Shortfall ratio (SF Ration) or z-score = [E(Rp) - RT] / σP
When making comparison across portfolios, we can just look at their SF ratios and conclude that the higher the value of the SF Ratio, the better the risk-return tradeoff the portfolio offers, given the investor’s threshold level
LOS 9o: Explain the relationship between normal and lognormal distributions and why the lognormal distribution is used to model asset prices
A random variable, Y, follows the lognormal distribution if its natural logaruthm, ln Y, is normally distributed. The reverse is also true.
Three important features differentiate the lognormal distribution from the normal distribution:
- It is bounded by zero on the lower end
- The upper end of its range is unbounded
- it is skewed to the right (positively skewed)
The HPR on any asset can range between -100% to infinity
Instead of using the return on an investment as the random variable, if we were to use the ratio of the ending value of the investment to its beginning value (Vt/V0), the distribution will still be skewed, but with a lower bound of zero and no upper bound. Notice that (Vt/V0) equals (1 + holding period return)
Finally if we were to take the distribution of the natrual logarithm of (Vt/V0) we would find that the distribution is unbounded at both ends. These quantities represent the continuously compounded rate of return
Takeaway While the distribution of the variable follows the lognormal distribution, the distribution of the natural logarithm is normally distributed
Conclusion the lognormal distribution can be used to model the distribution of asset prices because it is bounded from below by zero. The normal distribution on the other hand, can be used as an approximation for returns. If a stock’s contunuously compounded return is normally distributed, then future stock price must be lognormally distributed
LOS 9p: Distinguish between discretely and continuously compounded rates of return, and calculate and interpret a continuously compounded rate of return, given a specific holding period return
Discretely compounded returns are based on discrete or define compounding periods, such as 12 months or 6 months. As the compounding periods get shorter and shorter, the effective annual rate (EAR) rises. With continuous compounding the EAR is given as:
- EAR = ercc - 1 where rcc= continuously compounded annual rate
By reorganizing the EAR equation above, we can define an expression for calculating the continuously compounded stated annual rate:
- EAR = ercc - 1
- ln(EAR + 1) = ln ercc
- ln(EAR + 1) = rcc x ln e ,therefore;
- rcc= ln(EAR + 1)
We can use the method described above to calculate the continuously compounded state annual rate of return given a HPR:
- rcc = ln(1 +HPR), and since HPR = (Vt/V0), then:
- rcc= ln (Vt/V0)
The relationship between HPR and the continuously compounded rate is given as:
- HPRt = ercc x t -1
LOS 9q: Explain Monte Carlo simulation and describe its major applications and limitations
LOS 9r: Compare Monte Carlo simulation and historical simulation
Monte Carlo Simulation generates random numbers and operator inputs to synthetically create probability distributions for variables. It is used to calculate expected values and dispersion measure of random variables, which are then used for statistical inference.
Specifying the Simulation
- Step 1- Specify the quantity of interest along with the underlying variables
- Step 2 - specify a time period and break it down into a number of subperiods
- Step 3 - specify the distributional assumptions for the risk factors affecting the underlying variables
Running The Simulation
- Step 4- Use a computer program or a spreadsheet function to draw random variables for each risk factore
- Step 5- Based on the random values generate, calculate the values for the underlying variables and then calculate the average of those values
- Step 6- Bsed on the average values of the underlying variables calculated in step 5 calculate the value of the stock option
- Step 7- iteratively go back to step 4 until a sufficinet number of trials have been performed
Investment Applications
- To experiment with a proposed policy before actually implementing it
- To provide a probability distribution that is used to eliminate investment risk
- To provide expected values of investments that can be difficult to price
- To test models and investment tools and strategies
Limitations
- Answers are only as good as the assumptions and model used
- Does not provide cause and effect relationships
Historical simulation assumes that the distribution of the random variable going forwad depends on its distribution in the past. It has an advantage in that the distribution of risk factors does not have to be estimated. However there are limitations:
- a risk factor that was not represented in historical data will not be considered in the simulation
- It does not facilitate what if analysis whn the if factor has not occurred in the past
- It assumed that the future will be similar to the past
- It does not provide any cause and effect relationship information
