Module 4

Definitions: Chi-Square Distribution when do we use Chi Square? - we're comparing standard deviation to a benchmark. You'll see why in portfolio management. We want to make sure you never exceed the level of risk the client was willing and able to assume.

+ The chi-square distribution is asymmetrical. + The chi-square distribution with k degrees of freedom is the distribution of the sum of the squares of k independent standard normally distributed random variables. Therefore, like the t-distribution, the chi-square distribution is a family of distributions. + The chi-square distribution does not take on negative values. + A different distribution exists for each possible value of degrees of freedom.

Calculation: The Binomial Distribution (most practice question)

+ an experiment has only two possible outcomes which are labeled "success" and "failure." + 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 P(X=x)=nCx(p)^x(1−p)^n-x p = success rate 1-p = failure rate x = success times n-x = failure times

Definitions: Z value

+ number of standard deviations away from the population mean, a given observation lie + (observed value - population mean) / standard deviation = (x - μ)/σ

Definitions: Univariate distribution Multivariate distribution

- A distribution that specifies the probabilities for a single random variable. (up to this point we're based on univariate distribution only) - specify probabilities associated with a group of random variables taking into account the interrelationships that may exist between them

Definitions/Facts: Lognormal Distribution:

- It is frequently 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 (and is frequently) used as an approximation for returns. If a stock's continuously compounded return is normally distributed, then future stock price must be lognormally distributed. - so HPR isn't normally distributed, what should portfolio manager do? you can convert holding period returns to continuously compounded returns which are approximately normally distributed.

L1QMR09-LIC004-2107 If the mean of a normal distribution is 0 and the standard deviation is 20, the probability of an observation lying between -40 and 20 is closest to: 40.9%. 81.8%. 95.4%.

1) find -40 z score: (-40 - 0)/20 = -2. -> Z score, on z table 2 = 97.725%, since it's negative, we look from left to right. 1 - 97.725% = 2.3% on your left tail. 97.725 - 2.3 = 95.425. 95.425/2 = 47.7... 2) 20 / 20 = 1. -> Z score, on z table 1 = 84.1%. We look from right to left. we have 15.9% on your right tail and left tail. = 84.1 - 15.9 = 68.2%. 68.2% /2 = 34.1% 3) 47.7.. + 34.1 = 81.8%

Simple Practice( that's why i wrote it up here) The average annual return on a stock is 15% and the standard deviation of returns equals 5%. Given that the stock's returns are distributed normally, calculate the 90% confidence interval for the return in any given year.

15 ± 1.65(5)= 6.75% to 23.25% interpretation: The probability that the return on the stock for a given year lies between 6.75% and 23.25% equals 0.90, or P(6.75% ﹤ x ﹤ 23.25%) = 0.90.

L1QMR09-LIC012-1510 A stock market rises by an average of 10% a year, and the standard deviation of returns is 6%. The probability of the stock market falling by more than 2% in a year is closest to: 2.3%. 4.6%. 15.9%.

2.3%. A fall of 2% is two standard deviations from the mean; 95.4% of observations fall between the mean plus or minus two standard deviations, so 4.6% lies outside this range. Therefore, 2.3% will be below -2%, and 2.3% will be above 22%.

During a one-year holding period, a non-dividend paying stock has seen its price rise from $50 to $70. An investment in this stock during the one-year period will earn a continuously compounded return that is closest to: 34 percent. 37 percent. 40 percent.

34 percent. The continuously compounded rate of return is LN(1 + HPR), HPR = 70/50 − 1 = 0.4, so LN(1 + HPR) = LN(1.4) = 0.34 = 34%.

*** The total number of parameters that fully characterizes a multivariate normal distribution for the returns on two stocks is: 3. 4. 5.

5 C is correct. A bivariate normal distribution (two stocks) will have two means, two variances, and one correlation. A multivariate normal distribution for the returns on n stocks will have n means, n variances, and n(n - 1)/2 distinct correlations.

Definitions: Multivariate distribution

multivariate normal distribution for the return on a portfolio with n stocks is completely defined by: + mean returns on all the n individual stocks + variances of returns of all n individual stocks + return correlations between each possible pair of stocks. There will be n(n − 1)/2 pairwise correlations EX: 4 assets in a porfolio: 4(4-1)/2= 6, we need 6 pairs.

Example 2.5. ***** Continuously Compounded Returns A stock that was purchased for $65 one year ago was sold for $86 today. What was the continuously compounded return on this investment?

step 1- ( 86/ 65) -1 = .323 step 2- skip, it's only 1 year. HPR = EAR step 3- In(1+0.323) = 0.279

Example 2.6. ***** Continuously Compounded Returns An investment of $1,000 appreciates to a value of $1,250 in 2 years. Calculate the continuously compounded return on this investment.

step 1- (1250/1000) -1 = 0.25 step 2- EAR = (1.025) ^ 1/2 -1 = 0.118034 step 3- In(1+0.118) = 0.1115

Definitions: t distribution

( it's a basically z distribution, but for smaller size) - It's a reliability factor. It's used to calculate dispersion around a sample mean when variance is unknown an n < 30. - t-distribution properties: + It is symmetrical. + It is defined by a single parameter, the degrees of freedom (df), where degrees of freedom equal sample size minus one (n − 1). + It has a lower peak than the normal curve, but fatter tails. + As the degrees of freedom increase, the shape of the t-distribution approaches the shape of the standard normal curve. + When you look at the t table, there will be one tail and 2 tail numbers. youll get it moving on, thus, just be familiar. Lets say we have CI 90% 10% is your alpha, we look at 0.1 at two-tailed test or 0.05 in one tail let's say n = 32, degree of freedom: 32 -1 = 31 with 0.1, your t is 1.684 on the table

Calculation: Part 2 Lognormal Distribution: (important)

(note: V end/V beg) = 1 + HPR + If that holding period is more than a year, calculate an effective annual return. If it's not, you can skip step 2 + Continuous compounded return => EAR = e^rcc - 1 , rcc = continuous compounded return [Check ex] ex: If the stated rate equals 5%, EAR continuous compounding is simply (e^0.05−1) = 5.127% (using calculator. "0.05" [2nd] [ln] [-] "1") How about converting EAR to Continuously compounded return is given: lets say 5.127% is given and we're looking for EAR: + it's simply "In(EAR +1) or In(1.05217) = 5%, -> it gives you the stated rate.

Definitions: f distribution when do we use f distribution? You're going to use an F-distribution when you're comparing standard deviation of two populations. You're comparing the standard deviation of portfolio manager number one to portfolio manager number two.

+ F-distribution is a family of asymmetrical distributions bounded from below by 0. +Each F-distribution is defined by two values of degrees of freedom, called the numerator and denominator degrees of freedom. +If χ2-1 is one chi-square random variable with m degrees of freedom and χ2-2 is another chi-square random variable with n degrees of freedom, then F=(χ2-1/m)/(χ2-2/n) follows an F-distribution with m numerator and n denominator degrees of freedom. (i dont get the last +, ill choose to ignore for test sake)

Definitions: Monte Carlo Simulation (for sure just recognize on the exam)

- 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 measures of random variables, which are then used for statistical inference. - INVESTMENT APPLICATIONS + To experiment with a proposed policy before actually implementing it. + To provide a probability distribution that is used to estimate investment risk (e.g., VAR). + To provide expected values of investments that can be difficult to price. + To test models and investment tools and strategies. - LIMITATION +answers are only as good as the assumptions and model used. +Does not provide cause-and-effect relationships.

Definitions: Roy's safety-first criterion

- States that the optimal portfolio minimizes the probability that the return of the portfolio falls below some minimum acceptable level or threshold level. (*other words, minimizes P of being in the left tail or maximize the probability of not being in the left tail)

Tracking Error (ps: there weren't really much questions on this, maybe less chances to be on the test) ((I will choose to skip the example , doubt will be on the test)(sort of know the formula, it's really similar for the materials later on)

- a measure of how closely a portfolio's returns match the returns of the index to which it is benchmarked. It is the difference between the total return on the portfolio (before deducting fees) and the total return on the benchmark index. + Tracking error = Gross return on portfolio − Total return on benchmark index +(watch the example will be better)

Definitions: Historical simulation

- assumes that the distribution of the random variable going forward depends on its distribution in the past. This method of forecasting has an advantage in that the distribution of risk factors does not have to be estimated.

Definitions: Binomial Tree ((I will choose to skip , doubt will be on the test)

- binomial tree is the best way to model the price of an asset over varying periods and chances of rising or falling each period.

Calculation: cumulative distribution function (cdf)

- computes probabilities less than or equal to a specified value or ( not different definitions, just different wording) - The cumulative distribution function, F(x), gives the probability that a random variable X is less than or equal to a particular value x. - Know how to calculate this type of question(it's quite simple)

Definitions: Normal distribution

- continuous random variable that follows the normal distribution has a bell-shaped probability distribution Characteristics: + completely described by its mean (μ) and variance (σ^2) + skewness of 0, Kurtosis equals 3 and excess kurtosis equals 0. +tail on either side extends to infinity

Calculation: Continuous Uniform Distribution

- described by a lower limit, a, and an upper limit, b. These limits serve as the parameters of the distribution - Know how to calculate this type of question(it's quite simple) -total range = denominator, subset = numerator

Facts: mutually exclusive collectively exhaustive

- mutually exclusive, which means that the occurrence of one outcome precludes the occurrence of the other. - collectively exhaustive, which means that no other outcomes are possible.

Definitions/Components: Lognormal Distribution:

- not normally distributed. - 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) + Basically, holding period return (HPR) on any asset can range between −100% and + ∞ + Asset prices/ values have lognormal distribution. However, the distribution of the natural log of our holding period returns is approximately normally distributed. Thus, we use our formula to change it back to normal distribution.

Calculation: Discrete Uniform Distribution

- probability of each of the possible outcomes is the same - Know how to calculate this type of question(it's quite simple) + The discrete uniform distribution is also known as the "equally likely outcomes" distribution. (Facts from question)

Definition: Confidence Interval

- represents the range of values within which a certain population parameter is expected to lie in a specified percentage of the time Example 2.1 Confidence Intervals The average annual return on a stock is 15% and the standard deviation of returns equals 5%. Given that the stock's returns are distributed normally, calculate the 90% confidence interval for the return in any given year. Solution The 90% confidence interval is calculated as: 15 ± 1.65(5) = 6.75% to 23.25% Interpretation: The probability that the return on the stock for a given year lies between 6.75% and 23.25% equals 0.90, or P(6.75% ﹤ x ﹤ 23.25%) = 0.90.

Definitions: Shortfall risk:

- 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. (Basically, probability of being in the left tail) (We also call it: Minimal acceptable return)

L1QMR09-LIC005-2107 A normal distribution has a mean of 25 and a standard deviation of 5. What is the standardized normal random variable representing an observation of 15? -2.00. 1.67. 3.00.

-2.00. Z = ( X − μ )/σ= 15 − 255 = -2.0, which is the number of standard deviations of the observation below the mean.

Definitions: 1) random variable 2) discrete random variable 3) continuous random variable

1) random variable. - variable whose outcome cannot be predicted (e.g., the number of cars that will cross a traffic light during a given 10-minute period). 2) discrete random variable. - countable number of values. Each outcome has a specific probability of occurring, which can be measured (e.g: toss of a coin, fair die, number of cars sold by a salesman in a week; x = 0, 1, 2 .... number of customers waiting in line for a bank clerk at a point in time; x = 0, 1, 2 .... ) 3) continuous random variable - number of possible outcomes cannot be counted (there are infinite possible outcomes) (eg: For example, consider the length of time it takes to get served at a particular restaurant (a random variable), which can be anywhere between 25 and 40 minutes (possible outcomes). While the probability of the waiting period being between 25 and 30 minutes can be measured, the probability of waiting for exactly 25 minutes and 15 seconds is zero because time can be measured in seconds, half seconds, even thousandths of seconds)

Calculation: Shortfall Ratio(SF ratio or Roy's ratio)(or Z score) This is for Roy's safety first criterion formula (Must memorize) (This is the first slide I just started adding pink notes, all info prior to this should be memorized too You know which ones!)

= [E(Rp) - Rt]/σ E(R) = Expected return Rt = target value return(or minimum acceptable return) This is basically a Z score, with a bit of a concept twist + The higher the ratio, lower the probability to attain return lower than the threshold level( for test purpose, highest ratio is the winner) (Must memorize) + examples: (from pictures) (skip all the essential calculation) we got Portfolio A: 0.3, Portfolio B: 0.466 Portfolio B wins. How? z score 0.466 = 0.6808, which is 32% on the left and right tail. Portfolio A loses. How? z score 0.3 = 0.6179, which is 38% on the left and right tail. B wins with lower probability to be lower than the threshold.

Which of the following statements concerning a multivariate distribution is least accurate? The distribution describes the probability of different outcomes for a group of random variables. A multivariate distribution describes the outcomes for a single random variable under different scenarios. If the random variables are normally distributed it is usually assumed the multivariate distribution is normally distributed.

A multivariate distribution describes the outcomes for a single random variable under different scenarios. If it is a single random variable it is a univariate distribution.

Facts: (from question) Which of the following statements about the cumulative distribution function (CDF) are most accurate? Statement I: The CDF lies between 0 and 1 for any value of x. Statement II: As x increases, the CDF remains the same. Statement III: The sum of the PDFs of all possible outcomes equals 1. A I and II. B II and III. C I and III. +The CDF lies between 0 and 1 for any value of x. +As x increases, the CDF either increases or remains constant. +The sum of the PDFs of all possible outcomes equals 1.

Ans) C +The CDF lies between 0 and 1 for any value of x. +As x increases, the CDF either increases or remains constant. +The sum of the PDFs of all possible outcomes equals 1. The cumulative distribution function, F(x), gives the probability that a random variable X is less than or equal to a particular value x.

Facts/Calculation: Mean plus or minus one standard deviation. ? Mean plus or minus two standard deviations. ? Mean plus or minus three standard deviations. ?

Approximately 68% Approximately 95% Approximately 99%

Facts: (important, because my bumbass forgot)( just always check ex 2.3 to remind yourself) If Z value is 1, we get 0.84 from Z table. It means 16% on one tail. 16 * 2 = 32%, 32% is the alpha or level of significance. 1 - 32% = 68%, confidence interval. Sometime they will give you the Z value to fine confidence interval or sometime they give you confidence interval to find Z value.

Ex 2.3 (pg 80) is a very important example for the future me if I forget about z score again.

Calculation: Part 1. Lognormal Distribution: (important)

Formula:(important) Step 1) (V end/ V beg) - 1 = HPR step 2) => ( 1 + HPR)^ 1/# of years - 1 = EAR (Convert HPR to EAR). [If HPR is more than 1 year] Step 3) => In(1 +EAR) = rcc (Convert EAR to continuously compounded return) ---------------------------------------------------------------------- (note: V end/V beg) = 1 + HPR + If that holding period is more than a year, calculate an effective annual return. If it's not, you can skip step step 2 + Discrete compounded return = 12 or 6 months + Continuous compounded return = EAR = e^rcc - 1 , rcc = continuous compounded return ex: If the stated rate equals 5%, EAR continuous compounding is simply (e^0.05−1) = 5.127% (using calculator. "0.05" [2nd] [ln] [-] "1") How about converting EAR to Continuously compounded return is given: lets say 5.127% is given and we're looking for EAR: + it's simply "In(EAR +1) or In(1.05217) = 5%, -> it gives you the stated rate.

L1R10TB-BW007-1612 Which of the following is not a characteristic of a lognormal distribution? Its farthest point on the left side is bounded by 0. The probability distribution function starts at zero, increases to its mode, and decreases thereafter. It is skewed to the left.

It is skewed to the left.

Test Banks: (Questions that's worthy/good practice)(I just started, rest of them should slowly cumulate)

M4L1: L1QMR09-LIC006-2107* L1R09TB-AC015-1512* M4L2: L1R09TB-AC027-1512

Calculation: (great reminder for myself) Z score= step 1) find Z score or (x - μ)/σ step 2) use the Z score to find number on z table. The value you find are the left side of the distribution. If you're looking for the right side of the distribution, you 1 - (values or probabilities on the table). step 3) if Z score is negative, it goes to the right. It's mirror image of the positive value. Here's an example for negative:

Reminder Example: mean = 1500 STD = 500 Observed value = 1200, 3000 z scores = (1200 - 1500)/500 = -0.6 z scores = (3000- 1500)/500 = 3 What's Confidence interval? go to the table: 0.6 => approximately 0.73 or 73% Usually, if z is positive, you look from right to left. it will give you .27 on the right as the tail. But, this is negative, thus we look from left to right, This leaves you .27 on the left tail. ( I hope my future self gets it, if you forgot look back to the notes)