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二、Investment Tools: Quantitative Methods

1.A.: Time Value of Money

a: Calculate the future value (FV) and present value (PV) of a single sum of money.

Future Value:

FV = PV(1 + I/Y)N

Where PV = the amount of money invested today, I/Y = the rate of return, and N = the length of the holding period.?

Example:Using a financial calculator, here's an example of how you would find the FV of a $300 investment (PV), given you earn a compound rate of return (I/Y) of 8% over a 10-year (N) period of time:

N = 10, I/Y = 8, PV = 300; CPT FV = $647.68 (ignore the sign).

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Present Value:

PV = FV / (1 + I/Y)N

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Example:Using a financial calculator, here's an example of how you'd find the PV of a $1,000 cash flow (FV) to be received in 5 (N) years, given a discount rate of 9% (I/Y).

N = 5, I/Y = 9, FV = 1,000; CPT PV = $649.93 (ignore the sign).

b: Calculate an unknown variable, given the other relevant variables, in single-sum problems.

Example 1:Solving for I/Y

In this example, you want to find the rate of return (I/Y) that you'll have to earn on a $500 investment (PV) in order for it to grow to $2,000 (FV) in 15 years (N). This very same problem could also be set up in terms of growth rates - e.g., what rate of growth (I/Y) is necessary for a company's sales to grow from $500 per year (PV) to $2,000 per year (FV) in 15 years (N).?

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N = 15, PV = -500, FV = 2,000; CPT I/Y = 9.68%

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Example 2:Solving for N

In this example, you want to find out how many years (N) it will take for a $500 investment (PV) to grow to $1,000 (FV), given that we can earn 7% annually (I/Y) on your money.

?

I/Y = 7, PV = -500, FV = 1,000; CPT N = 10.24 years.

c: Calculate the FV and PV of an regular annuity and an annuity due.

Calculate the FV of an ordinary annuity:

Example:Find the FV of an ordinary annuity that will pay $150 per year at the end of each of the next 15 years, given the investment is expected to earn a 7% rate of return.

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N = 15, I/Y = 7%, PMT = $150; CPT FV = $3,769.35 (ignore the sign).

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Calculate the FV of an annuity due:

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Example:Find the FV of an annuity due that will pay $100 per year for each of the next three years, given the cash flows can be invested at an annual rate of 10%.?

Note:When solving for a FV of an annuity due, you MUST put your calculator in the beginning of year mode (BGN), otherwise you'll end up with the wrong answer.

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N = 3, I/Y = 10%, PMT = $100; CPT FV = $364.10 (ignore the sign).

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Calculate the PV of an ordinary annuity:

Example:Find the PV of an annuity that will pay $200 per year at the end of each of the next 13 years, given a 6% rate of return.

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N = 13, I/Y = 6, PMT = 200; CPT PV = $1,770.54

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Calculate the PV of an annuity due:

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Example:Find the PV of a 3-year annuity due that will make a series of $100 beginning of year payments, given a 10% discount rate.

Note:There are two ways to approach this question. The first is to put your calculator in BGN mode and then input all the variables as you normally would. The second is to shorten the annuity by one year (N - 1) and find the PV of that shortened annuity as if it were an ordinary annuity, then add the first annuity payment (PMT0) to it to come up with the PV of this annuity due. In this second alternative, you will leave your calculator in the END mode.

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1. BGN mode: N = 3, I/Y = 10, PMT = 100; CPT PV = $273.55

2. END mode: N = 2, I/Y = 10, PMT = 100; CPT PV = $173.55 + 100 = PV = $273.55

d: Calculate an unknown variable, given the other relevant variables, in annuity problems.

Example:Find the PMT required to fund a retirement program of $3,000 at the end of 15 years, given a rate of return of 7%.

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N = 15, I/Y = 7%, FV = 3,000; CPT PMT = $119.38 (ignore the sign).

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Example:Suppose that you will deposit $100 at the end of each year for 5 years into an investment account. At the end of 5 years, the account will be worth $600. What is the rate of return?

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N = 5, FV = 600, PMT = 100; CPT I/Y = 7 years.

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Example:Solve for the PMT given a 13-year annuity with a discount rate of 6%, and a PV of $2,000.

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N = 13, I/Y = 6, PV = 2,000; CPTPMT = $225.92.

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Example:Suppose that you have $1,000 in the bank today. If the interest rate is 8%, how many annual, end-of-year payments of $150 can you withdraw?

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I/Y = 8, PMT = 150, PV = -1,000; CPTN = 9.9 years.

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Example:What rate of return will you earn on an annuity that costs $700 today and promises to pay you $100 per year for each of the next 10 years?

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N = 10, PV = 700, PMT = 100; CPT I/Y = 7.07%.

p

e: Calculate the PV of a perpetuity.

Example:Assume a certain preferred stock pays $4.50 per year in annual dividends (and they're expected to continue indefinitely). Given an 8% discount rate, what's the PV of this stock?

PVperpetuity = PMT / I/Y

PVperpetuity = 4.50 / .08 = $56.25

This means that if the investor wants to earn an 8% rate of return, she should be willing to pay $56.25 for each share of this preferred stock.

f: Calculate an unknown variable, given the other relevant variables, in perpetuity problems.

Example:Continuing with our example from LOS 1.A.e, what rate of return would the investor make if she paid $75.00 per share for the stock?

I/Y = PMT / PVperpetuity

4.50 / 75.00 = 6.0%

g: Calculate the FV and PV of a series of uneven cash flows.

FV Example:Given: I = 9%; PMT1 is $100; PMT2 is $500; and PMT3 is $900. How much is this future stream worth at the end of the 3rd year?

Solve:enter PMT1=$100 as PV;I=9%n=2:Compute FV1=$118.81

enter PMT2=$500 as PV;I=9%n=1:Compute FV2=$545.00

enter PMT3=$900 as PV;I=9%n=0:Compute FV3=$900.00

Sum of FVs=$1,563.81

PV Example:Given: I = 10%; PMT1 is $100; PMT2 is $200; and PMT3 is $300. Solve for the PV of this cash flow stream.

Solve:enter PMT1=$100 as PV;I=10%n=1:Compute PV1=$90.91

enter PMT2=$200 as PV;I=10%n=2:Compute PV2=$165.29

enter PMT3=$300 as PV;I=10%n=3:Compute PV3=$225.39

Sum of PVs=$481.59

h: Calculate time value of money problems when compounding periods are other than annual.

Example: PV = $100, N = 1 year, I = 12%. Find the FV for various compounding periods.

Annual:N = 1I = 12%PV = $100compute FV = $112.00

Semi-annual:N = 2I = 6%PV = $100compute FV = $112.36

Quarterly:N= 4I = 3%PV = $100compute FV = $112.55

Monthly:N = 12I = 1%PV = $100compute FV = $112.68

Daily:N = 365I = .03287PV = $100compute FV = $112.747

Continuous:FV = (PV)(e(i rate)(n))=100(e)(.12)(1)App6A compute FV = $112.75

In the continuous compounding equation, the interest rate is the stated or nominal annual rate.

Example:Given: a 10% annual rate paid quarterly; PV = 500; time is 5 years; compute FV.

Solve: I = 10/4 = 2.5; N = 5 * 4 = 20; PV = 500: compute FV = 819.31.

i: Distinguish between the stated annual interest rate and the effective annual rate.

The stated rate of interest is known as the nominal rate, and represents the contractual rate. The periodic rate, in contrast, is the rate of interest earned over a single compound period - e.g., a stated (nominal) rate of 12%, compounded quarterly, is equivalent to a periodic rate of 12/4 = 3%. Finally, the true rate of interest is known as the effective rate and represents the rate of return actually being earned, after adjustments have been made for different compounding periods.

j: Calculate the effective annual rate, given the stated annual interest rate and the frequency of compounding.

Example: Compute the effective rate of 12%, compounded quarterly. Given m = 4, and periodic rate = 12/4 = 3%.

Effective rate = (1 + periodic rate)m - 1

Where m = the number of compounding periods in a year.

(1 + .03)4? - 1 = 1.1255 - 1 = 12.55%

k: Draw a time line, specify a time index, and solve problems involving the time value of money as applied to mortgages, credit card loans, and saving for college tuition or retirement.

Example: Paying off a Loan (or Mortgage)

A company wants to borrow $50,000 for five years. The bank will lend the money at a 9% rate of interest and will require that the loan be paid off in five equal, annual (end-of-year) installment payments. What are the annual loan payments that this company will have to make in order to pay off this loan?

N = 5, I/Y = 9, PV = 50,000; CPT PMT = $12,854.62

This loan can be paid off in five equal annual payments of $12,854.62.

Example:Loan Amortization

An individual borrows $10,000 at 10% today amortized over 5 years. What are his payments?

PV = 10,000, N = 5, I/Y = 10; CPT PMT = $2,637.97

He will pay $2,637.97 at the end of each of the next 5 years.

Example:Funding a Retirement Program

A 35-year old investor wants to retire in 25 years at age 60. Given he expects to earn 12.5% on his investments prior to his retirement, and then 10% thereafter, how much must he deposit annually (at the end of each year) for the next 25 years in order to be able to withdraw $25,000 per year (at the beginning of each year) for the next 30 years?

This is a two-part problem. First, use PV to compute the present value of the 30-year, $25,000 annuity due and second, use FV to find the amount of the fixed annual deposits that must be made at the end of the first 25-year period to come up with the needed funds.

Step 1:N = 29, I/Y = 10, PMT = 25,000; CPT PV = 234,240 + 25,000 = $259,240

Step 2:N = 25, I/Y = 12.5, FV =259,240; CPTPMT = $1,800.02

The investor will need a nest egg of $259,240. He will then have to put away $1,800 per year at the end of each of the next 25 years in order to accumulate a nest egg worth $259,240 - which will enable him to withdraw $25,000 per year for the following 30 years.

1.B: Statistical Concepts and Market Returns

a: Differentiate between a population and a sample.

A population is defined as all members of a specified group. Any descriptive measure of a population characteristic is called a parameter. Populations can have many parameters, but investment analysts are usually only concerned with a few, such as the mean return, or the standard deviation of returns.

A sample is defined as a portion, or subset of the population of interest. Even if it is possible to observe all members of a population, it is often too expensive or time consuming to do so. Once the population has been defined, we can take a sample of the population with the view of describing the population as a whole.

b: Explain the concept of a parameter.

Any descriptive measure of a population characteristic is called a parameter.

c: Explain the differences among the types of measurement scales.

Nominal scale:this represents the weakest level of measurement. Observations are classified or counted with no particular order. An example would be assigning the number one to a large cap value fund, the number two to a large cap growth fund, etc.

Ordinal scale:this is a higher level of measurement. All observations are placed into separate categories and the categories are placed in order with respect to some characteristic. An example would be ranking 100 large cap growth mutual funds by performance and assigning the number one to be the 10 best performing funds and the number ten to the 10 worst performing funds.

Interval scale:this scale provides ranking and assurance that differences between scale values are equal. Measuring temperature is a prime example.

Ratio scale:these represent the strongest level of measurement. In addition to providing ranking and equal differences between scale values, ratio scales have a true zero point as the origin. Money is a good example.

d: Define and interpret a frequency distributions.

A frequency distributionis agrouping of raw data into categories (called classes) so that the number of observations in each of the nonoverlapping classes can be seen and tallied. The purpose of constructing a frequency distribution is to group raw data into a useable visual framework for analysis and presentation.

e: Define, calculate, and interpret a holding period return.

Holding period return (HPR)measures the total return for holding an investment over a certain period of time, and can be calculated using the following formula:

HPR = Pt - Pt - 1 + Dt / Pt - 1

Where: Pt = price per share at the end of time period t, and Dt = cash distributions received during time period t.

Example:A stock is currently worth $60. If you purchased the stock exactly one year ago for $50 and received a $2 dividend over the course of the year, what is your HPR?

(60 - 50 + 2) / 50 = 24%

f: Define and explain the use of intervals to summarize data.

An interval is the set of return values within which an observation falls. Each observation falls into only one interval, and the total number of intervals covers the entire population. It is important to consider the number of intervals to be used. If too few intervals are used, too much data may be summarized and we may lose important characteristics; if too many intervals are used, we may not summarize enough. Each interval has a lower limit and an upper limit. Intervals must be all-inclusive and non-overlapping.

After intervals have been defined, you must tally the observations and assign each observation to its respective interval.

Once the data set has been tallied, you should count the number of observations that were placed in each interval. The actual number of observations in a given interval is called the absolute frequency, or simply the frequency.

g: Calculate relative frequencies, given a frequency distribution.

Another useful way to present data is the relative frequency. Relative frequency is calculated by dividing the frequency of each return interval by the total number of observations. Simply, relative frequency is the percentage of total observations falling within each interval.

h: Describe the properties of data presented as a histogram or a frequency polygon.

A histogram is the graphical equivalent of a frequency distribution. It is a bar chart of continuous data that has been grouped into a frequency distribution. The advantage of a histogram is that we can quickly see where most of the observations lie. To construct a histogram, the class intervals are scaled on the horizontal axis and the absolute frequencies are scaled on the vertical axis.

A second graphical tool used for displaying data is the frequency polygon. To construct a frequency polygon, we plot the midpoint of each interval on the horizontal axis and the absolute frequency for that interval on the vertical axis. Each point is then connected with a straight line.

i: Define, calculate, and interpret measures of central tendency, including the population mean, sample mean, arithmetic mean, geometric mean, weighted mean, median, and mode.

A population meanis the entire group of objects that are being studied. To find the population's mean, sum up all the observed values in the population (sum X) and divide this sum by the number of observations (N) in the population.

A sample meanis sum of all the values in a sample of a population divided by the number of values in the sample. The sample mean is used to make inferences about the population mean.

Example:A stock you and your research partner are analyzing has 12 years of annualized return data. The returns are 12%, 25%, 34%, 15%, 19%, 44%, 54%, 33%, 22%, 28%, 17%, and 24%. Your research partner is exceedingly lazy and has decided to collect data based on only five years of returns. Given this data, calculate the population mean and calculate the sample mean. (Your partner's data set is shown above as bold).

Population mean = 12 + 25 + 34 + 15 + 19 + 44 + 54 + 33 + 22 + 28 + 17 + 24 / 12 = 27.25%

Sample mean = 25 + 34 + 19 + 54 + 17 / 5 = 29.8%

Arithmetic meanis the sum of the observation values divided by the number of observations. It is the most widely used measure of central tendency, and is the only measure where the sum of the deviations of each value from the mean is always zero.

Example:A data set contains the following numbers: 5, 9, 4, and 10. The mean of these numbers is: ( 5 + 9 + 4 + 10) / 4 = 7. The sum of the deviations from the mean is: (5 - 7) + (9 - 7) + (4 - 7) + (10 - 7) = -2 + 2 - 3 + 3 = 0.

Geometric meanis often used when calculating investment returns over multiple periods, or to find a compound growth rate.

Example:For the last three years the return for Acme Corporation common stock have been -9.34%, 23.45%, and 8.92%. Find the geometric mean.

Take the cube root of (-.0934 + 1)(.2345 + 1)(.0892 + 1) = The cube root of 1.21903. On your TI calculator, enter 1.21903 and hit the yx key, then enter .3333 = to get of 1.06825. Now you must subtract this number from one to get an answer of 6.825%.

Weighted meanis a special case of the mean that allows different weights on different observations.?

Example:A portfolio consists of 50% common stocks, 40% bonds, and 10% cash. If the return on common stocks is 12%, the return on bonds is 7%, and the return on cash is 3%, what is the return to the portfolio?

Weighted mean = [(0.50 *0.12) + (0.40 * 0.07) + (0.10 * 0.03)] = 0.091, or 9.1%

The medianis the mid-point of the data when the data is arranged from the largest to the smallest values. Half the observations are above the median and half are below the median. To determine the median, arrange the data from highest to the lowest and find the middle observation.

Example:The five-year annualized total returns for five investment managers are 30%, 15%, 25%, 21%, and 23%. Find the median return for the managers.

First, arrange the returns from hi to lo: 30, 25, 23, 21, 15.

The return observation half way down from the top is 23%.

The modeof a data set is the value of the observation that appears most frequently.

Example:In the following set of numbers, 30%, 28%, 25%, 23%, 28%, 15%, and 5%, 28% is the most frequently occurring value.

j: Distinguish between arithmetic mean and geometric means.

The value for the arithmetic mean is higher. The geometric mean will always be less than or equal to the arithmetic mean. In general, the difference between the two means increases with the variability between period-by-period observations. The only time when the two means will be equal is when there is no variability in the observations (e.g., all observations are 10%).

k: Define, calculate, and interpret (1) a portfolio return as a weighted mean, (2) a weighted average or mean, (3) a range and mean absolute deviation, and (4) a sample and a population variance and standard deviation.

Refer to LOS 1.B.i for a review of weighted mean and weighted average.

Rangeis the distance between the largest and the smallest value in the data set.

Example:The five-year annualized total returns for five investment managers are 30%, 12%, 25%, 20%, and 23%. What is the range of the data? Range = 30 - 12 = 18%.

Mean absolute deviation (MAD)is the average of the absolute values of the deviations of individual observations from the arithmetic mean. Remember that the sum of all of the deviations from the mean is equal to zero. To get around this zeroing out problem, the mean deviation uses the absolute values of each deviation.

Example:Continuing from above, what is the mean deviation of investment returns and how is it interpreted?

MAD = {I (30 - 22) I + I (12 - 22) I + I (25 - 22) I + I (20 - 22) I + I (23 - 22) I } / 5

MAD = [ 8 + 10 + 3 + 2 + 1] / 5 = 4.8%

Population varianceis the mean of the squared deviations from the mean. The population variance is computed using all members of a population.

Example:Assume the five-year annualized total returns for the five investment managers used in the earlier example represent all of the managers at a small investment firm. What is the population variance?

μ = {30 + 12 + 25 + 20 + 23} / 5 = 22%

ó = { (30 - 22)2 + (12 - 22)2 + (25 - 22)2 + (20 - 22)2 + (23 - 22)2 } / 5 = 35.60%2

Population standard deviationis the square root of the population variance.

Example:Continuing with our example, take the square root of 35.60 = 5.97%

Sample varianceapplies when we are dealing with a subset, or sample of the total population.

Example:Assume the five-year annualized total returns for the five investment managers used in the earlier example represent only a sample of the managers at a large investment firm. What is the sample variance?

sample mean = {30 + 12 + 25 + 20 + 23} / 5 = 22%

s2 = { (30 - 22)2 + (12 - 22)2 + (25 - 22)2 + (20 - 22)2 + (23 - 22)2 } / 5 - 1 = 44.5%2

Sample standard deviationcan be found by taking the positive square root of the sample variance.

Example:Continuing with our example, take the square root of 44.50 = 6.67%

l: Calculate the proportion of items falling within a specified number of standard deviaitons of the mean, using Chebyshev's inequality.

Chebyshev's inequality states that for any set of observations (sample or population, regardless of the shape of the distribution), the proportion of the observations within k standard deviations of the mean is at least 1 - 1/k2 for all k > 1. If we know the standard deviation, we can use Chebyshev's inequality to measure the minimum amount of dispersion, regardless of the shape of the distribution.

Chebyshev's inequality states that for any distribution, approximately:

36% of observations lie within 1.25 standard deviations of the mean

56% of observations lie within 1.50 standard deviations of the mean

75% of observations lie within 2 standard deviations of the mean

89% of observations lie within 3 standard deviations of the mean

94 of observations lie within 4 standard deviations of the mean

m: Define, calculate, and interpret the coefficient of variation.

The coefficient of variation expresses how much dispersion exists relative to the mean of a distribution and allows for direct comparison of dispersion across different data sets.

CV = [standard deviation of returns]/[Expected rate of return]

Example:

Investment A has an ER of 7% and a s of .05.

Investment B has an ER of 12% and a s of .07.

Which is riskier?

A’s CV is .05/.07 = .714

B’s CV is .07/.12 = .583

A has .714 units of risk for each unit of return while B has .583 units of risk for each unit of return. A is riskier, it has more risk per unit of return.

n: Define, calculate, and interpret the Sharpe measure of risk-adjusted performance.

The Sharpe measureseeks to measure excess return per unit of risk. The numerator of the Sharpe measure recognizes the existence of a risk-free return. Portfolios with large Sharpe ratios are preferred to portfolios with smaller ratios because it is assumed that rational investors prefer return and dislike risk. The Sharpe ratio is also called the reward-to-variability ratio.

Example:The mean monthly return on T-bills is 0.25%. The mean monthly return on the S&P 500 is 1.30% with a standard deviation of 7.30%. Calculate the Sharpe measure for the S&P 500 and interpret the results.

Sharpe measure = (1.30 - 0.25) / 7.30 = 0.144

The S&P 500 earned 0.144% of excess return per unit of risk, where risk is measured by standard deviation.

o: Describe the relative locations of the mean, median, and mode for a nonsymmetrical distribution.

For a symmetrical distribution, the mean, median, and mode are equal.

For a positively skewed distribution, the mode is less than the median, which is less than the mean. Recall that the mean is affected by outliers. In a positively skewed distribution, there are large, positive outliers which will tend to "pull" the mean upward.

For a negatively skewed distribution, the mean is less than the median, which is less than the mode. In this case, there are large, negative outliers which tend to "pull" the mean downward.

p: Define and interpret skewness and explain why a distribution might be positively or negatively skewed.

Skewnessrefers to a distribution that is not symmetrical.

A positively skeweddistribution is characterized by many outliers in its upper or right tail. Recall that an outlier is defined as an extraordinarily large outcome in absolute value. Positively skewed distributions have long right tails.

A negatively skeweddistribution is the opposite of a positively skewed distribution. A negatively skewed distribution has a disproportionately large amount of outliers on its left side. In other words, a negatively skewed distribution is said to have a long tail on its left side.

q: Define and interpret kurtosis and explain why a distribution might have positive excess kurtosis.

Kurtosisdeals with whether or not a distribution is more or less "peaked" than a normal distribution.

A distribution that is more peaked than normal is leptokurtic. A leptokurtic return distribution will have more returns clustered around the mean and more returns with large deviations from the mean (fatter tails).

A distribution that is less peaked, or flatter than normal is said to be platykurtic.

For all normal distributions, kurtosis is equal to three. Statisticians, however, sometimes report excess kurtosis, which is defined as kurtosis minus three. A normal distribution has excess kurtosis equal to zero, a leptokurtic distribution has excess kurtosis greater than zero, and platykurtic distribution will have excess kurtosis less than zero.

r: Explain why a semi-logarithmic scale is often used for return performance graphs.

Semi-logarithmic scalesuse an arithmetic scale on the horizontal axis, but use a logarithmic scale on the vertical axis. Hence, values on the vertical axis are spaced according to their logarithms. On a semi-logarithmic scale, equal movements on the vertical axis reflect equal percentage changes.

1.C: Probability Concepts

a: Define a random variable.

A random variable is a quantity whose outcomes are uncertain. A realized random variable is a number associated with the outcome of an experiment. When rolling a conventional six-sided die, the random variable might be the number that faces up with the die stops rolling.

b: Explain the two defining properties of probability.

The probability of any event "i" is between zero and one.

If a set of events: E1, E2, ....En, are mutually exclusive and exhaustive, then the sum of the probabilities of those events equals one.

Mutually exclusive means that the events do not share any outcomes. Knowing that you have an outcome in one event excludes the possibility of an outcome in another event.

Exhaustive means that a given list of events represent all possible outcomes.

c: Distinguish among empirical, a priori, and subjective probabilities.

We can assign probabilities to events three ways:

We calculate an empirical probability by analyzing past data.

We calculate an a priori probabilityby using formal reasoning and inspection.

A subjective probability is less formal and involves personal judgment.

d: Describe the investment consequences of probabilities that are inconsistent.

With respect to investment opportunities, when two assets are price based upon different probabilities being assigned to the same event, this is called inconsistent probabilities. It is best defined by a general example.

Example:Event E will increase the return of both stock A and B. The price of stock A incorporates a higher probability of E than does stock B. All other things equal, stock A is overpriced when compared to stock B. Therefore, an investor should lower holdings of stock A and increase holdings of stock B. An investor that is not too risk averse might engage in a pairs arbitrage trade, where he/she short sells A and uses the proceeds to buy stock B.

e: Distinguish between unconditional and conditional probabilities.

An unconditional probability is also called a marginal probability, and it is the most basic type of probability. It is the probability of an event where the occurrence of other events is not important. We might be concerned with the probability of an economic recession where we do not care about interest rates, inflation, etc. In such a case, we would be concerned with the unconditional probability of a recession.

A conditional probability is one where the knowledge of some other event is important. We might be concerned about the probability of a recession given that the monetary authority increases interest rates. This is a conditional probability. The key thing to look for is "the probability of A givenB." This is noted by a vertical bar symbol.

f: Define a joint probability.

An joint probability is the probability that both events occur at the same time, but neither is certain or a given. We write the probability of A and B as P(AB). Unless both A and B occur, it does not qualify as the event "A and B."

g: Calculate, using the multiplication rule, the joint probability of two events.

There is a relationship between the expressions P(AB) and P(A I B). It is called the multiplication rule for probabilities:

P(AB) = P(A I B) * P(B)

In words this is: "the probability of A and B is the probability of A given B times the unconditional probability of B."

We can manipulate this to give the following representation for a conditional probability: P (A I B) = P(AB) / P(B)

Example:We will assume the probabilities in the list below:

The probability of the monetary authority increasing interest rates "I" is 40%: P(I) = .4

The probability of a recession "R" given an increase in interest rates is 70%: P(R given I) = .7

The probability of "R" without an increase in interest rates is 10%: P(R given IC) = .1

Without additional information, we can assume that the events "increase in interest rates" and "no increase in interest rates" are the only possible events. They are mutually exclusive and exhaustive, and since there are only two events, they are called complements. The superscript "C" stands for complement.

P(IC) = 1 - P(I) = .60

What is the probability of "recession and an increase in interest rates?"

P(RI) = P(R given I) * P(I) = .7 * .4 = .28

h: Calculate,using the addition rule, the probability that at least one of two events will occur.

The general rule of additionstates that if two events A and B are not mutually exclusive then you must account for the joint probability of events. That is the possibility that the two events will occur at exactly the same time. Joint probability is shown by the overlap of the occurrence circles in the traditional Venn diagram shown below.

P (A or B) = P (A) + P(B) – P(A and B),where P(A and B) is the joint probability of A and B.

The joint probability [P(A and B)]is defined as the probability that measures the likelihood that 2 or more events will happen concurrently.

P(A and B) = P(A)*P(B) for independent events, or

P(A and B) = P(A)*P(B given that A occurs) for conditional events.

i: Distinguish between dependent and independent events.

Independent eventsare a list of events where knowledge of one has no influence on the other. That is easily expressed using conditional probabilities. A and B are independent if:

P (A I B) = P (A), and P(B I A) = P(B)

The best examples of independent events are found with the a priori probabilities of dice throws or coin flips. A die has no memory; therefore, the event of a "4" on a second throw of a die is independent of a "4" on the first throw.

j: Calculate a joint probability of any number of independent events.

The multiplication rule for independent events is:

P (A I B) * P(B) = P(A) * P(B) = P(AB)

P (B I A) * P(A) = P(B) * P(A) = P(AB)

Example:On the roll of two dice, the probability of getting two "4s" is:

P(4 on first die and 4 on second die) = P(4 on first die) * P(4 on second die)

P(4 on first die and 4 on second die) = (1/6) *(1/6) = 1/36 = .0278.

k: Calculate, using the total probability rule, an unconditional probability.

The total probability rule is used to demonstrate how joint probabilities tie in with unconditional probabilities. If we continue with our example from LOS 1.C.g about interest rates and recession, and assume that the events "I" and "IC" are mutually exclusive and exhaustive, then a recession can only occur with either of these two events. In that case, the sum of these two joint probabilities is the unconditional probability of a recession:

P(R) = P(R given I) * P(I) + P(R given IC) * P(IC)

P(R) = P(RI) + P(RIC)

P(R) = .28 + .06 = .34

l: Define and calculate expected value.

The expected value is the probability-weighted average of the possible outcomes of the random variable.?

E(X) = ∑xi*P(xi) = x1*P(x1) + x2*P(x2) + … + xn*P(xn)