Lectures Skeleton LTU

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Topic 1:

Time Value of Money

(Part I)

Associate Professor Ishaq Bhatti

La Trobe Business School

E-Mail: [email protected]

Slides have been drafted by the La Trobe University, School of Business based on DeFusco et al (2007)

Statistics for Business and Finance

Chapter 1 Time Value of Money

1.2

1. INTRODUCTION

What is the time value of money?

Is $1 today equal to $1 tomorrow?

Would you agree to pay $500 to a friend and receive $500 back 1 year from now?

Would you agree to pay $500 to a friend and receive $1000 back 1 year from now?

Time Value of Money

1.3

2. INTEREST RATES

Would you agree to receive $9,500 now and pay $10,000 now?

What if you receive $9,500 now and pay $10,000 one year from now?

Time Value of Money

1.4

2. INTEREST RATES

An interest rate is a rate of return that reflects the relationship between

differently dated cash flows.

How much is your return in the previous example?

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Time Value of Money

1.5

2. INTEREST RATES

Interest rate may be referred to as:

Required rate of return: The minimum rate of return an investor must receive in

order to accept the investment.

Discount rate: The rate we use to discount future cash flows.

Opportunity cost: Value that investors forgo by choosing a particular course of

action.

Time Value of Money

1.6

3. FUTURE VALUE OF A SINGLE CASH FLOW

You invest $9,500 now and receive $10,000 one year from now.

This $10,000 includes the initial $9,500 plus $500 interest on that.

Future Value is equal to the Present Value of the investment plus interest on the investment.

rPVFV

PVrPVFV

1

Time Value of Money

1.7

3. FUTURE VALUE OF A SINGLE CASH FLOW

Now what if you invest that money for one more year?

Hence, formula for future value of a single cash flow after N periods is:

21

11

rPVFV

rPVrrPVFV

NrPVFV 1

Time Value of Money

1.8

Example 1: FV of a Lump Sum

An institution promises to pay you a lump sum, six years from now at an 8% annual interest

rate, if you invest $2,500,000 today.

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Time Value of Money

1.9

3.1 The Frequency of Compounding

Some investments pay interest more than once a year.

Financial institutions often quote an annual interest rate.

If your bank states the annual interest rate is 8%, compounded monthly, how much is the

monthly interest rate?

Time Value of Money

1.10


With more than one compounding period per year, the future value formula can be

expressed as:

mN

s

m

rPVFV

1

Time Value of Money

1.11

Example 2: FV of a Lump Sum with Monthly Compounding

An investment has a six-year maturity and annual quoted interest rate is 8% compounded

monthly. FV if you invest $2,500,000 is:

Time Value of Money

1.12

3.2 Continuous Compounding

If the number of compounding periods per year becomes infinite, then interest is said to

compound continuously.

Formula for FV of a sum in N years with continuous compounding is:

7182818.2in which

e

PVeFVNrs

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Time Value of Money

1.13

Example 3: FV of a Lump Sum with Continuous

Compounding

An investment has a six-year maturity and annual quoted interest rate is 8% compounded

continuously. FV if you invest $2,500,000 is:

Time Value of Money

1.14

3.3 Stated and Effective Rates

In all of the examples 1-3 stated interest rate and maturity were 8% and 6 years. But they all

have different future values:

PV = $2,500,000

FV Compounded annually = $3,967,186

FV Compounded Monthly = $4,033,755

FV Compounded Continuously = $4,040,186

Time Value of Money

1.15

3.3 Stated and Effective Rates

Examples 1-3 illustrate that when interest rate is compounded, monthly or continuously,

effective rate is more than the stated rate.

The effective annual rate is calculated as:

Or for continuous compounding:

1 1 mateInterest RPeriodicEAR

1 sreEAR

Time Value of Money

1.16

Example 4: Effective Annual Rate

An investment has a six-year maturity and annual quoted interest rate is 8%. What is the EAR if interest rate is compounded monthly? What if it is compounded continuously:

For continuous compounding:

8%r

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Time Value of Money

1.17

4. FUTURE VALUE OF A SERIES OF

CASH FLOWS

Common terms used in this topic:

Annuity: a finite set of sequential cash flows

Ordinary annuity: first cash flow occurs one year from now

Annuity due: first cash flow occurs immediately (at t=0)

Perpetuity: an infinite set of cash flows beginning one year from now.

Time Value of Money

1.18


CASH FLOWS

Consider an ordinary annuity paying 5% annually. Suppose we have 5 annual deposits

of $100 starting one year from now.

We are interested in FV of this ordinary annuity.

Now: t=0 2 1 3 4 5

$100 $100 $100 FV= ?

$100 $100

Time Value of Money

1.19


CASH FLOWS

Total FV is equal to the sum of FV of every single payment:

FV of 1st payment :

FV of 2nd payment :

FV of 3rd payment :

FV of 4th payment :

FV of 5th payment :

105.01100$ FV 205.01100$ FV 305.01100$ FV 405.01100$ FV 505.01100$ FV

Time Value of Money

1.20


CASH FLOWS

Total FV is equal to the sum of FV of every single payment:

But if all the payments are equal, we can arrive at a general annuity formula:

r

rAFV

N11

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Time Value of Money

1.21


CASH FLOWS

Hence, In this example we have:

Time Value of Money

1.22

5. PRESENT VALUE OF A SINGLE CASH FLOW

Present value is the discounted value of a future cash flow.

PV of a lump sum can be found through the following equation:

N

NrFV

rFVPV

1

1

1

Time Value of Money

1.23

Example 5: PV of a Lump Sum

An institution promises to pay you $100,000 in six years with an 8% annual interest rate. How

much should they invest today to have this

money at the end of 6th year?

Time Value of Money

1.24


With more than one compounding period per year, the present value formula can be

expressed as:

mN

s

m

rFVPV

1

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Time Value of Money

1.25

Example 6: PV of a Lump Sum with Monthly Compounding

A company must make a $5 million payment 10 years from now. How much should they

invest today if the annual interest rate is 6%,

compounded monthly?

Time Value of Money

1.26

6. PRESENT VALUE OF A SERIES OF

CASH FLOWS

Consider an ordinary annuity paying 5% annually. Suppose we have 5 annual deposits

of $100 starting one year from now.

We are interested in PV of this ordinary annuity.

Now: t=0 2 1 3 4 5

$100 $100 $100

PV= ? $100 $100

Time Value of Money

1.27


CASH FLOWS

Total PV is equal to the sum of PV of every single payment:

PV of 1st payment :

PV of 2nd payment :

PV of 3rd payment :

PV of 4th payment :

PV of 5th payment :

105.01100$ PV 205.01100$ PV 305.01100$ PV 405.01100$ PV 505.01100$ PV

Time Value of Money

1.28


CASH FLOWS

Total PV is equal to the sum of PV of every single payment:

But if all the payments are equal, we can arrive at a general annuity formula:

r

rAPV

N11

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Time Value of Money

1.29


CASH FLOWS

Hence, In this example we have:

Time Value of Money

1.30

Present Value of an Infinite Series of Equal

Cash Flows

Present value of an infinite series of equal cash flows can be calculated as:

r

APV

Time Value of Money

1.31

Example 6: An Infinite Series of Equal Cash Flows

A type of British government bonds pays $100 per year in perpetuity. What would it be worth

today if interest rate were 5%?

Time Value of Money

1.32

Thank You!

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Topic 2:

Time Value of Money (Part II) Discounted Cash Flow Applications






Chapter 2 Discounted Cash Flow Applications

1. INTRODUCTION

Key Time Value of Money concepts: NPV and IRR

Making investment decision

Portfolio return measurement

Calculation of money market yields

Discounted Cash Flow Applications

2.1 Net Present Value (NPV)

Invest or Not?

You first need to know the present value of the cash flows

Second, you need to know how much the project cost you

If the project costs less than the PV of the cash flows you will invest; otherwise you will not


2.1 Net Present Value (NPV)

NPV is a method for choosing among alternative investments. The Net

Present Value is the present value of

cash inflows, minus the present value of

cash outflows.

0

1 0(1 ) (1 )

N Nt t

t tt t

CF CFNPV CF

r r

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Example 1: NPV

A project generates $100m next year, $150m

in year 2 and $120m in year 3. If the project

costs $310m and you can finance the

investment with an opportunity cost of 10%,

will invest in the project?


Example 1: NPV

What if the opportunity cost is 5%?

Would you invest in this project?


2.2 Internal Rate of Return (IRR)

Internal Rate of Return is the rate of return that makes the NPV equal to zero

IRR can be calculated using financial software or financial calculators, or trial and error method!

0)1(

...)1()1( 2

2

1

10

N

N

IRR

CF

IRR

CF

IRR

CFCFNPV


2.2 Internal Rate of Return (IRR)

Investment decision is to invest if IRR is more than the interest rate (opportunity cost) and not invest if IRR is

lower than the interest rate.

If discount rate is more than IRR, NPV will be negative and if interest rate is lower that IRR, NPV will be positive

Therefore, NPV and IRR will always lead to the same decision

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Example 2: IRR

A project generates $100m next year, $150m

in year 2 and $120m in year 3. If the project

costs $310m. What is the internal rate of

return? Is it 8.34%, 9.12% or 10.27%?


Example 2: IRR

Will you invest if the opportunity cost is 10%?

What if the interest rate is 5%?


2.3 Problems with IRR rule

IRR and NPV always lead to the same invest/not invest decision, but sometimes they rank the projects differently

If the scale of the projects differs

If projects have different timing of future cash flows

If there is a conflict between IRR and NPV, we should follow NPV as it reflects the real change in investors wealth


2.3 Problems with IRR rule

IRR and NPV always lead to the same invest/not invest decision, but sometimes they rank the projects differently

If the scale of the projects differs

If projects have different timing of future cash flows

If there is a conflict between IRR and NPV, we should follow NPV as it reflects the real change in investors wealth

If the sign of the cash flows changes more than once, we may get more than one IRR.

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3. Portfolio Return Measurement

Holding Period Return (HPR), the fundamental concept

Return that an investor earns over a specified holding period

0

101

P

DPPHPR

0

1

1

is the initial investment

is the price at the end of the period

is the cash paid by the investment

P

P

D


3.1 Money Weighted Rate of Return

IRR is called Money Weighted Rate of Return because it depends on timing and the dollar value of cash flows

IRR is not a good measure for investment managers

Usually, client decides when and how much to invest or withdraw

An evaluation tool should only judge the investment manager only for his own decisions, not for the clients


3.2 Time Weighted Rate of Return

The preferred measurement tool in investment management industry

Measures the compound rate of growth of each $1 of initial investment over the period

Does not depend on the dollar value of the investment

Not affected by withdrawals or additions to the portfolio


3.2 Time Weighted Rate of Return

Calculation of Time Weighted Rate of Return

Price the portfolio before any additions or withdrawals, breaking the period into subperiods

Calculate the HPR for each subperiod

Take the geometric mean of the calculated Holding Period Returns (HPR)

n1 2 nTWRR= (1+r ) (1+r ) ... (1+r ) 1

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Example 3: Money Weighted Rate of Return

At t = 0, an investor buys one share at $200. At time t = 1, he purchases an additional

share at $225,

At the end of Year 2, t = 2, he sells both shares for $235 each.

During both years, the share pays a per-share dividend of $5.

Calculate the Money Weighted Rate of Return


Example 3: Money Weighted Rate of Return

t = 0 t = 1 t = 2

$200 $225 $10 2 x $235 $5


Example 4: Time Weighted Rate of Return

t = 0 t = 1 t = 2

$200 $225 $10 2 x $235 $5



First period:

t = 0 t = 1 t = 2

$200 $225 $10 2 x $235 $5

P0=$200 P1=$225

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t = 0 t = 1 t = 2

$200 $225 $2 x 5 2 x $235 $5

P0=$225 P1=$235



n1 2 nTWRR= (1+r ) (1+r ) ... (1+r ) 1

t = 0 t = 1 t = 2

$200 $225 $2 x 5 2 x $235 $5


4. Money Market Yields

Consider two 1-year bonds of a company:

One with a $100 face value and $10 coupon at maturity;

The other with a $110 face value but no coupon

Are they selling at the same price today?

What is the interest of the second bond?

Many short-term debts (one-year maturity or less) pay no explicit coupon but they are sold at a discount.

Pure discount instruments


4. Money Market Yields

Pure discount instruments such as T-bills are quoted on a Bank Discount basis, rather than

on a price basis:

rBD: annualized yield on a Bank Discount basis

D: dollar discount = face value purchase price

F: face value

T: actual number of days remaining to maturity

360: bank convention of the number of days in a year

tF

DrBD

360

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Thank You!

Topic 3:

Statistical Concepts and Market

Returns





Statistics for Business and Finance Chapter 3

Statistical Concepts and Market Returns

2.59

INTRODUCTION

Statistical methods provide a powerful set of tools for analyzing data.

Descriptive statistics includes basics of describing and analyzing data.

We explore four properties of return distributions: Where the returns are centered (central tendency)

How far returns are dispersed from their center (dispersion)

Whether the distribution of returns is symmetrically shaped or not (skewness)

Whether extreme outcomes are likely (kurtosis)


2.60

Descriptive and Inferential Statistics

Descriptive statistics is to summarize a small set of data (sample) effectively to describe the important aspects of a larger dataset (population)

Statistical inference is to make forecasts. estimations or judgments about a larger dataset (population) from a smaller group (sample) actually observed.

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2.61

Frequency Distributions

The simplest way of summarizing data is the frequency distribution.

A frequency distribution is a tabular display of data summarized into a

relatively small number of intervals.


2.62

Example: Frequency Distributions

Banna Insurance Pty Ltd - 4 sales incentive

programs A, B, C and D. 40 Salespeople

asked for their opinion of the preferred

program.

Fill the following table:

B A D C A C D B D B

D D B A D B D A D C

D B C D A D B D B C

B A D B A B A C D B


2.63

Example 1: Frequency Distributions

Program Frequency Relative Frequency Cumulative

Frequency Cumulative Relative

Frequency

A

B

C

D


2.64

Histogram & Bar Chart

Histogram consist of adjacent rectangles whose bases are marked off by class width and their heights are

proportional to the frequencies they possessed.

Bar chart is special form of histograms where bars are not adjacent and the data have been grouped into a

frequency distribution.

The following two slides display the bar chart of absolute and relative frequency distributions of example

1. Similarly you can compute histogram of ASX200

returns; Text page

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2.65

Bar Chart of Banna Insurance Example


2.66

Bar Chart of Banna Insurance Example


2.67

Measures of Central Tendency

Arithmetic Mean:

The (arithmetic) mean is the sum of the observations divided by the number

of observations.

The population mean is given by

The sample mean looks at the arithmetic average of the sample of data:

The mean return of ASX200 is 0.70%

NXN

i

i

1

nXXn

i

i

1


2.68

Example: Arithmetic Mean

Find the mean of 46, 54, 42, 46, 32:.

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2.69

Median

Median is the value of the middle item of a set of items, sorted by ascending or descending

order.

If n is an odd number it occupies the (n+1)/2 position.

If n is an even number it is the average of items in the n/2 and (n+2)/2 positions.

Unlike the mean, the median is not affected by a few large observations.


2.70

Example: Median

Find the median of 46, 54, 42, 46, 32:.


2.71

Mode

The mode is the most frequently occurring value in a distribution.

A distribution can have more than one mode or even no mode.

Stock returns or other data from continuous distribution may not have a modal outcome but we often find the modal interval (intervals) Which internal in our Banna Insurance example is the

modal interval? (Hint: see the Histogram in slide 9)


2.72

Example: Mode

Find the mode of 46, 54, 42, 46, 32:.

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2.73

Weighted Mean

The weighted mean allows us to place greater importance on different observations.

For example, we may choose to give larger companies greater weight in our computation of an

index. In this case, we would weight each

observation based on its relative size.

The arithmetic mean is a special case where each observation is given the same weight.

i

i

n

i

iiw wXwX 1 where,1


2.74

Geometric Mean

The geometric mean is most frequently used to average rates of change over time

or to compute the growth rate of a

variable.

which can also be calculated by

n, . . . , , i XXXXG in

n 21for 0 with ,]...[/1

21

n

i

iXn

G1

ln1

ln


2.75

Geometric Mean Return

Geometric mean requires all data are positive It cannot be applied to observations with negative data like return.

The geometric mean return allows us to compute the average return when there is

compounding.

1)1(

)1)...(1)(1)(1(1

1

1

1

321

TT

t tG

TTG

RR

RRRRR


2.76

Quartiles and Percentiles

If your lecturer tells you that your exam mark is in top 10%, what does it mean?

Median divides the data in half. The dataset can be also divided into:

Quartiles; Quintiles; Deciles; Percentiles

To find the value of yth percentile with n observations:

first organize data in ascending order

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2.77

Quartiles and Percentiles

then the location of the yth percentile is at

Ly = (n + 1) * y%

however, some software package like Excel uses location Ly = [(n - 1)*y%] +1

if Ly is an integer, the value of yth percentile, Py, is equal to the observation at Ly

if Ly is not an integer, it can be written as Ly = k.d, where . is the decimal point, and

Py =Vk + .d (Vk+1 - Vk),

where Vk and Vk+1 are the values of kth and (k + 1)th

observations.


2.78

Example: Percentiles

Find the values of 25th and 40th percentiles of the 46, 54, 42, 46, 32:

Note, median = P50, = Q2 = D5, P75 = Q3, , here Q and D denote Quartile

and Decile.


2.79

Measures of Dispersion

One of simplest measures of dispersion is the range, which is the difference between

the maximum and minimum values in a

dataset:

Range = Maximum value Minimum value.


2.80

Mean Absolute Deviation

Mean Absolute Deviation (MAD) measures the average distance that each observation

is from the mean:

nXXMADn

i

i

1

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2.81

Population Variance and Standard

Deviation

Variance measures the average squared deviation from the mean:

Because the variance is not in the same units as the mean, sometimes we prefer the standard deviation, the

square root of variance, which is in the same units as

the mean

NXN

i

i

1

22 )(

NXN

i

i

1

2)(


2.82

Sample Variance and Standard Deviation

Sample Variance and standard deviation slightly differ from population variance and

standard deviation:

)1()(

1

22

nXXsn

i

i)1()(

1

2

nXXsn

i

i


2.83

Example: MAD and Standard Deviation

Find the MAD for 46, 54, 42, 46, 32:

iX iX XX iX X


2.84

Example: MAD and Standard Deviation

Find Std Dev. for 46, 54, 42, 46, 32:

iX iX XX 2

iX X

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2.85

Semivariance

Often times, observations above the mean are good the variance is not a good measure of risk. Semivariance looks at

the average squared deviations below the

mean:

where n* is the number of observations

greater than the average of observations.

XX

i

in

XX

allfor *

2

)1(

)(


2.86

Coefficient of Variation

The coefficient of variation is the ratio of the standard deviation to their mean value.

measure of relative dispersion

can compare the dispersion of data with different scales

What is the coefficient of variation of the dataset in previous example?

XsCV


2.87

Sharpe Ratio

The Sharpe ratio is the ratio of mean excess return to riskan application of mean and standard deviation analysis.

Risk averse investors who make decisions only in terms of mean and standard deviation prefer

portfolios with larger Sharpe ratios:

Assuming monthly risk-free interest rate is 0.3%, the Sharpe ratio of ASX200 is 0.121.

p

Fp

hs

RRS


2.88

Skewness

Skewness measures the symmetry of a distribution.

A symmetric distribution has a skewness of 0.

Positive skewness indicates that the mean is greater than the median (more than half the deviations from

the mean are negative)

Negative skewness indicates that the mean is less than the median (less than half the deviations from

the mean are negative)

3

1

3)(

)2)(1( s

XX

nn

nS

n

i i

K

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2.89

Graphic Illustration of Skewness


2.90

Sample Excess Kurtosis

Kurtosis measures how peaked the distribution is relative to the normal distribution.

Using the sample excess kurtosis formula

KE 0 is called Mesokurtic, which means the distribution is normally distributed

KE > 0 is called Leptokurtic, which means the distribution is more peaked.

KE < 0 is called Platykurtic means the is less peaked.

)3)(2(

)3(3)(

)3)(2)(1(

)1( 2

4

1

4

nn

n

s

XX

nnn

nnK

n

i i

E


2.91

Leptokurtic: Fat Tailed


2.92

Thank You!

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Topic 4:

Probability Concepts






Chapter 4 Probability Concepts

4.94

Uncertainty and Probability

A random variable is a quantity whose outcomes are uncertain.

An event is a specified set of outcomes.

Probability: the likelihood or chance that something is the case or will happen

The probability of any event, E, is a number between .

The sum of the probabilities of any set of mutually exclusive & exhaustive events equals.


4.95


A random variable is a quantity whose outcomes are uncertain.

An event is a specified set of outcomes.

Probability: the likelihood or chance that something is the case or will happen The probability of any event, E, is a number between 0 and 1: 0 P(E)

1 The sum of the probabilities of any set of mutually exclusive &

exhaustive events equals 1.

Probability of an event A is equal to:


4.96

Example

An experiment is conducted in which a coin is tossed three times - the uppermost face recorded on each toss. Draw the tree diagram.

First throw Second throw Third throw

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4.97

Example

If A is the event of throwing 2 heads and 1 tail in any order, then:

A=

and P(A) =


4.98


Mutually exclusive events are those only one of which can occur at a time.

Exhaustive events are the events that cover all possible outcomes.

How to estimate probability? An empirical probability is estimated by relative frequency of occurrence

based on historical data

A subjective probability is one drawing on personal or subjective judgment.

A priori probability is one based on logical analysis rather than on observation or personal judgment.

A priori or an empirical probability is also called objective probability.


4.99

Example

A die is thrown. Determine the probability of obtaining a number 2 or a 5.

AB

S


4.100

Example

A die is thrown. Determine the probability of obtaining a multiple of 2 or a multiple of 3.

A

B

S

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4.101

Example

A die is tossed twice. Determine the probability of obtaining a 3 on the first toss and number > 5 on the second toss.

A

B

S


4.102

Example

A die is tossed twice. Determine the probability of obtaining a 3 on the first toss and a total of 5 on both tosses.

A

B

S


4.103

Multiplication and Addition Rules for

Independent Events

When two events are independent, the joint probability is the product of two probabilities

Consequently, addition rule becomes

Example: What is the probability of tossing two heads in a row?


4.104

Independent Events

Two events are independent if and only if

)B(P)A|B(Ply equivalentor )A(P)B|A(P

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4.105

Multiplication Rule for Probability

The joint probability can be found using the multiplication rule:

On the other hand, if joint probability P(AB) and unconditional probability P(B) are known,

the conditional probability is

)B(P)B|A(P)AB(P

0P(B) ,)B(P

)AB(P)B|A(P


4.106

Unconditional and Conditional

Probabilities

Unconditional or marginal probability answers question, What is the probability of event A.

Conditional probability answers the question, What is the probability of event A, given that event B occurs.

Joint probability answers the question, What is the probability of both events A and B

happening.


4.107

Example

A die is tossed twice. Determine the probability of obtaining a total of 5 on both tosses if a 3 is obtained on the first toss.

A BS

3,1 3,3 3,4 3,5 3,6

3,2

2,3 1,4 4,1


4.108

Example:

A sample of Business Degree evening students was surveyed in order to investigate the relationship between age and marital status. The results of

the survey are tabled below:

Answer the following questions

Marital Status

S M

Age

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4.109

Example:

What is the sample size?

Are the events being single and being < 30 independent?

( 30)P Age

( )P Being Single

( AND < 30)P Being Single

( GIVEN < 30)P Being Single


4.110

The Total Probability Rule

)()|()()|(

)()()( .1

CC

C

SPSAPSPSAP

ASPASPAP

)()|(...)()|()()|(

)(...)()()( .2

2211

21

nn

n

SPSAPSPSAPSPSAP

ASPASPASPAP

eventsor scenarios exhaustive

and exclusivemutually are ,...,S where 21 nSS

where S is an even and SC is the even not-S or the

complement of S


4.111

Expected Value

The expected value of a random variable is the probability weighted average of the

possible outcomes of the random variable.

n

i

ii

nn

XXP

XXPXXPXXPXE

1

2211

)(

)(...)()()(


4.112

Variance

The variance of a random variable is the expected value of squared deviations from

the random variables expected value:

Note, a better notation is

2 2 2 2( ) {[ ( )] } ( ) ( )X E X E X E X E X

n

i

ii XEXXPX1

22 )]()[()(

2

X

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4.113

Variance

(i) Var (c) = 0

(ii) Var (cX) = c2 Var (X)

(iii) Var (c + X) = Var (X)

(iv) Var (X + Y) =Var (X) + Var (Y), if X and Y are independent

=Var (X) + Var (Y) + 2 COV (X,Y) if X and Y

are not independent


4.114

Standard Deviation

Standard deviation is the positive square root of variance.

2


4.115

Example

A random variable, X, has the following probability distribution:

xi

2P( xi)X=xi P(xi) or

P(X=xi)

xiP(xi)

0 0.1

1 0.6

2 0.3

Total 1.0


4.116

Example

Find the following:

E(X)

E(X2)

E(Y), if Y = aX cX2, where a = 1, c = 2

E(W), if W = (d c) X + a, where d = 8

Var(X)

Var(W)

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4.117

Conditional Expected Value

A conditional expected value is the expected value of a random variable X

given an event or scenario S, is denoted

E(X|S):

Total probability rule for expected value

)()|(...)()|()()|()|( 2211 nn SPSXESPSXESPSXESXE

nn XSXPXSXPXSXPSXE )|(...)|()|()|( 2211


4.118

Conditional Variance

Since variance is the expected value of random variable

we can define conditional variance

accordingly

SSXEXESXVar |)]|([)|( 2

2)]([ XEXE


4.119

Portfolio Expected Return and Variance

Investment diversification and portfolio

Portfolio is the profile of the investment

If there are n assets and you invest wi (i =1, 2, .., n) portion of your wealth in asset i, then the investment

portfolio is (w1, w2 , wn). The portfolio return is

where Ri is the return of asset i.

Modern portfolio theory often uses expected return as the measure of reward and the

variance of returns as a measure of risk.

nnp RwRwRwR ...2211


4.120

Properties of Expected Value

The expected value of a constant times a random variable equals the constant times

the expected value of the random variable

Expected value of the sum of random variables is equal to the sum of expected

values of the random variables:

)(...)()(

)...(

2211

2211

nn

nn

REwREwREw

RwRwRwE

)()( iiii REwRwE

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4.121

Calculation of Portfolio Expected Return

Given a portfolio with n securities, the expected return on the portfolio is a

weighted average of the expected

returns on the component securities.

)(...)()()( 2211 nnP REwREwREwRE


4.122

Covariance and Correlation

)])([(),( jjiiji ERRERRERRCov

)()(),( jijiij RRRRCov

Covariance of two random variables is defined as

Correlation coefficient of two random variables is defined as


4.123

Interpretation of Return Covariance

If the covariance is 0, the returns on the assets are unrelated.

If the covariance is negative (positive), when the returns on one asset is above its expected

value, the returns of the other asset tend to be

below (above) its expected value; i.e the two

returns tends to move in the same (opposite)

direction.

The covariance of a random with itself is its own variance.


4.124

Interpretation of Correlation Coefficient

Correlation is a scaled covariance that falls between -1 and +1.

A correlation of +1 means the variables are perfectly positively correlated.

A correlation of -1 means the variables are perfectly negatively correlated.

A correlation of 0 means the variables are uncorrelated.

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4.125

Portfolio Variance

Unlike portfolio expected return, portfolio variance is not a weighted average of the

variances of the securities in the portfolio.

To compute portfolio variance, we need to incorporate the interaction between each

pair of variables (correlation or

covariance).


4.126

Portfolio Variance

Portfolio variance for a two-security portfolio.

Portfolio variance for an n-security portfolio.

211221

2

2

2

2

2

1

2

1

2121

2

2

2

2

2

1

2

1

2

2

),(2)(

wwww

RRCovwwwwRP

),()(1 1

2

n

i

n

j

jijiP RRCovwwR


4.127

Thank You!

Topic 5:

Common Probability Distributions






Chapter 5

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5.129

Random Variables and Distributions

A probability distribution specifies the probabilities of the possible outcomes of a random variable.

There are two types of random variables:

A discrete random variable can take on at most a countable number of possible values;

A continuous random variable takes infinitely many values, on an interval, say between [0, 1] or (-, +)


5.130

Random Variables and Distributions

For discrete random variable, the probability function specifies the probability that the random

variable takes on a specific value.

For continuous variable,

The probability density function p(x) specifies the probability density the random variable takes on the

value x or the approximate probability the random

variable takes on values around x of a unit length.

The cumulative distribution function P(x) gives the probability that the random variable is less than or equal

to x.


5.131

Bernoulli Random Variable

Sometimes a random variable can only take on two values, success or failure. This is referred to

as a Bernoulli random variable.

A Bernoulli trial is an experiment that produces only two outcomes.

Y = 1 for success and Y = 0 for failure.

p1)0Y(P)0(pp)1Y(P)1(p


5.132

Binomial Distribution

A binomial random variable X is defined as a number Bernoulli trials.

The probability of x successes out of n trials is

The mean and variance of B(n,p) are:

= np

2 = np(1-p).

n21 YYYX

xnxxnx ppxxn

npp

x

nxXPxp

)1(

!)!(

!)1()()(

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5.133

Binomial Distribution

General notations (pages 166-168):

n factorial: n! n(n-1)(n-2)1 and 0! 1

Combination (x choices out of n options)

Binomial distribution assumes

The probability, p, of success is constant for all trials

The trials are independent

!)!(

!

xxn

n

x

nCrn


5.134

Example: Binomial Distribution

Flipping a fair coin: Probability of head = 50% and probability of tail = 50%

If you flip three coins in a row, what is the probability you have two heads and one tail?

So, can we answer the question?


5.135

Example: Binomial Distribution

Three customers enter a clothing store. The probability that a customer will make a purchase p(s) is 0.30. investigate the probability distribution.


5.136

Example: A Binomial Model of Stock

Price Movements

If the probability of stock price moving up is 60% and down is 40%, what is the probability that the stock price goes up in exactly two years?

Find the probability of an upward movement in the first two years followed by a fall in the price in the third year.

What is the probability that the price goes down at least twice?

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5.137

Example: A Binomial Model of Stock

Price Movements


5.138

Continuous Uniform Distribution

Probability density function (pdf) and cumulative distribution function (cdf) of uniform distribution on [a, b] are:

The mean and variance of a continuous distribution:

Mean: E(X)= = (a+b)/2

Variance: Var (x) = 2 = (a+b)2/12

otherwise 0

for 1

)(bxa

abxf

1

for

for 0

)(

bfor x

bxaab

ax

ax

xF


5.139

Normal Distribution

Random variable X follows a normal distribution with mean and variance 2 (X ~ N(, 2)) if it has a probability density function as:

There is not a closed-form cdf for a normal distribution and we have to use a table of cumulative probabilities for a normal distribution

A normal distribution can be determined using its mean and variance.

x

xxf for

2

)(exp

2

1)(

2

2


5.140

Normal Distribution

A normal distribution has a skewness of 0 (it is symmetric)

its mean, median and mode (slightly abuse the term) are equal

It has a kurtosis of 3 (or excess kurtosis of 0).

A linear combination of two or more normal random variables is also normally distributed

This property is vary useful to determine the distribution of portfolio return, given each assets return.

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5.141

Graphic Illustration of Two Normal

Distributions


5.142

Units of Standard Deviation


5.143

Units of Standard Deviation

Approximately 50 percent of all observations fall in the interval (2/3).

Approximately 68 percent of all observations fall in the interval .

Approximately 95 percent of all observations fall in the interval 2.

Approximately 99 percent of all observations fall in the interval 3.


5.144

Confidence Intervals for Values of a Normal

Random Variable X

We expect

90 percent of the values of X to lie within the interval



These intervals are called 90%, 95% and 99% confidence intervals for X.

s96.1X

s65.1X

s58.2X

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5.145

Standard Normal Distribution

Standard normal distribution has a mean of zero and a standard deviation of 1.

If X is a normal random variable that X ~ N(, 2), then Z follows the standard normal distribution (i.e., Z ~ N(0, 1)), if:

XZ


5.146

Example: Normal Distribution

Find the following probabilities: (Z table is available in the next slide)

1. P (0 z 1.4)=

2. P (0 z 1.46)=

3. P (-1.5 z 1.5)=


5.147


5.148

Example: Normal Distribution

A portfolio has an estimated mean return of 12% and standard deviation of return of 22%.

What is the probability that portfolio return will exceed 20%?

What is the probability of that portfolio return will be between 5.5% and 20%?

What is the returns 90% confidence interval?

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5.149

Application: Safety-First Rule

Roy demonstrates that if portfolio return, RP, is normal then minimizing the probability of RP falling below, RL,

require maximizes

PLP RRE /])([SFRatio


5.150

Application: Safety-First Rule

You are managing an $800,000 portfolio for an investor whose objective is long-term growth. But she may want to liquidate $30,000

at the end of a year. If that need arises, she hopes the liquidation of

$30,000 would not invade the initial capital of $800,000. If return on

the portfolio is 8% with 3% std deviation:

To protect the initial investment, portfolio managers rank the investents based on the SFRatios


5.151

Application: Safety-First Rule (Cont.)

There are three investment alternatives : A B C

Expected annual return (%) 25 11 14

Standard deviation (%) 27 8 20

What is the shortfall level (RL)?

According to safety-first criterion, which of the three allocations is the best?

What is the probability that the return on the safety-first optimal portfolio will be less than the shortfall level?


5.152

Lognormal Distribution

The lognormal distribution is widely used for modeling asset prices.

A random variable Y follows a lognormal distribution if and only if X = lnY is normally distributed.

Note,

If X has mean and variance 2, then Y have mean exp( + 0.5 2) and variance exp(2 + 2)[exp(2) 1].

XY e

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5.153

Two Lognormal Distributions


5.154

Thank You!

Topic 6:

Sampling and Estimation






Chapter 6 Sampling and Estimation

6.156

Sampling

In statistics we are often interested in obtaining information about the value of some parameters of a population.

To obtain this information we usually take a small subset of the population and try to draw some conclusions from this sample.

A sampling plan is the set of rules used to select a sample.

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6.157

Simple Random Sampling

A simple random sample is a subset of a larger population created in such a way that each element

of the population has an equal probability of being

selected.


6.158

Stratified Random Sampling

Stratified random sampling occurs when the population is divided into subpopulations

(strata) and a simple random sample is drawn

from each strata.

It guarantees that population subdivisions of interests are represented in the sample.

It generates more accurate estimates (smaller variance) than simple random sampling


6.159

Types of Sample Data

Cross-sectional data represent observations over individual units at a point in time;

Time series data is a set of observations on a variables outcomes in different time periods;

Panel data have both time-series and cross-sectional aspects and consist of observations

through time on a single characteristics of

multiple observational units.


6.160

Sampling Error and Statistic

Sampling error is the difference between the observed value of a statistic and the quantity it is intended to estimate.

Sampling distribution of a statistic is the distribution of all the distinct possible values that the statistic can assume when computed from samples of the same size randomly drawn from the same population.

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6.161

Example: Distribution of Sample Mean

Suppose we have a 'population' of 5 elements with values 1, 2, 3, 4, 5

What are the average and standard deviation of this population?

Now, consider all possible samples of size 3 to provide a point estimate of the population mean, and find the average of each

sample.


6.162

Example: Distribution of Sample Mean

Now, consider all possible samples of size 3 to provide a point estimate of the population mean, and find the average of each

sample. What is the average and standard deviation of the new

distribution? x Possible Samples, Size 3

1, 2, 3

1, 2, 4

1, 2, 5

1, 3, 4

1, 3, 5

1, 4, 5

2, 3, 4

2, 3, 5

2, 4, 5

3, 4, 5


6.163

Shape of the Sampling Distribution of Sample Mean

If n is large enough (>30) the sampling distribution of will be a normal distribution regardless of the distribution

type exhibited by the population.

If the population distribution is normal the sampling distribution of will be normal regardless of the sample

size.

X

X


6.164

Standard Error of the Sample Mean

When we use the sample mean to estimate the population mean, there are some errors.

The standard error of the sample mean is the standard deviation of the difference between the sample mean and the population mean.

For a sample mean calculated from a sample generated from a population with standard deviation , the standard error of the

sample mean is

when population standard deviation () is known.

nX

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6.165

Standard Error of the Sample Mean

In practice, the population variance is almost always unknown. The standard error of the sample mean is

estimated by,

Note,

XXs

)1( where,1

22

nXXsnssn

i

iX


6.166

Central Limit Theorem

The central limit theorem: Given a population described by any probability distribution having

mean and finite variance 2, the sampling distribution of the sample mean computed

from samples of size n from this population will

be approximately normal with mean (the population mean) and variance 2/n (the population variance divided by n) when the

sample size n is large.

X


6.167

Example: Central Limit Theorem

Electronics Associates Industry has 2500 managers on salaries such that = $31,800 and s = $4000.

What is the probability that a random sample of 30 managers will have a

mean salary that lies within $1000 of the population mean?


6.168

Confidence Intervals

Any estimate has errors. But we know that the estimated parameter must be around the

estimate with high probability.

A confidence interval is an interval for which we can assert with a given probability 1 , called the degree of confidence, that it will contain the

parameter it is intended to estimate.

Note, here we move from a point estimation to internal estimation.

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6.169

Confidence Intervals

A (1 )% confidence interval for a parameter has the following structure:

Point estimate Reliability factor Standard error the reliability factor is a number based on the

assumed distribution of the point estimate and the degree of confidence (1 ) for the confidence interval

standard error is the standard error of the sample statistic providing the point estimate.


6.170

Confidence Intervals for the Population Mean

For normally distributed population with known variance.

For large sample, population variance unknown.

nzX 2/

n

szX 2/


6.171

Confidence Intervals for the Population Mean

For population variance unknown, we have to use t-distribution

The t-distribution is a symmetrical probability distribution defined by a single

parameter known as degrees of freedom

(df).

n

stX 2/


6.172

(Students) t-Distribution versus the Standard Normal Distribution

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6.173

Choose Which Reliability Factor?

Sampling from Small

Sample Size

Large

Sample Size

Normal, Known Var z z

Normal, Unknown Var t t (or z)

Nonnormal, Known Var N/A z

Nonnormal, unknown Var N/A t (or z)


6.174

Example: Confidence Interval for

Population Mean

Now reconsidering that the population variance of the distribution of Sharpe ratio

is unknown, the analyst decides to

calculate the confidence interval using the

theoretically correct t-statistic.

Compare the result obtained under normal distribution.


6.175


Population Mean

ABC insurance surveys 36 policyholder in order to obtain an estimate of the average age of all policy holders. If average age of a sample of policy

holders 39.5 years with 1.8 years standard deviation, determine a 90%

confidence interval for the average policy holders age.

.


6.176


Population Mean

Banana Bank is interested in introducing a new computer based training program for use by its employees. A sample of 15 employees is selected to undergo training on the new system and on completion it is found that the duration of training is 53.87 days and s= 6.82 days. Determine a 95% confidence interval estimate.

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6.177


Population Mean

City insurance has found that from a sample of size 36, with mean age 39.5 years and standard deviation 1.8 years, we can say with 90% confidence that the sampling error associated with the sample mean is 0.5. What sample size would we need to obtain this sampling error with 95% confidence.


6.178

Selection of Sample Size

What conclusion can we draw from the previous example?

All else equal, a larger sample size decreases the width of the confidence interval because

and reliability factor (or t-critical value) declines in degree of freedom.

size Sample

deviation standard Sample mean sample theoferror Standard


6.179 5.179

Thank You!

Topic 7:

Hypothesis Testing






Chapter 7

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46

Hypothesis Testing

Hypothesis Testing

Statistical inference as two subdivisions: estimation and hypothesis testing

Estimation addresses the question: what is this parameters value? For example, what is the population mean of annual returns for

company XYZ? The answer is usually a confidence interval built around a point

estimate.

A hypothesis testing question is:" Is the value of that parameter equal to ? For example, is the average annual return of XYZ equal to 10%

The statement Average annual return of XYZ is equal to 10% is called a hypothesis

Hypothesis Testing

1.Stating the hypotheses.

2.Identifying the appropriate test statistic

and its probability distribution.

3.Specifying the significance level.

4.Stating the decision rule.

5.Collecting the data and calculating the

test statistic.

6.Making the statistical decision.

Steps in Hypothesis Testing

Hypothesis Testing

Null vs. Alternative Hypothesis (Step 1)

The null hypothesis is the hypothesis to be tested. Null hypothesis can never be accepted. We can either reject the null

(hence accepting the alternative) or not reject the null (and not

accepting the alternative either!)

The alternative hypothesis is the hypothesis accepted when the null hypothesis is rejected.

Eg: H0: Population average risk premium for Canadian equities is less

than or equal to zero.

HA: Population average risk premium for Canadian equities is greater than zero.

Hypothesis Testing

Formulation of Hypotheses (Step 1)

There three ways to form hypothesis

1. H0: = 0 versus Ha: 0

2. H0: 0 versus Ha: > 0 3. H0: 0 versus Ha: < 0

The first formulation is a two-sided test. The other two are one-sided tests.

Eg. we may want to test

H0: Population average risk premium for Canadian equities is less than or equal to zero.

HA: Population average risk premium for Canadian equities is greater than zero.

0 : 0H

: 0aH

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Hypothesis Testing

Test Statistic (Step 2)

A test statistic is a quantity, calculated based on a sample, whose value is the

basis for deciding whether or not to reject

the null hypothesis.

statistic sample theoferror Standard

Hunder parameter population theof Valuestatistic SamplestatisticTest 0

Hypothesis Testing

Test Statistic (Step 2)

Note, if we test a hypothesis of population mean, the standard error of the sample statistic is calculated by

the same formulas as we used the last topic:

In our example, we are testing whether the average can be 0. The test

statistic for our example is

unknown is population theofdeviation standard theif ,nssX

known is population theofdeviation standard theif ,nX

0 0 or

X X

X X

s

Hypothesis Testing

Two types of Errors (Step 3)

In reaching a statistical decision, we can make two possible errors:

We may reject a true null hypothesis (a Type I error), or

Probability of type I error is denoted by the Greek letter alpha,

We may fail to reject a false null hypothesis (a Type II error).

Probability of type II error is denoted by the Greek letter beta,

Hypothesis Testing

Level of Significance (Step 3)

The level of significance of a test is the probability of a Type I error that we are

preparing to accept in conducting a hypothesis

test, is denoted by .

The standard approach to hypothesis testing involves specifying a level of significance

(probability of Type I error) only.

Conventional significance levels: 0.1 (some evidence), 0.05 (strong evidence), 0.01 (very

strong evidence).

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48

Hypothesis Testing

Level of Significance (Step 3)

Trade-off: all else equal, if we decrease the probability of a type I error by increasing specifying a smaller significance level, we increase the probability of making a Type II error.

The power of a test is the probability of correctly rejecting the null (rejecting the null when it is false).

It is equal to 1 minus the probability of a Type II error.

Hypothesis Testing

Rejection Points (Step 4)

A rejection point (critical value) for a test statistic is a value with which the computed test statistic is compared to decide whether to reject the null hypothesis or not.

For a one-tailed test, we indicate a rejection point using the symbol for the test statistic with a subscript of significance level (e.g., z, t)

For a two-tailed test, the subscript is a half of the significance level (e.g., z/2, t/2)

Hypothesis Testing

Example: Rejection Points of a One-

Sided z-test

If Canadian equity risk premium is normal and its variance is known, then we can use a z-test to test the

hypotheses at the 0.05 level of significance:

H0: 0 (average risk premium is less than or equal to zero) versus

HA: > 0 (average risk premium is greater than zero)

One rejection point exists: z0.05 = 1.65

Hypothesis Testing

Example: Rejection Points of a One-

Sided z-test

So if our sample data yield

we reject the null hypothesis.

Otherwise, we do not reject the null hypothesis.

We cant accept it either!

This is illustrated by the following slide.

Note, H0: 0 versus Ha: < 0, the rejection point is z0.05 = -1.645 and we reject null hypothesis if z < -1.645

01.645

X

X

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49

Hypothesis Testing

Rejection Point, 0.05 Significance Level, One-Sided Test of the

Population Mean Using a z-Test

Hypothesis Testing

Example: Rejection Point of a Two-Sided

z-test

On the other hand, if the hypotheses are

H0: = 0 (The average Canadian equity risk premium is zero) versus

Ha: 0 (The average Canadian equity risk premium is not equal to zero)

There exists two rejection points. If significance level is still 0.05, the rejection points are:

z0.025 = 1.96 and -z0.025 = -1.96

from the normal distribution table.

Hypothesis Testing

Rejection Points, 0.05 Significance Level, Two-Sided Test of

the Population Mean Using a z-Test

Hypothesis Testing

Example: Rejection Points of a Two-

Sided z-test

So if sample data yield either

we reject the null hypothesis as illustrated by the next

slide. But we do not reject the null hypothesis if

0 01.96 or 1.96

X X

X X

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50

Hypothesis Testing

Confidence Interval

The (1 ) confidence interval represents the range of values of the test statistic for

which the null hypothesis will not be rejected

at an significance level.

In our previous examples, the 95% confidence intervals are, respectively

0 1.96 , 0 1.96X X

, 0 1.645X

Hypothesis Testing

Collecting Data and Calculating the Test

Statistic (Step 5)

Data collection issues:

Measurement errors

Sample selection bias and time-period bias

Test statistic calculation has shown in the previous examples.

Hypothesis Testing

Making Statistical Decision (Step 6)

Comparing the calculated test statistic with corresponding critical value to

decide whether reject the null

hypothesis or not.

Although we will meet other tests below, the basic principal of statistical decision

making is the same.

Hypothesis Testing

p-Value

An alternative approach is called p-value approach.

The p-value is the smallest level of significance at which the null hypothesis can be rejected.

The smaller the p-value, the stronger the evidence against the null hypothesis and in favor of the alternative hypothesis.

FIN5SBF

51

Hypothesis Testing

Hypothesis Tests Concerning the Mean:

t-tests

Can test that the mean of a population is equal to or differs from some hypothesized value.

Can test to see if the sample means from two different populations differ.

Hypothesis Testing

Tests Concerning a Single Mean: t-test

A t-test is used to test a hypothesis concerning the value of a population

mean, if the variance is unknown and

the sample is large, or

the sample is small but the population is normally distributed, or approximately

normally distributed.

Hypothesis Testing

Tests Concerning a Single Mean

deviation standard sample

mean population theof valueedhypothesiz the

mean sample

freedom of degrees 1n with statistic

where,

/

1

1

s

X

tt

ns

Xt

n

n

Hypothesis Testing

Example: Testing Rio Tinto Mean Return

Sender Equity Fund has achieved a mean monthly return of 1.50% with a sample st. dev. of 3.60% during a 24 months period. Given its level of systematic risk, the share is expected to have earned a 1.10% mean monthly return. Assuming return is normally distributed, is the actual result consistent with an underlying or population mean monthly return of 1.10% with 10% level of significance?

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52

Hypothesis Testing

Example: Testing Rio Tinto Mean Return

cont.

The hypothesis statement is: H0: the average return is equal to 1.10% The alternative hypothesis is Ha: mean monthly return 0 We use a 2-sided t-test to test for 24 months period (n=24)

Step 1: H0: =0 Ha: 0

Step 2: test statistic =

Step 3: a two tailed test has two rejection points: t/2,df=t0.05,23=1.714 and -t/2,df=-t0.05,23=-1.714

Step 4: reject if 0.544 < -1.714 or 0.544 > 1.714

Can we reject?

1

1.50 1.100.544

/ 3.60 / 24n

Xt

s n

Hypothesis Testing

Example: Testing average Days of

Receivables

FashionDesigns is concerned about a possible slowdown in payments from its customers. The rate of payment is measured by the average number of days in receivables. FashionDesigns has generally maintained an average of 45 days in receivables. A recent random sample of 50 accounts shows a mean number of days in receivables of 49 with a standard deviation of 8 days. Determine whether the evidence supports the suspected condition that customer payments have slowed at 5% level of significance.

Hypothesis Testing

Example: Testing average Days of

Receivables

The hypothesis statement is: H0: number of days in receivable is less than or equal 45 The alternative hypothesis is Ha: number of days in receivable is more than 45 We use a 1-sided t-test to test for 50 accounts (n=50)

Step 1: H0: 45 Ha: >45


Step 3: a tone tailed test has one rejection point: t,df=t0.05,49=1.677

Step 4: reject if 3.536 > 1.677

Can we reject?

1

49 453.536

/ 8 / 50n

Xt

s n

Hypothesis Testing

The z-Test Alternative

If the population sampled is normally distributed with known variance, then the

test statistic for a hypothesis test

concerning a single population mean, , is

deviation standard populationknown

mean population theof valueedhypothesiz the

where,

/

n

Xz

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53

Hypothesis Testing

The z-Test Alternative

If the population sampled has unknown variance and the sample is large, in place

of a t-test, an alternative statistic is

deviation standard populationknown

where

/

s

ns

Xz

Hypothesis Testing

Rejection Points for a z-Test For

= 0.10

1.H0: = 0 versus Ha: 0

Reject the null hypothesis if z > 1.645 or if z < -1.645.

2.H0: 0 versus Ha: > 0

Reject the null hypothesis if z > 1.282

3.H0: 0 versus Ha: < 0

Reject the null hypothesis if z < -1.282

Hypothesis Testing

Rejection Points for a z-Test

For = 0.05


Reject the null hypothesis if z > 1.96 or if z < -1.96.


Reject the null hypothesis if z > 1.645

3.H0: 0 versus Ha: < 0

Reject the null hypothesis if z < -1.645

Hypothesis Testing

Rejection Points for a z-Test

For = 0.01


Reject the null hypothesis if z > 2.576 or if z < -2.576


Reject the null hypothesis if z > 2.326.

3.H0: 0 versus Ha: < 0 Reject the null hypothesis if z < -2.326

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Hypothesis Testing

t-test or z-test?

Population normal and variance known z-test, for both small and large sample.

Variance unknown

Large Sample

(n30) Small Sample

(n 0

3.H0: 1 - 2 0 versus HA: 1 - 2 < 0

Hypothesis Testing

Test Statistics for Difference between Two Population

Means: equal variances

For normally distributed populations with unknown variances, but if the variances

can be assumed to be equal, the t-statistic

is

and degrees of freedom is n1 + n2 - 2

2/1

2

2

1

2

2121

n

s

n

s

XXt

pp

2

)1()1( where

21

2

22

2

112

nn

snsnsp

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Hypothesis Testing

Test Statistics for Difference between Two Population

Means: unequal variances

For normally distributed populations, if population variances are unequal and unknown, the t-statistic is

and degrees of freedom is given by

2/1

2

2

2

1

2

1

2121

n

s

n

s

XXt

2

2

1

2

2

1

2

1

2

1

2

2

2

2

1

2

1

//

n

ns

n

ns

n

s

n

s

df

Hypothesis Testing

Example: Mean Return on S&P 500

The realized mean monthly return on the S&P 500 in the 1980s appears to have been substantially different from that in the 1970s. Was the difference statistically significant?

The data, shown on the next slide, indicate that assuming equal population variances for returns in the two decades is not unreasonable. But if you assumed unequal variances, would you reach a different conclusion?

Assume 5% level of significance

Hypothesis Testing

S&P 500 Monthly Return and Standard

Deviation

Decade No. of months Mean return St. Dev.

1970s 120 0.580 4.598

1980s 120 1.470 4.738

Hypothesis Testing

Example: Mean Return on S&P 500 cont.

We assume equal variance to answer the question: Was the difference statistically significant?

This means, is the difference between the average in 80s and 70s significantly different

from zero?

In other words,80s 70s=0 or 80s 70s0

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Hypothesis Testing

Example: Mean Return on S&P 500 cont.

Step 1: H0: 80s 70s =0 Ha: 80s 70s =0


Note that you must find Sp first and then calculate test statistic using the Sp

Step 3: a two tailed test has two rejection points: t/2,df=t0.025,238=1.97 and -t/2,df=-t0.025,238=-1.97

Step 4: reject if 1.477 < -1.96 or 1.477 > 1.96

Can we reject?

1 2 1 21/2

2 2

1 2

1.4767

p p

X Xt

s s

n n

Hypothesis Testing

Mean Differences Samples Not Independent

Reminder: In the previous two t-tests, samples are assumed to be independent

If the samples are not independent, a test of mean difference is done using paired observations.

1. H0: d = d0 versus HA: d d0

2. H0: d d0 versus HA: d > d0 3. H0: d d0 versus HA: d < d0

where d stands for the population mean difference and d0 is a hypothesis value for the population mean difference

Hypothesis Testing

t-statistic for Mean Differences Samples Not Independent

To calculate the t-statistic, we first need to find the sample mean difference:

where di is the difference between two paired

observations (the ith pair)

The sample variance is

n

i

idn

d1

1

)1(2

1

2

nddsn

i

id

Hypothesis Testing

t-statistic for Mean Differences Samples Not Independent

The standard error of the mean difference is

The test statistic, with n 1 df, is,

n

ss d

d

d

d

s

dt 0

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Hypothesis Testing

Hypothesis Tests Concerning Variance

We examine two types:

tests concerning the value of a single population variance and

tests concerning the differences between two population variances.

Hypothesis Testing

Tests Concerning a Single Population

Variance

We can formulate hypotheses as follows:

2

0

2

a

2

0

2

0

2

0

2

a

2

0

2

0

2

0

2

a

2

0

2

0

:H versus:H .3

:H versus:H .2

:H versus:H .1

Hypothesis Testing

Test-Statistic for Tests Concerning a Single

Population Variance

If we have n independent observations from a normally distributed population,

the appreciate test statistic is chi-squared

statistic

where s2 is sample variance,

freedom of degrees 1n with ,)1(

2

0

22

sn

)1(1

22

nXXsn

i

i

Hypothesis Testing

Rejection Points for Tests Concerning a

Single Population Variance

1. Equal to H0: Reject the null if the statistic is greater than or smaller

than

2. Not greater than H0: Reject the null if the statistic is greater than

3. Not less than H0: Reject the null if the statistic is less than

2

2/2

2/1

2

2

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Hypothesis Testing

Tests Concerning the Equality (Inequality) of

Two Variances

Suppose we want to know the relative values of the variances of two populations,

we can formulate one of the following

hypotheses:

2

2

2

1a

2

2

2

10

2

2

2

1a

2

2

2

10

2

2

2

1a

2

2

2

10

:H versus:H .3

:H versus:H .2

:H versus:H .1

Hypothesis Testing

Test Statistic for Tests Concerning the

Equality (Inequality) of Two Variances

Suppose we have two samples, the first has n1 observations with sample

variance and the second has n2

observations with sample variance . If

both populations are normal, the test-

statistic is

freedom of degrees )1(n and )1( with , 212

2

2

1 ns

sF

2

1s2

2s

Hypothesis Testing

Rejection Points for Tests Concerning the

Equality (Inequality) of Two Variances

Convention: Let the sample with larger variance be sample 1 and the other sample 2 F-statistic is always greater than or equal to 1.

Thus, decision rule is:

1. Equal to H0: Reject the null if the statistic is greater than F/2 .

2. Not greater than and Not less than H0: Reject the null if the statistic is greater than F .

Hypothesis Testing

5.232

Thank You!

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233 Slide

Correlation and Simple Linear Regression,p-283-300

The Simple Linear Regression Model

The Least Squares Method

The Coefficient of Determination

Model Assumptions

Testing for Significance

Using the Estimated Regression

Equation for Estimation and Prediction

Residual Analysis: Validating Model Assumptions

Residual Analysis: Outliers and Influential

Observations

Chapter 8

234 Slide

The Simple Linear Regression Model

Simple Linear Regression Model

y = 0 + 1x +

Simple Linear Regression Equation

E(y) = 0 + 1x

Estimated Simple Linear Regression Equation

y = b0 + b1x

y = dept var

^

235 Slide


Least Squares Criterion

min S(yi - yi)2

where

yi = observed value of the dependent variable

for the i th observation

yi = estimated value of the dependent variable

for the i th observation

^

^

236 Slide

Slope for the Estimated Regression Equation

y -Intercept for the Estimated Regression Equation

b0 = y - b1x

where

xi = value of independent variable for i th observation

yi = value of dependent variable for i th observation

x = mean value for independent variable

y = mean value for dependent variable

n = total number of observations

_ _

bx y x y n

x x n

i i i i

i i1 2 2

( ) /

( ) /b

x y x y n

x x n

i i i i

i i1 2 2

( ) /

( ) /

_

_


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237 Slide

Example: Reed Auto Sales

Reed Auto periodically has a special week-long sale. As part of the advertising campaign Reed runs one or more television commercials during the weekend preceding the sale. Data from a sample of 5 previous sales showing the number of TV ads run and the number of cars sold in each sale are shown below.

Number of TV Ads Number of Cars Sold

1 14

3 24

2 18

1 17

3 27

238 Slide

Slope for the Estimated Regression Equation

b1 = 220 - (10)(100)/5 = 5

24 - (10)2/5

y -Intercept for the Estimated Regression Equation

b0 = 20 - 5(2) = 10

Estimated Regression Equation

y = 10 + 5x ^


239 Slide

The Coefficient of Determination

Relationship Among SST, SSR, SSE

SST = SSR + SSE

Coefficient of Determination

r 2 = SSR/SST

where

SST = total sum of squares

SSR = sum of squares due to regression

SSE = sum of squares due to error

( ) ( ) ( )y y y y y yi i i i 2 2 2( ) ( ) ( )y y y y y yi i i i 2 2 2^ ^

240 Slide

Coefficient of Determination

r 2 = SSR/SST = 100/114 = .88

The regression relationship is very strong since

88% of the variation in number of cars sold can be

explained by the linear relationship between the

number of TV ads and the number of cars sold.


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241 Slide

The Correlation Coefficient & Hypothesis Testing

The sample correlation coefficient is plus or minus the square root of the coefficient of determination.

Sample Correlation coefficient

Testing for Sample Correlation coefficient p297 r rxy

2r rxy 2

242 Slide

Model Assumptions

Assumptions About the Error Term

The error is a random variable with mean of zero.

The variance of , denoted by 2, is the same for all values of the independent variable.

The values of are independent.

The error is a normally distributed random variable.

243 Slide

Testing for Significance: F Test

Hypotheses

H0: 1 = 0

Ha: 1 = 0

Test Statistic

F = MSR/MSE

Rejection Rule

Reject H0 if F > F

where F is based on an F distribution with 1 d.f. in

the numerator and n - 2 d.f. in the denominator.

244 Slide

Testing for Significance: t Test (p.312)

Hypotheses

H0: 1 = 0

Ha: 1 = 0

Test Statistic

Rejection Rule

Reject H0 if t < -tor t > t

where t is based on a t distribution with

n - 2 degrees of freedom.

tb

sb 1

1

tb

sb 1

1

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245 Slide

Using the Estimated Regression Equation for Estimation and Prediction

Confidence Interval Estimate of E (yp)

Prediction Interval Estimate of yp yp + t/2 sind

where the confidence coefficient is 1 - and t/2 is

based on a t distribution with n - 2 d.f.

/ y t sp yp 2 / y t sp yp 2

246 Slide

F Test

Hypotheses H0: 1 = 0

Ha: 1 = 0

Rejection Rule

For = .05 and d.f. = 1, 3: F.05 = 10.13

Reject H0 if F > 10.13.

Test Statistic

F = MSR/MSE = 100/4.667 = 21.43

Conclusion

We can reject H0.


247 Slide

t Test

Hypotheses H0: 1 = 0

Ha: 1 = 0

Rejection Rule

For = .05 and d.f. = 3, t.025 = 3.182

Reject H0 if t > 3.182

Test Statistics

t = 5/1.08 = 4.63

Conclusions

Reject H0: 1 = 0


248 Slide

Point Estimation

If 3 TV ads are run prior to a sale, we expect the mean number of cars sold to be:

y = 10 + 5(3) = 25 cars

Confidence Interval for E (yp)

95% confidence interval estimate of the mean number of cars sold when 3 TV ads are run is:

25 + 4.61 = 20.39 to 29.61 cars

Prediction Interval for yp

95% prediction interval estimate of the number of cars sold in one particular week when 3 TV ads are run is: 25 + 8.28 = 16.72 to 33.28 cars

^


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249 Slide

Residual Analysis

Residual for Observation i

yi - yi

Standardized Residual for Observation i

where

^

y y

si i

y yi i

y y

si i

y yi i

^

^

s s hy y ii i 1s s hy y ii i 1^

250 Slide

Detecting Outliers

An outlier is an observation that is unusual in comparison with the other data.

Minitab classifies an observation as an outlier if its standardized residual value is < -2 or > +2.

This standardized residual rule sometimes fails to identify an unusually large observation as being an outlier.

This rules shortcoming can be circumvented by using studentized deleted residuals.

The |i th studentized deleted residual| will be larger than the |i th standardized residual|.

Residual Analysis

251 Slide

The End of Chapter 8

252 Slide

Multiple Regression & Issues in Regression Analysis p. 325 text

The Multiple Linear Regression Model


The Multiple Coefficient of Determination

Model Assumptions

Testing for Significance

Using the Estimated Regression Equation

for Estimation and Prediction

Qualitative Independent Variables

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