Quantitative Methods - I

© EduPristine CFA - Level – I © EduPristine – www.edupristine.com

Quantitative Methods - I

© EduPristine CFA - Level – I 1

Mapping to Curriculum

Reading 5: Time Value of Money

Reading 6: Discounted Cash Flow Applications

Reading 7: Statistical Concepts and Market Returns

© EduPristine CFA - Level – I

Reading 5: Time Value of Money

2


Coverage of Reading 5

1. Understand the Concept of NPV and IRR

2. Calculate NPV, IRR

3. Calculating various yield measures

4. Return Calculations

Time weighted return calculations

Money weighted return calculations

3


Time is Money

Given a $1 million, would u prefer to consume goods worth of $1 mn now or save it for a future?

Should we deposit that money with a bank or keep cash in hand?

Do we need any interest or returns on investments at all?

4


Simple interest

Rationale:

Value of money accrues over a period of time

Value of Rs 100 today is greater than Rs 100 received after 5 years due to the presence of “interest” accruing to this sum over periods

Simple Interest:

Refers to the payment of interest on the principal amount of an investment.

Interest payments are made at a constant absolute rate

Interest = Principal * Rate * Time

E.g. Mr. Sharma invests Rs100000 in a fund earning simple interest 10% p.a.

Interest after 1 year = 100000*10% *1

= Rs10000

5


Compound interest

Rationale:

Each interest payment received is re-invested at the same rate as that of original principal. Thus, the original principal changes year-on-year

where Rate = interest rate (r)/100 & N = number of years

6

nrpA )1(

1,000.00

6,000.00

11,000.00

16,000.00

21,000.00

26,000.00

31,000.00

36,000.00

41,000.00

46,000.00

51,000.00

0 5 10 15 20 25 30 35 40

Hu

nd

red

s

Simple Interest

Compound Interest


Present Value and future value (Single Sum)

Assume can invest PV at interest rate i to receive future sum, FV

Similar reasoning leads to Present Value of a Future sum today.

7

FV1 = (1+i)PV

FV3 = (1+i)3PV

PV

FV2 = (1+i)2PV

PV = FV1/(1+i)

FV1

PV = FV2/(1+i)2

FV2

PV = FV3/(1+i)3

FV3

Present Value

Future Value

1 2 3 0

1 2 3 0


Present Value and future value (Series of CF)

8

FVo= (1+i)3CF

FV2= (1+i)CF

FV1 = (1+i)2CF

PV = FV1/(1+i)

FV1

PV = FV2/(1+i)2

FV2

PV = FV3/(1+i)3

FV3

Present Value

Future Value

1 2 3 0

1 2 3 0

This is called as the CASH FLOW ADDITIVE PRINCIPLE


Increase in frequency of Compounding

Compounding can be done any number of times in a year like twice in a year or 4 times in a year or even it can be continuous

Impact on CI calculations

• Rate of interest (which is Annual) needs to be reduced to make it per period

• No. of period (T used in the formula) need to be increased

Continuous compounding (Fastest compounding possible)

• Extends the notion of compounding periods to a point where the number of periods becomes infinitely large and the length of each period is correspondingly small

• Future amount = Principal * e(Rate * Time)

• where e ≈ 2.7182818

• E.g., : Rs 100,000 is invested to earn a continuous return of 7% for 1 year.

• Future value= 100,000 * e(0.07*1) = Rs 107251

9

nm

m

rpA )1(

m = number of periods in a year n = Number of years


Required interest rate

Real interest rate is the interest rate on a loan which does not take into account expected inflation

Nominal rate = real rate + expected inflation rate

Risk-free rate = interest rate that is available on government securities & bonds where the risk of default is very less i.e. close to nil

Other securities may have following three types of risks associated with its cash flow

a) Default risk: borrower fails to make the promised repayment

b) Liquidity risk: associated with a marketable value of a security being less than its fair value

c) Maturity risk: more risk on money which we will get later. E.g., long term bonds are more volatile than short term bonds, and thus demand higher maturity risk premium

Required interest rate on a security = Risk Free Rate + Inflation Rate + default risk premium + Liquidity premium + maturity risk premium

10


Present Value Factor and Cumulative Discount Factor

PVF is the present value of Re. 1 which is expected to be received after N years

CDF is the present value of series of Re. 1 which is expected to be received every year up to Nth year

11

0

$100

1

$100

2

$100

3

i%

0 1 2

$100

3

i%


Annuity

Equal Payment of Cash flow every year (Annual). Annuity is of two types:

1. Regular or ordinary annuity: a finite set of sequential cash flows, all with the same value A, which has a first cash flow that occurs one period from now. In simple terms, constant cash flows occur at the end of each period for a finite period.

2. Annuity due: a finite set of sequential cash flows, all with the same value A, which has a first cash flow that is paid immediately. In simple terms, constant cash flows occur at the beginning of each period for a finite period.

E.g. Time line for an ordinary annuity of $100 for 3 years.

12

0

$100

1

$100

2

$100

3

i%


Perpetual + Annuity

Perpetuity is a series of constant payments, A, each period forever

Intuition: Present Value of a perpetuity is the amount that must invested today at the interest rate i to yield a payment of A each year without affecting the value of the initial investment

PVperpetuity = A/i

PV1 = A/(1+r)

PV2 = A/(1+r)2

PV3 = A/(1+r)3

PV4 = A/(1+r)4

etc.

Perpetuities

13

1 0 2 3 4 5 6 7 8

etc.


Interest Rate Definitions

Periodic interest rate, ip = is/m

where is the stated annual interest rate divided by the number of periods in a year, m.

used in calculations, shown on time lines (in later slides)

Stated Annual interest rate or quoted interest rate, is = m * ip

where ip is the periodic interest rate times the number of periods in a year, m

stated in contracts

does not account for effects of compounding within the year

Effective Annual interest rate:

the amount to which a $1 grows to in year with compounding taken into account

use EAR only for comparisons when payment periods differ between investments

given a stated annual interest rate, iS, the periodic rate is iP = iS/m, where m is the number of periods a year

Effective annual interest rate is computed as = (1 + ip)m – 1

E.g. calculate the effective annual interest rate on a loan, which charge 12% p.a. rate on a monthly basis

EAR = {1+(12%/12)}12 – 1 = 12.68%

14

Note: Before applying any formula, just ensure that you have calculated the effective annual interest rate


Perpetuity of A per period in Period 0 – PV1 = A/i

Perpetuity of A per period in Period 8 – PV8 = [1/(1+i)8] *(A/i)

Annuity of A for 8 periods – PV + PV1 – PV8 =(A/i)* {1-[1/(1+i)8]}

Intuition: Formula for a N-period annuity of A is: PV of a Perpetuity of A today minus PV of a Perpetuity of A in period N`

Let us warm up on Present Value concept

2 4 6 8 10 12 14 0

2 4 6 8 10 12 14 0

2 4 6 8 10 12 14 0

15


Practical Applications of TVM………….VV Important

1. Finding the maturity amount in a FD

2. Finding the maturity amount in a recurring deposit

3. Finding the value of EYI which is required to pay off a loan in a specific time period

1. When 1st installment is due at the end of each period

2. When 1st installment is due at the beginning of each period

4. Computing the value of (irregular) installment required to repay the loan

5. Calculating the return earned on an investment

6. Finding the Maximum price that can be paid for an asset (equity or debt or Machinery)

7. Amount to be deposited each year to withdraw XX amount each year after retirement (Funding a retirement plan)

16


Practice Question

Question 1: A person wants to retire after five years and withdraw Rs. 10,000 each year for three years post retirement. Rate of interest is 12%.

A) How much he should deposit today to fund the post-retirement withdrawals

B) If he does not have enough money today, how much he should deposit each year to fund the post-retirement withdrawals

17


Present Value mechanics

18

Cash Flow Type Discounting Formula Compounding Formula

Simple CF CFn / (1+r)n CF0 (1+r)n

Annuity

Growing Annuity

Perpetuity

Growing Perpetuity (cash flow increases by a constant rate every year)

Expected Cash flow next year/(r-g)

r

1 - nr)+(1A

g-r

r)+(1

g)+(1 - 1

A n

n

rA /

r

r)(1

1-1

A n


Questions

1. If you deposit $250 a month, beginning next month, for 20 years into an account paying 7% per year, compounded monthly, how much is in your account after that last deposit?

A. $58,205.58

B. $308,663.09

C. $130,231.66

2. If you need $25,000 in 10 years, how much must you deposit today, if your money will earn 6% per year, compounded annually?

A. $25,000

B. $13,959.87

C. $2,320.01

3. What is the present value today of these annual cash flows: $5,000, $4,000, $3,000, $2,000, $1,000? Assume the first cash flow occurs 1 year from today and an interest rate of 8% per year, compounded annually.

A. $12,854.32

B. $15,000

C. $12,591.14

19


Questions

4. If the stock’s initial price is $30 & its year-end price is $35, then its continuously compounded annual rate of return is:

A. 15.4%

B. 14.5%

C. 16.4%

5. The yield to maturity on otherwise identical option-free bonds issued by the U.S. Treasury and a large industrial corporation is 6 percent and 8 percent, respectively. If annual inflation is expected to remain steady at 2.5 percent over the life of the bonds, the most likely explanation for the difference in yields is a premium due to:

A. maturity

B. inflation

C. default risk

6. A consumer is shopping for a home. His budget will support a monthly payment of $1,300 on a 30- year mortgage with an annual interest rate of 7.2 percent. If the consumer puts a 10 percent down payment on the home, the most he can pay for his new home is closest to:

A. $191,518.

B. $210,840.

C. $212,800

20


Solutions

1. C. On the BAII Plus, press 240 N, 7 divide 12 = I/Y, 0 PV, 250 PMT, CPT FV. On the HP12C, press 240 n, 7 ENTER 12 divide i, 0 PV, 250 PMT, FV. On the BAII Plus, make sure the value of P/Y is set to 1. Note that the answer is displayed as a negative number.

2. B. On the BAII Plus, press 10 N, 6 I/Y, 0 PMT, 25000 FV, CPT PV. On the HP12C, press 10 n, 6 i, 0 PMT, 25000 FV, PV.Note that the answer will be shown as a negative number.

3. C.

4. A. ln(35/30) = 0.154 = 15.4%

5. C An interest rate is the sum of a real risk-free rate, expected inflation, and premiums that compensate investors for distinct types of risk. The difference in yield on otherwise identical U.S Treasury and corporate bonds is attributed to default risk.

6. C The consumer’s budget will support a monthly payment of $1,300. Given a 30-year mortgage at 7.2 percent, the loan amount will be $191,517.76 (N = 360, %I = 0.6, PMT = 1,300, solve for PV). If he makes a 10% down payment, then the most he can pay for his new home = $191,517.76 / (1 – 0.10) = $212,797.51 ≈ $212,800.s

21


Reading 6: Discounted Cash Flow Applications

22


Coverage of the reading 6

1. Concept of NPV and IRR

2. Calculating various yield measures

a) Holding Period Yield

b) Effective Annual Yield

c) Bank Discount Yield

d) Money market Yield (CD equivalent yield)

e) Bond Equivalent Yield

3. Return Calculations

a) Time weighted return calculations

b) Money weighted return calculations

23

To be done along with Capital Budgeting


Holding Period Yield (HPY)

24

The total return earned over the period of the investment

Total Return = Cash Return + Capital Appreciation

HPY = (P1 – P0 + D1) / P0

Important Points

Period of Holding can be more than one year or less than one year

HPY is never per annum

This is extensive used by various categories of investors to calculate return % over short term investments

1

valuebeginning

valueending

valuebeginning

valuebeginningvalueendingHPR


Effective Annual Yield (EAY)

The yield that converts holding period yield to a compound annual yield

Unlike Money market Yield and Bank Discount Yield, EAY is based on a 365 day year

It is exactly similar to calculating Effective Annual interest Rate in TVM calculations

T = Holding period (in no. of days)

HPY = Holding Period Return

Important Points

It is based on 365 days a year

Fundamentals of Compound Interest is used (in place of simple interest)

25

11 /365 tHPYEAY

For annualizing


Bank Discount Yield (rBD)

26

The BDY expresses the dollar discount from the face (par) value as a fraction of the face value (not the market price of the instrument)

It implied at what discount rate has the bank issued the bond

Treasury bills (T-bills) are quoted on a Bank Discount Basis rather than on a price basis (purchase price)

D = Dollar discount (Face Value – Issue Price)

F = Face value

T = No. of days remaining until maturity

Limitations of BDY (why it is not useful for investors)

Yield is based on face value of the bond and not its purchase price

Yield is based on a 360 day a year (instead of 365 day year)

BDY annualizing assumes no compounding (i.e. simple interest)

tF

Dr BD

360

For annualizing


Money Market Yield (rMM)

This convention makes the quoted yield on a T-bill comparable to yield quotations on interest bearing money market instruments that pay interest on a 360 day basis

Important Points

It is based on 360 days a year

Return is calculated over purchase price of investment rather than face value (maturity price)

Fundamentals of Simple Interest is used (in place of Compound interest)

Rather than memorizing 2nd formula, convert r BD into HPY and apply 1st formula

27

T = Holding period (in no. of days) HPY = Holding Period Return


r BD

= Bank Discount Yield

For annualizing

1 2

tHPYrMM

360 BD

BDMM

rt

rr

360

360


Bond Equivalent Yield (BEY)

Interest payments on the US treasury bills is paid on a semi-annual basis.

BEY = 2 X semi-annual discount rate

Important Points

No. of days is no more a concern since it is based on half yearly frequency

Return is calculated over purchase price of investment rather than face value (maturity price)

Fundamentals of Simple Interest is used (it is multiplied by 2, not raised to the power of 2)

Corporate Finance has different calculation for BEY


HPY = Holding Period Return

28

BEY = HPY X 365

t


Types of returns of a portfolio

There are different ways to calculate return of an investment portfolio

These concepts apply more when there are multiple transactions of buying and selling of securities

The two methods most often used are:

• Time weighted return calculations (TWR)

• Money weighted return calculations (MWR)

29


Money weighted return - Basics

It enable you to determine your rate or return relative to the length of time you have invested your capital

It gives more importance (weightage) to the dollar amount of investment and measure it against the length of growth period

Money Weighted return = the Internal Rate of Return (IRR) on a portfolio taking into account all cash inflows and outflows

IRR is that rate that equates satisfies the equation:

Where:

Inflows: Investment into the portfolio (input from investor)

Outflows: Withdrawal from the portfolio (returns to the investor)

30

PV (inflows) = PV (outflows)


Money-weighted return - Steps

Step 1: Identify the cash flows at each time zone

a) Inflows: Investment into the portfolio (input from investor)

b) Outflows: Withdrawal from the portfolio (returns to the investor)

Step 2: Compute a rate at which the present value of Inflows equals the present value of the Outflows using:

a) IRR function on Financial Calculator

b) Hit and Trial

31


Time-weighted return - Basics

Time weighted return (TWR) eliminates the effect of additions and withdrawals that can distort Money weighted returns

TWR is the average annual compound rate of return over the entire holding period

It is the rate at which $1 compounds over a specified performance horizon

TWR is the preferred method of performance measurement, because it is not affected by the timing of the cash inflows and outflows

32

TWR is the geometric mean of HPR for the sub-periods


Time-weighted return - Steps

Step 1: Form sub-periods over the evaluation period that correspond to the dates of deposits and withdrawals:

a) Form a sub-period just after a sale (Withdrawal)

b) Form a sub-period just before a purchase (addition or investment)

Step 2: Compute the holding period returns (HPR) of the portfolio for each sub-period

Step 3: Compute the geometric mean of the (1+HPR) for each sub-period to obtain a total return for the entire measurement period.

33

TWR eliminates the effect of additions and subtractions to the portfolio, therefore is the correct measure for measuring the performance of a portfolio manager


Summary and conclusion

Comparison of MWR and TWR

If funds are contributed to a portfolio just before the start of the growth phase, the MWR will be higher than TWR

If funds are contributed to a portfolio just before the start of the poor-growth phase, the MWR will be lower than TWR

Preferable method

If a manager has complete control over money flow into and out of an investment account - ONLY then MWR is preferable.

Otherwise – TWR is preferable since it eliminates the effect of additions and subtractions to the portfolio (weight of dollar amount of money)

34


Question

1. A 180-day U.S. Treasury bill has a holding period yield (HPY) of 2.375%. The bank discount yield (in %) is closest to:

A. 4.640

B. 4.750

C. 4.875

2. The bond-equivalent yield for a semi-annual pay bond is most likely:

A. equal to the effective annual yield

B. more than the effective annual yield

C. equal to double the semi-annual yield to maturity

3. A 182-day U.S. Treasury bill has a face value of $100,000 and currently sells for $98,500. Which of the following yields is most likely the lowest?

A. Bank discount yield

B. Money market yield

C. Holding period yield

35


Question

4. If the stated annual interest rate is 9% and the frequency of compounding is daily, the effective annual rate is closest to:

A. 9.00%.

B. 9.42%

C. 9.88%

36


Solutions

1. A rMM = (HPY) ×(360/t) = .02375 × (360 / 180) = 0.0475. Use the money market yield to find the bond discount yield: rMM = (360 rBD) / ((360 – (t)( rBD)). In this case: 0.0475 = (360 rBD) / ((360 – (180)( rBD)). rBD = 0.046398

2. C The bond equivalent yield for a semi-annual pay bond is equal to double the semiannual yield to maturity

3. C The holding period yield is: (100 – 98.5) / 98.5 = 0.015228. This is less than the bank discount yield: ((100-98.5) / 100) × (360 / 182) = 0.02967. It is also less than the money market yield: (360 × 0.02967) / (360 – 182 × 0.02967) = 0.030122

4. B Solve for effective annual rate using: EAR = (1 + periodic interest rate)m -1 = (1 + (.09 / 365))365 – 1 = 0.094162 ~ 9.42%

37


Reading 7: Statistical Concepts and Market Returns

38


Coverage of the reading 7

Descriptive and inferential statistics

Mean, Median & Mode

Variance & Standard Deviation

Chebyshev’s Inequality

Skewness & Kurtosis

Arithmetic Mean, geometric Mean & harmonic mean

39


Descriptive & Inferential Statistics

Modern storage and analytical capabilities have tremendously increased the amount of complexity of data

• Difficult to analyze & take decisions

Descriptive Stats: Simple summaries about the sample and the measures

• Graphical displays of the data in which graphs summarize the data or facilitate comparisons

• Tabular description in which tables of numbers summarize the data

• Summary statistics (single numbers) which summarize the data

Steps in descriptive statistics:

• Collect data

• Classify data

• Summarize data

• Present data

• Proceed to inferential statistics if there are enough data to draw a conclusion

Inferential Statistics – Making Inference about the larger group based on actually observed smaller group

• Making Forecast, Estimates etc.

• Usage of probability theory for making inference

40


Statistical Concepts

Population: It is defined as all members of a specified group

Sample: It is a subset of a defined population

For example, the population can represent the daily closing stock prices of Reliance, since inception.

Population = { P0, P1, P2, …… Pn}, with a mean µ, and standard deviation s

If we want to make an inference about the population, we can take a sample, which can be the weekly closing prices of the stock for the past 2 years.

Sample = {Px, Px+7, Px+14, …… Pn} with a mean µ, and standard deviation s

We draw our conclusions about parameters of the population from the characteristics of the sample.

41


Measurement Scales

Nominal – Only classifies according to types, does not say which is better or worse e.g. classification of different types of hedge funds based on hedging strategy or instruments

Ordinal - Ranks them according to some scale. For e.g. Bond Ratings like AAA, AA rank the bonds in their order of creditworthiness

Interval – It classifies elements in an interval. The Celsius temperature scale is such a thing. It tells us by how much is something more or less than another thing. Though a temperature of zero degree does not mean absence of temperature

Ratio – This is the strongest level of measurement. For. e.g. length, returns on stocks

Examples: Identify the type of scale for each of the following:

Your names: Aakash, Anirudh, Pallavi, Tanu.

Weight of IAS aspirants

The average temperature on 20 successive days in January in Chicago

Interest rates on corporate bonds each year for 60 years

Heavy Trucks, Medium Trucks, or Light Trucks

42


Statistical Concepts

Parameter: A measure used to describe a characteristic of a population is referred to as a parameter

Sample Statistic: It is used to measure the characteristic of a sample

Frequency Distribution: It is a tabular display of data summarized into a relatively small number of intervals.

Frequency distribution is the list of intervals together with the corresponding measures of frequency for the variable of interest.

A histogram - graphical equivalent of a frequency distribution; it is a bar chart where continuous data on a random variable’s observations have been grouped into intervals

A frequency polygon is the line graph equivalent of a frequency distribution; it is a line graph that joins the frequency for each interval, plotted at the midpoint of that interval

43


Frequency Distribution Table

Classify the following daily stock prices of Reliance Industries into intervals

24, 26, 24, 21, 27, 27, 30, 41, 32, 38

Frequency Distribution Table Steps:

Determine Range

Select Number of Classes

Usually Between 5 & 15 Inclusive

Compute Class Intervals (Width)

Determine Class Boundaries (Limits)

Compute Class Midpoints

Count Observations & Assign to Classes

44

Class Frequency

15 but <25 3

25 but <35 5

35 but <45 2

0

2

4

6

15 but <25 25 but <35 35 but <45


Frequency Distribution Table (Cont…)

Modal Interval: The interval that has the greatest frequency is called as the modal interval

Relative Frequency: It is calculated by dividing the absolute frequency of each return interval by total number of observations. It is the %age of total observations falling within each interval

Cumulative absolute frequency: It is calculated by summing the absolute frequency starting from the lowest interval & progressing through the highest

Cumulative relative frequency: It is calculated by summing the relative frequencies starting from the lowest interval & progressing through the highest

45


Calculate Relative and Cumulative Frequency

46

Class Frequency Relative Frequency Cumulative

Frequency

15 but <25 3 3/10 = 30% 30%

25 but <35 5 5/10 = 50% 30%+50%=80%

35 but <45 2 2/10 = 20% 30%+50%+20%

=100%

Total 10 10/10 = 100%


Histogram

Histogram: It is a graphical presentation of the absolute frequency distribution. It is simply a bar chart of continuous data that has been classified into a frequency distribution

47

0

1

2

3

4

5

Frequency

Relative Frequency

Percent

0 15 25 35 45 55

Lower Boundary

Bars Touch

Count Class Frequency

15 but <25 3

25 but <35 5

35 but <45 2


Frequency Polygon

Frequency Polygon: It is constructed by plotting the mid-point of each interval on the horizontal axis & absolute frequency for that interval on the vertical axis

48

0

1

2

3

4

5

Frequency

Relative Frequency

Percent

0 10 20 30 40 50 60

Midpoint

Count

Fictitious Class

Class Frequency

15 but <25 3

25 but <35 5

35 but <45 2


Case for Central Tendency

You are looking to invest in a Mutual Fund. There are 20-30 mutual funds available to chose from. How would you decide which mutual fund is better?

What are the basic questions you might have during Mutual Fund selection?

What is the fund’s average return?

What is its Variation?

Mutual Fund Returns are Random Variables.

What is a random variable?

Variation is irrespective of direction. It does not give the inclination or the tendency of that variable.

49


Numerical Data Properties

50

Central Tendency (Location)

Variation (Dispersion)

Shape


Measures of Central Tendency

Measures of Central: Tendency summarize the location on which the data are centered

Population Mean: calculated as

Where there are N members in the population and each observation is Xi i =1, 2, …N.

All the members are equally likely to occur

Sum of all the deviations is zero.

Sample Mean: calculated as

Where there are n observations in the sample and each observation is Xi, i =1, 2, …n. It is also the arithmetic mean of the sample observations.

All the members are equally likely to occur

51

N

i

iX XN 1

1

n

i

iXn

X1

1


Measures of Central Tendency

Median: calculated as the middle observation in a group that has been ordered in either ascending or descending order.

In an odd-numbered group this is the (n+1)/2 position.

In an even numbered group it is the average of the values in the n/2 & (n+2)/2 positions.

Mode: is the most frequently occurring value in the distribution. A distribution may have one, more than one, or no mode.

52


Other Definitions for Means

Measures of central tendency summarize the location on which the data are centered.

Arithmetic Mean: The following formula is used when all observations are equally likely:

Geometric Mean: is calculated as

Where there are n observations and each observation is Xi.

Compound Annual Growth Rate(CAGR): It’s the geometric mean of the returns.

Example: The returns on the equity portfolio had been 10%, 15% and 20% from last 3 years. Calculate CAGR.

Harmonic Mean: It is calculated as = (N/(∑1/xi ) where i = 1 to N

It is used for calculations as the cost of shares purchased over time

53

n

i

iweighted Xn

X1

1

nnXXXXG ...321


Use of Arithmetic Mean or Geometric Mean

Geometric mean of past annual returns is the appropriate measure of past performance. It gives the average annual compound return

Example: With annual returns of 6% & 13% over two years, the geometric mean of return is,

[(1.06)(1.13)]1/2 – 1 = 1.0944 – 1 = 0.0944 = 9.44%

Arithmetic mean is appropriate for forecasting single period returns in future periods

Geometric mean is appropriate for forecasting future compound returns over multiple periods

54


Question

1. A portfolio of non-dividend-paying stocks earned a geometric mean return of 5 percent between January 1, 1995, and December 31, 2001. The arithmetic mean return for the same period was 6 percent. If the market value of the portfolio at the beginning of 1995 was $100,000, the market value of the portfolio at the end of 2001 was closest to:

A. $140,710

B. $142,000

C. $150,363

2. Which measure of scale has a true zero point as the origin?

A. Nominal scale

B. Ordinal scale

C. Ratio scale

3. When calculated for the same data and provided there is variability in the observations, the geometric mean will most likely be:

A. equal to the arithmetic mean

B. less than the arithmetic mean

C. greater than the arithmetic mean

55


Solution

1. C Identify what you are being asked for – Portfolio Ending value P12/31/2001

Given the following:

• Portfolio Beginning value = P1/1/1995 =$100,000

• Geometric mean return = 5%, Arithmetic mean return = 6%

• Number of periods = 7, Non-dividend paying stocks in portfolio.

• Identify correct approach – use geometric mean return and formula

• Pt+7 = (1+r)7 Pt

• Pt+7 = (1.05)7* $100,000 = $140,710

2. C Ratio scales are the strongest level of measurement; they quantify differences in the size of data and have a true zero point as the origin.

3. B AM>=GM>=HM (always)

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Quantile (or Fractile)

Value at or below which a stated proportion of the data in the distribution lies

Quartile (4 Classes): 25%

Quintile (5 Classes): 20%

Decile (10 Classes): 10%

Percentile (100 Classes): 1%

Question:

What is the 3rd Quartile for the following data (in %age):

8, 10, 0, 0, 0, 15, 13, 12, 11, 9, 7, 9

Approach for the solution:

Arrange in Increasing Order

3rd Quartile would be (Number of data-points + 1) * 3/4

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Question

1. Below are the returns on 22 industry groups of stocks over the past year: 5%, 6%, 8%, 19%, 7%, -2%, 3%, 15%, 28%, 4%, -31%, 22%, 9%, 28%, -23%, 2%, 11%, -13%, -7%, 17%, 21%, -22%. What is the 40th percentile of this array of data?

A. 4.6%

B. 6.4%

C. 6.6%

2. What does it mean to say that an observation is at the eighty-fifth percentile?

A. 85% of all the observations are above that observation

B. it is 85% above the lowest number in series

C. 85% of all the observations are below that observation

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Solution

1. B. The location of the 40th percentile is (22+1)(40/100) = 9.2. The 9th & 10th lowest returns are 4% & 5%, so the 40th percentile is 4 + 0.2 ( 5 – 4 ) = 4.2%

2. C. If the observation falls at the eighty-fifth percentile, 85% of all the observations fall below that observation.

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Measures of Dispersion

Range: It is the difference between the maximum & minimum values in a dataset

Mean Absolute Deviation (MAD): It is the average of the data’s absolute deviations from the mean

Population Variance: It is the average of the population’s squared deviations from the mean

The above formula holds true for discrete distributions in which all observations are equally likely.

The population standard deviation is simply the square root of the population variance

Sample Variance: is the average of the sample data’s squared deviations from the sample mean

The sample standard deviation is simply the square root of the sample variance

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n

i

i XXn

MAD1

1

N

i

iXN 1

22 1s

n

i

i XXn

s1

22

1

1


Questions

Use the following frequency distribution for questions 1 to 3:

1. The number of intervals in the frequency table is: A) 6 B) 5 C) 7

2. The sample size is: A) 20 B) 22 C) 24

3. The relative frequency for the fourth interval is: A) 10.0% B) 16.0% C) 43.8%

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Return, R Frequency

-10% up to 0% 5

0% up to 10% 6

10% up to 20% 4

20% up to 30% 3

30% up to 40% 2

40% to 50% 2


Questions (Cont…)

4. ABC Corp. Annual Sock Prices are given as below. What is the Mean Absolute Deviation for the same?

5. What is the mean absolute deviation for ABC stock returns?

A. 9.8%

B. 8.0%

C. 9.0%

6. Which of the following statements about standard deviation is TRUE?

A. It is always denominated in the same units as that of the original data.

B. Standard deviation = square of the variance

C. It can be a positive or a negative number.

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2005 2006 2007 2008 2009 2010

13% 7% 9% -15% 6% -8%


Solution

1. A. An interval is the set of return values that an observation falls within. Simple count the return intervals on the table – there are six of them

2. B. The sample size is the sum of all the frequencies in the distribution, or 5+6+4+3+2+2 = 22

3. C. The relative frequency is found by dividing the frequency of the interval by the total number of frequencies: 3/22 = 13.63%

4. C. The mean absolute deviation is found by taking the mean of the absolute values of the deviations from the mean. Here mean = 2, MAD = 54/6 = 9%

5. C. It can be calculated as:

0.5*0.5*0.18*0.18 + 0.5*0.5*0.25*0.25 + 2*0.4*0.5*0.5*0.18*0.25 = 0.3272

Taking square root, we get standard deviation = 0.189 = 18.9%

6. A. It is always denominated in the same units as that of the original data.

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Chebyshev’s Inequality

For any set of observations

Regardless of the shape of distribution

%age of observations that lie between k standard deviation is at least

(1 – 1/ k2) for k > 1

Implying

36% of observations within +/- 1.25 sigma

75% within +/- 2 sigma

History of the Inequality

The inequality is named after the Russian mathematician Pafnuty Chebyshev, who first stated the inequality without proof in 1874. Ten years later the inequality was proved by Markov in his Ph.D. dissertation.

Due to variances in how to represent the Russian alphabet in English, it is Chebyshev is also spelled as Tchebysheff

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Measures of Risk vs. Return

Coefficient of variation

CV shows relative dispersion. If X is returns on an asset then CV shows the amount of risk (measured by sample standard deviation s) for every % of mean return on the asset. The lower an asset’s CV, the more attractive it is in risk per unit of return.

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X

sCV

Sharpe measure

Sharpe Measure is a more precise return-risk measure as it takes into account that an investor can earn the risk-free rate (rf), without bearing any risk. Hence a portfolio’s risk (measured by its standard deviation (σp) must be compared to its return in excess of the risk-free rate . The higher is Sharpe ratio, the better the return-risk tradeoff on the portfolio for an investor

p

fp rrSM

s

)(

You will learn more about the Sharpe Ratio in Capital Market Theory

M

CML E(Rp)

σp

Efficient Frontier

RFR

Borrowing Portfolio

Lending Portfolio


Shortfall Risk & Safety-first Criteria

Shortfall Risk:

Probability that portfolio value will fall below a target value over a given time

Roy’s Safety-first Criteria:

Optimal portfolio minimizes the probability that the return of the portfolio falls below minimum acceptable level (RL)

Minimize P (Rp<RL)

This ratio is very similar to Sharpe Ratio

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p

RL-E(RP) RatioFirst Safety

s


Questions

1. Which of the following portfolios provides the optimal “safety first” return if the minimum acceptable return is 9 percent?

A. Either Portfolio 1 or 2 or 3

B. Portfolio 4

C. Portfolio 2

2. A stock with a coefficient of variation of 0.50 has a(n):

A. Variance equal to half the stock's expected return.

B. Expected return equal to half the stock's variance.

C. Standard deviation equal to half the stock's expected return.

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Portfolio Expected Return (%) Standard Deviation (%)

1 12 4

2 13 5

3 11 3

4 9 2


Solution

1. Roy’s safety-first criterion requires the maximization of the SF Ratio: SF Ratio = (expected return – threshold return) / standard deviation Portfolio #2 has the highest safety-first ratio at 0.80

2. C. Standard deviation equal to half the stock's expected return.

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XsCV

Portfolio Expected Return (%) Standard Deviation (%) SF Ratio

1 12 4 0.75

2 13 5 0.80

3 11 3 0.67

4 9 2 0.00


Skewness

Describes How Data Are Distributed

Measures of Shape

Kurtosis = How Peaked or Flat

Skewness = Measure of Asymmetry

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Symmetric

Mean = Median = Mode

Positive-Skewed

Mode Median Mean

Negative-Skewed

Mean Median Mode


Skewness

Frequency distribution that is not symmetric is skewed

Positively-skewed distribution is characterized by many small losses but a few extremely large gains. It has a long tail on the right side of the distribution

Negatively-skewed distribution is characterized by many small gains but a few extremely large losses. It has a long tail on the left-hand side of the distribution

Skewness arises as a result of the properties of asset prices and returns. A share price can never be negative – there is a lower limit on the asset’s returns (-100%) but no theoretical limit on its upper limit – so an asset’s return may be positively-skewed

Symmetrical distribution: Mean = Median = Mode

Positively-skewed distribution: Mean > Median > Mode

Negatively-skewed distribution: Mean < Median < Mode

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Kurtosis

A frequency distribution that is more or less peaked than a Normal distribution is said to exhibit kurtosis. If the distribution is more peaked than a Normal (i.e. exhibits “fat tails”) it is leptokurtic. If it is less peaked than a Normal it is called platykurtic.

A distribution is mesokurtic if it has the same kurtosis as a normal distribution

Kurtosis describes the degree of “flatness” of a distribution, or width of its tails.

Excess Kurtosis = Kurtosis - 3

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4

1

4)(

s

n

i

ix

K

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-4 -3 -2 -1 0 1 2 3 4

Platykurtic K<3 Mesokurtic

K=3

Leptokurtic K>3


Kurtosis (Cont…)

Because of the fourth power, large observations in the tail will have large weight and hence create large kurtosis. Such a distribution is called leptokurtic, or fat tailed.

Positive excess kurtosis, i.e. a leptokurtic distribution, means that large positive and negative deviations from the mean have higher probabilities for occurring than they would under a Normal distribution.

If an portfolio’s returns are leptokurtic then its true risk is higher than the risk suggested by an analysis that assumes returns are Normally distributed. This is important for Value at Risk (VAR) calculations that must assume distributions for asset returns in a portfolio.

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Question

1. Which of the following statements about the arithmetic mean is FALSE?

A. The arithmetic mean is the only measure of central tendency where the sum of the deviations of each observation from the mean is always zero.

B. All interval data sets have an arithmetic mean.

C. If the distribution is skewed to the left then the mean will be greater than the median.

2. The correlation coefficient for a series of returns on a pair of investments is equal to 0.80. The covariance of returns is 0.06974 . Which of the following are possible variances for the returns on the two investments?

A. 0.02 and 0.44

B. 0.03 and 0.28

C. 0.04 and 0.19

3. A distribution with a mode of 40 and a range of 20 to 55 would most likely be:

A. positively skewed.

B. negatively skewed

C. normally distributed

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Question (Cont…)

Use the following data for question 3 & 4: The annual returns for RSB’s common stock over the years 2007, 2008, 2009 & 2010 are 12%, 11%, 14% & -10%.

4. What is the arithmetic mean return for RSB’s common stock?

A. 6.75%

B. 8.75%

C. 9.25%

5. What is the geometric mean return for RSB’s common stock?

A. 6.27%

B. 10.62%

C. It cannot be determined because the 2009 return is negative

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Question (Cont…)

6. A portfolio manager manages two different sector specific portfolios with the following annual returns: Portfolio M : 7%, 5%, 12% ||Portfolio N: 15%, 22%, 24% The best measure for the comparative performance of the two portfolios can be obtained from which of the following measures of dispersion:

A. Standard deviation

B. Compounded annual rate of return

C. Coefficient of variation

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Questions (Cont…)

7. The following values about two portfolios have been obtained The risk free rate is 5%. Which of the following statements about the two portfolios is true?

A. Portfolio A has a risk not commensurate with its returns as compared to portfolio B

B. A risk neutral investor would prefer portfolio B over A

C. Given the risk in the two portfolios a risk neutral investor would invest in A rather than B

8. Which of the following is most accurate regarding a distribution of returns that has a mean greater than its median?

A. It is positively skewed

B. It is a symmetric distribution

C. It has positive excess kurtosis

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Expected returns Standard deviation

Portfolio A 15% 4%

Portfolio B 22% 5%


Questions (Cont…)

9. What is the coefficient of variation for a distribution with a mean of 30 and a variance of 9?

A. 40%

B. 25%

C. 10%

10. Which of the following statements about the Chebyshev’s Theorem is most likely to be correct for a population with mean µ and standard deviation σ?

A. At least 75% of the data lies within µ±2σ

B. At least 65% of the data lies within µ±3σ

C. At least 94% of the data lies within µ±1σ

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Solution

1. C. If the distribution is skewed to the left then the mean will be greater than the median

2. A. The correlation coefficient is:

0.06974 / [(Std Dev A) (Std Dev B)] = 0.8.

(Std Dev A) (Std Dev B) = 0.08718.

Since the standard deviation is equal to the square root of the variance, each pair of variances can be converted to standard deviations and multiplied to see if they equal 0.08718.

√0.04 = 0.20 and √0.19 = 0.43589.

The product of these equals 0.08718.

3. B. The distance to the left from the mode to the beginning of the range is 20. The distance to the right from the mode to the end of the range is 15. Therefore, the distribution is skewed to the left, which means that it is negatively skewed.

4. A. (12% + 11% + 14% - 10%)/4 = 6.75%

5. A. (1.12*1.11*1.14*0.9)^0.25 - 1 = 6.27%

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Solution

6. C. Here, Coefficient of variation is the best measure for comparing the two portfolios

7. A. Compare sharpe ratio of both portfolios and portfolio A's sharpe is 2.5 - less as compared to B's 3.4

As the sharpe ratio of Portfolio A is lower than that of B, it means portfolio A gives lower returns for the amount of risk involved in it. A risk averse investor will always choose B over A and it makes no difference to a risk neutral investor whether portfolio A has more risk or portfolio B

8. A. A distribution with a mean greater than its median is positively skewed, or skewed to the right. The skew “pulls” the mean. Kurtosis deals with the overall shape of the distribution & not its skewness.

9. C. Coefficient of variation, CV = standard deviation / mean. The standard deviation is the square root of the variance, or 9½ = 3. So, CV = 3 / 30 = 20%.

10. A. At least 75% of the data lies within µ±2σ

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Quantitative Methods - I