L Pch6

Levy and Post, Investments © Pearson Education Limited 2005

Slide 6.1

Investments

Chapter 6: Rates of Return


Slide 6.2

Different Methods yield Different Rates of Return

Investment managers, who are rated on their performance, have an incentive to use those methods that make them look best.


Slide 6.3

Standardisation of Methods

• In response to this incentive the Association of Investment Management and Research (AIMR) has established guidelines to calculate rates of return.

• A global standard is also evolving (the Global Investment Performance Standards).


Slide 6.4

Ex-post and Ex-ante Rates of Return

• Ex-post rates of return

Rates of return that has already been earned.

• Ex-ante rates of return

Rates of return expected to occur in the future.


Slide 6.5

Simple Rate of Return

• Two components included:

1. Capital gains or losses.

2. Cash flow yield.

• Calculated over the investment period and depicted as a percentage of the dollar amount invested.


Slide 6.6

Simple Returns• Dollar Returns

– the sum of the cash received and the change in value of the asset, in dollars.

Time 0 1

Initial investment

Ending market value

Dividends

•Percentage Returns

–the sum of the cash received and the change in value of the asset divided by the original investment.


Slide 6.7

Dollar Return = Dividend + Change in Market Value

Simple Returns

yield gains capitalyield dividend

uemarket val beginning

uemarket valin change dividend

uemarket val beginning

returndollar return percentage


Slide 6.8

Adjusted Rate of Return: I

• The simple rate of return does not account for the timing of the cash flows received.

• Two approaches to compute the adjusted rate of return:

1. Linking method.

2. Index method.


Slide 6.9

Sample Mean

Two methods:

1. Arithmetic average.2. Geometric average.

The arithmetic average doesn’t compound the rates of return. Hence, if the returns are not constant, the arithmetic average is upwardly biased.


Slide 6.10 Holding-Period Returns (Linking Method)

• The holding period return (linking method) is the return that an investor would get when holding an investment over a period of n years, when the return during year i is given as ri:

1)1()1()1(

return period holding

21

nrrr


Slide 6.11

Holding Period Return: Example

• Suppose your investment provides the following returns over a four-year period:Year Return

1 10%2 -5%3 20%4 15% %21.444421.

1)15.1()20.1()95(.)10.1(

1)1()1()1()1(

return period holdingYour

4321

rrrr


Slide 6.12


• An investor who held this investment would have actually realized an annual return of 9.58%:

Year Return

1 10%2 -5%3 20%4 15% %58.9095844.

1)15.1()20.1()95(.)10.1(

)1()1()1()1()1(

return average Geometric

4

43214

g

g

r

rrrrr

• So, our investor made 9.58% on his money for four years, realizing a holding period return of 44.21%

4)095844.1(4421.1


Slide 6.13


• Note that the geometric average is not the same thing as the arithmetic average:Year Return

1 10%2 -5%3 20%4 15%

%104

%15%20%5%104

return average Arithmetic 4321

rrrr


Slide 6.14

Arithmetic or Geometric?

• Start with 100, remain 100 first year, decrease to 50 next year and increase to 100 the third year.

Year Return1 0%2 -50%3 100% %0

1)11()5.01()1(

)1()1()1()1(

return average Geometric

3

3213

g

g

r

rrrr

!!%!67.163

%100%50%03

return average Arithmetic 321

rrr


Slide 6.15

Adjusted Rate of Return: IIBesides timing of the cash flows, rates of return also need to be adjusted for other developments:

1. Stock splits.

2. Stock dividends.

3. Rights acquired.

4. Accrued interest on bonds (accrues daily).

5. Changes in the composition of the portfolio.


Slide 6.16

Other Complications

Taxes, inflation and (for foreign investors) exchange rates generally change the return of an investor.

Solutions:

1. After-tax rates of return.

2. Inflation-adjusted rates of return.

3. Exchange rate-adjusted rates of return.


Slide 6.17

INDICES

An index measures the performance of a certain segment of the financial markets.


Slide 6.18

Stock Indexes

Can be categorized as:

1. Price-weighted (Example: Dow Jones Industrial Average.)

2. Value-weighted(Example: S & P 500.)

3. Equally-weighted


Slide 6.19

Bond Indices: I

The set of bonds in a bond index is always changing.

Reasons:

1. Bonds have finite maturities.

2. Call features within bonds.


Slide 6.20

Sample Return Statistics

Describe or summarize a sample of past rates of return

Note: (statistics describing a sample of ex-ante rates of return are discussed in chapter 7).


Slide 6.21

Risk Statistics

• There is no universally agreed-upon definition of risk.

• The measures of risk that we discuss are variance and standard deviation.– The standard deviation is the standard statistical

measure of the spread of a sample, and it will be the measure we use most of this time.

– Its interpretation is facilitated by a discussion of the normal distribution.


Slide 6.22Return Statistics

• The history of capital market returns can be summarized by describing the – average return

– the standard deviation of those returns

– the frequency distribution of the returns.

T

RRR T )( 1

1

)()()( 222

21

T

RRRRRRVARSD T


Slide 6.23

Sample Covariance andSample Correlation

Give information on the dependency between various assets and how they move together:

• Covariance (measure of dependence in units such as dollars or percentages).

• Correlation (measure of dependence on a scale of –1 to +1, thus more comparable).


Slide 6.24

Holding Period Returns

• A famous set of studies dealing with the rates of returns on common stocks, bonds, and Treasury bills was conducted by Roger Ibbotson and Rex Sinquefield.

• They present year-by-year historical rates of return starting in 1926 for the following five important types of financial instruments in the United States:– Large-Company Common Stocks– Small-company Common Stocks– Long-Term Corporate Bonds– Long-Term U.S. Government Bonds– U.S. Treasury Bills


Slide 6.25

The Future Value of an Investment of $1 in 1926

0,1

10

1000

1930 1940 1950 1960 1970 1980 1990 2000

Common StocksLong T-BondsT-Bills

$40.22

$15.64

63.845,2$)1()1()1(1$ 199919271926 rrr

Source: © Stocks, Bonds, Bills, and Inflation 2000 Yearbook™, Ibbotson Associates, Inc., Chicago (annually updates work by Roger G. Ibbotson and Rex A. Sinquefield). All rights reserved.


Slide 6.26 Historical Returns, 1926-1999


– 90% + 90%0%

Average Standard Series Annual Return Deviation Distribution

Large Company Stocks 13.0% 20.3%

Small Company Stocks 17.7 33.9

Long-Term Corporate Bonds 6.1 8.7

Long-Term Government Bonds 5.6 9.2

U.S. Treasury Bills 3.8 3.2

Inflation 3.2 4.5


Slide 6.27 Average Stock Returns and Risk-Free Returns

• The Risk Premium is the additional return (over and above the risk-free rate) resulting from bearing risk.

• One of the most significant observations of stock market data is this long-run excess of stock return over the risk-free return.– The average excess return from large company common

stocks for the period 1926 through 1999 was 9.2% = 13.0% – 3.8%

– The average excess return from small company common stocks for the period 1926 through 1999 was 13.9% = 17.7% – 3.8%

– The average excess return from long-term corporate bonds for the period 1926 through 1999 was 2.3% = 6.1% – 3.8%


Slide 6.28

Risk Premiums

• Rate of return on T-bills is essentially risk-free.• Investing in stocks is risky, but there are

compensations.• The difference between the return on T-bills and

stocks is the risk premium for investing in stocks.• An old saying on Wall Street is “You can either

sleep well or eat well.”


Slide 6.29

Risk Premia

• Suppose that The Wall Street Journal announced that the current rate for on-year Treasury bills is 5%.

• What is the expected return on the market of small-company stocks?

• Recall that the average excess return from small company common stocks for the period 1926 through 1999 was 13.9%

• Given a risk-free rate of 5%, we have an expected return on the market of small-company stocks of 18.9% = 13.9% + 5%


Slide 6.30

The Risk-Return Tradeoff

2%

4%

6%

8%

10%

12%

14%

16%

18%

0% 5% 10% 15% 20% 25% 30% 35%

Annual Return Standard Deviation

Ann

ual R

etur

n A

vera

ge

T-Bonds

T-Bills

Large-Company Stocks

Small-Company Stocks


Slide 6.31

Rates of Return 1926-1999

-60

-40

-20

0

20

40

60

26 30 35 40 45 50 55 60 65 70 75 80 85 90 95

Common Stocks

Long T-Bonds

T-Bills



Slide 6.32

Stock Market Volatility

0

10

20

30

40

50

60

1926

1935

1940

1945

1950

1955

1960

1965

1970

1975

1980

1985

1990

1995

1998


The volatility of stocks is not constant from year to year.

Business

L Pch6