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Slide 6.1

Investments

Chapter 6: Rates of Return

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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.

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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).

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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.

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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.

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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.

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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

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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.

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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.

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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

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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

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Slide 6.12

Holding Period Return: Example

• 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

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Slide 6.13

Holding Period Return: Example

• 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

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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

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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.

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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.

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Slide 6.17

INDICES

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

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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

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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.

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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).

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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.

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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

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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).

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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

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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.

Levy and Post, Investments © Pearson Education Limited 2005

Slide 6.26 Historical Returns, 1926-1999

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.

– 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

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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%

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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.”

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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%

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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

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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

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.

Levy and Post, Investments © Pearson Education Limited 2005

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

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.

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