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1-1 1 A Brief History of Risk and Return

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Page 1: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-1

1

A Brief History of

Risk and Return

Page 2: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-2

A Brief History of Risk and Return

• Two key observations:

1. There is a substantial reward, on average, for bearing risk.

2. Greater risks accompany greater returns.

Page 3: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-3

Dollar & Percent Returns

• Total dollar return = the return on an investment measured in dollars• $ return = dividends + capital gains

• Total percent return is the return on an investment measured as a percentage of the original investment.• % Return = $ return/$ invested• The total percent return is the return for each

dollar invested.

Page 4: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-4

Percent Return

t

ttt

t

tt

t

t

P

PPD

CGYDY

P

PPCGY

P

DDY

11

1

1

Return %

Return %

Dividend Yield

Capital Gains Yield

Page 5: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-5

Example: Calculating Total Dollar and Total Percent Returns

• You invest in a stock with a share price of $25. • After one year, the stock price per share is $35. • Each share paid a $2 dividend.

• What was your total return?

Dollars Percent

Dividend $2.00 $2/25 = 8%

Capital Gain $35 - $25 = $10 $10/25= 40 %

Total Return $2 + $10 = $12 $12/$25 = 48%

Page 6: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-6

Annualized Returns

Effective Annual Rate (EAR)

1)HPR1(EAR M

Where:

HPR = Holding Period Return

M = Number of Holding Periods per year

Page 7: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-7

Annualized Returns – Example 1

P0 = $20 P.33 = $22 t = “.33” since 4 months is 1/3 of a year

4-month HPR = 3 periods per year

Holding Period Return (HPR)

Annualized Return

(EAR)

%$

$$

.

1020

2022

P

PP

0

0330

%.

).(

133

1101 3

You buy a stock for $20 per share on January 1.

Four months later you sell for $22 per share.

No dividend has been paid yet this year.

Page 8: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-8

Annualized Returns – Example 2

P0 = $20 P2 = $28

HPR = 2 years (t = 2)

HPR per year = ½ (0.50)

Holding Period Return

(HPR)

Annualized Return

(EAR)

%$

$$

4020

2028

P

PP

0

02

%.

).( .

3218

1401 500

Suppose the $20 stock you bought on January 1 is selling for $28 two years later

No dividends were paid in either year.

Page 9: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-9

A $1 Investment in Different Typesof Portfolios, 1926—2006

Page 10: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-10

Financial Market History

Page 11: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-11

The Historical Record:Total Returns on Large-Company Stocks

Page 12: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-12

The Historical Record: Total Returns on Small-Company Stocks

Page 13: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-131-13

The Historical Record: Total Returns on U.S. Bonds.

Page 14: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-141-14

The Historical Record: Total Returns on T-bills.

Page 15: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-15

The Historical Record: Inflation

Page 16: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-16

Historical Average Returns

• Historical Average Return = simple, or arithmetic average.

• Using the data in Table 1.1: • Sum the returns for large-company stocks from 1926

through 2006, you get about 984 percent.

• Divide by the number of years (80) = 12.3%.

• Your best guess about the size of the return for a year

selected at random is 12.3%.

n

return yearly Return AverageHistorical

n

1i

Page 17: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-171-17

Average Annual Returns for Five Portfolios

Page 18: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-18

Average Returns: The First Lesson

• Risk-free rate:

• Rate of return on a riskless investment• Risk premium:

• Extra return on a risky asset over the risk-free rate

• Reward for bearing risk• The First Lesson: There is a reward, on

average, for bearing risk.

Page 19: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-19

Average Annual Risk Premiums for Five Portfolios

Page 20: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-20

Risk Premiums

• Risk is measured by the dispersion or spread of returns

• Risk metrics:• Variance • Standard deviation

• The Second Lesson: The greater the potential reward, the greater the risk.

Page 21: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-21

Return Variability Review and Concepts

• Variance (σ2)• Common measure of return dispersion • Also call variability

• Standard deviation (σ) • Square root of the variance• Sometimes called volatility• Same "units" as the average

Page 22: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-22

Return Variability: The Statistical Tools for Historical Returns

• Return variance: (“N" =number of returns):

• Standard Deviation

1N

RR σ VAR(R)

N

1i

2

i2

VAR(R) σ SD(R)

Page 23: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-23

Example: Calculating Historical Variance and Standard Deviation

• Using data from Table 1.1 for large-company stocks:(1) (2) (3) (4) (5)

Average Difference: Squared:Year Return Return: (2) - (3) (4) x (4)1926 11.14 11.48 -0.34 0.121927 37.13 11.48 25.65 657.821928 43.31 11.48 31.83 1013.021929 -8.91 11.48 -20.39 415.831930 -25.26 11.48 -36.74 1349.97

Sum: 57.41 Sum: 3436.77

Average: 11.48 Variance: 859.19

29.31Standard Deviation:

Page 24: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-24

Return Variability Review and Concepts

• Normal distribution: • A symmetric, bell-shaped frequency

distribution (the bell-shaped curve)• Completely described with an average

and a standard deviation (mean and variance)

• Does a normal distribution describe asset returns?

Page 25: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-25

Frequency Distribution of Returns on Common Stocks, 1926—2006

Page 26: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-261-26

Historical Returns, Standard Deviations, and Frequency Distributions: 1926—2006

Page 27: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-271-27

The Normal Distribution and Large Company Stock Returns

Page 28: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-28

Arithmetic Averages versusGeometric Averages

• The arithmetic average return answers the question: “What was your return in an average year over a particular period?”

• The geometric average return answers the question: “What was your average compound return per year over a particular period?”

Page 29: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-29

Geometric Average Return: Formula

1R1(...)R1()R1(GAR /N1N)21

Where:

Ri = return in each period

N = number of periods

Equation 1.5

Page 30: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-30

Geometric Average Return

1)R1(GARN/1N

1ii

Where:

Π = Product (like Σ for sum)

N = Number of periods in sample

Ri = Actual return in each period

Page 31: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-31

Example: Calculating a Geometric Average Return• Using the large-company stock data from Table 1.1:

Percent One Plus CompoundedYear Return Return Return:1926 11.14 1.1114 1.11141927 37.13 1.3713 1.52411928 43.31 1.4331 2.18411929 -8.91 0.9109 1.98951930 -25.26 0.7474 1.4870

1.0826

8.26%

(1.4896)^(1/5):

Geometric Average Return:

Page 32: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-32

Geometric Average Return

Year % Return $$ Invested1.0000$

1926 11.14 1.1114$ 1927 37.13 1.5241$ 1928 43.31 2.1841$ 1929 -8.91 1.9895$ 1930 -25.26 1.4870$

N 5I/Y CPT = 8.26%PV (1.0000)$

PMT 0FV 1.4870$

Page 33: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-33

Arithmetic Averages versusGeometric Averages

• The arithmetic average tells you what you earned in a typical year.

• The geometric average tells you what you actually earned per year on average, compounded annually.

• “Average returns” generally means arithmetic average returns.

Page 34: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-34

Geometric versus Arithmetic Averages

• For forecasting future returns:• Arithmetic average "too high" for long forecasts

• Geometric average "too low" for short forecasts

Page 35: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-35

Blume’s Formula

• Form a “T” year average return forecast from arithmetic and geometric averages covering “N” years, N>T.

Average Arithmetic1-N

T-N AverageGeometric

1N

1TTR

)(

Page 36: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-36

Check This 1.5aCompute the Average Returns

Year % Return $$ Invested1.0000$

1926 10 1.1000$ 1927 16 1.2760$ 1928 -5 1.2122$ 1929 -8 1.1152$ 1930 7 1.1933$

4.00

N 5I/Y CPT = 3.60%PV (1.0000)$

PMT 0FV 1.1933$

Arithmetic Average

Geometric Average

Page 37: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-37

Check This 1.5b

BLUME'S FORMULA

Arithmetic Average 4.0%Geometric Average 3.6%

N 25

For T = 5

AverageArithmeticAverageGeometric N

TTR

1-N

T-N

1

1

)(

%...

%).)(.(%).)(.(

%)(%).(

%%.)(

9333332360010

04833306316670

424

2063

24

4

41-25

5-25 63

125

155

R

Page 38: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-38

Check This 1.5bBLUME'S FORMULA

Arithmetic Average 4.0%Geometric Average 3.6%

N 25

For T = 10

AverageArithmeticAverageGeometric N

TTR

1-N

T-N

1

1

)(

%...

%).)(.(%).)(.(

%)(%).(

%%.)(

853502351

04625006337500

424

1563

24

9

41-25

10-25 63

125

11010

R

Page 39: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-39

Risk and Return

• The risk-free rate represents compensation for the time value of money.

• First Lesson: • If we are willing to bear risk, then we can

expect to earn a risk premium, at least on average.

• Second Lesson: • The more risk we are willing to bear, the

greater the expected risk premium.

Page 40: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

1-40

Historical Risk and Return Trade-Off

Page 41: 1-1 1 A Brief History of Risk and Return. 1-2 A Brief History of Risk and Return Two key observations: 1. There is a substantial reward, on average, for

41

Chapter 1 End

A Brief History of

Risk and Return


Risk & Return ~ Full (1)

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