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Chapter 1

Brief History of Risk and Return

Ayşe Yüce – Ryerson University

38.pdf

VALUATION AND MANAGEMENT

Investments

JORDAN MILLER DOLVIN YÜCE

third canadian edition

fundamentals of

To become a wise investor (maybe even one with too much money), you need to know:

1. How to calculate the return on an investment using different methods.

2. The historical returns on various important types of investments.

3. The historical risk on various important types of investments.

4. The relationship between risk and return.

Ayşe Yüce – Ryerson University

Learning Objectives

You can retire with One Million Dollars (or more).
How? Suppose:
You invest $300 per month.
Your investments earn 9% per year.
You decide to take advantage of deferring taxes on your investments.
It will take you about 36.25 years. Hmm. Too long.

Example I:

Who Wants To Be A Millionaire?

Ayşe Yüce – Ryerson University

Example II:

Who Wants To Be A Millionaire?

Instead, suppose:

You invest $500 per month.
Your investments earn 12% per year
you decide to take advantage of deferring taxes on your investments

It will take you 25.5 years.

Realistic?

$250 is about the size of a new car payment, and perhaps your employer will kick in $250 per month
Over the last 84 years, the S&P 500 Index return was about 12%

Try this calculator: cgi.money.cnn.com/tools/millionaire/millionaire.html

Our goal in this chapter is to see what financial market history can tell us about risk and return.

There are two key observations:
First, there is a substantial reward, on average, for bearing risk.
Second, greater risks accompany greater returns.

These observations are important investment guidelines.

Ayşe Yüce – Ryerson University

A Brief History of Risk And Return

Total dollar return is the return on an investment measured in dollars, accounting for all interim cash flows and capital gains or losses.

Example:

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Dollar Returns

Total percent return is the return on an investment measured as a percentage of the original investment.

The total percent return is the return for each dollar invested.
Example, you buy a share of stock:

Percent Returns

Ayşe Yüce – Ryerson University

Example: Calculating Total Dollar And Total Percent Returns

Suppose you invested $1,400 in a stock with a share price of $35.

After one year, the stock price per share is $49.

Also, for each share, you received a $1.40 dividend.

What was your total dollar return?

$1,400 / $35 = 40 shares

Capital gain: 40 shares times $14 = $560

Dividends: 40 shares times $1.40 = $56

Total Dollar Return is $560 + $56 = $616

What was your total percent return?

Dividend yield = $1.40 / $35 = 4%

Capital gain yield = ($49 – $35) / $35 = 40%

Total percentage return = 4% + 40% = 44%

Note that $616 divided by $1,400 is 44%.

Ayşe Yüce – Ryerson University

Annualizing Returns, I

You buy 200 shares of Lowe’s Companies, Inc. at $18 per share. Three months later, you sell these shares for $19 per share. You received no dividends. What is your return? What is your annualized return?

Return: (Pt+1 – Pt) / Pt = ($19 - $18) / $18

= .0556 = 5.56%

Effective Annual Return (EAR): The return on an investment expressed on an “annualized” basis.

Key Question: What is the number of holding periods in a year?

This return is
known as the holding period percentage return.

Ayşe Yüce – Ryerson University

Annualizing Returns, II

1 + EAR = (1 + holding period percentage return)m

m = the number of holding periods in a year.

In this example, m = 4 (12 months / 3 months). Therefore:

1 + EAR = (1 + .0556)4 = 1.2416.

So, EAR = .2416 or 24.16%.

Ayşe Yüce – Ryerson University

$1 Investment in Canadian S&P/TSX Index

Ayşe Yüce – Ryerson University

A $1 Investment in Different Types of Portfolios, 1926-2009

Ayşe Yüce – Ryerson University

Financial Market History

Ayşe Yüce – Ryerson University

The Historical Record: Total Returns on Large-Company Stocks

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The Historical Record: Total Returns on Small-Company Stocks

Ayşe Yüce – Ryerson University

The Historical Record: Total Returns on Long-term U.S. Bonds

Ayşe Yüce – Ryerson University

The Historical Record: Total Returns on U.S. T-bills

Ayşe Yüce – Ryerson University

The Historical Record: Inflation

Ayşe Yüce – Ryerson University

Historical Average Returns

A useful number to help us summarize historical financial data is the simple, or arithmetic average.

Using the data in Table 1.1, if you add up the returns for large-company stocks from 1926 through 2009, you get about 987 percent.

Because there are 84 returns, the average return is about 11.75%. How do you use this number?

If you are making a guess about the size of the return for a year selected at random, your best guess is 11.75%.

The formula for the historical average return is:

Ayşe Yüce – Ryerson University

Average Annual Returns for Five Portfolios and Inflation

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Average Annual Risk Premiums for Five Portfolios

Ayşe Yüce – Ryerson University

Average Returns: The First Lesson

Risk-free rate: The rate of return on a riskless, i.e., certain investment.

Risk premium: The extra return on a risky asset over the risk-free rate; i.e., the reward for bearing risk.
The First Lesson: There is a reward, on average, for bearing risk.

By looking at Table 1.3, we can see the risk premium earned by large-company stocks was 7.9%!
Is 7.9% a good estimate of future risk premium?
The opinion of 226 financial economists: 7.0%.
Any estimate involves assumptions about the future risk environment and the risk aversion of future investors.

Ayşe Yüce – Ryerson University

World Stock Market Capitalization

International Equity Risk Premiums

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Why Does a Risk Premium Exist?

Modern investment theory centers on this question.

Therefore, we will examine this question many times in the chapters ahead.

We can examine part of this question, however, by looking at the dispersion, or spread, of historical returns.
We use two statistical concepts to study this dispersion, or variability: variance and standard deviation.
The Second Lesson: The greater the potential reward, the greater the risk.

Ayşe Yüce – Ryerson University

The Bear Growled and Investors Howled

Ayşe Yüce – Ryerson University

Return Variability: The Statistical Tools

The formula for return variance is ("n" is the number of returns):

Sometimes, it is useful to use the standard deviation, which is related to variance like this:

Ayşe Yüce – Ryerson University

Return Variability Review and Concepts

Variance is a common measure of return dispersion. Sometimes, return dispersion is also call variability.

Standard deviation is the square root of the variance.

Sometimes the square root is called volatility.

Standard Deviation is handy because it is in the same "units" as the average.

Normal distribution: A symmetric, bell-shaped frequency distribution that can be described with only an average and a standard deviation.

Does a normal distribution describe asset returns?

Ayşe Yüce – Ryerson University

Frequency Distribution of Returns on Common Stocks, 1926-2009

Ayşe Yüce – Ryerson University

Example: Calculating Historical Variance and Standard Deviation

Let’s use data from Table 1.1 for Large-Company Stocks.

The spreadsheet below shows us how to calculate the average, the variance, and the standard deviation (the long way…).

Ayşe Yüce – Ryerson University

Sheet1


(1)	(2)	(3)	(4)	(5)
		Average	Difference:	Squared:
Year	Return	Return:	(2) - (3)	(4) x (4)
1926	11.14	11.48	-0.34	0.12
1927	37.13	11.48	25.65	657.92
1928	43.31	11.48	31.83	1013.15
1929	-8.91	11.48	-20.39	415.75
1930	-25.26	11.48	-36.74	1349.83
Sum:	57.41		Sum:	3436.77

Average:	11.48		Variance:	859.19

		Standard Deviation:		29.31

Historical Returns, Standard Deviations, and Frequency Distributions: 1926-2009

Ayşe Yüce – Ryerson University

The Normal Distribution and Large Company Stock Returns

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Returns on Some “Non-Normal” Days

Ayşe Yüce – Ryerson University

Arithmetic Averages versus Geometric 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?”

When should you use the arithmetic average and when should you use the geometric average?

First, we need to learn how to calculate a geometric average.

Ayşe Yüce – Ryerson University

Example: Calculating a Geometric Average Return

Let’s use the large-company stock data from Table 1.1.

The spreadsheet below shows us how to calculate the geometric average return.

Ayşe Yüce – Ryerson University

Sheet1


	Percent	One Plus	Compounded
Year	Return	Return	Return:
1926	11.14	1.1114	1.1114
1927	37.13	1.3713	1.5241
1928	43.31	1.4331	2.1841
1929	-8.91	0.9109	1.9895
1930	-25.26	0.7474	1.4870

(1.4870)^(1/5):			1.0826

Geometric Average Return:			8.26%

Arithmetic Averages versus Geometric 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.

When we talk about average returns, we generally are talking about arithmetic average returns.
For the purpose of forecasting future returns:
The arithmetic average is probably "too high" for long forecasts.
The geometric average is probably "too low" for short forecasts.

Ayşe Yüce – Ryerson University

Geometric versus Arithmetic Averages

Ayşe Yüce – Ryerson University

Risk and Return

The risk-free rate represents compensation for just waiting.
Therefore, this is often called 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: Further, the more risk we are willing to bear, the greater the expected risk premium.

Ayşe Yüce – Ryerson University

Historical Risk and Return Trade-Off

Ayşe Yüce – Ryerson University

Dollar-Weighted Average Returns, I

There is a hidden assumption we make when we calculate arithmetic returns and geometric returns.

The hidden assumption is that we assume that the investor makes only an initial investment.

Clearly, many investors make deposits or withdrawals through time.

How do we calculate returns in these cases?

Ayşe Yüce – Ryerson University

Dollar-Weighted Average Returns, II

If you only make an initial investment at the start of year one:
The arithmetic average return is 2.50%.
The geometric average return is 2.23%.

Suppose you makes a $1,000 initial investment and a $4,000 additional investment at the beginning of year two.
At the end of year one, the initial investment grows to $1,100.
At the start of year two, your account has $5,100.
At the end of year two, your account balance is $4,845.
You have invested $5,000, but your account value is only $4,845.

So, the (positive) arithmetic and geometric returns are not correct.

Ayşe Yüce – Ryerson University

Dollar-Weighted Average Returns and IRR

Ayşe Yüce – Ryerson University

The US stock market in 2000s was one of the worst decades ever in investment history, investors were better off investing in anything else, including bonds and gold.
Many investors were lured to the stock market by the bull market in 1980s through the 1990s with over 17% average returns.
For investors counting on stocks for retirement plans, the most recent decade means many have fallen behind retirement goals.
Decline in dividends presented another hurdle for stock market, playing an important role in helping achieve a 9.5% average annual return since 1926 with a yield of 4%.

A Look Ahead

This textbook focuses exclusively on financial assets: stocks, bonds, options, and futures.
You will learn how to value different assets and make informed, intelligent decisions about the associated risks.
You will also learn about different trading mechanisms and the way that different markets function.

Ayşe Yüce – Ryerson University

Useful Internet Sites

cgi.money.cnn.com/tools/millionaire/millionaire.html (millionaire link)

finance.yahoo.com (reference for a terrific financial web site)
www.globalfinancialdata.com (reference for historical financial market data—not free)
www.robertniles.com/stats (reference for easy to read statistics review)
www.tsx.com (reference for the Toronto Stock Exchange)
www.osc.gov.on.ca (reference for the Ontario Securities Commission)

Ayşe Yüce – Ryerson University

Returns

Dollar Returns
Percentage Returns

The Historical Record

A First Look
A Longer Range Look
A Closer Look

Average Returns: The First Lesson

Calculating Average Returns
Average Returns: The Historical Record
Risk Premiums

Ayşe Yüce – Ryerson University

Chapter Review

75.pdf

VALUATION AND MANAGEMENT

Investments

JORDAN MILLER DOLVIN YÜCE

third canadian edition

fundamentals of

Chapter Review

Return Variability: The Second Lesson

Frequency Distributions and Variability
The Historical Variance and Standard Deviation
The Historical Record
Normal Distribution
The Second Lesson

Arithmetic Returns versus Geometric Returns

The Risk-Return Trade-Off

Ayşe Yüce – Ryerson University

Loss)

(or

Gain

Capital

Income

Dividend

Stock

Return

Dollar

Total

)

Investment

Beginning

(i.e.,

Price

Stock

Beginning

Stock

Return

Dollar

Total

Return

Percent

Price

Stock

Beginning

Loss)

(or

Gain

Capital

Income

Dividend

Stock

Return

Percent

return

yearly

Return

Average

Historical

VAR(R)

SD(R)

(

)

VAR(R)

(1)(2)(3)(4)(5)

AverageDifference:Squared:

YearReturnReturn:(2) - (3)(4) x (4)

192611.1411.48-0.340.12

192737.1311.4825.65657.92

192843.3111.4831.831013.15

1929-8.9111.48-20.39415.75

1930-25.2611.48-36.741349.83

Sum:57.41Sum:3436.77

Average:11.48Variance:859.19

29.31Standard Deviation:

PercentOne PlusCompounded

YearReturnReturnReturn:

192611.141.11141.1114

192737.131.37131.5241

192843.311.43312.1841

1929-8.910.91091.9895

1930-25.260.74741.4870

1.0826

8.26%

(1.4870)^(1/5):

Geometric Average Return: