chapter 1 corporate finance

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RISK AND RETURN LESSONS FROM MARKET HISTORY

Chapter 10

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KEY CONCEPTS AND SKILLS• Calculate the return on an investment• Compute the standard deviation of an investment’s

returns• Explain the connection between historical returns and

risks on various types of investments• Describe the importance of the normal distribution

and its relationship to investment return• Contextualize the US market risk premium in global

terms• Discern the impact on returns of the 2008 financial

crisis• Differentiate between arithmetic and geometric

average returns

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CHAPTER OUTLINE

10.1 Returns10.2 Holding Period Returns10.3 Return Statistics10.4 Average Stock Returns and Risk-Free

Returns10.5 Risk Statistics10.6 The U.S. Equity Risk Premium10.7 2008: A Year of Financial Crisis10.8 More on Average Returns

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10.1 RETURNS

• Returns have two components:

• Current Income (e.g., interest or dividends); and,

• Capital Gains (or Losses)

• Returns can be expressed in dollar or percentage terms.

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GRAPHIC REPRESENTATION OF RETURNS

• Dollar Returns:the sum of the current income received plus the change in value of the asset, in dollars.

Time 0 1

Initial investment

Ending market value

Dividends

Percentage Returns

the sum of the current income received plus the change in value of the asset, divided by the initial investment.

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FORMULAIC DEFINITION OF RETURNSDollar Return = Dividend + Change in Market Value

yield gains capitalyield dividend

uemarket val beginninguemarket valin change dividend

uemarket val beginningreturndollar Return Percentage

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RETURNS: EXAMPLE• Suppose you bought 100 shares of Wal-

Mart (WMT) one year ago today at $50. Over the last year, you received $100 in dividends ($1 per share × 100 shares). At the end of the year, the stock sells for $60. How did you do?• Quite well. You invested $50 × 100 =

$5,000. At the end of the year, you have stock worth $6,000 and cash dividends of $100. Your dollar gain was $1,100 = $100 + ($6,000 – $5,000).• Your percentage gain for the year is:• 1,100/5,000 = 22%

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RETURNS: EXAMPLE

Dollar Return:$1,100 gain

Time 0 1

-$5,000

$6,000

$100

Percentage Return:

22% = $5,000$1,100

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10.2 HOLDING PERIOD RETURNS

• The holding period return 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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HOLDING PERIOD RETURN: EXAMPLE

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

Year Return1 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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HOLDING PERIOD RETURNS• A famous set of studies dealing with 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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ILLUSTRATION: HOLDING PERIOD RETURNS

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UPS AND DOWNS: A FACT OF INVESTING LIFE

• Investments don’t offer their returns on a consistent basis:• Some years are higher;• Some years are lower: and, • Some years are losses

• Historically:• The most rewarding investments have the most

volatile returns• The least rewarding investments have the least

volatile returns

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RETURNS AND VOLATILITY: A COMPARISON

Small Company Stocks U.S. Treasury Bills

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10.3 RETURN STATISTICS

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

• Where R1…RT are the individual observed returns• Where T is the time period analyzed

• the frequency distribution of the returns

TRRR T )( 1

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EXAMPLE FREQUENCY DISTRIBUTION• Frequency distribution is a histogram of yearly returns

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INTERPRETATION OF THE EXAMPLE FREQUENCY DISTRIBUTION

• In the example above notice that:• Small company stocks have a higher average return

but a wider array (or variance) of actual returns• Large company stocks have a lower average return

but a more narrow array (or variance) of actual returns.

• Conclusion: • In any given observation period, it is more likely that

large company stocks will produce a return near the mean than small company stocks.

• On average, small company stocks produce a greater return than large company stocks, but also present a much greater likelihood of a loss in any given year

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Historical Returns, 1926-2012

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10.4 AVERAGE STOCK RETURNS AND RISK-FREE RETURNS

• U.S. government securities have very low volatility, or risk• U.S. debt is considered risk free because the

government can raise taxes to repay it• Consider U.S. Treasury Bills, discount bonds

that mature in less than a year• The return of such a security is often

considered the “risk-free rate.”• The Risk Premium is the added return (over

and above the risk-free rate) resulting from bearing risk.

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RISK PREMIUM

• Suppose that The Wall Street Journal announced that the current rate for one-year Treasury bills is 5%. • What is the expected return of small-company

stocks?• The average return on large company common

stocks for the period 1926 through 2012 was 11.8%.• Given a risk-free rate of 5%, and average large

company return of 11.8% the return premium attributable solely to the risk of small company stocks is 6.8%

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THE RISK-RETURN TRADEOFF

• The general rule:

• The more volatile (or risky) a return is, the greater the return will be expected to be.

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10.5 RISK STATISTICS• There is no universally agreed-upon definition

of risk.• The measures of risk that we discuss are

variance and standard deviation.• Variance and standard deviation measure the

dispersion of actual returns around the security’s mean return

• The standard deviation is the standard statistical measure of the spread of a sample, and it is the measure used in most cases.

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

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NORMAL DISTRIBUTION• A large enough sample drawn from a normal

distribution looks like a bell-shaped curve.

Probability

Return on large company common stocks

99.74%

– 3 – 48.8%

– 2 – 28.6%

– 1 – 8.4%

011.4%

+ 1 32.0%

+ 2 52.2%

+ 3 72.4%

The probability that a yearly return will fall within 20.2 percent of the mean of 11.8 percent will be approximately 2/3.

68.26%

95.44%

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NORMAL DISTRIBUTION

• The return of large company stocks has a standard deviation of 20.2 from 1926 through 2012.

• That standard deviation can be interpreted as follows:• if stock returns are approximately normally distributed,

the probability that a yearly return will fall within 20.2% of the mean of 11.8% will be approximately 2/3.

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EXAMPLE – RETURN AND VARIANCE

Year Actual Return

Average Return

Deviation from the Mean

Squared Deviation

1 .15 .105 .045 .002025

2 .09 .105 -.015 .000225

3 .06 .105 -.045 .002025

4 .12 .105 .015 .000225

Totals .00 .0045

Variance = .0045 / (4-1) = .0015 Standard Deviation = .03873

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10.6 THE U.S. EQUITY RISK PREMIUM

• The past U.S. stock market risk premium has been substantial• Did U.S. investors earn particularly large

returns?• Comparison to international markets is useful:• Greater than 50% of tradable stock is not in the US• 17 have an average risk premium of 6.9%• US premium is 7.2%

• U.S. Investors did well, but not dramatically better than investors internationally

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GLOBAL STOCK MARKET RISK PREMIUMS

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CONCLUSION: U.S. MARKET RISK PREMIUM

• Apparent global MRP mean is 6.9%• Given U.S. 7.2% mean MRP with a 19.8%

standard deviation, there is a 95% probability that actual MRP is between 3.4 and 11%• 226 financial economists independently

estimated MRP at 7%• Conclusion: U.S. MRP estimate of 7% is

reasonable barring unforeseen changes in risk environment.

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10.7 2008: A YEAR OF FINANCIAL CRISIS

• 2008 was one of the worst years for stock investors• S&P Index decreased 37%• Of 500 stocks in index, 485 were down for the year

• The beginning of 2009 was also bad with a 25.1% further S&P decline• From November 2007 through March 9, 2009 the

S&P lost 56.8% of its value• Things turned around dramatically in the

remainder of 2009, with the market gaining about 65%

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2008 CRISIS: LESSONS LEARNED

• Global Phenomenon – No economy unscathed• Some securities performed well:• Treasury Bonds• High Quality Corporate Bonds• “Flight to Quality” Phenomenon

• Reminder that stocks have significant risk and can under- as well as over-perform the mean• Argues for careful diversification• Suggests that MRP may be greater than 7% right

now

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10.8 MORE ON AVERAGE RETURNS: ARITHMETIC VS. GEOMETRIC MEAN

• Arithmetic average – return earned in an average period over multiple periods

• Geometric average – average compound return per period over multiple periods

• The geometric average will be less than the arithmetic average unless all the returns are equal

• Which is better?• The arithmetic average is overly optimistic for long

horizons.• The geometric average is overly pessimistic for

short horizons.

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GEOMETRIC RETURN: EXAMPLE

Year Return1 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 an average of 9.58% per year, realizing a holding period return of 44.21%.

4)095844.1(4421.1

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GEOMETRIC RETURN: EXAMPLE

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

Year Return1 10%2 -5%3 20%4 15% %10

4%15%20%5%10

4return average Arithmetic 4321

rrrr

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FORECASTING RETURN• To address the time relation in forecasting

returns, use Blume’s formula:

AverageArithmeticNTN

verageGeometricANTTR

1

11)(

where, T is the forecast horizon and N is the number of years of historical data we are working with. T must be less than N.

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QUICK QUIZ

• Which of the investments discussed has had the highest average return and risk premium?

• Which of the investments discussed has had the highest standard deviation?

• Why is the normal distribution informative?• What is the difference between arithmetic and

geometric averages?