Chapter

1 A Brief History of

Risk and Return

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A Brief History of Risk and Return

• 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.

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Computing Returns

• To start us off, let us look at two measures of returns:

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

• Do we include capital gains if we do not sell the stock?

Loss) (or Gain Capital

Income Dividend Stock a on Return Dollar Total

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Computing 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.

priceStock Beg.

pricestock beg - priceStock End.

priceStock Beg.

income Dividend

yield gains capital yield dividend (Stock)Return Percent

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

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Financial Market History

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Financial Market History

• So what do we learn from the graphs?

•There is a strong financial incentive for long-term investing but…. recent market behavior may scare off even the most ardent believers in long-run investing.

• Maybe Keynes’ “In the long run, we are all dead”.. is more appropriate??

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The Historical Record:Total Returns on Large-Company Stocks.

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The Historical Record: Total Returns on Small-Company Stocks.

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

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The Historical Record: Total Returns on T-bills.

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The Historical Record: Inflation.

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Historical Average Returns

• What measures can we use to summarize the history of financial market returns?

• We can compute average returns in two ways:– Arithmetic average returns

– Geometric average returns:

n

returnyearly Return Average Arithmetic

n

1i

1)]R1)...(R1)(R[(1 Return Average Geometric /1N21 N

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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?”

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Arithmetic Averages versusGeometric Averages

• The arithmetic average tells you what you earned in a typical year and what way may expect to earn in any given year in the future.

• 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.

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Blume’s Formula For Estimating Returns

• If the geometric average tends to be too high, and the

arithmetic average too low, how can we best estimate

returns?

• Blume’s formula gives an unbiased estimate.

• Suppose we calculate averages from N years of data and we wish to forecast future returns over T years . Then forecasted returns R(T) are estimated as:

R(T) = geometric mean*(T-1)/(N-1) + arithmetic mean*(N-T)/(N-1)

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What Can We Learn From Average Returns?

• 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.

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How Do We Compute the Risk Premium?

• In order to calculate an appropriate risk premium, we must answer two questions:– What risk-free security do we use?– Do we use arithmetic or geometric average returns?

• Both questions can be answered by examining why we are calculating the risk premium. For example, if we are calculating it as an input to stock valuation, then it makes sense to use a longer-term risk-free security and a geometric average since stock valuation tends to be an analysis of long-term (technically infinite) stream of cash flows.

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Why Does a Risk Premium Exist?

• Modern investment theory centers on this question.

• A risk premium exists because certain securities are more risky than others therefore investors should be compensated with higher return for bearing this greater risk.

• We can start our examination of risk by looking at variance and standard deviation of returns.

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Return Variability Review and Concepts

• Variance is a common measure of return dispersion. Sometimes, return dispersion is also called 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.

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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:

1N

RR σ VAR(R)

N

1i

2

i2

VAR(R) σ SD(R)

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Frequency Distribution of Returns on Common Stocks, 1926—2005

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Historical Returns, Standard Deviations, and Frequency Distributions: 1926—2005

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The Normal Distribution and Large Company Stock Returns

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Useful Rule of Thumb

• In any given year, your investment has an approximately 1/3 chance of generating a return that is outside of the mean + standard deviation for that investment.

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Risk and Return

• 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.

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Historical Risk and Return Trade-Off

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Readings

• All of Chapter 1