1. Risk and Return: Lessons From Capital Market History
MSBC 5060 Chapter 10
Risk and Return:
Lessons From Capital Market History

2. We already know : We also know :
Capital budgeting requires calculating the NPV:
Discounting future Cash Flows (Numerator)
At the Require Rate of Return (Denominator)
We also know :
Which Cash Flows to use…
Use the Stand-Alone Principle
Use Incremental Cash Flows associated with:
Operations (OCF)
Capital Spending (NCS)
Working Capital (DNWC)
Test the CF forecasts used to calculate the NPV
Sensitivity and Scenario analysis
Now we will start looking at the Required Rate of Return 

3. The General Idea:
The appropriate discount rate for a project (or a company)
Reflects the project’s risk
The riskier the project, the higher the required return
Why? Investors are RISK AVERSE
So how do we measure the project’s risk?
And once we know the risk, what is the correct rate of return for that risk?

4. Start with this Assumption:
The new project has the SAME risk as the firm’s current projects
Then we can use the rate or return firm is currently paying
But how do we calculate that?
Later: What if the new project’s risk is different,
We can adjust the use the rate or return firm is currently paying to account for difference in risk
So we will look at:
The general historic risk and return for all companies (the market)
The risk and return for different types of companies
The risk for the company we’re analyzing

5. Goals The Mechanics of Calculating Returns
Return Variability (aka Risk) – Standard Deviation
Look at the Historical Record for various investment types
Understand the Normal Distribution
Arithmetic vs. Geometric Returns
Longer holding periods

6. Chapter Outline 10.1 Returns 10.2 Holding-Period Returns
10.3 Return Statistics (Arithmetic Average Return)
10.4 Average Stock Returns and Risk-Free Returns
10.5 Risk Statistics
(Variance and Standard Deviation)
10.6 More on Average Returns
(Arithmetic Mean vs. Geometric Mean)
10.7 The U.S. Equity Risk Premium: Historical and International Perspectives
: A Year of Financial Crisis

7. 10.1 Returns Dollar Returns: Bond Example:
You bought a bond for $950 1 year ago.
You have received two coupons of $30 each.
You can sell the bond for $975 today.
Calculate your total dollar return:
Income = $30 + $30 = $60
Capital gain = $975 – $950 = $25
Total dollar return = $60 + $25 = $85

8. Dollar Returns: Stock Example: Calculate your total dollar return:
1 year ago, you bought stock for $50 per share.
You received 4 dividends of $1.25 each
Today the price of the stock is $48
Calculate your total dollar return:
Income = 4($1.25) =$5.00
Capital gain = $48 – $50 = -$2.00
Total dollar return = $5.00 – $2.00 = $3.00

9. Of course dollar returns aren’t very useful!
Percent Returns:
Of course dollar returns aren’t very useful!
New Stock Example:
1 year ago, you bought stock for $100 per share.
You received 4 dividends of $1.25 each
Today the price of the stock is $98
Calculate your total dollar return:
Income = 4($1.25) =$5.00
Capital gain = $98 – $100 = -$2.00
Total dollar return = $5.00 – $2.00 = $3.00
Dollar returns are the same for $50 and $100 stocks!
Make $3.00 on $100 vs. Make $3 on $50
We can see this is 3% vs. 6%

10. Percent Return Formula
A little Algebra:

11. Percent Return Formula
Total Return = (P1 + D1)/ P0 - 1
A stock costs $100 today.
One year ago is was $90.
It paid a $3 dividend today
Calculate the return over the last year:
Total Return = R = ($100 + $3)/$90 - 1
= – 1 = = 14.44%
Note: $90(1 + R) = $90(1.1444) = $103

12. HPR = (1 + R1) x (1 + R2) x … x (1 + RT) - 1
10.2 Holding Period Returns
The Cumulative return
Earn 10% per year for 3 years:
HPR = ( ) x (1 + .1) x (1+ .1) – 1 = %
This is equal to (1 + .1)3 – 1 = 33.10%
Instead earn 10%, -5%, 20%
HPR = ( ) x ( ) x (1+ .2) -1 = %
HPR = (1 + R1) x (1 + R2) x … x (1 + RT) - 1

13. 10.3 Return Statistics
Measures of Dispersion: Variance and Standard Deviation
Variance is the sum of the squared deviations from the mean
Standard Deviation is the square-root of the Variance
Why divided the sum by T – 1 and not T when calculating Variance?

14. Deviation from the Mean
How to Calculate Variance (s2) and Stdev (s)
Mean Return = 0.42/4 = = 10.5%
s2 = (Sum of Squared Deviations)/(T – 1) = /(4 – 1) =
s = Square Root of Variance = (0.0015)½ = = 3.87%
The Standard Deviation is in the SAME units as the variable
In this case % return, so it can be expresses as a %
Variance is NOT, so it can not be expressed as a %
Year
Return
Average Return
Deviation from the Mean
Squared Deviation 1
.15
.105
.045 2
.09
-.015 3
.06
-.045 4
.12
.015
Sum
.42
.00
.0045

15. How to Calculate Variance (s2) and Stdev (s)
We’ll use Excel:
Year
Return 1
15% 2 9% 3 6% 4
12%
=average()
10.5%
=var.s()
.0015
=stdev.s()
3.87%

16. The Historical Record Why do we look at the historical record?
Do we care (directly) about what happened in the past?
Maybe…
What we do care about is the future.
So what will happen in the future?
Maybe our best guess is what has happened in the past.
So we do care - indirectly - about the past

17. So lets look at the Historical record:
What has happened to:
Large Stocks
Small Stocks
Corporate Bonds
Long-Term Government debt (T-Bonds)
Short-Term Government debt (T-Bills)
Securities trade in financial markets
And market prices allow us to measure past returns and risk for different securities
So we’ll look at:
the historic returns
the variation in historic returns
Variance and Standard Deviation

18. Financial Markets: Match SAVERS of funds with USERS of funds
Savers of funds INVEST in financial assets
They can defer consumption (Save!)
And earn a return to compensate for the deferred consumption
Users have access to unused capital
They can invest in productive assets (Invest!)
IF… They can earn enough from those assets to pay the return required by savers
We’ll examine financial markets to provide us with information about the returns savers require for various levels of risk
Based on the type of security

19. We’ll look at 5 types of Securities:
Large-Company Stocks
The S&P 500
Small-Company Stocks
Bottom 20% of NYSE by market cap
So really “smaller” stocks
These are still NYSE stocks, so not that small
Long-Term, High-Quality Corporate Bonds
20 years to maturity
Long-Term US Government Bonds
20 year T-bonds
US Treasury Bills
3 month T-bills

21. Linear Scale vs Log Scale
What’s the difference?
Log Scale shows percent changes with equal vertical size
From 100 to 110 is the same as 1000 to 1100
See FF Spreadsheet…

23. Figures 10.5 and 10.6 Large Stocks Returns Small Stocks Returns
Note the Different Scales
Note the correlation between returns
Small Stocks Returns

24. Figure 10.7 T-Bond Returns T-Bill Returns
Again note the Different Scales
T-Bonds -10% to 50%, T-bills 0 to 16% (never negative)
Again note the correlation between returns
T-Bill Returns

25. Figure 10.8
Annual Inflation
Again note Scale

26. Average Returns of Investment Categories
RReal = (1 + RNom)/(1 + i) – 1 = ( )/(1.0300) - 1 = 13.30%
Risk Premium = RNom – Risk-Free = – = 13.20%
(T-Bills are risk-free)
Category
Nominal
Return
Real
Risk Premium
Small Stocks
16.70%
13.30%
13.20%
Large Stocks
12.10%
8.83%
8.60%
Long-term Corporate Bonds
6.40%
3.30%
2.90%
Long-term Government Bonds
6.10%
3.01%
2.60%
U.S. Treasury Bills
3.50%
0.49%
Inflation
3.00%

27. Average Returns RReal = (1 + RNom)/(1 + Inflation) – 1
For Small Stocks:
RNom = 16.7%
RReal = 13.3%
Money Doubled in Small Stocks:
Rule of 72’s: 72/16.7 = 4.31 years
=nper(rate, pmt, pv, [fv],[type])
=nper(.167,0,-1,2) = 4.49

28. Average Returns RReal = (1 + RNom)/(1 + Inflation) – 1
For Small Stocks:
RNom = 16.7%
RReal = 13.3%
Purchasing Power Doubled in Small Stocks:
Rule of 72’s: 72/13.3 = 5.41 years
=nper(rate, pmt, pv, [fv],[type])
=nper(.133,0,-1,2) = 5.55

29. Risk Premium is a RETURN measure, not a risk measure
10.4 Average Stock Returns and Risk-Free Returns
The Risk-Premium is the return earned above the risk-free rate
T-bills are risk free (Why?)
Risk Premium = Average Return - Average T-bill Return
This is the compensation for incurring risk associated with this investment class
Make sure you can distinguish Risk Premium
from Real Return
Average Risk Premium:
Small stocks % – 3.50% = 13.20%
Large stocks % – 3.50% = 8.60%
Risk Premium is a RETURN measure, not a risk measure

30. Return Variability - Risk
Recall Figure 10.4:
Why would you own anything but small stocks?
Since the return is higher
Because the variability is also higher
If you have a short investment horizon…
You could lose large amount in that short period
How much could you lose in T-Bills over a short period?
Nothing (or not very much?)
This is the essence of the risk-return trade-off

31. Risk: A Picture of the Dispersion of Returns
Start with a Frequency Distribution for Large Stocks (Fig 10.9):
Calculate Returns since 2009:
Date
S&P 500
Ann Return
12/31/2008
903.25
12/31/2009
1,115.10
12/31/2010
1,257.64
12/30/2011
1,257.60
12/31/2012
1,426.19
12/31/2013
1,848.36
12/31/2014
2,058.90
12/31/2015
2,043.94
12/30/2016
2,238.83

32. Risk: A Picture of the Dispersion of Returns
Start with a Frequency Distribution for Large Stocks (Fig 10.9):
Calculate Returns since 2009:
Date
S&P 500
Ann Return
12/31/2008
903.25
12/31/2009
1,115.10
23.45%
12/31/2010
1,257.64
12/30/2011
1,257.60
12/31/2012
1,426.19
12/31/2013
1,848.36
12/31/2014
2,058.90
12/31/2015
2,043.94
12/30/2016
2,238.83
2009

33. Risk: A Picture of the Dispersion of Returns
Start with a Frequency Distribution for Large Stocks (Fig 10.9):
Calculate Returns since 2009:
Date
S&P 500
Ann Return
12/31/2008
903.25
12/31/2009
1,115.10
23.45%
12/31/2010
1,257.64
12.78%
12/30/2011
1,257.60
12/31/2012
1,426.19
12/31/2013
1,848.36
12/31/2014
2,058.90
12/31/2015
2,043.94
12/30/2016
2,238.83
2010
2009

34. Risk: A Picture of the Dispersion of Returns
Start with a Frequency Distribution for Large Stocks (Fig 10.9):
Calculate Returns since 2009:
Date
S&P 500
Ann Return
12/31/2008
903.25
12/31/2009
1,115.10
23.45%
12/31/2010
1,257.64
12.78%
12/30/2011
1,257.60
0.00%
12/31/2012
1,426.19
12/31/2013
1,848.36
12/31/2014
2,058.90
12/31/2015
2,043.94
12/30/2016
2,238.83
2010
2011
2009

35. Risk: A Picture of the Dispersion of Returns
Start with a Frequency Distribution for Large Stocks (Fig 10.9):
Calculate Returns since 2009:
2012
Date
S&P 500
Ann Return
12/31/2008
903.25
12/31/2009
1,115.10
23.45%
12/31/2010
1,257.64
12.78%
12/30/2011
1,257.60
0.00%
12/31/2012
1,426.19
13.41%
12/31/2013
1,848.36
12/31/2014
2,058.90
12/31/2015
2,043.94
12/30/2016
2,238.83
2010
2011
2009

36. Risk: A Picture of the Dispersion of Returns
Start with a Frequency Distribution for Large Stocks (Fig 10.9):
Calculate Returns since 2009:
2012
Date
S&P 500
Ann Return
12/31/2008
903.25
12/31/2009
1,115.10
23.45%
12/31/2010
1,257.64
12.78%
12/30/2011
1,257.60
0.00%
12/31/2012
1,426.19
13.41%
12/31/2013
1,848.36
29.60%
12/31/2014
2,058.90
12/31/2015
2,043.94
12/30/2016
2,238.83
2010
2013
2011
2009

37. Risk: A Picture of the Dispersion of Returns
Start with a Frequency Distribution for Large Stocks (Fig 10.9):
Calculate Returns since 2009:
2014
2012
Date
S&P 500
Ann Return
12/31/2008
903.25
12/31/2009
1,115.10
23.45%
12/31/2010
1,257.64
12.78%
12/30/2011
1,257.60
0.00%
12/31/2012
1,426.19
13.41%
12/31/2013
1,848.36
29.60%
12/31/2014
2,058.90
11.39%
12/31/2015
2,043.94
12/30/2016
2,238.83
2010
2013
2011
2009

38. Risk: A Picture of the Dispersion of Returns
Start with a Frequency Distribution for Large Stocks (Fig 10.9):
Calculate Returns since 2009:
2014
2012
Date
S&P 500
Ann Return
12/31/2008
903.25
12/31/2009
1,115.10
23.45%
12/31/2010
1,257.64
12.78%
12/30/2011
1,257.60
0.00%
12/31/2012
1,426.19
13.41%
12/31/2013
1,848.36
29.60%
12/31/2014
2,058.90
11.39%
12/31/2015
2,043.94
-0.73%
12/30/2016
2,238.83
2010
2015
2013
2011
2009

39. Risk: A Picture of the Dispersion of Returns
Start with a Frequency Distribution for Large Stocks (Fig 10.9):
Calculate Returns since 2009:
2014
2012
Date
S&P 500
Ann Return
12/31/2008
903.25
12/31/2009
1,115.10
23.45%
12/31/2010
1,257.64
12.78%
12/30/2011
1,257.60
0.00%
12/31/2012
1,426.19
13.41%
12/31/2013
1,848.36
29.60%
12/31/2014
2,058.90
11.39%
12/31/2015
2,043.94
-0.73%
12/30/2016
2,238.83
9.54%
2010
2015
2013
2011
2016
2009

40. Distributions and s’s 1926 to 2014

41. Mean and Standard Deviation of Historic Returns
Category
Mean Return
Standard Deviation
Small Stocks
16.70%
32.10%
Large stocks
12.10%
20.10%
Long-term Corporate Bonds
6.40%
8.40%
Long-term Government Bonds
6.10%
10.00%
U.S. Treasury Bills
3.50%
3.10%
Inflation
3.00%
4.10%

42. Interpreting the Distribution Measure (s)
What does s mean?
It depends…
If we assume that the data is from a Normal Distribution
Then s tells us a lot
Recall for the Normal Distribution:
Mean +/- 1s contains 68.27% of the observations
Mean +/- 2s contains 95.45% of the observations
Mean +/- 3s contains 99.73% of the observations

43. Interpreting the Distribution Measure (s)
For the Large Stock Portfolio:
Mean Return = 12.1%
Standard Deviation (s) = 20.1%
68.27% of observations between 12.1% +/- 1 x 20.1%
Between -8.0% and 32.2%
95.45% of observations between 12.1% +/- 2 x 20.1%
Between -28.1% to 52.3%
99.73% of observations between 12.1% +/- 3 x 20.1%
Between -48.2% to 72.4%
Notice you can be MORE CONFIDENT about a WIDER interval!

44. Normal Distribution
A large enough sample drawn from a normal distribution looks like a bell-shaped curve.
Probability
The probability that an annual return will fall within -8.0% and 32.2% is 68.26% (approximately 2/3).
- 3s %
- 2s %
- 1s %
0 12.1%
+ 1s %
+ 2s %
+ 3s %
Return on large company common stocks
68.27%
95.45%
99.73%

45. Return Distributions (Risk) for Different Invest Classes:
-2σ
-1σ
Mean
+1σ
+2σ SS
-47.50%
-15.40%
16.70%
48.80%
80.90% LS
-28.10%
-8.00%
12.10%
32.10%
52.30%
T-Bonds
-13.90%
-3.90%
6.10%
16.10%
26.10%
95%

46. 27.36% chance of a negative return next year in large stocks
Return Probabilities Assuming Normality
Another thing we can do:
Large Stock Mean = 12.1% and σ = 20.1%
Calculate the probability of a negative return in large stocks
P(R < 0) = ?
How far from the mean is 0?
How many stdevs to the left of the mean to get to 0?
z = (x – mean)/σ = (0 – 0.121)/0.201 =
So a little more than 60% of one stdev from the mean to 0
How much of the curve is to the left of the mean σ?
=NORM.DIST(x, mean, standard_dev, cumulative)
=NORM.DIST(0,0.1210,0.201,1) = = 27.36%
27.36% chance of a negative return next year in large stocks

47. Return Probabilities Assuming Normality
Another way to do it:
Large Stock Mean = 12.1% and σ = 20.1%
Calculate the probability of a negative return in large stocks
How far from the mean is 0?
How many stdevs to the left of the mean to get to 0?
z = (x – mean)/σ = (0 – 0.121)/0.201 =
How much of the curve is to the left of the mean σ?
=NORM.S.DIST(z)
=NORM.S.DIST(-.602) = = 27.36%
NORM.DIST: Enter x, mean and σ
NORM.S.DIST: Enter only z

48. Return Probabilities Assuming Normality
One more example:
Large Stock Mean = 12.1% and σ = 20.1%
Calculate the probability of a return of at least 50% (R > 50%)
How far from the mean is 50%?
How many stdevs to the right of the mean to get to 50%?
z = (x – mean)/σ = (0.50 – 0.121)/0.201 = 0.379/0.201 = 1.886
How much of the curve is to the RIGHT of the mean σ?
=NORM.S.DIST(z)
=NORM.S.DIST (1.886) = = 97.03%
97.03% to the left or right?
So what is the probability of a return greater then 50%?
P(R > 50%) = 1 – = 2.97%

49. P(R > 10% and R < 30%) = 81.34% - 45.84% = 35.50%
Return Probabilities Assuming Normality
Last example:
Large Stock Mean = 12.1% and σ = 20.1%
Calculate the probability earning between 10% and 30%
How far from the mean is 10%?
z = (x – mean)/σ = (0.10 – 0.121)/0.201 = /0.201 = -.104
=NORM.S.DIST (-0.104) =
P(R < 10%) = = 45.84%
How far from the mean is 30%?
z = (x – mean)/σ = (0.30 – 0.121)/0.201 = 0.179/0.201 = 0.891
=NORM.S.DIST (0.891) =
P(R < 30%) = = 81.34%
P(R > 10% and R < 30%) = 81.34% % = 35.50%

50. Larger Left Tail: “Fatter Tails”
How Good is the Normal Distribution Assumption?
Are security returns Normally distributed?
It’s okay, but not great.
The actual distribution tends to have:
Larger Left Tail:
Large negative returns are more likely than predicted by the Normal Dist
This is measure by the “negative skew”
“Fatter Tails”
Large returns are more likely than predicted by the Normal Dist
This is measured by the “kurtosis”

51. How Good is the Normal Distribution Assumption?
Recall for large stocks:
Mean 12.1% and Stdev 20.1%
So If you assume Normality:
=normsinv(0.05) =
5% of the curve is to the left of:
12.1% + (-1.645) x 20.1% = %
So assuming Normality, we estimate that there is only a 5% chance you will lose more than 20.96% in large stocks in a year
This is called the 5% Value at Risk
But if the distributional assumption IS NOT CORRECT…
Then the probability of a loss greater than 20.96%
is greater than 5%

52. What do we do with these Values?
Suppose we have a project we believe has the same risk as the risk associated with “large stocks”
The historic Risk Premium for large stocks is 8.60%
(See slide 26)
On 7/27/2016, 1 year T-Bills were paying 0.53%
So we can earn 0.53% over the next year without risk
So on this project we might expect to earn:
The risk free rate plus a Premium for incurring risk
Risk Free = 0.53%
Premium for incurring the risk = 8.60%
Expected Return = 0.53% % = 9.13%
This might be a reasonable Expected Return for a project of this risk

53. What do we do with these Return and Risk Values?
What if the proposed project is 25% riskier than large stocks?
This means the volatility of the project’s return will be 25% greater than large stock volatility
So the project needs to earn more than 0.63% % = 9.13%
But how much more?
Increase the Risk Premium by 25%:
Multiply the risk premium by ( ) = 1.25
Then add the risk-free rate
Expected Return = 0.53% (8.60%)
= 0.53% % = 11.28%
This is an application of something called the CAPM
We’ll talk about how to get the 25% riskier (or the 1.25) soon

54. Geometric Mean vs. Arithmetic Mean Returns
Two Different Questions:
What can I expect to earn next year?
Was the average of the annual returns?
Calculate the return for an average year
In any year, what can I expect the return to be?
This is the Arithmetic mean
What was the average annual return over the holding period?
In annualized terms, what return did I earn over the holding period?
What return, that is same in each year, replicates the performance?
This is the Geometric Mean
Arithmetic Mean = (R1 + R2… + RN)/N
Geometric Mean = [(1 + R1)(1 + R2)…(1 + RN)]1/N – 1
Example Calculations:
R1 = -20% and R2 = 25%
Arithmetic Mean = ( )/2 = = 2.5%
Geometric Mean = [(1 +(-0.20))( )]1/2
=[(0.80)(1.25)]1/2 – 1 = 0%

55. Geometric vs. Arithmetic (Continued 1)
Let’s see why these answers are not that same:
Recall the 2 Different Questions:
Arithmetic Mean:
What can I expect the return to be next year?
Geometric Mean:
In annual terms, what return did I earn over the holding period?
Example: P0 = $100, P1 = $80, P2 = $100
Calculate the two annual returns and then the averages:
R1 = $80/$100 – 1 = = -20%
R2 = $100/$80 – 1 = 0.25 = 25%
Arithmetic Mean = (-20% + 25%)/2 = 2.5%
Geometric Mean = [(1 +(-0.20))( )]1/2 =[(0.80)(1.25)]1/2 – 1 = 0%

56. Review Question: Two years ago, a manager lost 40%.
Last year the manager made 25%
What was the annualized return over the two years?
What return do you expect the manager to earn this year?

57. Review Answer: Two years ago, a manager lost 40%.
Last year the manager made 25%
What was the Annualized Return over the two years?
Geometric Mean: ( )( )½ - 1 = %
What return do you Expect the manager to earn this year?
Arithmetic Mean: ( )/2 = -7.50%

58. Lose 40%, then make 25% same as lose 13.40% twice
Extra Explanation:
Note that $100 after the first year would be worth:
$100(1 + R) = $100( ) = $100(0.60) = $60
After the second year it would be worth:
$60(1 + R) = $60( ) = $60(1.25) = $75
A 40% loss followed by a 25% gain is the same as two 13.40% losses:
$100(1 – ) = $86.60
$86.60(1 – ) = $75
So the Geometric Mean Return is the ONE annual return that is equivalent to the observed series
Lose 40%, then make 25% same as lose 13.40% twice

59. Geometric vs. Arithmetic
Note that the Geometric Mean is smaller than the Arithmetic Mean
Always true (unless they are equal)
The difference is a function of the level of volatility of the returns
If there is no volatility (which all the returns are equal) then Geometric Mean equals Arithmetic Mean
An approximate relationship:
Geometric Mean ≈ Arithmetic Mean – σ2/2
Large Stocks: – (0.2010)2/2 = = 10.08% ≈ 10.12%
Small Stocks: – (0.3210)2/2 = = 11.55% ≈ 12.17%
Category
Arithmetic
Geometric σ
Large stocks
12.10%
10.12%
20.10%
Small Stocks
16.70%
12.17%
32.10%

60. Geometric vs. Arithmetic
Recall Figure 10.4
Use these values to calculate Annualized Holding Period Returns:
Large Stocks:
$1 to $5, over 89 yrs
($5,316.85/$1)(1/89) – 1 = 10.12%
Small Stocks:
$1 to $27, over 89 yrs
($27,419.32/$1)(1/89) – 1 = 12.17%
Geometric Return also equal to:
R = (FV/PV)(1/N) – 1
Same as:
R = (PN/P0)(1/N) – 1

61. Geometric Mean
From Table 10.1 Page 310

62. Geometric Mean
Earning all of these returns, results in $1 growing to $5, in 89 years
Earning 10.12% in each year also results in $1 growing to $5, in 89 years
So 10.12% is the geometric mean of this series of returns.
Go to “FF Small Big ” spreadsheet 

63. Geometric vs. Arithmetic
Think about risk and return and how it effects holding period return:
Look at the arithmetic mean and s of the historic returns for two investments
Think about what the annualized holding period return might be in the future
Category Arith Mean s
Investment % %
Investment % %
Annualized Holding Period Return:
1: Geometric Mean ≈ Arithmetic Mean – s2/2 = 0.18 – 0.302/2 = 13.5%
2: Geometric Mean ≈ Arithmetic Mean – s2/2 = 0.21 – 0.402/2 = 13.0%
Investment 2 has the higher expected return in any given year
Investment 1 has the higher expected holding period return
Why?

64. Arithmetic Mean of 20 yr Holding Periods
Longer Holding Periods:
Look at 20 year holding periods for Small Stocks and Large Stocks
I used a “Bootstrap Simulation” to estimate a large number of 20 year holding period returns for each asset class.
Next I calculate the arithmetic mean of the 20yr holding-period returns
This is the arithmetic mean of the geometric means
What does this tell us about holding Stocks for 20 years?
Lets look at holding Small Stocks for 1 Year vs. 20 Years 
Arithmetic Mean of 20 yr Holding Periods σ SS
12.1%
5.1% LS
9.5%
4.1%

65. Return Distributions (Risk) for 1yr and 20yr
-2σ
-1σ
Mean
+1σ
+2σ
SS 1yr
-47.5%
-15.4%
16.7%
48.8%
80.09%
SS 20yr
1.9%
7.0%
12.1%
17.2%
22.3%
95%

66. Other Common Stock Categories:
Besides large and small
A common way to categorized stocks by:
Market Value aka Market Cap
Categories are Large, Mid and Small
Multiply the price per share by number of shares
This is what is costs to buy the company
The relative position of the company in the range of PE and Market-to-Book ratios
Categories are Growth, Balanced, Value
Are you buying a company’s stock because
You expect it to grow or because it has current profits?
These two factors (3x3) create nine categories
Called “styles”
Represented by “Style Boxes”
See Morningstar.com

67. Indices and ETF tickers that Track Each Style:
But for now we’ll stick to just two categories of stocks (Small and Large ) as well as bonds
Value
Balanced
Growth
Large
S&P 500 Value
(IVE)
S&P 500
(IVV)
S&P 500 Growth
(IVW)
Mid
S&P 400 Value
(IJJ)
S&P 400
(IJH)
S&P 400 Growth
(IJK)
Small
S&P 600 Value
(IJS)
S&P 600
(IJR)
S&P 600 Growth
(IJT)

68. Where do we go from here? MORE Risk and Return
We have an idea of how to measure TOTAL risk
Can we break risk down into categories?
What happens if we combine stocks into a portfolio?
What is the portfolio’s risk?
For what types of risk are you compensated?
We’ll introduce a basic model to look at this…
The CAPM