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

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 

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?

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

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

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 10.8 2008: A Year of Financial Crisis

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

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

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%

Percent Return Formula A little Algebra:

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 = 1.1444 – 1 = 0.1444 = 14.44% Note: $90(1 + R) = $90(1.1444) = $103

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 = (1 + .10) x (1 + .1) x (1+ .1) – 1 = 33.10% This is equal to (1 + .1)3 – 1 = 33.10% Instead earn 10%, -5%, 20% HPR = (1 + .10) x (1 + -.05) x (1+ .2) -1 = 25.40% HPR = (1 + R1) x (1 + R2) x … x (1 + RT) - 1

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?

Deviation from the Mean How to Calculate Variance (s2) and Stdev (s) Mean Return = 0.42/4 = 0.105 = 10.5% s2 = (Sum of Squared Deviations)/(T – 1) = 0.0045/(4 – 1) = 0.0015 s = Square Root of Variance = (0.0015)½ = 0.0387 = 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 .002025 2 .09 -.015 .000225 3 .06 -.045 4 .12 .015 Sum .42 .00 .0045

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%

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

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

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

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

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…

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

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

Figure 10.8 Annual Inflation Again note Scale

Average Returns of Investment Categories RReal = (1 + RNom)/(1 + i) – 1 = (1 +0.1670)/(1.0300) - 1 = 13.30% Risk Premium = RNom – Risk-Free = 0.1670 – 0.0350 = 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%

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

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

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 16.70% – 3.50% = 13.20% Large stocks 12.10% – 3.50% = 8.60% Risk Premium is a RETURN measure, not a risk measure

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

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

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

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

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

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

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

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

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

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

Distributions and s’s 1926 to 2014

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%

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

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!

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 - 48.2% - 2s - 28.1% - 1s - 8.0% 0 12.1% + 1s 32.2% + 2s 52.3% + 3s 72.4% Return on large company common stocks 68.27% 95.45% 99.73%

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%

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 = -0.602 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 - 0.602σ? =NORM.DIST(x, mean, standard_dev, cumulative) =NORM.DIST(0,0.1210,0.201,1) = 0.2736 = 27.36% 27.36% chance of a negative return next year in large stocks

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 = -0.602 How much of the curve is to the left of the mean - 0.602σ? =NORM.S.DIST(z) =NORM.S.DIST(-.602) = 0.2736 = 27.36% NORM.DIST: Enter x, mean and σ NORM.S.DIST: Enter only z

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 + 1.886σ? =NORM.S.DIST(z) =NORM.S.DIST (1.886) = 0.9703 = 97.03% 97.03% to the left or right? So what is the probability of a return greater then 50%? P(R > 50%) = 1 – 0.9703 = 2.97%

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 = -0.021/0.201 = -.104 =NORM.S.DIST (-0.104) = 0.4584 P(R < 10%) = 0.4584 = 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) = 0.8134 P(R < 30%) = 0.8134 = 81.34% P(R > 10% and R < 30%) = 81.34% - 45.84% = 35.50%

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”

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) = -1.645 5% of the curve is to the left of: 12.1% + (-1.645) x 20.1% = -20.96% 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%

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% + 8.60% = 9.13% This might be a reasonable Expected Return for a project of this risk

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% + 8.60% = 9.13% But how much more? Increase the Risk Premium by 25%: Multiply the risk premium by (1 + 0.25) = 1.25 Then add the risk-free rate Expected Return = 0.53% + 1.25(8.60%) = 0.53% + 10.75% = 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

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 = (-0.20 + 0.25)/2 = 0.025 = 2.5% Geometric Mean = [(1 +(-0.20))(1 + 0.25)]1/2 =[(0.80)(1.25)]1/2 – 1 = 0%

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 = -0.20 = -20% R2 = $100/$80 – 1 = 0.25 = 25% Arithmetic Mean = (-20% + 25%)/2 = 2.5% Geometric Mean = [(1 +(-0.20))(1 + 0.25)]1/2 =[(0.80)(1.25)]1/2 – 1 = 0%

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?

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: (1 - 0.40)(1 + 0.25)½ - 1 = -13.40% What return do you Expect the manager to earn this year? Arithmetic Mean: (-0.40 + 0.25)/2 = -7.50%

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(1 - 0.40) = $100(0.60) = $60 After the second year it would be worth: $60(1 + R) = $60(1 + 0.25) = $60(1.25) = $75 A 40% loss followed by a 25% gain is the same as two 13.40% losses: $100(1 – 0.1340) = $86.60 $86.60(1 – 0.1340) = $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

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: 0.1210 – (0.2010)2/2 = 0.1008 = 10.08% ≈ 10.12% Small Stocks: 0.1670 – (0.3210)2/2 = 0.1155 = 11.55% ≈ 12.17% Category Arithmetic Geometric σ Large stocks 12.10% 10.12% 20.10% Small Stocks 16.70% 12.17% 32.10%

Geometric vs. Arithmetic Recall Figure 10.4 Use these values to calculate Annualized Holding Period Returns: Large Stocks: $1 to $5,316.85 over 89 yrs ($5,316.85/$1)(1/89) – 1 = 10.12% Small Stocks: $1 to $27,419.32 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

Geometric Mean From Table 10.1 Page 310

Geometric Mean Earning all of these returns, results in $1 growing to $5,316.85 in 89 years Earning 10.12% in each year also results in $1 growing to $5,316.85 in 89 years So 10.12% is the geometric mean of this series of returns. Go to “FF Small Big 2016-06” spreadsheet 

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 1 18% 30% Investment 2 21% 40% 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?

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%

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%

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

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)

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