A history of risk

1. A history of risk What have previous market conditions taught us? Some math tools to help us Was the US stock market on an long winning streak?

2. Application: Spirit Airlines • How does Spirit Airlines’ risk compare to other airlines? • How about when it first started operations? • When this company first opened, the idea of an ultra low cost airlines was not well-known among many people

3. Who can predict the future? • Many people try to predict the future • Will the stock market go up or down today? • Who will win the NBA championship? • Will tomorrow’s temperature be higher than today’s? • What color will be on the roulette wheel next time? • Just for fun/expert status • Earning money with predictions • Who’s an expert? • Listen to arguments behind the prediction

4. Why can’t we just get good advise to get ahead of others? • Because everyone else knows if it’s on the news! • The market reacts to revealed information • People that can interpret information better than others can benefit • Of course, never act on news in an illegal way • Example: Insider trading • You may get asked “What will happen to the stock market?” • How would you answer? What if you were a good predictor of the stock market?

5. Getting back to the stock market • In previous lectures, we looked at simple ways to try to come up with stock value • Perpetuity • Growing perpetuity • High growth rate, followed by growing perpetuity (properly discounted)

6. The real stock market • In reality, stocks go up and down in value on a daily basis • Market conditions change constantly • Companies regularly distribute information to the public • Sales • Earnings and dividends • Forecasts

7. Calculating yearly returns in stocks • Assume that we buy a stock immediately after a dividend is paid • We want to find out what our yearly return to holding the stock for a year is • As before, we ignore tax issues

8. Two ways we can calculate returns • Total dollar return • How much money do we gain (or lose) in a given year? • Percentage returns • What percentage of our stock’s value do we gain (or lose) in a year • I will typically refer to “returns” as “percentage returns” later on unless mentioned otherwise

9. Two components to returns • Dividend paid one year from now • Could be nothing, or a positive amount • Change in the price of the stock

10. Change in the price of the stock • A capital gain is the change in the percentage of a stock’s value in a given year • Some people refer to capital gains as simply the change in price (not percentage change) • In this class, unless I mention otherwise, a capital gain should be expressed as a percentage or fraction • Example: 14% or 0.14 • Of course, a capital loss can occur, in which the price of the stock goes down from one year to the next

11. Calculating total dollar returns • Suppose we buy a share today for $50 • Dividend was just paid, so the next dividend is paid one year from now • The next dividend payment (one year from now) is $1.25 • The value of a share of the stock one year from today is $53.75

12. Calculating total dollar returns • The total dollar returns (yearly) for each share we hold are the sum of the dividend paid and the change in value of the stock • Dividend is $1.25 • The change in value of the stock over the year is $53.75 – $50, or $3.75 • Total dollar return for this share is $5

13. Calculating total dollar returns • Notice that the total dollar return for a stock is dependent on the number of shares we own • 1 share: Total dollar return is $5 • 2 shares: $10 • 1,000 shares: $5,000 • X shares: $5X

14. Calculating percentage returns • Percentage returns have two components • Dividend yield • The dividend yield will help us to determine percentage returns • The dividend yield is simply next year’s dividend divided by the stock’s price today • Capital gains • I may refer to capital losses as negative capital gains

15. Calculating percentage returns of our example • Buy a share for $50 today, $1.25 dividend one year from now, price of the stock one year from now is $53.75 • Dividend yield is $1.25 / $50 = 0.025 = 2.5% • Capital gain is ($53.75 – $50) / $50 = 0.075 = 7.5% • Percentage return is 0.025 + 0.075 = 0.10 = 10%

16. A note about percentage returns • If we buy 1,000 shares in our previous example, note that the percentage return is still the same • Dividend yield is $1,250 / $50,000 = 0.025 • Capital gain is ($53,750 - $50,000) / $50,000 = 0.075 • Percentage return is still 10%, no matter how many shares of the stock we own

17. Long-run returns • In the long run, we would like to make investments that give us the best expected return possible • However, we also want to make sure we do not have too much risk • Remember that investments with the highest expected payouts are also usually the riskiest • More on this in upcoming lectures • This is a trade-off that thousands of people make on a daily basis

18. What is a typical investment strategy? • Many people invest in high-risk, high-expected-return investments in their 20s and 30s • Many years to smooth out risk • In your 40s and 50s, you may decide to start putting some of your money in safer investments • Preservation starts to become important • By the time you reach 60, you will likely be retired or be thinking about when you want to retire • Inflation becomes an issue once you retire

19. What has happened? • Many baby boomers (those born between 1946-1964) have already begun to retire • Many in this generation have not retired • Why? • 2008: Financial crisis • High rates of return often assumed did not materialize

20. An example • Henry is 60 years old on Jan. 1, 2006 • He has $2,000,000 invested in large-company stocks • He assumes that he will need $2,500,000 to retire • He assumes that the stock market will go up 8% each year until he retires • He will then convert his investments into safe government bonds Assume that all numbers are in real terms

21. 8% assumption seems reasonable to many • Let’s look at 1995-2006: Large-company stock returns • Relatively low inflation (1.55-3.42%) • Returns over 20% each year 1995-1999 • Drops in 2000-2002 • 29% gain in 2003 • 11% gain in 2004 • 5% gain in 2005 • 16% gain in 2006 • Overall the stock market almost quadrupled over these 12 years • Equivalent to an 11.8% annual return compounded over the 12 years

22. Henry • How long will it take until Henry retires? • Assume his assumption is correct • $2,000,000 in 2006 • $2,160,000 in 2007 • $2,332,800 in 2008 • $2,519,424 in 2009 • He will retire in 2009 • …or will he?

23. Unheard of drop in stock prices • Let’s see what actually would have happened to Henry’s stock value (real values) on Jan. 1 of… • $2,000,000 in 2006 • $2,257,400 in 2007 • $2,287,649 in 2008 • $1,434,814 in 2009 • $1,775,582 in 2010 • $2,009,071 in 2011 • $1,991,391 in 2012 • $2,268,422 in 2013 • $2,960,064 in 2014 Calculated using data from: http://www.moneychimp.com/features/market_cagr.htm

24. When will Henry retire? • He could decide to retire now and have a lower standard of living • He could work for two or three more years and then decide to retire unless the market goes down • Remember that in 2011 he was only 65 • “Normal” retirement age for Social Security is currently 66

25. Why buy stocks? • Many people buy stocks since they have historically out-performed bonds and inflation • $1 in 1925 = • $27,419 in 2014 if invested in small-company stocks • $5,317 in 2014 if invested in large-company stocks • $135 in 2014 if invested in long-term government bonds • $61 in 2014 if invested in gold at New York market price • $21 in 2014 if invested in short-term government bonds • $13 in 2014 if indexed to inflation See Figure 10.4 for more information on wealth indices All calculations from the textbook, except for gold (http://www.measuringworth.com/datasets/gold/result.php)

26. How do we calculate these values? • Assume that any dividends and coupons are re-invested • We can calculate the value of $1 invested today into the future • Stock purchased for $50 today • $1 dividend each year • $1 increase in the value of the stock each year

27. Yearly rates of return • Year 1: $2 / $50 = 0.04 • Year 2: $2 / $52 = 0.038462 • Year 3: $2 / $54 = 0.037037 • Year 4: $2 / $56 = 0.035714 • Our four-year holding period return is • (1.04 * 1.038462 * 1.037037 * 1.035714) – 1 • 16% • Another way to look at this • $58 / $50 = 1.16  16% return

28. Averaging returns over long periods of time • Two ways of doing this • Arithmetic average • Useful when doing statistical analysis • Often misleading when a lot of the returns are far from zero • Geometric average • Usually the preferred way for other types of analysis in this course

29. Suppose that annual percentage returns on stocks are as follows over a five-year period 10% -5% 4% 3% 8% The arithmetic mean is .1-.05+.04+.03+.08 5 Arithmetic average return is 4% Calculating average returns

30. Example • Case 1: Four years, annual percentage returns of 16%, 0%, 0%, 0% • Arithmetic average is 4% • $1 invested today is worth $1.16 in four years • Case 2: Four years, annual percentage returns of 4%, 4%, 4%, 4% • Arithmetic average is 4% • $1 invested today is worth about $1.17 in four years

31. Calculating geometric average • The geometric average is the return that, when compounded over a time period, gets us from starting value to ending value • Example: Case 2 from the last slide • 4% is also the geometric average, since the annual percentage returns are 4% each year • (1.04)4≈ 1.17

32. When are arithmetic and geometric means different? • The two means are different when the annual percentage returns are not constant over time • Example: Our previous Case 1 • Geometric mean is 1.161/4 = 1.037802 • Geometric average return is 1.037802 – 1 • 3.7802%

33. How do we calculate geometric average returns in general? • Step 1: Find the product of (1 + return) each year • Step 2: Find the nth root of this product • Same as taking the (1/n)th power • Step 3: Subtract 1 to get the geometric average return

34. Example • A stock has annual returns of 10%, -5%, 4%, 3%, 8% • Recall that the arithmetic average is 4% • Geometric average calculation • Step 1: (1.1)(0.95)(1.04)(1.03)(1.08) = 1.2090 • Step 2: 1.20901/5 = 1.038681 • Step 3: Geometric average return is 3.8681%

35. Volatility • As we know, there is risk in almost any investment • For many types of investments, the distribution of returns is approximately normally distributed

36. The normal distribution Source: Mwtoews, from http://en.wikipedia.org/wiki/File:Standard_deviation_diagram.svg

37. The normal distribution and stock returns • If we know that the average return for a set of stocks is 11.7% and the standard deviation is 20.6%, then about 68.2% of stocks in this set will be expected to have a return between –8.9% and 32.3% Assume normal distributions of stock returns here

38. More on volatility:Calculating standard deviation • As we just saw, one measure of risk is using standard deviation • To get standard deviation, we must first find variance, and then take the square root

39. Steps to finding variance of a set of T stocks • Step 1: Find the arithmetic average • Step 2: For each stock, find the difference between the return and the average, and then square it • Step 3: Add up all of the numbers in Step 2 • Step 4: Divide by T – 1 We can also do this for yearly returns over many years instead of many stocks This method is the same as the formula in Example 10.3

40. Finding variance:An example • Five stocks have returns of 5%, 3%, 4%, 7%, and 6% • Step 1: Arithmetic average is 5% • Step 2: (.05 - .05)2 = 0, (.03 - .05)2 = .0004, (.04 - .05)2 = .0001, (.07 - .05)2 = .0004, (.06 - .05)2 = .0001 • Step 3: The total of the five bolded #’s is .001 • Step 4: .001 / 4 = .00025

41. Finding standard deviation • To get the standard deviation, we just take the square root of the variance • 0.000251/2 = 0.015811 • The standard deviation is 1.5811%

42. The equity risk premium • The equity risk premium is the difference between average returns of stocks and short-term US government bonds • Recall that short-term US government bonds have the lowest risk • Short term aspect minimizes the interest rate risk • US government bonds are the safest investment in the US • Think of the premium as to the amount of extra expected return you get in exchange for taking on the risk of stock ownership

43. Historical equity risk premiums • In the US, from 1900-2010, the historical equity risk premium was 7.2% • About 5-10% for most developed countries • How far back matters, however… • 1926-2011  Higher equity risk premium • 1802-2011  Lower equity risk premium • Equity risk premium from 1802-1925 was ~3½%

44. How certain are we about our estimates? • We can use simple statistical methods to determine a confidence interval • An X% confidence interval tells us that there is an X% probability that the true mean return is with a calculated interval • Assumptions: Normal and independent distribution • Look at the 1900-2005 period

45. 1900-2005 returns • Historical average: 7.41% • Standard deviation: 19.64% • Standard error of the average is calculated by taking the standard deviation and dividing by the square root of the number of observations (N) • SD = 19.64% and N = 106 • Let X = 95.4 here • We need to be within two standard errors 1900-2010: Lower average, higher s.d. How does this affect the confidence interval?

46. Calculating standard error • Our 95.4% confidence interval is • 7.41 ± 2 * 19.64 / sqrt(106) = 7.41 ± 3.82 • Note that on page 320 of RWJ 9/e, the authors forget to multiply the fraction by 2 • Our 95.4% confidence interval for 1900-2005 • Lower bound: 3.59% • Upper bound: 11.23%

47. Doing the same for 1926-2008 • Average, 7.71%; standard dev., 20.6% • 95.4% confidence interval is • 7.71 ± 2 * 20.6 / sqrt(83) = 7.71 ± 4.52 • Lower bound: 3.19% • Upper bound: 12.23% • Smaller sample size and including an extra outlier (2008) increases the size of our confidence interval

48. Why do we see a premium for risk? • Recall that people are almost always risk averse when the stakes are high enough • You probably want an amount of money with certainty instead of a risky investment with the same expected value • The equity risk premium tells us how much additional return that the market wants (in equilibrium) to take on that risk • How do reward and risk relate to one another?

49. Reward-to-risk ratio • It may be intuitive that the higher the risk, the greater the reward that we want • We will use the Sharpe ratio • Divide reward by risk • Reward measure: Yearly risk premium • Risk measure: Yearly standard deviation

50. Calculating Sharpe ratios • US, 1900-2005 • Reward: 7.41% equity risk premium • Risk: 19.64% standard deviation • Sharpe ratio = 7.41 / 19.64 = .3773 • Other developed countries • A typical Sharpe ratio is in the 0.25-0.4 range • Higher Sharpe ratio is better… Why?