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# BCOR 2200 Chapter 10

BCOR 2200 Chapter 10. Lessons From Capital Market History. We already 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

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## BCOR 2200 Chapter 10

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1. BCOR 2200Chapter 10 Lessons From Capital Market History

2. We already 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 isdifferent, • 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. Chapter Outline: • The Mechanics of Calculating Returns • Dollar Returns • Percent Returns • The Historical Record • Calculating Average Returns • Return Variability (Risk) • More about Calculating Average Returns • Market Efficiency

6. 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

7. Dollar Returns: Stock Example: • 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

8. 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%

9. Percent Return Formula A little Algebra:

10. 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

11. Clicker Question: • A stock’s current market price is \$50. It just paid a \$2.00 dividend. One year ago, the stock cost \$40 • Calculate: • the stock’s Dividend Yield • the stock’s Capital Gain Yield • the stock’s Total Return

12. Clicker Answer: • P1 = \$50 D1 = \$2.00 P0 = \$40 • Div Yld = D1/P0 = \$2.00/\$40 = 5% • Cap Gain Yld = P1/P0 – 1 = \$50/\$40 – 1 = 25% • Total Return = (P1 + D1)/P0 – 1 = (\$50 + \$2)/\$40 – 1 = 30% -OR- • Total Return = Div Yld + Cap Gain Yld = 5% + 25% = 30% The Answer is D

13. Clicker Question: • A stock’s current price is \$80. It is expected to pay a \$4.00 dividend in one year. The expected price in one year is \$70 • Calculate: • the stock’s expectedDividend Yield • the stock’s expectedCapital Gain Yield • the stock’s expected Total Return

14. Clicker Answer: • P1 = \$70 D1 = \$4.00 P0 = \$80 • Div Yld = D1/P0 = \$4.00/\$80 = 5% • Cap Gain Yld = P1/P0 – 1 = \$70/\$80 – 1 = -12.5% • Total Return = (P1 + D1)/P0 – 1 = (\$70 + \$4)/\$80 – 1 = -7.5% -OR- • Total Return = Div Yld + Cap Gain Yld = 5% + -12.5% = -7.5% The Answer is B

15. 10.2 The Historical Record • Why do we look at the historical record? • Do we care (directly) about what happened in the past? • Not really • 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

16. 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

17. 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 • Theycan 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

18. 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 • Reread pages 217 to 219 and the box on page 221 • 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

19. Historic Returns for four of the items:

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

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

22. Figure 10.8 • Annual Inflation • Again note Scale Inflation

23. Further discussion on the historical record: • Read Page 315 through 321 • See Figures 10.5 through 10.8 and Tables 10.1 through 10.3 • Asset Classes: • Large stocks • Small Stocks • Long-term Corporate Bonds • Long-term Government Bonds • U.S. Treasury Bills • Inflation • Get an idea of large and small changes for each of these categories • How is inflation measured?

24. 10.3 Average Returns of Investment Categories RReal = (1 + RNom)/(1 + i) – 1 = (1 +0.1180)/(1.0310) - 1 = 8.44% Risk Premium = RNom– Risk-Free = 0.1180 – 0.0360 = 8.20% (T-Bills are risk-free) Data is shown in Tables 10.2 and 10.3

25. Average Returns RReal = (1 + RNom)/(1 + Inflation) – 1 For Small Stocks: RNom = 16.5% RReal = 13.0% MoneyDoubled in Small Stocks: • Rule of 72’s: 72/16.5 = 4.36 years • PV = -1; FV = 2; I/Y = 16.5; N = 4.54 • =nper(rate, pmt, pv, [fv],[type]) • =nper(.165,0,-1,2) = 4.54

26. Average Returns RReal = (1 + RNom)/(1 + Inflation) – 1 For Small Stocks: RNom = 16.5% RReal = 13.0% Purchasing Power Doubled in Small Stocks: • Rule of 72’s: 72/13.0 = 5.54 years • PV = -1; FV = 2; I/Y = 13.0; N = 5.67 • =nper(rate, pmt, pv, [fv],[type]) • =nper(.130,0,-1,2) = 5.67

27. Risk Premium • The amount of return earned over 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: Large stocks 11.8% – 3.6% = 8.2% Small stocks 16.5% – 3.6% = 12.9% L-T corporate bonds 6.4% – 3.6% = 2.8% L-T government bonds 6.1% – 3.6% = 2.5% Risk Premium is a RETURN measure, not a risk measure

28. 10.4 Variability of Returns (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

29. Risk: A Picture of the Dispersion of Returns Start with a Frequency Distribution for Large Stocks (Fig 10.9): Calculate Returns since 2009:

30. Risk: A Picture of the Dispersion of Returns Start with a Frequency Distribution for Large Stocks (Fig 10.9): Calculate Returns since 2009: 2009

31. Risk: A Picture of the Dispersion of Returns Start with a Frequency Distribution for Large Stocks (Fig 10.9): Calculate Returns since 2009: 2010 2009

32. Risk: A Picture of the Dispersion of Returns Start with a Frequency Distribution for Large Stocks (Fig 10.9): Calculate Returns since 2009: 2010 2009 2011

33. Risk: A Picture of the Dispersion of Returns Start with a Frequency Distribution for Large Stocks (Fig 10.9): Calculate Returns since 2009: 2012 2010 2009 2011

34. Risk: A Picture of the Dispersion of Returns Start with a Frequency Distribution for Large Stocks (Fig 10.9): Calculate Returns since 2009: 2012 2010 2013 2009 2011

35. Risk: Dispersion of Returns Calculating a Number • Measures of Dispersion: Variance and Standard Deviation Recall: • 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?

36. 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 %

37. Example 10.2 page 315 How to Calculate Variance (s2) and Stdev (s): Mean Return = 0.16/4 = 0.04 = 4% s2 = (Sum of Squared Deviations)/(T – 1) = 0.0270(4 – 1) = 0.009 s = Square Root of Variance = (0.009)½ = 0.0949 = 9.49%

38. Mean and Standard Deviation of Historic Returns Date is shown in Figure 10.10 on page 327

39. 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 • See the Normal Distribution and Statistics Review on the website • Recall for the Normal Distribution: • Mean +/- 1s contains approximately 68% of the observations • Really 68.26% • Mean +/- 2s contains approximately 95% of the observations • Really 95.44% • Mean +/- 3s contains approximately 99% of the observations • Really 99.73%

40. Interpreting the Distribution Measure (s) For the Large Stock Portfolio: • Mean Return = 11.8% • Standard Deviation (s) = 20.3% • 68% of observations between 11.8% +/- 1 x20.3% • Between -8.5% and 32.1% • 95% of observations between 11.8% +/- 2x 20.3% • Between -28.8% to 52.4% • 99% of observations between 11.8% +/- 3x 20.3% • Between -49.1% to 72.7% Notice you can be MORE CONFIDENT about a WIDER interval!

41. 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 -8.5% and 32.1% is 68.26% (approximately 2/3). + 3s72.7% - 3s- 49.1% - 2s- 28.8% - 1s - 8.5% 011.5% + 1s32.1% + 2s52.4% Return onlarge company commonstocks 68% 95% 99%

42. Clicker Question: • The expected return of a stock is 15% and it’s standard deviation is 18% • What will be the range of returns with 68% confidence? • What will be the range of returns with 95% confidence?

43. Clicker Answer: • 68% confidence is within the mean plus or minus one standard deviation: 15% - 18% = -3% 15% + 18% = 33% • 95% confidence is within the mean plus or minus two standard deviation: 15% - 2(18%) = -21% 15% + 2(18%) = 51% The Answer is A

44. Return Probabilities Assuming Normality Another thing we can do: • Large Stock Mean = 11.8% and σ = 20.3% • 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 – 0.118)/0.203 = -0.581 • So a little more than ½ a stdev from the mean to 0 How much of the curve is to the left of the mean - 0.568σ? • =NORMDIST(x, mean, standard_dev, cumulative) • =NORMDIST(0,0.118,0.203,1) = 0.2805 = 28.05% 28.05% chance of a negative return next year in large stocks

45. Return Probabilities Assuming Normality Another way to do it: • Large Stock Mean = 11.8% and σ = 20.3% • 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 – 0.118)/0.203 = -0.581 How much of the curve is to the left of the mean - 0.581σ? • =NORM.S.DIST(z,1) • =NORM.S.DIST(-.581,1) = 0.2806 = 28.05% NORM.DIST: Enter x, mean and σ NORM.S.DIST: Enter only z

46. Return Probabilities Assuming Normality One more example: • Large Stock Mean = 11.8% and σ = 20.3% • Calculate the probability earning 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 – 0.118)/0.203 = 0.382/0.203 = 1.88 How much of the curve is to the RIGHT of the mean + 1.88σ? • =NORM.S.DIST(z,1) • =NORM.S.DIST(1.88,1) = 0.9701 = 97.01% • 97.01% to the left or right? • So what is the probability of earnings more then 50%? • P(R > 50%) = 1 – 0.9701 = 2.99%

47. Return Probabilities Assuming Normality Last example: • Large Stock Mean = 11.8% and σ = 20.3% • Calculate the probability earning between10% and 30% How far from the mean is 10%? • z = (x – mean)/σ = (0.10 – 0.118)/0.203 = -0.018/0.203 = -.0887 • =NORM.S.DIST(-0.0887) = 0.4647 • P(R < 10%) = 46.47% How far from the mean is 30%? • z = (x – mean)/σ = (0.30– 0.118)/0.203 = 0.182/0.203 = 0.8966 • =NORM.S.DIST(0.8966) = 0.8150 • P(R < 30%) = 81.50% P(R < 10%) = 46.47% P(R < 30%) = 81.50% P(R > 10%andR < 30%) =81.50%-44.47%= 35.03%

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