Ch 6 Risk and Rates of Return 2002

Ch 6: Risk and Rates of Return 2002, Prentice Hall, Inc. Risk

Chapter 6: Objectives • Inflation and rates of return • How to measure risk (variance, standard deviation, beta) • How to reduce risk (diversification) • How to price risk (security market line, CAPM)

Inflation, Rates of Return, and the Fisher Effect Interest Rates

Conceptually: Interest Rates

Conceptually: Nominal risk-free Interest Rate krf Interest Rates

Conceptually: Nominal risk-free Interest Rate krf = Interest Rates

Interest Rates Conceptually: Nominal risk-free Interest Rate krf = Real risk-free Interest Rate k*

Interest Rates Conceptually: Nominal risk-free Interest Rate krf = Real risk-free Interest Rate k* +

Interest Rates Conceptually: Nominal risk-free Interest Rate krf = Real risk-free Interest Rate k* + Inflationrisk premium IRP

Interest Rates Conceptually: Nominal risk-free Interest Rate = Real risk-free Interest Rate krf Mathematically: k* + Inflationrisk premium IRP

Interest Rates Conceptually: Nominal risk-free Interest Rate = Real risk-free Interest Rate krf k* + Inflationrisk premium IRP Mathematically: (1 + krf) = (1 + k*) (1 + IRP)

Interest Rates Conceptually: Nominal risk-free Interest Rate = Real risk-free Interest Rate krf k* + Inflationrisk premium IRP Mathematically: (1 + krf) = (1 + k*) (1 + IRP) This is known as the “Fisher Effect”

Interest Rates • Suppose the real rate is 3%, and the nominal rate is 8%. What is the inflation rate premium? (1 + krf) = (1 + k*) (1 + IRP) (1. 08) = (1. 03) (1 + IRP) = (1. 0485), so IRP = 4. 85%

Term Structure of Interest Rates • The pattern of rates of return for debt securities that differ only in the length of time to maturity.

Term Structure of Interest Rates • The pattern of rates of return for debt securities that differ only in the length of time to maturity. yield to maturity time to maturity (years)

Term Structure of Interest Rates • The yield curve may be downward sloping or “inverted” if rates are expected to fall. yield to maturity time to maturity (years)

For a Treasury security, what is the required rate of return?

For a Treasury security, what is the required rate of return? Required rate of return =

For a Treasury security, what is the required rate of return? Required rate of return = Risk-free rate of return Since Treasuries are essentially free of default risk, the rate of return on a Treasury security is considered the “risk-free” rate of return.

For a corporate stock or bond, what is the required rate of return?

For a corporate stock or bond, what is the required rate of return? Required rate of return =

For a corporate stock or bond, what is the required rate of return? Required rate of return = Risk-free rate of return

For a corporate stock or bond, what is the required rate of return? Required rate of return = Risk-free rate of return + Risk premium How large of a risk premium should we require to buy a corporate security?

Returns • Expected Return - the return that an investor expects to earn on an asset, given its price, growth potential, etc. • Required Return - the return that an investor requires on an asset given its risk and market interest rates.

Expected Return State of Probability Return Economy (P) Orl. Utility Orl. Tech Recession. 20 4% -10% Normal. 50 10% 14% Boom. 30 14% 30% For each firm, the expected return on the stock is just a weighted average:

Expected Return State of Probability Return Economy (P) Orl. Utility Orl. Tech Recession. 20 4% -10% Normal. 50 10% 14% Boom. 30 14% 30% For each firm, the expected return on the stock is just a weighted average: k = P(k 1)*k 1 + P(k 2)*k 2 +. . . + P(kn)*kn

Expected Return State of Probability Return Economy (P) Orl. Utility Orl. Tech Recession. 20 4% -10% Normal. 50 10% 14% Boom. 30 14% 30% k = P(k 1)*k 1 + P(k 2)*k 2 +. . . + P(kn)*kn k (OU) =. 2 (4%) +. 5 (10%) +. 3 (14%) = 10%

Expected Return State of Probability Return Economy (P) Orl. Utility Orl. Tech Recession. 20 4% -10% Normal. 50 10% 14% Boom. 30 14% 30% k = P(k 1)*k 1 + P(k 2)*k 2 +. . . + P(kn)*kn k (OI) =. 2 (-10%)+. 5 (14%) +. 3 (30%) = 14%

Based only on your expected return calculations, which stock would you prefer?

Have you considered RISK?

What is Risk? • The possibility that an actual return will differ from our expected return. • Uncertainty in the distribution of possible outcomes.

What is Risk? • Uncertainty in the distribution of possible outcomes.

What is Risk? • Uncertainty in the distribution of possible outcomes. Company A return

What is Risk? • Uncertainty in the distribution of possible outcomes. Company A Company B return

How do we Measure Risk? • To get a general idea of a stock’s price variability, we could look at the stock’s price range over the past year. 52 weeks Yld Vol Net Hi Lo Sym Div % PE 100 s Hi Lo Close Chg 134 80 IBM. 52. 5 21 143402 98 95 9549 -3 115 40 MSFT … 29 558918 55 52 5194 -475

How do we Measure Risk? • A more scientific approach is to examine the stock’s standard deviation of returns. • Standard deviation is a measure of the dispersion of possible outcomes. • The greater the standard deviation, the greater the uncertainty, and therefore , the greater the risk.

Standard Deviation n s = S (ki i=1 2 k) P(ki)

s= n S (ki i=1 Orlando Utility, Inc. 2 k) P(ki)

s= n S (ki - 2 k) i=1 Orlando Utility, Inc. ( 4% - 10%)2 (. 2) = 7. 2 P(ki)

s= n S (ki - 2 k) i=1 Orlando Utility, Inc. ( 4% - 10%)2 (. 2) = 7. 2 (10% - 10%)2 (. 5) = 0 P(ki)

s= n S (ki - 2 k) i=1 Orlando Utility, Inc. ( 4% - 10%)2 (. 2) = 7. 2 (10% - 10%)2 (. 5) = 0 (14% - 10%)2 (. 3) = 4. 8 P(ki)

s= n S (ki - 2 k) i=1 Orlando Utility, Inc. ( 4% - 10%)2 (. 2) = (10% - 10%)2 (. 5) = (14% - 10%)2 (. 3) = Variance = 7. 2 0 4. 8 12 P(ki)

s= n S (ki - 2 k) i=1 Orlando Utility, Inc. ( 4% - 10%)2 (. 2) = 7. 2 (10% - 10%)2 (. 5) = 0 (14% - 10%)2 (. 3) = 4. 8 Variance = 12 Stand. dev. = 12 = P(ki)

s= n S (ki - 2 k) P(ki) i=1 Orlando Utility, Inc. ( 4% - 10%)2 (. 2) = 7. 2 (10% - 10%)2 (. 5) = 0 (14% - 10%)2 (. 3) = 4. 8 Variance = 12 Stand. dev. = 12 = 3. 46%

s= n S (ki - 2 k) i=1 Orlando Technology, Inc. P(ki)

s= n S (ki - 2 k) i=1 Orlando Technology, Inc. (-10% - 14%)2 (. 2) = 115. 2 P(ki)

s= n S (ki - 2 k) i=1 Orlando Technology, Inc. (-10% - 14%)2 (. 2) = 115. 2 (14% - 14%)2 (. 5) = 0 P(ki)

s= n S (ki - 2 k) i=1 Orlando Technology, Inc. (-10% - 14%)2 (. 2) = 115. 2 (14% - 14%)2 (. 5) = 0 (30% - 14%)2 (. 3) = 76. 8 P(ki)

s= n S (ki - 2 k) i=1 Orlando Technology, Inc. (-10% - 14%)2 (. 2) = 115. 2 (14% - 14%)2 (. 5) = 0 (30% - 14%)2 (. 3) = 76. 8 Variance = 192 P(ki)

s= n S (ki - 2 k) i=1 Orlando Technology, Inc. (-10% - 14%)2 (. 2) = 115. 2 (14% - 14%)2 (. 5) = 0 (30% - 14%)2 (. 3) = 76. 8 Variance = 192 Stand. dev. = 192 = P(ki)

s= n S (ki - 2 k) P(ki) i=1 Orlando Technology, Inc. (-10% - 14%)2 (. 2) = 115. 2 (14% - 14%)2 (. 5) = 0 (30% - 14%)2 (. 3) = 76. 8 Variance = 192 Stand. dev. = 192 = 13. 86%

Which stock would you prefer? How would you decide?

Summary Orlando Utility Expected Return Standard Deviation Orlando Technology 10% 14% 3. 46% 13. 86%

It depends on your tolerance for risk! Remember, there’s a tradeoff between risk and return.

It depends on your tolerance for risk! Return Risk Remember, there’s a tradeoff between risk and return.

Portfolios • Combining several securities in a portfolio can actually reduce overall risk. • How does this work?

Suppose we have stock A and stock B. The returns on these stocks do not tend to move together over time (they are not perfectly correlated). rate of return time

Suppose we have stock A and stock B. The returns on these stocks do not tend to move together over time (they are not perfectly correlated). k. A rate of return time

Suppose we have stock A and stock B. The returns on these stocks do not tend to move together over time (they are not perfectly correlated). k. A rate of return k. B time

What has happened to the variability of returns for the portfolio? k. A rate of return k. B time

What has happened to the variability of returns for the portfolio? k. A rate of return k. B time kp

Diversification • Investing in more than one security to reduce risk. • If two stocks are perfectly positively correlated, diversification has no effect on risk. • If two stocks are perfectly negatively correlated, the portfolio is perfectly diversified.

• If you owned a share of every stock traded on the NYSE and NASDAQ, would you be diversified? YES! • Would you have eliminated all of your risk? NO! Common stock portfolios still have risk.

Some risk can be diversified away and some cannot. • Market risk (systematic risk) is nondiversifiable. This type of risk cannot be diversified away. • Company-unique risk (unsystematic risk) is diversifiable. This type of risk can be reduced through diversification.

Market Risk • Unexpected changes in interest rates. • Unexpected changes in cash flows due to tax rate changes, foreign competition, and the overall business cycle.

Company-unique Risk • A company’s labor force goes on strike. • A company’s top management dies in a plane crash. • A huge oil tank bursts and floods a company’s production area.

As you add stocks to your portfolio, company-unique risk is reduced.

As you add stocks to your portfolio, company-unique risk is reduced. portfolio risk number of stocks

As you add stocks to your portfolio, company-unique risk is reduced. portfolio risk Market risk number of stocks

As you add stocks to your portfolio, company-unique risk is reduced. portfolio risk companyunique risk Market risk number of stocks

Do some firms have more market risk than others? Yes. For example: Interest rate changes affect all firms, but which would be more affected: a) Retail food chain b) Commercial bank

• Note As we know, the market compensates investors for accepting risk - but only for market risk. Companyunique risk can and should be diversified away. So - we need to be able to measure market risk.

This is why we have Beta: a measure of market risk. • Specifically, beta is a measure of how an individual stock’s returns vary with market returns. • It’s a measure of the “sensitivity” of an individual stock’s returns to changes in the market.

The market’s beta is 1 • A firm that has a beta = 1 has average market risk. The stock is no more or less volatile than the market. • A firm with a beta > 1 is more volatile than the market.

The market’s beta is 1 • A firm that has a beta = 1 has average market risk. The stock is no more or less volatile than the market. • A firm with a beta > 1 is more volatile than the market. – (ex: technology firms)

The market’s beta is 1 • A firm that has a beta = 1 has average market risk. The stock is no more or less volatile than the market. • A firm with a beta > 1 is more volatile than the market. – (ex: technology firms) • A firm with a beta < 1 is less volatile than the market.

The market’s beta is 1 • A firm that has a beta = 1 has average market risk. The stock is no more or less volatile than the market. • A firm with a beta > 1 is more volatile than the market. – (ex: technology firms) • A firm with a beta < 1 is less volatile than the market. – (ex: utilities)

Calculating Beta

Calculating Beta XYZ Co. returns 15 10 S&P 500 returns 5 -10 -5 -5 -10 -15 5 10 15

Calculating Beta XYZ Co. returns 15 S&P 500 returns -15 . . . . 10. . . 5. -10 -5 -5 10. . . . -10. . . . -15. 15

Calculating Beta XYZ Co. returns 15 S&P 500 returns -15 . . . Beta = slope = 1. 20 . . 10. . . 5. -10 -5 -5 10. . . . -10. . . . -15. 15

Summary: • We know how to measure risk, using standard deviation for overall risk and beta for market risk. • We know how to reduce overall risk to only market risk through diversification. • We need to know how to price risk so we will know how much extra return we should require for accepting extra risk.

What is the Required Rate of Return? • The return on an investment required by an investor given market interest rates and the investment’s risk.

Required rate of return =

Required rate of return = Risk-free rate of return +

Required rate of return = Risk-free rate of return + Risk premium

Required rate of return = Risk-free rate of return market risk + Risk premium

Required rate of return = Risk-free rate of return market risk + Risk premium companyunique risk

Required rate of return = Risk-free rate of return market risk + Risk premium companyunique risk can be diversified away

Required rate of return Let’s try to graph this relationship! Beta

Required rate of return 12% . Risk-free rate of return (6%) 1 Beta

Required rate of return 12% . security market line (SML) 1 Beta Risk-free rate of return (6%)

This linear relationship between risk and required return is known as the Capital Asset Pricing Model (CAPM).

Required rate of return SML . 12% Risk-free rate of return (6%) 0 1 Beta

Required rate of return Is there a riskless (zero beta) security? SML . 12% Risk-free rate of return (6%) 0 1 Beta

Required rate of return Is there a riskless (zero beta) security? . 12% Risk-free rate of return (6%) 0 1 SML Treasury securities are as close to riskless as possible. Beta

Required rate of return Where does the S&P 500 fall on the SML? SML . 12% Risk-free rate of return (6%) 0 1 Beta

Required rate of return Where does the S&P 500 fall on the SML? . 12% Risk-free rate of return (6%) 0 1 SML The S&P 500 is a good approximation for the market Beta

Required rate of return SML Utility Stocks 12% . Risk-free rate of return (6%) 0 1 Beta

Required rate of return High-tech stocks SML . 12% Risk-free rate of return (6%) 0 1 Beta

The CAPM equation:

The CAPM equation: kj = krf + b j (km - krf )

The CAPM equation: kj = krf + b j (km - krf ) where: kj = the required return on security j, krf = the risk-free rate of interest, b j = the beta of security j, and km = the return on the market index.

Example: • Suppose the Treasury bond rate is 6%, the average return on the S&P 500 index is 12%, and Walt Disney has a beta of 1. 2. • According to the CAPM, what should be the required rate of return on Disney stock?

kj = krf + b (km - krf ) kj =. 06 + 1. 2 (. 12 -. 06) kj =. 132 = 13. 2% According to the CAPM, Disney stock should be priced to give a 13. 2% return.

Required rate of return SML . 12% Risk-free rate of return (6%) 0 1 Beta

Required rate of return Theoretically, every security should lie on the SML . 12% Risk-free rate of return (6%) 0 1 Beta

Required rate of return Theoretically, every security should lie on the SML . 12% If every stock is on the SML, investors are being fully compensated for risk. Risk-free rate of return (6%) 0 1 Beta

Required rate of return If a security is above the SML, it is underpriced. SML . 12% Risk-free rate of return (6%) 0 1 Beta

Required rate of return If a security is above the SML, it is underpriced. SML . 12% If a security is below the SML, it is overpriced. Risk-free rate of return (6%) 0 1 Beta

Simple Return Calculations

$50 t Simple Return Calculations $60 t+1

$50 Simple Return Calculations $60 t+1 t Pt+1 - Pt Pt = 60 - 50 50 = 20%

Simple Return Calculations $50 $60 t+1 t Pt+1 - Pt Pt Pt+1 Pt = -1 = 60 - 50 50 60 50 = 20% -1 = 20%

month Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec price $50. 00 $58. 00 $63. 80 $59. 00 $62. 00 $64. 50 $69. 00 $75. 00 $82. 50 $73. 00 $80. 00 $86. 00 (a) (b) monthly expected return (a - b)2

month Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec price $50. 00 $58. 00 $63. 80 $59. 00 $62. 00 $64. 50 $69. 00 $75. 00 $82. 50 $73. 00 $80. 00 $86. 00 (a) (b) monthly expected return 0. 160 (a - b)2

month Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec price $50. 00 $58. 00 $63. 80 $59. 00 $62. 00 $64. 50 $69. 00 $75. 00 $82. 50 $73. 00 $80. 00 $86. 00 (a) (b) monthly expected return 0. 160 0. 100 (a - b)2

month Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec price $50. 00 $58. 00 $63. 80 $59. 00 $62. 00 $64. 50 $69. 00 $75. 00 $82. 50 $73. 00 $80. 00 $86. 00 (a) (b) monthly expected return 0. 160 0. 100 -0. 075 (a - b)2

month Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec price $50. 00 $58. 00 $63. 80 $59. 00 $62. 00 $64. 50 $69. 00 $75. 00 $82. 50 $73. 00 $80. 00 $86. 00 (a) (b) monthly expected return 0. 160 0. 100 -0. 075 0. 051 (a - b)2

month Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec price $50. 00 $58. 00 $63. 80 $59. 00 $62. 00 $64. 50 $69. 00 $75. 00 $82. 50 $73. 00 $80. 00 $86. 00 (a) (b) monthly expected return 0. 160 0. 100 -0. 075 0. 051 0. 040 (a - b)2

month Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec price $50. 00 $58. 00 $63. 80 $59. 00 $62. 00 $64. 50 $69. 00 $75. 00 $82. 50 $73. 00 $80. 00 $86. 00 (a) (b) monthly expected return 0. 160 0. 100 -0. 075 0. 051 0. 040 0. 070 (a - b)2

month Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec price $50. 00 $58. 00 $63. 80 $59. 00 $62. 00 $64. 50 $69. 00 $75. 00 $82. 50 $73. 00 $80. 00 $86. 00 (a) (b) monthly expected return 0. 160 0. 100 -0. 075 0. 051 0. 040 0. 070 0. 000 (a - b)2

month Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec price $50. 00 $58. 00 $63. 80 $59. 00 $62. 00 $64. 50 $69. 00 $75. 00 $82. 50 $73. 00 $80. 00 $86. 00 (a) (b) monthly expected return 0. 160 0. 100 -0. 075 0. 051 0. 040 0. 070 0. 000 0. 087 (a - b)2

month Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec price $50. 00 $58. 00 $63. 80 $59. 00 $62. 00 $64. 50 $69. 00 $75. 00 $82. 50 $73. 00 $80. 00 $86. 00 (a) (b) monthly expected return 0. 160 0. 100 -0. 075 0. 051 0. 040 0. 070 0. 000 0. 087 0. 100 (a - b)2

month Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec price $50. 00 $58. 00 $63. 80 $59. 00 $62. 00 $64. 50 $69. 00 $75. 00 $82. 50 $73. 00 $80. 00 $86. 00 (a) (b) monthly expected return 0. 160 0. 100 -0. 075 0. 051 0. 040 0. 070 0. 000 0. 087 0. 100 -0. 115 (a - b)2

month Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec price $50. 00 $58. 00 $63. 80 $59. 00 $62. 00 $64. 50 $69. 00 $75. 00 $82. 50 $73. 00 $80. 00 $86. 00 (a) (b) monthly expected return 0. 160 0. 100 -0. 075 0. 051 0. 040 0. 070 0. 000 0. 087 0. 100 -0. 115 0. 096 (a - b)2

month Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec price $50. 00 $58. 00 $63. 80 $59. 00 $62. 00 $64. 50 $69. 00 $75. 00 $82. 50 $73. 00 $80. 00 $86. 00 (a) (b) monthly expected return 0. 160 0. 100 -0. 075 0. 051 0. 040 0. 070 0. 000 0. 087 0. 100 -0. 115 0. 096 0. 075 (a - b)2

month Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec price $50. 00 $58. 00 $63. 80 $59. 00 $62. 00 $64. 50 $69. 00 $75. 00 $82. 50 $73. 00 $80. 00 $86. 00 (a) (b) monthly expected return 0. 160 0. 100 -0. 075 0. 051 0. 040 0. 070 0. 000 0. 087 0. 100 -0. 115 0. 096 0. 075 0. 049 0. 049 (a - b)2

month Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec price $50. 00 $58. 00 $63. 80 $59. 00 $62. 00 $64. 50 $69. 00 $75. 00 $82. 50 $73. 00 $80. 00 $86. 00 (a) (b) monthly expected return 0. 160 0. 100 -0. 075 0. 051 0. 040 0. 070 0. 000 0. 087 0. 100 -0. 115 0. 096 0. 075 0. 049 0. 049 (a - b)2 0. 012321 0. 002601 0. 015376 0. 000004 0. 000081 0. 000441 0. 002401 0. 001444 0. 002601 0. 028960 0. 002090 0. 000676

month Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec price $50. 00 $58. 00 $63. 80 $59. 00 $62. 00 $64. 50 $69. 00 $75. 00 $82. 50 $73. 00 $80. 00 $86. 00 (a) (b) monthly expected return 0. 160 0. 100 -0. 075 0. 051 0. 040 0. 070 0. 000 0. 087 0. 100 -0. 115 0. 096 0. 075 0. 049 0. 049 (a - b)2 0. 012321 0. 002601 0. 015376 0. 000004 0. 000081 0. 000441 0. 002401 0. 001444 0. 002601 0. 028960 0. 002090 0. 000676 St. Dev: sum, divide by (n-1), and take sq root: 0. 0781

Calculator solution using HP 10 B: • Enter monthly return on 10 B calculator, followed by sigma key (top right corner). • Shift 7 gives you the expected return. • Shift 8 gives you the standard deviation.