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Risk and Return: Past and Prologue

5Bodie, Kane, and MarcusEssentials of Investments, 9th Edition

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5.1 Rates of Return

• Holding-Period Return (HPR)• Rate of return over given investment period

• HPR= [PS − PB + CF] / PB• PS = Sale price• PB = Buy price• CF = Cash flow during holding period

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5.1 Rates of Return• Measuring Investment Returns over Multiple Periods• Arithmetic average

• Sum of returns in each period divided by number of periods

• Geometric average• Single per-period return; gives same cumulative performance as

sequence of actual returns

• Compound period-by-period returns; find per-period rate that compounds to same final value

• Dollar-weighted average return• Internal rate of return on investment

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Table 5.1 Quarterly Cash Flows/Rates of Return of a Mutual Fund

1st Quarter

2nd Quarter

3rd Quarter

4th Quarter

Assets under management at start of quarter ($ million)

1 1.2 2 0.8

Holding-period return (%) 10 25 −20 20

Total assets before net inflows 1.1 1.5 1.6 0.96

Net inflow ($ million) 0.1 0.5 −0.8 0.6

Assets under management at end of quarter ($ million)

1.2 2 0.8 1.56

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5.1 Rates of Return• Conventions for Annualizing Rates of Return• APR = Per-period rate × Periods per year

• 1 + EAR = (1 + Rate per period)

• 1 + EAR = (1 + Rate per period)n = (1 + )n

• APR = [(1 + EAR)1/n – 1]n

• Continuous compounding: 1 + EAR = eAPR

APR n

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5.2 Risk and Risk Premiums• Scenario Analysis and Probability Distributions• Scenario analysis: Possible economic scenarios; specify likelihood and HPR

• Probability distribution: Possible outcomes with probabilities

• Expected return: Mean value of distribution of HPR

• Variance: Expected value of squared deviation from mean

• Standard deviation: Square root of variance

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Spreadsheet 5.1 Scenario Analysis for the Stock Market

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5.2 Risk and Risk Premiums 

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Figure 5.1 Normal Distribution with Mean Return 10% and Standard Deviation 20%

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5.2 Risk and Risk Premiums

• Normality over Time• When returns over very short time periods are normally distributed, HPRs up to 1 month can be treated as normal

• Use continuously compounded rates where normality plays crucial role

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5.2 Risk and Risk Premiums• Deviation from Normality and Value at Risk

• Kurtosis: Measure of fatness of tails of probability distribution; indicates likelihood of extreme outcomes

• Skew: Measure of asymmetry of probability distribution

• Using Time Series of Return• Scenario analysis derived from sample history of returns• Variance and standard deviation estimates from time

series of returns:

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Figure 5.2 Comparing Scenario Analysis to Normal Distributions with Same Mean and Standard Deviation

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5.2 Risk and Risk Premiums• Risk Premiums and Risk Aversion

• Risk-free rate: Rate of return that can be earned with certainty

• Risk premium: Expected return in excess of that on risk-free securities

• Excess return: Rate of return in excess of risk-free rate

• Risk aversion: Reluctance to accept risk• Price of risk: Ratio of risk premium to variance

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5.2 Risk and Risk Premiums

• The Sharpe (Reward-to-Volatility) Ratio• Ratio of portfolio risk premium to standard deviation

• Mean-Variance Analysis• Ranking portfolios by Sharpe ratios

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5.3 The Historical Record• World and U.S. Risky Stock and Bond Portfolios• World Large stocks: 24 developed countries, about 6000 stocks

• U.S. large stocks: Standard & Poor's 500 largest cap• U.S. small stocks: Smallest 20% on NYSE, NASDAQ, and Amex

• World bonds: Same countries as World Large stocks• U.S. Treasury bonds: Barclay's Long-Term Treasury Bond Index

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Figure 5.4 Rates of Return on Stocks, Bonds, and Bills

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5.4 Inflation and Real Rates of Return

• Equilibrium Nominal Rate of Interest• Fisher Equation

• R = r + E(i)• E(i): Current expected inflation• R: Nominal interest rate• r: Real interest rate

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5.4 Inflation and Real Rates of Return

• U.S. History of Interest Rates, Inflation, and Real Interest Rates• Since the 1950s, nominal rates have increased roughly in tandem with inflation

• 1930s/1940s: Volatile inflation affects real rates of return

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Figure 5.5 Interest Rates, Inflation, and Real Interest Rates 1926-2010

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5.5 Asset Allocation across Portfolios

• Asset Allocation• Portfolio choice among broad investment classes

• Complete Portfolio• Entire portfolio, including risky and risk-free assets

• Capital Allocation• Choice between risky and risk-free assets

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• The Risk-Free Asset• Treasury bonds (still affected by inflation)• Price-indexed government bonds• Money market instruments effectively risk-free• Risk of CDs and commercial paper is miniscule compared to most assets

5.5 Asset Allocation across Portfolios

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• Portfolio Expected Return and RiskP: portfolio compositiony: proportion of investment budgetrf: rate of return on risk-free assetrp: actual rate of returnE(rp): expected rate of returnσp: standard deviationE(rC): return on complete portfolioE(rC) = yE(rp) + (1 − y)rf

σC = yσrp + (1 − y) σrf

5.5 Asset Allocation Across Portfolios

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Figure 5.6 Investment Opportunity Set

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• Capital Allocation Line (CAL)• Plot of risk-return combinations available by varying allocation between risky and risk-free

• Risk Aversion and Capital Allocation• y: Preferred capital allocation

5.5 Asset Allocation across Portfolios

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5.6 Passive Strategies and the Capital Market Line

• Passive Strategy• Investment policy that avoids security analysis

• Capital Market Line (CML)• Capital allocation line using market-index portfolio as risky asset

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Table 5.4 Excess Return Statistics for S&P 500

Excess Return (%)Average Std Dev Sharpe Ratio 5% VaR

1926-2010 8.00 20.70 .39 −36.861926-1955 11.67 25.40 .46 −53.431956-1985 5.01 17.58 .28 −30.511986-2010 7.19 17.83 .40 −42.28

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• Cost and Benefits of Passive Investing• Passive investing is inexpensive and simple• Expense ratio of active mutual fund averages 1%

• Expense ratio of hedge fund averages 1%-2%, plus 10% of returns above risk-free rate

• Active management offers potential for higher returns

5.6 Passive Strategies and the Capital Market Line

Selected Problems

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Problem 1

V(12/31/2004) = V (1/1/1998) x (1 + GAR)7 = $100,000 x (1.05)7 =

$140,710.04

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Problem 2

a. The holding period returns for the three scenarios are:Boom: Normal: Recession: E(HPR) = 2(HPR)

(50 – 40 + 2)/40 = 0.30 = 30.00%(43 – 40 + 1)/40 = 0.10 = 10.00%

(34 – 40 + 0.50)/40 = –0.1375 = –13.75%[(1/3) x 30%] + [(1/3) x 10%] + [(1/3) x (–13.75%)] = 8.75%

]8.75%) – (–13.75% x [(1/3) ]8.75%) – (10% x [(1/3) ]8.75%) – (30% x [(1/3)σ 222(HPR)2 0.031979

17.88%σ(HPR)

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Problem 2 Cont.

b. E(r) =

= (0.5 x 8.75%) + (0.5 x 4%) = 6.375%

0.5 x 17.88% = 8.94%

Risky E[rp] = 8.75%Risky p = 17.88%

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Problems 3 & 4

3. For each portfolio: Utility = E(r) – (0.5 4 2 )

We choose the portfolio with the highest utility value, which is Investment 3.

Investment E(r) U1 0.12 0.30 -0.06002 0.15 0.50 -0.35003 0.21 0.16 0.15884 0.24 0.21 0.1518

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Problems 3 & 4 Cont.

4. When an investor is risk neutral, A = _ so that the portfolio with the highest utility is the portfolio with the _______________________.So choose ____________. highest expected return

0

Investment 4

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Problem 5

(95 – 90 + 4)/90 = 10.00%2004-2005

(90 – 110 + 4)/110 = –14.55%2003-2004

(110 – 100 + 4)/100 = 14.00%2002-2003

Return = [(capital gains + dividend) / price]

a. TWRYear

Dividends on four shares,plus sale of four shares at $95 per share

396

Dividends on five shares,plus sale of one share at $90 110

Purchase of two shares at $110,plus dividend income on three shares held

-208

Purchase of three shares at $100 per share-300

3

2

1

0

ExplanationCash flowTime

b. DWR

a. T

WR

3

10.00%14.55%14.00%AAR

110]0.1455)x1.[1.14x(1GAR 1/3 2.33%

3.15%

IRR)(1

$396IRR)(1

$110IRR)(1$208

IRR)(1$300$0 3210

-0.1661%

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