Risk and Return: Past and Prologue

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Risk and Return: Past and Prologue. 5. Bodie, Kane, and Marcus Essentials of Investments, 9th Edition. 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

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Risk and Return: Past and Prologue

5

Bodie, Kane, and Marcus

Essentials of Investments, 9th Edition

5.1 Rates of Return
• Holding-Period Return (HPR)
• Rate of return over given investment period
• HPR= [PS − PB + CF] / PB
• PS = Sale price
• CF = Cash flow during holding period
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
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

• 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
• 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
• 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:
Figure 5.2 Comparing Scenario Analysis to Normal Distributions with Same Mean and Standard Deviation
• 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
• The Sharpe (Reward-to-Volatility) Ratio
• Ratio of portfolio risk premium to standard deviation
• Mean-Variance Analysis
• Ranking portfolios by Sharpe ratios
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
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
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
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
5.5 Asset Allocation across Portfolios
• 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
• Portfolio Expected Return and Risk

P: portfolio composition

y: proportion of investment budget

rf: rate of return on risk-free asset

rp: actual rate of return

E(rp): expected rate of return

σp: standard deviation

E(rC): return on complete portfolio

E(rC) = yE(rp) + (1 − y)rf

σC = yσrp+ (1 − y)σrf

5.5 Asset Allocation across Portfolios
• 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
5.6 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

5.6 Passive Strategies and the Capital Market Line

• 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
Problem 1

\$140,710.04

• V(12/31/2004) = V (1/1/1998) x (1 + GAR)7

= \$100,000 x (1.05)7 =

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Problem 2

(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%

0.031979

a. The holding period returns for the three scenarios are:

Boom:

Normal:

Recession:

E(HPR) =

2(HPR)

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Problem 2 Cont.

Risky E[rp] = 8.75%Risky p = 17.88%

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

0.5 x 17.88% = 8.94%

b. E(r) =

 =

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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.

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Problems 3 & 4 Cont.

0

highest expected return

Investment 4

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

So choose ____________.

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Problem 5

Time

Cash flow

Explanation

0

1

2

3

b. DWR

a. TWR

Year

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

a. TWR

2002-2003

(110 – 100 + 4)/100 = 14.00%

-300

Purchase of three shares at \$100 per share

2003-2004

(90 – 110 + 4)/110 = –14.55%

-208

Purchase of two shares at \$110,

plus dividend income on three shares held

2004-2005

(95 – 90 + 4)/90 = 10.00%

3.15%

110

Dividends on five shares,

plus sale of one share at \$90

396

Dividends on four shares,

plus sale of four shares at \$95 per share

2.33%

-0.1661%

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