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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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slide1

Risk and Return: Past and Prologue

5

Bodie, Kane, and Marcus

Essentials of Investments, 9th Edition

5 1 rates of return
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
5 1 rates of return1
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 return2
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

5 2 risk and risk premiums
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
5 2 risk and risk premiums2
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
5 2 risk and risk premiums3
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:
slide12
Figure 5.2 Comparing Scenario Analysis to Normal Distributions with Same Mean and Standard Deviation
5 2 risk and risk premiums4
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
5 2 risk and risk premiums5
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
5 3 the historical record
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
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 return1
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
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 portfolios1
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 portfolios2
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 portfolios3
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
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
slide27

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
Problem 1

$140,710.04

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

= $100,000 x (1.05)7 =

5-29

problem 2
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)

5-30

problem 2 cont
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) =

 =

5-31

problems 3 4
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.

5-32

problems 3 4 cont
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 ____________.

5-33

problem 5
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%

5-34

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