Chapter 5 investments risk and return past and prologue

- Compound period-by-period returns; find per-period rate that compounds to same final value - Called a time-weighted average return... Normal DistributionE[r] = 10%  = 20% Average = M

Chapter 5 Risk and Return: Past and Prologue

5.1 Rates of Return

• PS = Sale price, PB = Buy price,

CF = Cash flow during holding period

- Compound period-by-period returns; find per-period rate

that compounds to same final value

- Called a time-weighted average return

Table 5.1 Quarterly Cash Flows/Rates of R eturn of a Mutual Fund

1st Quarter 2nd Quarter 3rd Quarter 4th Quarter

Assets under management at start of

Assets under management at end of

rg = 0.0719 or 7.19%

5.1 Rates of Return

Conventions for Annualizing Rates of Return

 Returns on assets with regular cash flows usually are quote

d as annual percentage rates, or APRs

- Mortgage: Monthly payments

- Bonds: Semiannual coupons

Conventions for Annualizing Rates of Return

Example:

• Suppose you have a 5% HPR on a 3 month

investment What is the annual rate of return with

and without compounding?

• Without:

• With:

n = 12/3 = 4, so HPRann(APR) = HPR*n = 0.05*4 = 20%

HPRann(EAR) = (1.054) - 1 = 21.55%

Example: Suppose you buy one share of a stock today for $45 and you ho

ld it for two years and sell it for $52 You also received $8 in dividends at the end of the two years.

5.2 Risk and Risk Premiums

Scenario Analysis and Probability tributions

Dis-• 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

Subjective expected returns

E(r) = Expected Return

2 p(s) [r E(r)]

σ Var(r)

2 p(s) [r E(r)]

σ

Characteristics of Probability Distributions

Arithmetic average & usually most likely

Dispersion of returns about the mean Long tailed distribution, either side

Too many observations in the tails

Characteristics 1 and 3

Middle observation

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Normal Distribution

E[r] = 10%

 = 20%

Average = Median

Risk is the possibility

of getting returns dif

-ferent from expected.

 measures deviations above the mean as well as below the mean

With symmetric distribution, it

is ok to use  to measure risk

• Central to the theory and practice of investment

• Bell-shaped plot is symmetric

• The probabilities are highest for outcomes near the mean

Normal Distribution

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Critical Simplifications

1 The return on a portfolio comprising two or more assets whose returns are normally distributed also will be normally distributed

2 The normal distribution is completely described by its mean and standard deviation

3 The standard deviation is the appropriate measure

of risk for a portfolio of assets with normally uted returns

distrib-Risk Premium & distrib-Risk Aversio n

• The risk free rate is the rate of return that can be earned with certainty.

• The risk premium is the difference between the expected return of a ris

ky asset and the risk-free rate.

• Risk aversion is an investor’s reluctance to accept risk.

• How is the aversion to accept risk overcome?

: By offering investors a higher risk premium.

The Sharpe(Reward-to-Volatility) M easure

• A statistic commonly used to rank portfolios in terms of risk-return trade-off is Sha rpe (or reward-to-volatility) measure

S = Portfolio risk premium / Standard deviation of portfolio excess retur

n

= (E(r p ) – r f) / p

(A risk-free asset: a risk premium=0, a standard deviation=0)

• Quantify the incremental reward for each increase of 1% in the standard deviation

of that portfolio.

• A higher sharp ratio indicates a better reward per unit of volatility (a more efficien

t portfolio)

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5.3 The Historical Record

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

5.4 Inflation and Real Rates of Ret

Nominal interest rate

: The interest rate in terms of nominal (not adjusted for purchasing power) dollars

Real interest rate

: The excess of the interest rate over the inflation rate

The growth rate of purchasing power derived from an investment

Real vs Nominal Rates

Real rate  Nominal rate - Inflation rate

The exact real rate is less than the approxi

mate real rate.

[(1 + r nom ) / (1 + i)] – 1 (r nom - i) / (1 + i)

(9% - 6%) / (1.06) = 2.83%

r real = real interest rate

r nom = nominal interest rate

i = expected inflation rate

Figure 5.5 Interest Rates, In flation, and Real Interest Rat

es 1926-2010

Historical Real Returns & Sharpe Ratios

Real Returns% Sharpe Ratio

• Real returns have been much higher for stocks than for bonds

• Sharpe ratios measure the excess return to standard deviation.

• The higher the Sharpe ratio the better.

• Stocks have had much higher Sharpe ratios than bonds.

5.5 Asset Allocation Across R

isky and Risk Free Portfolio

s

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Allocating Capital Between Risky & Risk-Fre

e Assets

 Possible to split investment funds between safe and

risky assets

 Risk free asset rf : proxy;

 Risky asset or portfolio rp: _

 Example Your total wealth is $10,000 You put $2,5

00 in risk free T-Bills and $7,500 in a stock portfolio i

nvested as follows:

• Stock A you put $2,500

• Stock B you put $3,000

• Stock C you put $2,000

T-bills or money market fund Risky portfolio

$7,500

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Allocating Capital Between Risky & Risk-Free Assets

• Issues in setting weights

- Examine risk & return tradeoff

- Demonstrate how different degrees of risk aversion w ill affect allocations between risky and risk free assets

Allocating Capital Between Risky & Risk-Free Assets

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Complete portfolio

 Varying y results in E[rC] and C that are linear combinations of E[rp] and rf and rp and rf respectively.

 The risk premium of the complete portfolio, C

= The risk premium of the risky asset X The fraction of the portfolio invested in the risky asset

E(r c ) – r f = y[E(r p ) – r f ]

E(rc) = yE(rp) + (1 - y)rf

c = yrp + (1-y)rf = yrp

Risk mium

Pre-Slope = [E(r p ) – r f ] /  rp

Combinations Without Leverag

y = 1 E(r c ) =

y = 0 E(r c ) =

(.75)(.14) + (.25)(.05) = 11.75%

(1)(.14) + (0)(.05) = 14.00%

(0)(.14) + (1)(.05) = 5.00%

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Using Leverage with Capital Allocati

Risk Aversion and Allocation

 Greater levels of risk aversion lead investors to choose larger

proportions of the risk free rate

 Lower levels of risk aversion lead investors to choose larger

proportions of the portfolio of risky assets

 Willingness to accept high levels of risk for high levels of returns would result in leveraged combinations

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5.6 Passive Strategies and th

e Capital Market Line

Table 5.4 Excess Return Statistics for S&P 5 00

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

1926-2010 8.00 20.70 39 −36.86

1926-1955 11.67 25.40 46 −53.43

1956-1985 5.01 17.58 28 −30.51

1986-2010 7.19 17.83 40 −42.28

Cost and Benefits of Passive

In-vesting

• Passive investing is inexpensive and simple

• Expense ratio of active mutual fund ages 1%

aver-• Expense ratio of hedge fund averages 2%,

plus 10% of returns above risk-free rate

• Active management offers potential for

higher

returns