INVESTMENTS | BODIE, KANE, MARCUS Risk, Return, and the Historical Record Copyright © 2014...

INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS

Risk, Return, and the Historical Record

• Interest rate determinants• Rates of return for different holding periods• Risk and risk premiums• Estimations of return and risk• Normal distribution

• Deviation from normality and risk estimation• Historic returns on risky portfolios

Chapter Overview

• Supply• Households

• Demand• Businesses

• Government’s net demand• Federal Reserve actions

Interest Rate Determinants

• Nominal interest rate (rn): • Growth rate of your money

• Real interest rate (rr): • Growth rate of your purchasing power

• Where i is the rate of inflation

Real and Nominal Rates of Interest

rr rn i 1

rn irr

Figure 5.1 Determination of the Equilibrium Real Rate of Interest

• As the inflation rate increases, investors will demand higher nominal rates of return

• If E(i) denotes current expectations of inflation, then we get the Fisher Equation:

Equilibrium Nominal Rate of Interest

rn rr E i

• Tax liabilities are based on nominal income• Given a tax rate (t) and nominal interest rate (rn),

the real after-tax rate is:

• The after-tax real rate of return falls as the inflation rate rises

Taxes and the Real Rate of Interest

1 1 1rn t i rr i t i rr t it

• Zero Coupon Bond:• Par = $100• Maturity = T• Price = P• Total risk free return

Rates of Return for Different Holding Periods

100( ) 1

( )fr T P T

Example 5.2 Annualized Rates of Return

• EAR: Percentage increase in funds invested over a 1-year horizon

Effective Annual Rate (EAR)

1 EAR 1 Tfr T

• APR: Annualizing using simple interest

Annual Percentage Rate (APR)

1 EAR 1APR

Table 5.1 APR vs. EAR

Table 5.2 T-Bill Rates, Inflation Rates,

and Real Rates, 1926-2012

• Moderate inflation can offset most of the nominal gains on low-risk investments

• A dollar invested in T-bills from 1926–2012 grew to $20.25 but with a real value of only $1.55

• Negative correlation between real rate and inflation rate means the nominal rate doesn’t fully compensate investors for increased in inflation

Bills and Inflation, 1926-2012

Figure 5.3 Interest Rates and Inflation, 1926-2012

• Rates of return: Single period

• HPR = Holding period return

• P0 = Beginning price

• P1 = Ending price

• D1 = Dividend during period one

Risk and Risk Premiums

HPR P P D

Ending Price = $110Beginning Price = $100Dividend = $4

Rates of Return: Single Period Example

$110 $100 $4HPR .14, or 14%

• Expected returns

• p(s) = Probability of a state• r(s) = Return if a state occurs• s = State

Expected Return and Standard Deviation

( ) ( ) ( )s

E r p s r s

State Prob. of State r in State Excellent .25 0.3100Good .45 0.1400Poor .25 -0.0675Crash .05 -0.5200

E(r) = (.25)(.31) + (.45)(.14) + (.25)(−.0675) + (0.05)(− 0.52) E(r) = .0976 or 9.76%

Scenario Returns: Example

• Variance (VAR):

• Standard Deviation (STD):

Expected Return and Standard Deviation

rEsrsp 22

• Example VAR calculation:σ2 = .25(.31 − 0.0976)2 + .45(.14 − .0976)2

+ .25(− 0.0675 − 0.0976)2 + .05(−.52 − .0976)2

= .038

• Example STD calculation:

Scenario VAR and STD: Example

σ .038

• True means and variances are unobservable because we don’t actually know possible scenarios like the one in the examples

• So we must estimate them (the means and variances, not the scenarios)

Time Series Analysis of Past Rates of Return

• Arithmetic Average

• Geometric (Time-Weighted) Average

= Terminal value of the investment

Returns Using Arithmetic and Geometric Averaging

srsprE11

)()()(

)1)...(1)(1( 21 nn rrrTV

1/ 1ng TV

• Estimated Variance• Expected value of squared deviations

• Unbiased estimated standard deviation

Estimating Variance and Standard

Deviation

rsrn 1

22 1̂

• Excess Return• The difference in any particular period between

the actual rate of return on a risky asset and the actual risk-free rate

• Risk Premium• The difference between the expected HPR on a

risky asset and the risk-free rate• Sharpe Ratio

The Reward-to-Volatility (Sharpe) Ratio

Risk premium

SD of excess returns

• Investment management is easier when returns are normal• Standard deviation is a good measure of risk when

returns are symmetric• If security returns are symmetric, portfolio returns will

be as well• Future scenarios can be estimated using only the

mean and the standard deviation• The dependence of returns across securities can be

summarized using only the pairwise correlation coefficients

The Normal Distribution

Figure 5.4 The Normal Distribution

Mean = 10%, SD = 20%

• What if excess returns are not normally distributed?• Standard deviation is no longer a complete

measure of risk• Sharpe ratio is not a complete measure of

portfolio performance• Need to consider skewness and kurtosis

Normality and Risk Measures

Figure 5.5A Normal and Skewed Distributions

Mean = 6%, SD = 17%

Figure 5.5B Normal and Fat-Tailed Distributions

Mean = .1, SD = .2

• Value at Risk (VaR)• Loss corresponding to a very low percentile of the

entire return distribution, such as the fifth or first percentile return

• Expected Shortfall (ES)• Also called conditional tail expectation (CTE),

focuses on the expected loss in the worst-case scenario (left tail of the distribution)

• More conservative measure of downside risk than VaR

• Lower Partial Standard Deviation (LPSD)and the Sortino Ratio• Similar to usual standard deviation, but uses only

negative deviations from the risk-free return, thus, addressing the asymmetry in returns issue

• Sortino Ratio (replaces Sharpe Ratio)• The ratio of average excess returns to LPSD

• The second half of the 20th century, politically and economically the most stable sub-period, offered the highest average returns

• Firm capitalization is highly skewed to the right: Many small but a few gigantic firms

• Average realized returns have generally been higher for stocks of small rather than large capitalization firms

Historic Returns on Risky Portfolios

• Normal distribution is generally a good approximation of portfolio returns• VaR indicates no greater tail risk than is characteristic

of the equivalent normal• The ES does not exceed 0.41 of the monthly SD,

presenting no evidence against the normality• However

• Negative skew is present in some of the portfolios some of the time, and positive kurtosis is present in all portfolios all the time

Historic Returns on Risky Portfolios

Figure 5.7 Nominal and Real Equity Returns Around the World, 1900-2000

Figure 5.8 SD of Real Equity & Bond Returns Around the World

Figure 5.9 Probability of Investment with a Lognormal Distribution

• When the continuously compounded rate of return on an asset is normally distributed, the effective rate of return will be lognormally distributed

Terminal Value with Continuous Compounding

INVESTMENTS | BODIE, KANE, MARCUS Risk, Return, and the Historical Record Copyright © 2014...

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