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Chapter FiveRisk, Return, and the Historical RecordCopyright 2014 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS1Interest rate determinantsRates of return for different holding periodsRisk and risk premiumsEstimations of return and riskNormal distribution Deviation from normality and risk estimationHistoric returns on risky portfolios

Chapter OverviewINVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS2SupplyHouseholdsDemandBusinessesGovernments net demandFederal Reserve actionsInterest Rate DeterminantsINVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS3Nominal interest rate (rn): Growth rate of your moneyReal interest rate (rr): Growth rate of your purchasing power

Where i is the rate of inflationReal and Nominal Rates of Interest

INVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS4Figure 5.1 Determination of the Equilibrium Real Rate of Interest

INVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS5As the inflation rate increases, investors will demand higher nominal rates of returnIf E(i) denotes current expectations of inflation, then we get the Fisher Equation:Equilibrium Nominal Rate of Interest

INVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS6Tax liabilities are based on nominal incomeGiven 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

INVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS7Zero Coupon Bond:Par = $100Maturity = TPrice = PTotal risk free return

Rates of Return for Different Holding Periods

INVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS8Example 5.2 Annualized Rates of Return

INVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS9EAR: Percentage increase in funds invested over a 1-year horizon

Effective Annual Rate (EAR)

INVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS10APR: Annualizing using simple interest

Annual Percentage Rate (APR)

INVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS11Table 5.1 APR vs. EAR

INVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS12Table 5.2 T-Bill Rates, Inflation Rates, and Real Rates, 1926-2012

INVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS13Moderate inflation can offset most of the nominal gains on low-risk investmentsA dollar invested in T-bills from 19262012 grew to $20.25 but with a real value of only $1.55Negative correlation between real rate and inflation rate means the nominal rate doesnt fully compensate investors for increased in inflation

Bills and Inflation, 1926-2012INVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS14Figure 5.3 Interest Rates and Inflation, 1926-2012

INVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS15Rates of return: Single period

HPR = Holding period returnP0 = Beginning priceP1 = Ending priceD1 = Dividend during period one

Risk and Risk Premiums

INVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS16Ending Price =$110Beginning Price = $100Dividend =$4

Rates of Return: Single Period Example

INVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS17Expected returns

p(s) = Probability of a stater(s) = Return if a state occurss = State

Expected Return and Standard Deviation

INVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS18State Prob. of Stater 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: ExampleINVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS19Variance (VAR):

Standard Deviation (STD):

Expected Return and Standard Deviation

INVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS20Example VAR calculation:2 = .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

INVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS21True means and variances are unobservable because we dont 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 ReturnINVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUSArithmetic Average

Geometric (Time-Weighted) Average

= Terminal value of the investment

Returns Using Arithmetic and Geometric Averaging

INVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUSEstimated VarianceExpected value of squared deviations

Unbiased estimated standard deviationEstimating Variance and Standard Deviation

INVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUSExcess ReturnThe difference in any particular period between the actual rate of return on a risky asset and the actual risk-free rate Risk PremiumThe difference between the expected HPR on a risky asset and the risk-free rateSharpe Ratio

The Reward-to-Volatility (Sharpe) Ratio

INVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS25Investment management is easier when returns are normalStandard deviation is a good measure of risk when returns are symmetricIf security returns are symmetric, portfolio returns will be as wellFuture scenarios can be estimated using only the mean and the standard deviationThe dependence of returns across securities can be summarized using only the pairwise correlation coefficients The Normal DistributionINVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS26Figure 5.4 The Normal Distribution

Mean = 10%, SD = 20%INVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS27What if excess returns are not normally distributed?Standard deviation is no longer a complete measure of riskSharpe ratio is not a complete measure of portfolio performanceNeed to consider skewness and kurtosisNormality and Risk MeasuresINVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS28Figure 5.5A Normal and Skewed Distributions Mean = 6%, SD = 17%

INVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS29Figure 5.5B Normal and Fat-Tailed Distributions

Mean = .1, SD = .2INVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS30Value at Risk (VaR)Loss corresponding to a very low percentile of the entire return distribution, such as the fifth or first percentile returnExpected 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

Normality and Risk MeasuresINVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS31Lower Partial Standard Deviation (LPSD)and the Sortino RatioSimilar to usual standard deviation, but uses only negative deviations from the risk-free return, thus, addressing the asymmetry in returns issueSortino Ratio (replaces Sharpe Ratio) The ratio of average excess returns to LPSDNormality and Risk MeasuresINVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS32The second half of the 20th century, politically and economically the most stable sub-period, offered the highest average returnsFirm capitalization is highly skewed to the right: Many small but a few gigantic firmsAverage realized returns have generally been higher for stocks of small rather than large capitalization firms

Historic Returns on Risky PortfoliosINVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS33Normal distribution is generally a good approximation of portfolio returnsVaR indicates no greater tail risk than is characteristic of the equivalent normalThe ES does not exceed 0.41 of the monthly SD, presenting no evidence against the normalityHoweverNegative skew is present in some of the portfolios some of the time, and positive kurtosis is present in all portfolios all the timeHistoric Returns on Risky PortfoliosINVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS34Figure 5.7 Nominal and Real Equity Returns Around the World, 1900-2000

INVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS35Figure 5.8 SD of Real Equity & Bond Returns Around the World

INVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS36Figure 5.9 Probability of Investment with a Lognormal Distribution

INVESTMENTS | BODIE, KANE, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS37When 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, MARCUS5-#INVESTMENTS | BODIE, KANE, MARCUS38