Professor John ZietlowMBA 621

Professor John ZietlowMBA 621

Risk And ReturnRisk And Return

Chapter 5Chapter 5

Chapter 5 OverviewChapter 5 Overview

• 5.1. Introduction to Risk and Return• 5.2. Risk and Return Fundamentals

– A Historical Overview of Risk and Return– Nominal and Real Returns– Risk Premium– Risk Aversion

• 5.3. Basic Risk and Return Statistics– Return on a Single Asset– Arithmetic and Geometric Averages– Risk of a Single Asset– Normal Distribution

Chapter 5 OverviewChapter 5 Overview

• 5.4. Risk and Return for Portfolios– Portfolio Returns– Portfolio Variance Example– Importance of Covariance– Variance of a Two-Asset Portfolio

• 5.5. Systematic and Unsystematic Risk– What Drives Portfolio Risk– The Systematic Risk of an Individual Security– Limitations of Beta

• 5.6. Summary

An Introduction To Risk & ReturnAn Introduction To Risk & Return

• Basic question in finance: “What is an asset worth?”– Valuing risky assets is fundamental to financial management

• Three-step procedure for valuing a risky asset– Determining the asset’s expected cash flows– Choosing a discount rate that reflects asset’s risk– Calculating present value (PV cash inflows - PV outflows)

• There is a trade-off between risk and expected return– Riskless investments (Treasury bills) offer low returns– Riskier investments (stocks) must promise higher returns

• An asset pricing model attempts to expressly model the trade-off between risk & return– Benefit: Defines risk & models risk/return trade-off rigorously– Drawback: unrealistic assumptions needed to build a model

The Historical Trade-Off Between Risk & Return, 1926-2000

The Historical Trade-Off Between Risk & Return, 1926-2000

• Ibbotson Associates annually publishes “Stocks, Bonds, Bills, and Inflation” for U.S. financial assets

• Investing $1.00 in these assets in Dec 1925, then re-investing dividends/interest, would have yielded at year-end 2000:– Small-company stocks: $6,402– Large-company stocks: $2,857– Long-term corporate bonds: $64– Intermediate-term govt bonds: $49– Long-term government bonds: $49– Treasury bills (short-term): $17– Basket of goods (inflation proxy): $10

• T-Bills (riskless, S-T investment) barely outpaced inflation– Above calculated using geometric mean returns

The Historical Trade-Off Between Risk & Return, 1926-2000

The Historical Trade-Off Between Risk & Return, 1926-2000

Series

Nominal return

Real (inflation-adjusted) return

Standard deviation

Small company stocks

12.4%

9.3%

33.4%

Large company stocks 11.0 7.9 20.2

Long-term corporate bonds

5.7 2.6 8.7

Long-term government bonds

5.3 2.2 9.4

Intermediate-term government bonds

5.3 2.2 5.8

U.S. Treasury bills 3.8 .07 3.2

Inflation 3.1 -- 4.4

Returns On U.S. Asset Classes, 1900-2000,In Nominal Terms

Returns On U.S. Asset Classes, 1900-2000,In Nominal Terms

Source: Dimson, Marsh &Staunton (ABN/AMRO),Millenium Book II (2001)

Total value of reinvestedreturns, year-end 2000

Annual returns

$10,000

119

70

24

10,000

100,000

1,000

100

10

1

Equities Bonds

Bills Inflation

Defining Financial Risk & ReturnDefining Financial Risk & Return

• Define risk as the variability of returns associated with a given asset.

• Define return as the total gain or loss experienced on an investment over a given period of time.

• Return measured as the change in an asset's value plus any cash distributions (dividends or interest payments).

t

tttt P

CPPR 11

1

• Where Pt+1 = price (value) of asset at time t+1;

Pt = price (value) of asset at time t;

Ct+1 = cash flow paid by time t+1

(Eq 5.1)

Realized Return Versus Expected ReturnRealized Return Versus Expected Return

• Realized (ex post) return easily computed with equation 4.1:– Calculate yearly, monthly, daily holding period returns (HPR)

• Real financial decisions, however, are based on expected (ex ante) returns, not realized returns:– Realized return (at best) useful in estimating expected return

• Can specify conditional or unconditional expected returns– Conditional expected return: “If the economy improves next

year, the asset’s return is expected to be 12%.” Or could be conditional on return on overall stock market.

– Unconditional expected return: “The asset’s return next year is expected to be 12%.”

• Usually generate expected return based on a specific asset pricing model, such as CAPM (Chapter 6).

Calculating Realized Returns On Two StocksCalculating Realized Returns On Two Stocks

• Both stocks purchased 12/31/02 and sold 12/31/03, so calculating one-year realized return for each investment

• Dynatech, bought for $60/share (P0), pays no dividends (Ct=0) in

2003, and is sold for $72/share (P1) 12/31/03.

• Utilityco, bought for $60/share (P0), pays $6/share dividend

(Ct=$6) in 2003, and is sold for $66/share 12/31/03.

%20 ]60$

12$[]

60$

0+60$-72$[ = R dyn

%20 ]60$

12$[]

60$

6$+60$-66$[ = R util

• Both have 20% return, one pure cap gains, one cap gains & dividends .

Nominal Versus Real ReturnsNominal Versus Real Returns

• The nominal return on a given investment has three components: the real rate of return, the expected inflation rate, and the risk premium.

Nominal return = real return + E(inflation) + risk premium• Treasury bills are virtually risk free, so the nominal return on

T-bills can be expressed:

  Nominal T-bill return = real return + E(inflation)• If the average annual rate of inflation is 3.2%, and the

average nominal return on T-bills is 3.8%, the real T-bill return is just 0.6% per year.

• Suppose that you expect 5% inflation next year. What nominal return would you expect on corporate bonds?

Nominal corporate bond return = 0.6% + 5% + risk premium

Arithmetic Versus Geometric ReturnsArithmetic Versus Geometric Returns

Year Return2000 -10.2%2001 -12.5%2002 +15.3%2003 +8.9%

• Average annual (periodic) returns can be computed as either arithmetic or geometric average returns.

• Average arithmetic return is the simple average of annual returns, and is best estimate of what return to expect each year.

• Geometric average return is the compound annual return earned by an investor who bought and held a stock for t years:

Geometric average return = [(1+R1)(1+R2)(1+R3)….(1+Rt)]1/t – 1

• Compute arithmetic (AAR) and geometric average returns (GAR) for series below:

AAR = [(-10.2%) + (-12.5%) + (15.3%) + (8.9%)] 4

= [-10.2% – 12.5% + 15.3% +8.9%] 4 = 0.375%

GAR = [(1-0.102)(1-0.125)(1+0.153)(1+0.089)]1/4 -1

= [(0.898)(0.875)(1.153)(1.089)]0.25 -1 = -0.33%

The Equity Risk Premium, 1900-2000The Equity Risk Premium, 1900-2000

19.85.67.5United States

19.94.76.5United Kingdom

19.44.36.1Switzerland

22.25.17.1Netherlands

28.06.810.0Japan

32.57.011.0Italy

35.34.910.3Germany

23.87.59.9France

16.74.66.0Canada

17.2%7.1%8.5%Australia

Standard deviation

Geometric mean

Arithmetic meanCountry

The higher return demanded by investors to hold stocks ratherthan less assets is the Equity Risk Premium. Table below showsERP, defined as stock return – bill returns for various countries.

Real And Nominal Rates of Return On U.S. Asset Classes, 1900-2000

Real And Nominal Rates of Return On U.S. Asset Classes, 1900-2000

Nominal rates of return (% per year)

Real rates of return (% per year)

Equities 10.1% Equities 6.7%

Bonds 4.8% Bonds 1.6

Bills 4.3% Bills 1.1

Inflation 3.2% Source: Dimson. Marsh & Staunton, Millenium Book II

Distribution of U.S. Risk Premia Arithmetic mean Geometric mean Std devEquity risk premium vs bills 7.5% / year 5.6% / year 19.8%Equity risk premium vs bonds 6.9% / year 5.0% / year 19.9%Bond maturity premium vs bills 0.8% / year 0.5% / year 7.4%

Risk Preferences: Comparing Two Assets With The Same Expected Return

Risk Preferences: Comparing Two Assets With The Same Expected Return

• Stocks 1 & 2 both have an expected return of 10%. – Both offer 10% return in an average economy– Stock 2 would have higher return if economy booms– Stock 1 has lower return variability; does better in bad times

• Whether an investor would consider them equally attractive depends on his/her degree of risk aversion (utility function)– Risk averse investor prefers lower variability for given R^

– Risk seeking investor prefers higher variability for given R^

– Risk neutral investor is indifferent about variability• Finance theory, common sense, and observed behavior all

suggest investors are risk averse– If two assets offer equal R^, will pick one with less variability– Must be offered higher R^ to accept higher variability

Two Assets With Same Expected Return But Different (Continuous) Probability Distributions

Two Assets With Same Expected Return But Different (Continuous) Probability Distributions

Stock 1

Stock 2

0 5 6 7 8 9 10 11 12 13 14 15

Return %

Pro

bab

ility

Den

sity

Risk Of A Single AssetRisk Of A Single Asset

• Can now calculate an asset’s return (expected and realized)– Next step to measure risk. – Simplest definition the likelihood of loss on an investment.

• Finance defines risk in terms of the variability of returns– Measure risk based on a probability distribution (known or

estimated) of expected returns.• Fig 5.1a shows histogram of returns on a portfolio of large

stocks; Fig 5.1b shows this for small stock portfolio– Small stock p/f shows higher mean return, higher variability– Both show returns clustering around mean value

• Following slide shows bell-shaped normal distribution– Great to use as a model of return distribution, if possible– Symmetric about mean, described fully by mean & variance

(2) or standard deviation, square root of variance ()– 68% of outcomes within 1 of mean; 95% within 2

-80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90

Histogram of Return on Portfolioof Large Company Stocks, 1926-1999

Histogram of Return on Portfolioof Large Company Stocks, 1926-1999

-80 -60 -40 -20 0 20 40 60 80 100-130

150

Histogram Of Returns On Portfolio Of Small Company Stocks, 1926-1999

Histogram Of Returns On Portfolio Of Small Company Stocks, 1926-1999

R-2 R-1 R+2R+1R

68%

95%95%

Normal Distribution

The Normal Probability Distribution:The Normal Probability Distribution:Area Under The Bell-Shaped CurveArea Under The Bell-Shaped Curve

The Normal Probability Distribution:The Normal Probability Distribution:Area Under The Bell-Shaped CurveArea Under The Bell-Shaped Curve

Calculating Variance And Standard Deviation Of Expected Returns

Calculating Variance And Standard Deviation Of Expected Returns

• The variance (2)of a distribution equals the expected value of squared deviations from the mean.– Can compute expected (ex ante) or historical variance

• Assume you predict that a stock has equally likelihood (p=0.167) of following six returns next year:– (-12%, -3%, 7%, 12%, 18%, 20%). Calculate expected return, E(R)– E(R) = (-12 - 3 + 7 + 12 + 18 + 20) ÷ 6 = 42 ÷ 6 = 7%

• Compute variance of these expected returns using Eq. 5.3:

= [(-12-7)2+(-3-7)2+(7-7)2+(12-7)2+(18-7)2 +(20-7)2]÷6

2 = [(-19)2+(-10)2+(0)2+(5)2+(11)2 +(13)2] ÷ 6 = [361 + 100 + 0 + 25 + 121+169] ÷ 6 = [776%2] ÷ 6 = 129.33%2.

• Note units of variance (%-squared). Units hard to interpret, so calculate standard deviation, square root of 2:

Standard deviation = σ = 129.33%2 = 11.37%

]))R(ER[(E 22

Calculating Variance And Standard Deviation Of Historical Returns

Calculating Variance And Standard Deviation Of Historical Returns

• It’s rarely feasible to specify the full distribution of possible returns and expected variance.– Must know all possible outcomes & associated probabilities

• Instead, analysts usually gather historical data and use these to generate expected return and variance– Historical variance computed using Eq 5.4:

• Where Rit = return on stock i during period t, R¯i = average return on stock i over sample period and N = number of periods in sample.• Denominator uses N-1 rather than N since one degree of

freedom used to compute average (mean) return.

1

)(1

2

2

N

RRVariance

N

t

iit

(Eq 5.4)

Monthly Return for Oracle CorporationJuly 1999 – July 2001

Monthly Return for Oracle CorporationJuly 1999 – July 2001

-19.697%Nov- 004.533%Oct-99

-10.558%Jul-00-4.842%Jul-0116.956%Jun-00

24.183%Jun-01-10.086%May-00-5.322%May-012.402%Apr-007.877%Apr-015.134%Mar-00

-21.158%Mar-0148.639%Feb-00-34.764%Feb-01-10.847%Jan-000.215%Jan-0165.25%Dec-999.670%Dec- 0042.575%Nov-99

-16.191%Oct- 0024.657%Sep-99-13.402%Sep- 00-4.105%Aug-9920.947%Aug-002.52%Jul-99

ReturnMonthReturnMonth

Variance Calculation for Oracle CorpVariance Calculation for Oracle Corp

• We can compute expected return, E(R), variance, 2, std dev, , for Oracle Corp stock from Jul 1999 to July 2001:– E(R) = 4.98% per month; 2 = 534.78%2, = 23.125%

• If Oracle’s returns are approximately normally distributed, can use this to find confidence intervals for E(R):– 68% probability returns will be within +/- one from E(R)– 95% probability returns will be within +/- two from E(R)

• Given E(R)=4.98%, =23.125%, then there is a 68% chance actual return will be between –18.145%% and +28.105%– 95% chance actual return between –41.27% and + 51.23%

• Clearly, Oracle is a risky stock!

• Can be generalized to n-asset p/f using Eq 5.6:

Calculating Expected Return For A PortfolioCalculating Expected Return For A Portfolio

• Have looked only at risk and return for single assets thus far, but most investors hold multiple asset portfolios (p/fs)– Asset pricing models all assume stocks held in p/fs

• Key insight of portfolio theory: Asset return adds linearly, but risk is (almost always) reduced in a portfolio– E(R) of p/f is a weighted average of individual asset E(R)– P/f variance is a non-linear function, based on covariance

(defined later) between assets’ return

• E(R) of a p/f calculated using Eq 5.5, where w1, w2 are weights of assets 1 & 2 in p/f:

)()()( 2211 REwREwRE p

)(...)()()()( 332211 NNp REwREwREwREwRE

Monthly Returns for Individual Stocks and Portfolios, 1998-2000

Monthly Returns for Individual Stocks and Portfolios, 1998-2000

-5.743%-3.176%-1.18%-10.30%-5.49%-0.87%May-00 -15.341%2.214%3.67%-34.35%6.76%-2.32%Apr-00 24.441%11.915%30.00%18.88%23.33%0.49%Mar-00

-11.374%-11.335%-14.06%-8.68%-16.80%-5.87%Feb-00 -12.451%-11.861%-8.73%-16.17%-19.38%-4.34%Jan-00 13.068%7.582%-2.09%28.23%12.75%2.42%Dec-99 -5.983%-2.010%-10.33%-1.64%-4.55%0.53%Nov-99 9.195%0.295%16.18%2.21%1.63%-1.04%Oct-99

-8.245%-0.237%-14.33%-2.16%-2.13%1.65%Sep-99 1.275%4.680%-5.317%7.87%1.90%7.46%Aug-99

-3.220%-2.298%-1.59%-4.85%-5.75%1.15%Jul-993.734%0.820%-4.306%11.77%0.26%1.38%Jun-99

-3.264%-4.664%-5.76%-0.77%-5.68%-3.65%May-99-1.136%34.648%7.00%-9.27%43.50%25.79%Apr-999.911%-0.626%0.42%19.40%3.22%-4.47%Mar-99

-2.415%1.767%9.38%-14.21%8.12%-4.59%Feb-999.520%0.403%-7.14%26.18%-8.33%9.14%Jan-99

50% Microsoft, 50% Berkshire

50% 3M, 50% Praxair

Berkshire Hathaway Inc

MicrosoftPraxair Inc3M CoDate

7.678%9.205%1.71%13.64%12.33%6.08%Oct-014.368%10.969%-1.69%10.42%12.17%9.77%Nov-015.589%3.785%8.00%3.18%4.40%3.17%Dec-01

-4.721%-8.123%0.86%-10.31%-10.77%-5.48%Sep-01-6.760%-1.569%0.29%-13.81%3.82%-6.95%Aug-01-4.808%-2.739%-0.29%-9.33%-3.53%-1.95%Jul-013.270%-5.160%1.02%5.52%-6.54%-3.78%Jun-011.570%2.946%1.03%2.11%6.25%-0.36%May-01

13.891%10.273%3.90%23.89%6.00%14.54%Apr-01-7.104%-3.869%-6.90%-7.31%0.11%-7.85%Mar-01-0.300%1.253%2.78%-3.38%0.61%1.90%Feb-0118.558%-4.138%-3.66%40.78%-0.10%-8.17%Jan-01-8.331%22.065%7.74%-24.40%23.48%20.65%Dec-00-6.622%-0.080%3.45%-16.70%-3.52%3.36%Nov-006.555%2.851%-1.09%14.20%-0.33%6.04%Oct-00

-0.998%-8.776%11.61%-13.61%-15.54%-2.02%Sep-002.359%7.555%4.72%0.00%11.85%3.26%Aug-00

-5.159%7.421%2.42%-12.73%5.68%9.17%Jul-009.841%-7.327%-8.191%27.87%-10.86%-3.79%Jun-00

50% Microsoft, 50% Berkshire

50% 3M, 50% Praxair

Berkshire Hathaway Inc

MicrosoftPraxair Inc3M CoDate

Monthly Returns for Individual Stocks and Portfolios, 1998-2000 (Cont.)

Monthly Returns for Individual Stocks and Portfolios, 1998-2000 (Cont.)

Calculating The Expected Return Of A Two-Asset Portfolio

Calculating The Expected Return Of A Two-Asset Portfolio

• Table 5.3 shows monthly and average returns (mean %) & standard deviations ( %) of four stocks over 3-yr period

– 3M (1.68%), Praxair (1.91%), Microsoft (1.17%), Berkshire (0.54%)

– 3M (7.56%), Praxair (12.01%), Microsoft (16.51%), Berkshire (8.43%)

• Also shows two p/fs with equal fractions of two stocks

– p/f #1: 50% 3M, 50% Praxair ; p/f #2: 50% Microsoft, 50% Berkshire

• E(R) of p/fs are weighted averages of individual stocks:

E(R) pf #1 = [(0.5)(1.68%)+(0.5)(1.91%)] = 1.80%

E(R) pf #2 = [(0.5)(1.17%)+(0.5)(0.54%)] = 0.86%

• But actual p/f standard deviations are not equal to weighted averages of individual std devs--less in both cases:

pf #1 = 9.00% [(0.5)(7.56 %)+(0.5)(12.01%)] = 9.78%

pf #2 = 8.95% [(0.5)(16.51 %)+(0.5)(8.43%)] = 12.47%

• Figure 5.3 shows risk/return tradeoff for 3M & Praxair Inc

Monthly Returns And Standard Deviations: Four Stocks And Two Portfolios

Monthly Returns And Standard Deviations: Four Stocks And Two Portfolios

8.950.8650% Microsoft, 50% Berkshire

9.001.8050% 3M, 50% Praxair

8.430.54Berkshire Hathaway Inc

16.511.17Microsoft

12.011.91Praxair Inc

7.56%1.68%3M Co

Standard deviation of monthly return, %

Average (mean) monthly return, %Company or portfolio

Average Return and Standard Deviation for Portfolios of 3M and Praxair Inc

0.016

0.0165

0.017

0.0175

0.018

0.0185

0.019

0.0195

0.07 0.08 0.09 0.1 0.11 0.12 0.13

Standard deviation

Average Monthly Return

100% Praxair

100% 3M

Calculating Variance And Standard Deviation Of Portfolio Expected Return

Calculating Variance And Standard Deviation Of Portfolio Expected Return

• In previous table, std dev of Microsoft-Berkshire p/f below weighted average of individual std dev

– Reason: returns on two stocks don’t co-move together

– Microsoft & Berkshire returns have negative covariance (Cov)

– 3M & Praxair have positive Cov, but don’t co-move perfectly

• To compute p/f variance account for Cov between p/f assets

– Calculate covariance of expected returns using Eq 5.7

– Calculate covariance of expected returns using Eq 5.7

))]())(([(),(var 22111221 RERRERERRianceCo (Eq 5.7)

1

))((),(var 1

2211

1221

N

RRRRRRianceCo

N

ttt

(Eq 5.8)

3M/Praxair correlation = 0.0059 ÷ (0.0756)(0.1201) = 0.65

Microsoft/Berkshire correlation = -0.0011 ÷ (0.1651)(0.0843) = -0.079

• Cov measures co-movement between assets 1 and 2, 12

– Units of Cov are %-squared, same problem as variance• Shown monthly for 3M & Praxair and Microsoft-Berkshire in

previous table– Positive Cov between 3M & Praxair, TP = +0.0059– Negative Cov between Microsoft & Berkshire, MB = -0.0011

• Besides awkward measurement units, Cov also unbounded– Would like a measure normalized between –1.0 and +1.0

• Correlation coefficient, 12, is unit-less and valued –1 to +1– Eq 5.9 is formula, then calculate for 3M/Praxair,

Microsoft/Berkshire:

Calculating And Using Covariance And Correlation Coefficients

Calculating And Using Covariance And Correlation Coefficients

(Eq 5.9)21

1212

tcoefficiennCorrelatio

Microsoft/Berkshire: p2 = (wM

2)(M2) + (wB

2)(B2) + 2wMwBMB

=(0.5)2(0.1651)2 +(0.5)2(0.0843)2 + 2(0.5)(0.5)(-0.0011)

=(0.25)(0.0273)+(0.25)(0.0071)-2(0.000275)=0.008 p=0.0895= 8.95%

3M/Praxair: p2 = (wT

2)(T2) + (wP

2)(P2) + 2wTwP TP

=(0.5)2(0.0756)2 +(0.5)2(0.1201)2 + 2(0.5)(0.5)(0.0059)

=(0.25)(0.0057)+(0.25)(0.0144)+2(0.0015)=0.0080 p=0.08936=8.94%

• Forming p/fs between 3M (T) and Praxair (P) and between Microsoft (M) and Berkshire (B) yields reduction in p/f variance and std dev

– Since TP > MB, combining 3M and Praxair yields less risk reduction than combining Microsoft and Berkshire

• Use Eq 5.10 to find variance of 50:50 3M/Praxair and 50:50 Microsoft/Berkshire p/fs using Cov (TP = +0.0053, MB = -0.0011)

Calculating And Using Covariance And Correlation Coefficients (Continued)

Calculating And Using Covariance And Correlation Coefficients (Continued)

(Eq 5.10)12212

22

22

12

12 2 wwwwariancePortfolioV p

Calculating And Using Covariance And Correlation Coefficients (Continued)

Calculating And Using Covariance And Correlation Coefficients (Continued)

• Use Eq 5.12 to find variance of 50:50 3M/Praxair and 50:50 Microsoft/Berksire p/fs using Cov (TP = +0.65, MB = -0.079)

– Remember that AB can be computed as AB = AB ÷ (A)(B)

– Or Cov can be computed from AB : AB = AB (A)(B)

2112212

22

22

12

12 2 wwwwariancePortfolioV p (Eq 5.12)

3M/Praxair : p2 = (wT

2)(T2) + (wP

2)(P2) + 2wTwP TPMP

=(0.5)2(0.0756)2 +(0.5)2(0.1201)2 + 2(0.5)(0.5)(0.65)(0.0756)(0.1201)

=(0.25)(0.0057)+(0.25)(0.0144)+2(0.0015)=0.007986 p=0.08936 = 8.94%

Microsoft/Berkshire : p2 = (wM

2)(M2) + (wB

2)(B2) + 2wMwB MBMB

=(0.5)2(0.1651)2 +(0.5)2(0.0843)2 + 2(0.5)(0.5)(-0.079)(0.1651)(0.0843)

=(0.25)(0.0273)+(0.25)(0.0071)-2(0.0003)=0.008041 p=0.0897= 8.97%

The Returns On Perfectly Positively and Perfectly Negatively Correlated Assets

The Returns On Perfectly Positively and Perfectly Negatively Correlated Assets

A

B

A

B

TimeTime TimeTime

Re

turn

Re

turn

Re

turn

Re

turn

Perfectly Positively CorrelatedPerfectly Positively Correlated Perfectly Negatively CorrelatedPerfectly Negatively Correlated

Imperfectly Correlated Assets And Portfolio Return Variability

Imperfectly Correlated Assets And Portfolio Return Variability

• Combining two imperfectly correlated assets into a portfolio reduces the variability of portfolio returns

Time Time Time

Return Return ReturnAsset MAsset M Asset NAsset N

Portfolio ofPortfolio ofAsset M and NAsset M and N

Demonstrating Positive & Negative CovarianceDemonstrating Positive & Negative Covariance

• Assume you can invest in three assets (stocks) with the same expected return, but are imperfectly correlated

– Stock 3: Retailing firm, does well in expansions, 3=5.7%

– Stock 4: Bankruptcy reseller, prospers in recessions, 4=5.7%

– Stock 5: Wholesale distributor, does very well in expansions, very poorly in recessions, 5=10.1%

• Stocks 3 & 5 have different return std dev, but historically move together (co-vary) as economy changes– Stock 5’s return follows 3’s, but with greater vigor– Stock 4 does well when other stocks do poorly & vice versa

• Explain co-movement as positive or negative covariance

– Stocks 3 & 4 have negative cov: 34 = -32.50

– Stocks 3 & 5 exhibit positive cov: 35 = +57.50

– Stocks 4 & 5 have negative cov: 45 = -57.50

• Calculate correlation coefficients between three matched pairs of stock using Eq 5.9:

Calculating Correlation Coefficients For Stocks 3, 4 And 5

Calculating Correlation Coefficients For Stocks 3, 4 And 5

• Returns on stocks 3 & 5 (retailer & wholesaler stocks) are perfectly positively correlated

• Stock 4's (bankruptcy reseller) returns are perfectly negatively correlated with assets 3 and 5:

1.00- = 0)(5.70)(5.7

32.50- =

Cov(3,4) =

433

4

1.00+ = 12)(5.70)(10.

57.50 =

)Cov(3, =

3

535

5

1.00- =12)(5.70)(10.

57.50- =

σσ

Cov(4,5) = ρ

5445

Constructing Portfolios Based On Correlation Coefficients

Constructing Portfolios Based On Correlation Coefficients

• Unless returns on all assets perfectly positively correlated, forming portfolios reduces p/f return variance– If =+1.0, forming p/f does not reduce return variability– For any <+1.0, forming p/f reduces variability

• Can form p/f with a standard dev of 0 (thus riskless), by combining assets that are perfectly negatively correlated– Works since 34= -1.0 or 45= -1.0 , but weights must be

carefully chosen• Combining assets with perfectly positively correlated

returns yields a weighted-average p/f variance of p/f (3,5) = (0.5)(3) + (0.5)(5)

= (5.7% + 10.12%) ÷ 2 = 7.91%

Computing And Using Correlation Coefficients In A Two-Asset Portfolio

Computing And Using Correlation Coefficients In A Two-Asset Portfolio

• The correlation between two assets’ returns can be used to construct an “efficient” two-asset portfolio– Minimize risk for given level of expected return & vice versa

• Will ultimately expand use of correlation to include an asset’s relationship with overall market– Allows creation of efficient multi-asset portfolio– Correlation is central to all modern asset pricing models

• Demonstrate using annual return data (presented on next slide) for two stocks and the S&P 500 stock index– Consolidated Consumer Corp (CCC): low-risk, low-return – Dynamic Technology Corp (DTC): high-risk, high-return

• Given mean historical return series, can compute each asset’s std dev, covariance & correlation with each other

Historical Returns And Standard Deviations For Two Stocks And The S&P 500 Index

Historical Returns And Standard Deviations For Two Stocks And The S&P 500 Index

12.315.0S&P 500 Index

20.020.0Dynamic Technology Corp (DTC)

9.5%13.0%Consolidated Consumer Corp (CCC)

Standard deviation of return, %

Mean historical return, %

Stock or Index

Computing Correlation Coefficients On Two Stocks And The S&P 500 Index

Computing Correlation Coefficients On Two Stocks And The S&P 500 Index

• Assume the following covariances are determined between:– CCC and DTC: Cov (c,d) = 112.7– CCC and S&P 500 [market]: Cov (c,m) = 76.0– DTC and S&P 500 [market]: Cov (d,m) = 236.9

• Can now use eq 5.9 to compute correlation coefficients and figure (next page) shows how to use these in p/f formation

cd = 112.7 [(9.5)(20.0)] = 0.59cm = 76.0 [(9.5)(12.3)] = 0.61dm = 236.9 [(20.0)(12.3)] = 0.90

21

1212

tcoefficiennCorrelatio

Portfolio Risk and Return For Combinations of CCC and DTCPortfolio Risk and Return For

Combinations of CCC and DTC

D

GF

E

C

0%

5%

10%

15%

20%

25%

0% 5% 10% 15% 20%

Standard Deviation of Portfolio Returns

Exp

ecte

d R

etu

rn o

n t

he

Po

rtfo

lio

100% DTC

100% CCC

The Risk-Return Trade-Off For Different Correlation Coefficients

The Risk-Return Trade-Off For Different Correlation Coefficients

• Figure (next slide) shows risk-return trade-off for p/fs of CCC & DTC stock with different jl between the stocks.

– The straight line CD represents a p/f assuming perfectly positively correlated returns between the two shares.

– Other curves represent assumed correlation coefficients of 0.00, -.50, and -1.00

• This figure shows that the lower the correlation between two assets’ returns, the greater the risk reduction from combining the assets into a portfolio– Perfectly negatively correlated assets yield the minimum

possible variance for any given level of expected return.

Correlation Coefficients And Risk ReductionCorrelation Coefficients And Risk Reduction

D

GF

EC

G

FE

10%

15%

20%

25%

0% 5% 10% 15% 20% 25%

Standard Deviation of Portfolio Returns

Exp

ecte

d R

etu

rn o

n t

he

Po

rtfo

lio

G

F

E

CEFGD assumes cd is +1.0CEFGD assumes cd is -1.0 < <1.0CEFGD assumes cd is -1.0

The Declining Importance Of Own Variance As The Number Of Assets In A Portfolio IncreasesThe Declining Importance Of Own Variance As The Number Of Assets In A Portfolio Increases

• Whatever the correlation between assets, increasing the number in a p/f reduces the impact of each one’s own variance

• Demonstrate with two assets, using eq , assuming equal weights of each stock (wj = wl = 0.5):

p2 = wj

2j2 + (1-wj)2l

2 + 2 wj (1-wj) Cov(j,l) = (0.5)2j2 + (0.5)2l

2

+ 2(0.5)(0.5)Cov(j,l)• Each asset’s own variance accounts for only 25% of total

p/f vraiance, and both own variances together only total half

• The formula for a three-asset (including stock q) portfolio’s return variance is given :

p2 = wj

2j2 + wl

2l2 + wq

2q2 + 2wjwl Cov(j,l) + 2wjwlCov(j,q) +

2wlwqCov(l,q)

• Consider the simplest case with equal asset amounts in the portfolio (wj=wl=wq=0.333=1/3).

  p2 = (0.333)2j

2 + (0.333)2l2 + (0.333)2q

2 + 2wjwlCov(j,l) +

2wjwqCov(j,q)+ 2wlwqCov (l,q)

= (0.111) j2 + (0.111) l

2 + (0.111) q2 + 2wjwlCov(j,l) +

2wjwqCov(j,q) + 2wlwqCov(l,q) • Each asset’s own variance only represents (1/3)2= 1/9 =

0.111, or 11.1% of total p/f variance; – Three collectively represent only 33.3% of total volatility. – Summed Cov terms represent other 66.7% of portfolio var.

Declining Importance Of Own Variance (Cont)Declining Importance Of Own Variance (Cont)

Diversifiable And Non-diversifiable RiskDiversifiable And Non-diversifiable Risk

• As the number of assets in a portfolio increases, the importance of own variance virtually disappears– In a 10-asset p/f, each own var accounts for only 1% of total– All own var collectively account for only 10% of p/f variance– In a 25-asset p/f, each own var is only 0.16% of p/f variance

• As number of assets increases, the importance of bilateral covariances also declines-- similarly to own variance– In a diversified p/f, an asset’s own var & cov matters little

• Only an asset’s covariance with all other assets contributes measurably to overall p/f return variance– Investor thus only looks at asset’s covariance with “market”

• Thus important to draw distinction between an asset’s total, diversifiable and non-diversifiable risk [Figure 5.7]– Diversifiable: unique, firm-specific risk (fire, flood, strike)– Nondiversifiable: systematic risk related to market or economy

The Impact Of Additional Assets On The Risk Of A Portfolio

The Impact Of Additional Assets On The Risk Of A Portfolio

Number of Securities (Assets) in PortfolioNumber of Securities (Assets) in Portfolio

Po

rtfo

lio R

isk,

k p

Nondiversifiable RiskNondiversifiable Risk

Diversifiable RiskDiversifiable Risk

Total riskTotal risk

1 5 10 15 20 25 1 5 10 15 20 25