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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