Chapter 8 The Rock Record Chapter 9 The Mesozoic and Cenozoic Eras.
Chapter 8. Record
Transcript of Chapter 8. Record
Chapter 8. RecordPortfolio Theory and Capital Market
Yiyang Yang
Department of Applied Mathematics and StatisticsState University of New York at Stony Brook
March 2012
Yiyang Yang (SUNY at Stony Brook) Record (AMS 512) March 2012 1 / 35
Outline
1 Introduction
2 Measuring Performance
3 Mutual-Fund Performance
4 The Shapes of Distributions
5 Using the Past to Predict the Future
Yiyang Yang (SUNY at Stony Brook) Record (AMS 512) March 2012 2 / 35
IntroductionPast vs. Future
portfolio theory and capital market theory: predictionempirical work: past recordMotivation: portfolio theory+predictions from past recordmeasuring tools:
probabilities; relative frequenciesexpected return; average returnvariability; relative frequencies of various deviations
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IntroductionThe Market Portfolio
Measure of market portfolio:Indexes: Dow-Jones’ Index, S&P Composite Indexaverage values for individual securites: RMt =
1N ∑N
i=1 Rit , whereRMt =return on the market portfolio in time period t, Rit =rate ofreturn on security i in time period t, N =number of securities
Figure: Rate of return on the market portfolio
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IntroductionThe Effectiveness of Diversification
Motivation: How many securities must be included to obtain a reasonablywell-diversified portfolio?
Figure: Variability and portfolio diversification
can be approximated by σp = 11.91+ 8.63n , where n =number of securities.
When n = 1, σT = 20.5, σS = 11.9, thus (σS )2
(σT )2 = 0.34.
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ItroductionMarket and Industry Factors
Table: Proportion of security risk attribute to market factors
Period Average proportion(%)June 1927-September 1935 58.4October 1935-Feburary 1944 55.7
March 1944-July 1952 41.2August 1952-December 1960 30.7
Remarks:The proportion decreases overtime.Analysis over the entire period indicates, market fluctuation accountsfor 52% variance of a typical security, a group of industry accountedfor another 11%.In an index model, the market can be represented by an index andadditional indexes can be added to represent industry factors.
Yiyang Yang (SUNY at Stony Brook) Record (AMS 512) March 2012 6 / 35
Measuring PerformanceReward-to-Variability
DefinitionCapital market line
re =EM − p
σM
where re =price of risk reduction for efficient portfolios, EM =expectedreturn, p =pure rate of interest, σM =standard deviation.
CorollaryThe past performance of any portfolio can be represented as:
(rv )p =
Ap − p′σ′p
where ( rv )p =reward-to-variability ratio for portfolio, Ap =actual average
return, p′ =actual pure interest rate, σ′p =actual variability of portfolio.Yiyang Yang (SUNY at Stony Brook) Record (AMS 512) March 2012 7 / 35
Measuring PerformanceReward-to-Variability
Figure: reward-to-variability
Remark:The slope of the line associated with the portfolio is thereward-to-variability ration.The steeper the line, the better the performance of portfolio.Used to measure the performance of portfolio.
Yiyang Yang (SUNY at Stony Brook) Record (AMS 512) March 2012 8 / 35
Measuring PerformanceReward-to-Volatility
DefinitionSecurity market line
rs =Ei − pbi
where rs =price of risk reduction for securities, Ei =expected return ofsecurity i, p =pure rate of interest, bi =volatility of security i.
CorollaryThe past performance of any portfolio can be represented as:
(rb )i =
Ai − p′b′i
where ( rb )i =reward-to-variability ratio of security i, Ai =actual average
return, p′ =actual pure interest rate, b′i =actual volatility of security i.Yiyang Yang (SUNY at Stony Brook) Record (AMS 512) March 2012 9 / 35
Measuring PerformanceReward-to-Volatility
Figure: Reward-to-volatility
Remark:The slope of the line is the reward-to-volatility ratio.The steeper the line, the better it is.Reward-to-volatility ratio is close related to characteristic line.
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Measuring PerformanceDifferential Return
DefinitionActual characteristic line represents the relationship of actual return of asecurity or portfolio and that of market portfolio. Thus, the line passesthrough the point at which both returns equal their actual average valueAi and AM ; and the slope is actual volatility b′i .
Figure: Actual characteristic line
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Measuring PerformanceDifferential Return
Figure: Differential return
Derive the actual characteristic line Y from Ai ,AM , b′i .Line Z represents the corresponding efficient portfolio with samevolatility.x = Ai−p′
b′iis the reward-to-volatility ratio, separate it:
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Measuring PerformanceDifferential Return
rb = dh + (AM − p′)
If dh > 0, performance of the security or portfolio was superior tothat of a market-based portfolio of comparable volatility; if dh < 0,itwas worse.dv = dh · b′i is the vertical distance from point P to the characteristicline.
Definitiondv is denoted as differential return. A positive differential return indicatesthat performance was superior to that of a market-based portfolio ofcomparable volatility; a negative differential return indicates that it wasworse.
Yiyang Yang (SUNY at Stony Brook) Record (AMS 512) March 2012 13 / 35
Measuring PerformanceDifferential Return
Figure: compare the performance of two securities
Remarks:Based on differential returns, line j was superior to that of i.Based on reward-to-volatility ratio, line i was superior to that of j.The reward-to-volatility ratio can compare securities mutually.The reward-to-volatility ratio and differential return both can comparethe performance of a security with that of market.
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Mutual-Fund PerformanceIntroduction
DefinitionAn open-end mutual fund is an institution designed to provide bothdiversification and professional management at relatively low cost.
Some facts about mutual fund:The managers are ready to issue new shares or retire old shares atvirtually any time.The net asset value per share=(current market value of the fund’sholding)/(number of shares)Load charge, typically 8 to 10 percent, goes to the sales organization.The managers of the fund are paid separately, usually 0.5% of thetotal net asset value.Most mutual funds hold over 100 different securities.
Yiyang Yang (SUNY at Stony Brook) Record (AMS 512) March 2012 15 / 35
Mutual-Fund PerformanceCorrelation with Market
Figure: Proportion of variance attribute to market
Remarks:The graph incudes the result of 115 mutual funds.On average, 85% of the variance of the mutual fund can be attributedto market fluctuations.In sum, most mutual funds are well-diversified.
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Mutual-Fund PerformanceVolatility
Figure: Volatility of corresponding 115 mutual funds
Remarks:The average volatility is 0.84.Most mutual funds perform more conservative than one made up ofthe securites in S&P Index.The differences in volatility may due to the objectives of the funds.
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Mutual-Fund PerformanceVolatility
Table: Volatility by type of fund
Classification Number offunds
Averagevolatility
Growth Funds: primary objective islong-term growth of capital
31 0.970
Growth-income Funds: emphasis onlong-term, consider income
30 0.941
Income-growth Funds: emphasiscurrent income, consider long-term
15 0.856
Income Funds: primary objective iscurrent income
9 0.674
Blanced Funds: relative stability andcontinuity of income
30 0.645
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Mutual-Fund PerformancePerformance of Mutual Fund
Figure: Reward-to-variability ratios, 1954-1963 (net returns)
Remarks:11 funds had ratios exceeding Dow-Jones’ portfolio; 23 are smaller.The results are based on net return.Apply same analysis on gross return, 19 had ratios larger than that ofDow-Jones’ portfolio; 15 had smaller ratios.
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Mutual-Fund PerformancePerformance of Mutual Fund
Figure: Average return and volatility 1955-1964 (115 mutual funds)
Remarks:M indicates the performance of market portfolio, based on S&P Index.Left graph indicates net performance, more funds plot below securitymarket line than above it.Right graph indicates gross performance, points scatter randomlyaround the line.Plausible explanation: excessive managing expenditures.Yiyang Yang (SUNY at Stony Brook) Record (AMS 512) March 2012 20 / 35
Mutual-Fund PerformancePerformance of Mutual Fund
Figure: Differential returns 1945-1964 (115 mutual funds)
Remarks:Based on net values, the average of differential return was -1.1%; 76had negative differential returns.Based on gross values, the average was -0.4%; 55 had negative values.Mutual funds do no better on average than market based portfolios ofcomparable volatility.
Yiyang Yang (SUNY at Stony Brook) Record (AMS 512) March 2012 21 / 35
Mutual-Fund PerformanceFund Managers got talent or luck?
Figure: Rank of reward-to-variability (net value)
Remarks:There are slight positive relationships in both graphs.Comparison suggests that difference in performance based on netreturns may be due more to cost of management than effectiveness.
Yiyang Yang (SUNY at Stony Brook) Record (AMS 512) March 2012 22 / 35
Mutual-Fund PerformanceFund Managers got talent or luck?
Well-managed funds with enjoy runs of successive superior performance.
Table: Frequency of successive superior
Length of run (numberof successive years ofsuperior performance)
Number ofinstances
Percent of instancesfollowed by another year ofsuperior performance (%)
1 574 50.42 312 52.03 161 53.44 79 55.8
Remarks:Typical fund obtained superior performance 50.2% of the time.Funds with one to four successive prior years of superior performancehad slightly (less than 6%) better success.
Yiyang Yang (SUNY at Stony Brook) Record (AMS 512) March 2012 23 / 35
Mutual-Fund PerformanceFund Managers got talent or luck?
Figure: Volatilities of 56 funds over two periods
Remarks:The relationship is clearly positive, though not perfect.Regardless of the overall performance, the target level of volatility iswell preserved.
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Mutual-Fund PerformanceSummary
The conclusions of this section can be summarized:Most funds diversified well.Most managers selected a general-risk class and maintained theirstated positions reasonably well.On the average, funds did no better, before expenses, thanmarket-based portfolios of comparable volatility.On the average, funds did worse, after expenses, than market-basedportfolios of comparable volatility.Few, if any, funds consistently performed better than market-basedportfolios of comparable volatility.Most funds appear to have spent too much searching for mispricedsecurities.
Yiyang Yang (SUNY at Stony Brook) Record (AMS 512) March 2012 25 / 35
Mutual-Fund PerformanceFurther Discussion
Other factors:Sales charge: no-load funds outperform load funds, switch fundsperform worst.Turnover: managers with low turnover outperform managers withhigh turnover.The ratio of expenses to assets: the less, the better.Fund size: unable to find any impact of size on performance.
Large funds have more to spend for information and analysis.Large funds have more impact on market, when they engage inpurchase and sales.
Yiyang Yang (SUNY at Stony Brook) Record (AMS 512) March 2012 26 / 35
The Shapes of DistributionsStable Paretian (Pareto-Levy) Distributions
Figure: rate of return on market portfolio from 1926 to 1965
Remark: Normal Distribution contradicts reallity.
Yiyang Yang (SUNY at Stony Brook) Record (AMS 512) March 2012 27 / 35
The Shapes of DistributionsStable Paretian (Pareto-Levy) Distributions
DefinitionRandom variables {Xn} are independent copies of X , then X is said tofollow Stable Distribution if:
X1 + X2 + · · ·+ Xn = CnX +Dn
where Cn = n 1α dictates the stability property, Dn is a real number.
Four parameters characterized Stable Distributionα the characteristic exponent; a measure of the height of the extremetail areas of the distribution α ∈ (0, 2)β an index of skewness β ∈ [−1, 1]γ a scale parameter γ ∈ (0,+∞)
δ a location parameter δ ∈ (−∞,+∞)
Yiyang Yang (SUNY at Stony Brook) Record (AMS 512) March 2012 28 / 35
The Shapes of DistributionsStable Paretian (Pareto-Levy) Distributions
Figure: Stable Distributions and parameters
Remarks:When α = 2, β = 0, it is a Gaussian Distribution withγ = σ2
2 , δ =expected valueEmpirical work suggests that actual rates of return are bestapproximated by 1.7 ≤ α ≤ 1.9When 1.7 ≤ α ≤ 1.9, E (X ) = δ,Var (X ) = ∞
Yiyang Yang (SUNY at Stony Brook) Record (AMS 512) March 2012 29 / 35
The Shapes of DistributionsIndex Model Based on Stable Distribution
DefinitionSingle-index model of return on security i can be represented as:
Ri = ai + bi I + ci
where ai , bi =parameters, ci =uncertain variable, I =level of index.
CorollaryAssume I and {ci} follow Stable Paretian Distribution with same α∗.Then, the risk of a portfolio can be represented as:
γp =N
∑i=1
X α∗i γci + bα∗
p γI
where γp = portfolio risk, γci =risk unique to security i, bp = ∑Ni=1 Xibi ,
γI =index risk.Yiyang Yang (SUNY at Stony Brook) Record (AMS 512) March 2012 30 / 35
The Shapes of DistributionsIndex Model Based on Stable Distribution
CorollaryAssume portfolios are well-diversified and half the risk of a typical securityis due to uncertainty of index, then:
γp = [n · (1n )α∗ + 1]bα∗
p γI
where n =the number of securities in the portfolio.
Figure: portfolio risk and number of security
Yiyang Yang (SUNY at Stony Brook) Record (AMS 512) March 2012 31 / 35
The Shapes of DistributionsSummary
Practical implications:Gaussian distrubtion fails to descirble the actual rate of return.Stable Paretian Distribution with 1.7 ≤ α ≤ 1.9 approximates reallitybest.Variance is an untrustworthy indicator of risk.Volatility can still be used.Index models are efficient in normative applications.
Yiyang Yang (SUNY at Stony Brook) Record (AMS 512) March 2012 32 / 35
Using the Past to Predict the FutureIs Future like Past?
The cause of difference between future and past:Chance eventsChanges in management
The price process might be greatly different.Diversification can stabilize volatilities of securities and portfolios.
Figure: Volatility in two periods
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Using the Past to Predict the FutureSolve the Problem
Based on above discussion, an investor could select the portfolio based onpast records:
maximize ∑Ni=1 XiEi
subject to ∑Ni=1 Xibi = b∗p and 0 5 Xi 5 1
n for security i.
where Ei is average past return of security i,bi is actual past volatility of security i,b∗p is some desired level of volatility,n is a number large enough to force adequate diversification.
Yiyang Yang (SUNY at Stony Brook) Record (AMS 512) March 2012 34 / 35
References
William F. Sharpe, Portfolio Theory and Capital Markets,McGraw-Hill Book Company 1790.Edward J. Elton, Martin J. Gruber, Stephen J. Brown, William N.Goetzmann, Modern Portfolio Theory and Investment Analysis, JohnWiley & Sons. Inc. 2009.Svetlozar T. Rachev, Young Shin Kim, Michele Leonardo Bianchi,Frank J. Fabozzi, Financial Models with Levy Process and VolatilityClustering, June 2010
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