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


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


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


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.


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.


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.


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.


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



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:



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.



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.


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.


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.


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.


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


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.



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


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.


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.



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.



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.


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.


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.


The Shapes of DistributionsStable Paretian (Pareto-Levy) Distributions

Figure: rate of return on market portfolio from 1926 to 1965

Remark: Normal Distribution contradicts reallity.



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 ∈ (−∞,+∞)



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 ) = ∞


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


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.


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


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.


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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