Chapter 8. Record
Portfolio Theory and Capital Market
Yiyang Yang
Department of Applied Mathematics and Statistics
State University of New York at Stony Brook
March 2012
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Record (AMS 512)
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Outline
1
Introduction
2
Measuring Performance
3
Mutual-Fund Performance
4
The Shapes of Distributions
5
Using the Past to Predict the Future
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Record (AMS 512)
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Introduction
Past vs. Future
portfolio theory and capital market theory: prediction
empirical work: past record
Motivation: portfolio theory+predictions from past record
measuring tools:
probabilities; relative frequencies
expected return; average return
variability; relative frequencies of various deviations
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Introduction
The Market Portfolio
Measure of market portfolio:
Indexes: Dow-Jones’ Index, S&P Composite Index
average values for individual securites: RMt = N1 ∑N
i =1 Rit , where
RMt =return on the market portfolio in time period t, Rit =rate of
return on security i in time period t, N =number of securities
Figure: Rate of return on the market portfolio
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Introduction
The Effectiveness of Diversification
Motivation: How many securities must be included to obtain a reasonably
well-diversified portfolio?
Figure: Variability and portfolio diversification
8.63
n , where
( σ )2
thus (σS )2 =
T
can be approximated by σp = 11.91 +
n =number of securities.
When n = 1, σT = 20.5, σS = 11.9,
0.34.
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Record (AMS 512)
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Itroduction
Market and Industry Factors
Table: Proportion of security risk attribute to market factors
Period
June 1927-September 1935
October 1935-Feburary 1944
March 1944-July 1952
August 1952-December 1960
Average proportion(%)
58.4
55.7
41.2
30.7
Remarks:
The proportion decreases overtime.
Analysis over the entire period indicates, market fluctuation accounts
for 52% variance of a typical security, a group of industry accounted
for another 11%.
In an index model, the market can be represented by an index and
additional indexes can be added to represent industry factors.
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Measuring Performance
Reward-to-Variability
Definition
Capital market line
re =
EM − p
σM
where re =price of risk reduction for efficient portfolios, EM =expected
return, p =pure rate of interest, σM =standard deviation.
Corollary
The past performance of any portfolio can be represented as:
Ap − p 0
r
( )p =
v
σp0
where ( vr )p =reward-to-variability ratio for portfolio, Ap =actual average
return, p 0 =actual pure interest rate, σp0 =actual variability of portfolio.
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Measuring Performance
Reward-to-Variability
Figure: reward-to-variability
Remark:
The slope of the line associated with the portfolio is the
reward-to-variability ration.
The steeper the line, the better the performance of portfolio.
Used to measure the performance of portfolio.
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Measuring Performance
Reward-to-Volatility
Definition
Security market line
rs =
Ei − p
bi
where rs =price of risk reduction for securities, Ei =expected return of
security i, p =pure rate of interest, bi =volatility of security i.
Corollary
The past performance of any portfolio can be represented as:
r
Ai − p 0
( )i =
b
bi0
where ( br )i =reward-to-variability ratio of security i, Ai =actual average
return, p 0 =actual pure interest rate, bi0 =actual volatility of security i.
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Measuring Performance
Reward-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 Performance
Differential Return
Definition
Actual characteristic line represents the relationship of actual return of a
security or portfolio and that of market portfolio. Thus, the line passes
through the point at which both returns equal their actual average value
Ai and AM ; and the slope is actual volatility bi0 .
Figure: Actual characteristic line
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Measuring Performance
Differential Return
Figure: Differential return
Derive the actual characteristic line Y from Ai , AM , bi0 .
Line Z represents the corresponding efficient portfolio with same
volatility.
0
x = Aib−0 p is the reward-to-volatility ratio, separate it:
i
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Measuring Performance
Differential Return
r
= dh + (AM − p 0 )
b
If dh > 0, performance of the security or portfolio was superior to
that of a market-based portfolio of comparable volatility; if dh < 0,it
was worse.
dv = dh · bi0 is the vertical distance from point P to the characteristic
line.
Definition
dv is denoted as differential return. A positive differential return indicates
that performance was superior to that of a market-based portfolio of
comparable volatility; a negative differential return indicates that it was
worse.
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Measuring Performance
Differential 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 compare
the performance of a security with that of market.
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Mutual-Fund Performance
Introduction
Definition
An open-end mutual fund is an institution designed to provide both
diversification and professional management at relatively low cost.
Some facts about mutual fund:
The managers are ready to issue new shares or retire old shares at
virtually any time.
The net asset value per share=(current market value of the fund’s
holding)/(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 the
total net asset value.
Most mutual funds hold over 100 different securities.
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Mutual-Fund Performance
Correlation 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 attributed
to market fluctuations.
In sum, most mutual funds are well-diversified.
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Mutual-Fund Performance
Volatility
Figure: Volatility of corresponding 115 mutual funds
Remarks:
The average volatility is 0.84.
Most mutual funds perform more conservative than one made up of
the securites in S&P Index.
The differences in volatility may due to the objectives of the funds.
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Mutual-Fund Performance
Volatility
Table: Volatility by type of fund
Classification
Growth Funds: primary objective is
long-term growth of capital
Growth-income Funds: emphasis on
long-term, consider income
Income-growth Funds: emphasis
current income, consider long-term
Income Funds: primary objective is
current income
Blanced Funds: relative stability and
continuity of income
Yiyang Yang (SUNY at Stony Brook)
Record (AMS 512)
Number of
funds
31
Average
volatility
0.970
30
0.941
15
0.856
9
0.674
30
0.645
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Mutual-Fund Performance
Performance 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 of
Dow-Jones’ portfolio; 15 had smaller ratios.
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Mutual-Fund Performance
Performance 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 security
market line than above it.
Right graph indicates gross performance, points scatter randomly
around the line.
Yiyang Yang
(SUNY at Stony
Brook)
Record (AMS
512)
Plausible
explanation:
excessive
managing
expenditures. March 2012 20 / 35
Mutual-Fund Performance
Performance of Mutual Fund
Figure: Differential returns 1945-1964 (115 mutual funds)
Remarks:
Based on net values, the average of differential return was -1.1%; 76
had 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 of
comparable volatility.
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Mutual-Fund Performance
Fund 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 net
returns may be due more to cost of management than effectiveness.
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Mutual-Fund Performance
Fund Managers got talent or luck?
Well-managed funds with enjoy runs of successive superior performance.
Table: Frequency of successive superior
Length of run (number
of successive years of
superior performance)
1
2
3
4
Number of
instances
574
312
161
79
Percent of instances
followed by another year of
superior performance (%)
50.4
52.0
53.4
55.8
Remarks:
Typical fund obtained superior performance 50.2% of the time.
Funds with one to four successive prior years of superior performance
had slightly (less than 6%) better success.
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Mutual-Fund Performance
Fund 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 is
well preserved.
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Mutual-Fund Performance
Summary
The conclusions of this section can be summarized:
Most funds diversified well.
Most managers selected a general-risk class and maintained their
stated positions reasonably well.
On the average, funds did no better, before expenses, than
market-based portfolios of comparable volatility.
On the average, funds did worse, after expenses, than market-based
portfolios of comparable volatility.
Few, if any, funds consistently performed better than market-based
portfolios of comparable volatility.
Most funds appear to have spent too much searching for mispriced
securities.
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Mutual-Fund Performance
Further Discussion
Other factors:
Sales charge: no-load funds outperform load funds, switch funds
perform worst.
Turnover: managers with low turnover outperform managers with
high 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 in
purchase and sales.
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The Shapes of Distributions
Stable Paretian (Pareto-Levy) Distributions
Figure: rate of return on market portfolio from 1926 to 1965
Remark: Normal Distribution contradicts reallity.
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The Shapes of Distributions
Stable Paretian (Pareto-Levy) Distributions
Definition
Random variables {Xn } are independent copies of X , then X is said to
follow Stable Distribution if:
X1 + X2 + · · · + Xn = Cn X + Dn
1
where Cn = n α dictates the stability property, Dn is a real number.
Four parameters characterized Stable Distribution
α the characteristic exponent; a measure of the height of the extreme
tail 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 ∈ (−∞, +∞)
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The Shapes of Distributions
Stable Paretian (Pareto-Levy) Distributions
Figure: Stable Distributions and parameters
Remarks:
When α = 2, β = 0, it is a Gaussian Distribution with
2
γ = σ2 , δ =expected value
Empirical work suggests that actual rates of return are best
approximated by 1.7 ≤ α ≤ 1.9
When 1.7 ≤ α ≤ 1.9, E (X ) = δ, Var (X ) = ∞
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The Shapes of Distributions
Index Model Based on Stable Distribution
Definition
Single-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.
Corollary
Assume I and {ci } follow Stable Paretian Distribution with same α∗ .
Then, the risk of a portfolio can be represented as:
N
γp =
∑ Xiα γc
∗
∗
i
+ bpα γI
i =1
where γp = portfolio risk, γci =risk unique to security i, bp = ∑N
i =1 Xi bi ,
γI =index risk.
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The Shapes of Distributions
Index Model Based on Stable Distribution
Corollary
Assume portfolios are well-diversified and half the risk of a typical security
is due to uncertainty of index, then:
∗
1 ∗
γp = [n · ( )α + 1]bpα γI
n
where n =the number of securities in the portfolio.
Figure: portfolio risk and number of security
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The Shapes of Distributions
Summary
Practical implications:
Gaussian distrubtion fails to descirble the actual rate of return.
Stable Paretian Distribution with 1.7 ≤ α ≤ 1.9 approximates reallity
best.
Variance is an untrustworthy indicator of risk.
Volatility can still be used.
Index models are efficient in normative applications.
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Using the Past to Predict the Future
Is Future like Past?
The cause of difference between future and past:
Chance events
Changes 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 Future
Solve the Problem
Based on above discussion, an investor could select the portfolio based on
past records:
maximize ∑N
i = 1 X i Ei
∗
subject to ∑N
i =1 Xi bi = bp 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,
bp∗ is some desired level of volatility,
n is a number large enough to force adequate diversification.
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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, John
Wiley & Sons. Inc. 2009.
Svetlozar T. Rachev, Young Shin Kim, Michele Leonardo Bianchi,
Frank J. Fabozzi, Financial Models with Levy Process and Volatility
Clustering, June 2010
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