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Mutual Fund and Security Selection

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Title: Mutual Fund and Security Selection


1
Mutual Fund and Security Selection
  • Through an Operations Research Lens

Committee Members Dr. Shaler Stidham, Jr. Dr.
George Fishman Dr. Bin Gao Dr. Gabor Pataki Dr.
David Rubin Dr. Jon Tolle
November 30, 2001
2
Investor Dilemma
  • Investors desire high returns.
  • To reduce risk, diversify.
  • Thousands of securities to choose from!
  • Use mutual funds or money managers?

3
Investor Types
  • Security Investor An investor who invests
    in individual securities on his own.
  • Fund Investor An investor who
    utilizes mutual funds or money managers.

4
Research Questions
  • Do fund and security investors obtain
    fundamentally different portfolios?
  • Who is better off?
  • By how much?
  • Are there other interesting results or patterns?

5
Three Models
  • Mean-variance
  • Preliminary results show the security investor is
    significantly better off, sometimes.
  • Downside risk
  • Challenging to compute risk.
  • Most likely place for significant contribution.
  • Multi-period models
  • Still in beginning stages.

6
Two Pronged Research Approach
  • Use analytical techniques to find and prove
    interesting results.
  • Use numerical portfolio optimization to find
    patterns we might have otherwise overlooked.
    Return to analytical techniques to explain these
    patterns.

7
Mean-Variance Models
  • Assumptions
  • Investors are risk-averse.
  • Portfolio selection is based solely on mean
    return and the risk measure variance.
  • Introduced by Markowitz in the 1950s.
  • We can formulate portfolio optimization problem
    as a quadratic program.

8
Mean-Variance Formulation
9
Short-Selling and Borrowing
  • xo lt 0 investor is borrowing at rate r.
  • xi lt 0 investor is short-selling (i gt0).
  • An investor who short-sells effectively sells a
    security he doesnt own, promising to buy it back
    later.
  • Prevent borrowing and short-selling with
    non-negativity constraints.

10
Implied Assumptions
  • No transaction costs.
  • If a risk-free asset is available and borrowing
    is allowed, the lending and borrowing rates are
    equal (r).
  • Any amount may be invested in any security.
  • These can be relaxed, and we may eventually
    consider doing so.

11
Benefits of Diversification
minimize
  • Say we have two independent investment
    opportunities, each with the same mean.
  • Standard deviations 0.2 and 0.3.
  • Invest completely in first security?
  • 70/30 combination does much better.
  • Standard Deviation

12
Efficient Frontier
  • Consists of all mean-standard deviation pairs
    such that no attainable portfolio has
  • same mean and lower standard deviation
  • same standard deviation and higher mean.
  • To trace out the frontier, set various values of
    mean and find standard deviation by optimizing.
  • Merton has given analytical formulas.

13
Efficient Frontier
Standard Deviation (Risk)
r
Mean (Return)
14
Recall Research Questions
  • Do fund and security investors obtain
    fundamentally different portfolios?
  • Who is better off?
  • By how much?
  • Are there other interesting results or patterns?

vs.
15
Portfolio Theory Assumption
  • Professionals and non-professionals
  • have the same skill level.
  • use mean-variance to select portfolios.
  • estimate same security parameters.
  • But, fund managers have advantages in reality
    more experience and time.
  • If fund investor is only slightly worse off than
    security investor, perhaps use funds.

16
Background Result 1Fund Investor Cannot be
Better Off
  • The fund investor cannot be better off than
    security investor.
  • By choosing to use only funds, he has limited his
    choices.
  • Cannot improve objective by adding constraints.

17
Background Result 2 Merton
  • Investor will be indifferent between
  • investing directly in N securities.
  • investing in two funds made up of the N
    securities.
  • Security investor has no advantage?
  • Merton assumes same pool of securities.
  • Funds are made up of different pools!
    Value/growth, domestic/international, etc.

18
New Question
  • Assuming funds consist of different pools, can
    security investor do strictly better?
  • Use numerical portfolio optimization to find out.
  • Without loss of generality, assume the fund
    investor has exactly two funds available to him.

19
Numerical Portfolio Optimization
  • Assume some universe of securities, solve
    security investors problem (QP) in CPLEX.
  • Split universe into two pools one for each
    fund. Optimize fund portfolios and calculate
    fund parameters (variances and correlation).
  • Assume investor will use funds. Determine
    optimal portfolio using parameters from step 2.
  • Compare variances from steps 1 and 3 to see how
    much worse off fund investor ends up.

20
Two Fund Types
  • We did numerical optimizations for
  • Symmetric Funds
  • Each funds securities have identical means and
    variances.
  • Constant pairwise correlation within a fund.
  • Asymmetric Funds
  • Different means, variances, covariances allowed.

21
Symmetric Funds
  • We found that the investor is indifferent between
    funds and securities. Why?
  • Because of symmetry, fund manager spreads capital
    evenly among all securities in his pool.
  • Then, investor has equivalent choice whether he
    uses the funds or not how much to invest in
    each type of security.
  • Once that decision is made, he would also spread
    money evenly within each pool.

22
Asymmetric Funds
  • We observe the security investor better off if
    correlation between managers is not zero.
  • If a risk-free asset is available
  • fund and security investors hold fundamentally
    different portfolios.
  • risk levels are not significantly different.
  • If no risk-free asset available to the fund
    managers, security investor does far better than
    fund investor even if correlation is zero between
    funds.

23
Example Candidate Securities
24
Example Candidate Securities
Standard Deviation
Minimum Mean Return
25
Other Parameters
  • Return on risk-free asset 5.
  • Investors minimum mean return 20.
  • Minimum mean returns
  • Fund 1 15
  • Fund 2 26
  • Internal correlation 0.7 in each fund.
  • External correlation 0.4.

26
Results I Risk-Free Asset Available to All
Fund investors portfolio Fund 1 0.5144,
Fund 2 0.4693, RF 0.0163, ?1,2 0.4668
27
Results IIR-F Unavailable to Fund Managers
Fund investors portfolio Fund 1 0.6673,
Fund 2 0.3942, RF -0.0616, ?1,2 0.4395
28
Possible Model Modifications
  • Upper and lower limits on investments
  • Binary-type investments
  • Cardinality constraints

29
III Cardinality Constraints
  • We can limit the number of securities in a
    portfolio by adding the constraints
  • Yields a quadratic integer program (QIP).
  • We will need to find a way to solve this.

30
Sample of Other Mean-Variance Questions
  • What patterns do we observe when we vary problem
    parameters?
  • Mean return
  • Internal and external correlation
  • Number of securities available
  • What will we see if we use real data? Will there
    be challenges to implementation?

31
Criticisms of Mean-Variance
  • Optimal M-V portfolios are only optimal in a
    utility theory sense if
  • security returns are from elliptical family or
  • the investors utility function is quadratic.
  • Variance penalizes the possibility of high
    portfolio returns as harshly as the possibility
    of low returns.

32
Downside-Risk Models
  • We only want to penalize the left hand side of
    the distribution.
  • This would not be a problem if return
    distributions were empirically symmetric.
  • But they are not, so downside-risk models emerged.

33
New Objective Function
  • ? is a target below which the investor does not
    want his portfolio return to fall, and f(x) is
    the portfolio return distribution.
  • Note constraints have not changed.

34
Possible ?-Values
  • If ? 2 , we obtain third order stochastic
    dominance results. The investor
  • prefers less to more
  • is risk-averse
  • displays decreasing absolute risk aversion.
  • This fits a real investor well.
  • If ? 2, downside-risk measure called target
    semivariance (TSV).

35
Obstacles to TSV Implementation
  • Even if portfolio return distribution is known,
    we must use numerical integration to obtain TSV.
  • Even if we know make-up of portfolio, the return
    distribution is not obvious.
  • Portfolio make-up is what we are trying to
    determine.
  • CPEX cant handle all this!

36
Can We Avoid These Obstacles?
  • Let ? be the discrete or continuous set of
    possible scenarios or outcomes.
  • ??? represents one possible outcome.

37
Is TSV Convex?
  • is convex for all ? since a convex increasing
    function of a convex function is convex.

is also convex since (1) is convex for all ?.
38
Can We Implement the TSV?
  • Use historical security price data.
  • Some authors have suggested using
  • Should we weight more recent observations more
    heavily?

39
Other Downside-Risk Issues
  • Can we overcome any of the three obstacles?
  • We have shown that TSV produces the same optimal
    portfolios as mean-variance if security returns
    are normal. Is it true for all symmetric
    distributions?
  • Perhaps we should consider measuring return with
    upside-potential.

40
Topics for Future Consideration
  • Multi-period models a real investor does not
    optimize his portfolio once and for all. He must
    periodically check to be sure it is still
    optimal.
  • What utility functions are implied by our various
    models? If we assume a particular utility
    function, what results will we obtain?

41
Questions?
42
Effects of Correlation
minimize
  • If securities are positively correlated,
    diversification will not have as much power to
    reduce risk.
  • If they are negatively correlated, we can make
    risk even smaller.
  • Investing in securities with low or negative
    correlation is good. Large price fluctuations
    will be buffered out.

43
Step 2 Correlation Calculation
  • The correlation between the two funds

44
Varying Correlation Values
  • Internal correlation External correlation
  • High internal correlation should
  • discourage diversification within funds.
  • encourage diversification between funds.
  • High external correlation may lead investors to
    choose one fund or the other.
  • If we allow negative correlation, we must be
    careful not to allow too much.

45
Real Data
  • Actual historical stock price data may give us
    interesting results.
  • We can download daily stock prices for all stocks
    on NYSE from Dreyfus website.
  • Calculating means and variances of N stocks will
    be easy, but covariances are estimated by doing
    N(N-1) regressions.
  • Can we get around this problem?

46
Factor Models
  • Assume the return on the ith security is given by
  • If we use a single factor model, we may assume F
    is the return on a market index.
  • Then, we need only run N regressions.
  • It is easy to modify our formulation.

47
I Limits on Investments
  • To limit proportion invested in security i, add
    the following constraint
  • To limit amount invested in security i,
  • let xi amount invested in security i.
  • change budget constraint to
  • Then, add the constraint above.

48
II Binary Type Investments
  • Let xi be binary.
  • Change budget from 1 to B.
  • Now our formulation is a quadratic integer
    program (QIP).
  • Can convert to an IP if we define new variables
    xi,j xi xj.
  • Must add appropriate forcing constraints.

49
Forcing Constraints
xi,j xi xi,j xj xi xj 1 xi,j
  • We have added N(N-1)/2 variables and three times
    that many constraints.
  • CPLEX will only solve small problems.

50
? Values of Zero and One
  • Some authors have used ? 0 or 1.
  • ? 0 corresponds to value at risk (VaR)
  • used by companies to determine the amount of
    money they stand to lose in a short period of
    time with some low probability.
  • ? 1 implies investor is risk-neutral.

51
Equal Weighting?
  • Imagine bringing the 1/T inside the sum.
  • Each portfolio return (rt(p)) is equally
    weighted.
  • Perhaps we really want to weight recent
    observations more heavily.

52
Acknowledgements
  • My advisor Dr. Stidham.
  • Doctoral committee Drs. Fishman, Gao, Pataki,
    Rubin, and Tolle.
  • AAUW and their fellowship deadline.
  • Friends, family, OR students.
  • God, who makes all things possible.
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