Mutual Fund and Security Selection

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Mutual Fund and Security Selection

• Through an Operations Research Lens

November 30, 2001
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Investor Dilemma

• Investors desire high returns.

• To reduce risk, diversify.
• Thousands of securities to choose from!
• Use mutual funds or money managers?

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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.

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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?

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Three Models

• Mean-variance
• Preliminary results show the fund 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.

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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.

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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.

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Mean-Variance Formulation
minimize
subject to
(min mean return)
(budget)
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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.

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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.

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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

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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.

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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.

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Efficient Frontier
Standard Deviation (Risk)
r
Mean (Return)
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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.
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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.

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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.

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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.

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New Question

• Assuming funds consist of different pools of
  securities, 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.

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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.

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Step 2 Correlation Calculation

• The correlation between the two funds

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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.

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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.

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Asymmetric Funds

• We observe that the security investor is better
  off if correlation between funds 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, security
  investor does far better than fund investor even
  if correlation is zero between funds.

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Example Candidate Securities
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Example Candidate Securities
Standard Deviation
Minimum Mean Return
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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.

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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
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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
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Sample of Other Mean-Variance Questions

• What happens when we vary the values of internal
  and external correlation?
• What will we see if we use real data? Will there
  be challenges to implementation?

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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.

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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?

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Factor Models

• Assume the return on the ith security is given by

• If we use a single factor model, we might assume
  F is the return on a market index.
• Then, we need only run N regressions.
• It is easy to modify our formulation.

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Possible Model Modifications

• Upper and lower limits on investments
• Binary-type investments
• Cardinality constraints

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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.

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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.

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Forcing Constraints
xi,j xi xi,j xj xi xj 1 xi,j

• This adds N(N-1)/2 variables and three times that
  many constraints.
• CPLEX will only solve small problems.

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III Cardinality Constraints

• We can limit the number of securities in a
  portfolio by adding the constraints

• This will give us an even larger problem and we
  may need to consider alternate solution
  techniques.

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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.

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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.

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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.

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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, the downside-risk measure is called
  target semivariance (TSV).

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? 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.

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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!

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Can We Avoid These Obstacles?

• Let ? be the discrete or continuous set of
  possible scenarios or outcomes.
• ??? represents one possible outcome.

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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 ?.
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Can We Implement the TSV?

• Use historical security price data.
• Some authors have suggested using

• Because of convexity, if we can find a local
  minimum, it must be a global minimum.

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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.

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Other Downside-Risk Issues

• Can we overcome any of the three obstacles?

• Authors suggest TSV produces the same optimal
  portfolios as mean-variance if security returns
  are normal.
• Is this true?
• Is it true for all symmetric distributions?
• Perhaps we should consider measuring return with
  upside-potential.

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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?
• Empirically, how does the fund investor compare
  to the security investor?

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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.