FINANCIAL INVESTMENTS Faculty:Bernard DUMAS

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• FINANCIAL INVESTMENTSFaculty Bernard DUMAS
• Investor goals and the benefits of
  diversification
• session 1-2

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Overview

• How people should behave a theory of rational
  behavior
• How some people do behave
• Implementing optimal portfolio choice
• Using Excel to optimize
• Risk accounting

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The investor

• Clients and managers decision making
• Rational or irrational? Behavior traits
• Currency of reference
• Risk aversion / Measurement of risk
• Non-traded risks liabilities and spending/income
  needs
• ?Time horizon
• Constraints
• Agency/delegation problems how client will look
  at performance

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Rational behavior?

• Rational probability beliefs
• Rational decision making
• Risk defined
• Horizon effects
• Non traded assets
• Constraints

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The theory of rational learning/beliefs

• The updating of probability beliefs is dictated
  by
• Bayes formula (the definition of conditional
  probability)
• Beliefs evolve Prior PA vs. Posterior
  PAB
• Initial beliefs remain unaccounted for (did we
  get them from our parents?)

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The theory of rational decision making

• In Economics, the only consequence we consider is
  the persons utility of total consumption ca,s
  (irrespective of where the spent income comes
  from)
• But definition of utility of consumption is open
• Example of habit formation utility of current
  consumption compared with past consumption
• Predictability of a persons behavior is based on
  the postulate that
• this person always maximizes expected value
  (i.e., probability weighted) of his/her utility
  of consumption, calculated the same way in all
  circumstances

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Risk defined

• Amount of risk the size of deviations (,-)
  from expected outcome
• A mean preserving spread is the definition of
  increased risk

x
EX
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Statistical analog Compare 80-year frequency
distribution of, e.g., bond rates of return and
rates of returns of stocks in the U.S.
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Comment IID assumption

• Counting beans is not innocuous
• Valid only when rates of return of successive
  periods of time can be assumed
• Independently and
• Identically distributed
• As in coin tossing
• (avoid fallacy heads do not have to come after
  many tails or vice versa)
• IID means that realized rates of return of
  successive periods differ only because
• they are independent random draws
• from the same probability distribution, time
  after time
• Opposite of IID assumption predictability
• They differ also because probability distribution
  moves over time
• Some state variables move the probability
  distribution from one period to the next
• (see later session)

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Time horizon vs. holding period

• Horizon T This is the length of time over which
  you will conduct your investment plan.
• Example for individuals, their lifetime or time
  to purchase of house (or college of kids)
• ? Holding period time after which you will
  consider re-balancing
• Except for transactions costs, natural holding
  period ?0 (continuous portfolio rebalancing)

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Horizon effect investing over a lifetime

• Generational investing
• Should older people (with shorter horizon) hold
  less equity and more bonds than younger people?
• The CARDIF plan
• Horizon matters for definition of riskless
  asset

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Non traded risks in a portfolio

• Assets
• Outside (labor) income/ Human capital
• Investment in residence high transactions costs
• Liabilities Consumption needs
• Future consumption of goods
• In particular, consumption needs after retirement
• Future consumption of housing
• You must consider that these are part and parcel
  of your portfolio but with weights that are fixed
  (i.e., that you cannot decide)
• This will produce matching of assets and
  liabilities
• For institution balance-sheet optimization also
  called Asset-liability management or ALM.

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Constraints

• Need to keep liquidity (cash)
• for household, immediate consumption needs
• for mutual fund redemptions
• Trading difficulties illiquid markets
• Regulations public or self imposed
• SEC FSA AMF etc..
• Pension funds Employee Retirement Income
  Security Act (ERISA) European directives
• Diversification rule no more than 5 in any one
  publicly traded company
• Mostly domestic assets
• Mutual funds
• No borrowing.
• CFA rules of behavior
• Taxes do not realize gains until you have to
• Organization specific restrictions (e.g., risk
  management)

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Irrational behavior?

• Irrational probability beliefs
• Irrational decision making

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Irrational beliefs

• Overconfidence
• Confidence intervals too narrow
• Incorrect probability estimates for highly likely
  and highly unlikely events
• Probabilities distorted underweight rare events
• ?Likely variations in equity returns are seen as
  narrow when they are not
• Optimism and wishful thinking risk is seen as
  controllable
• Ambiguity
• people have several models or probability
  distributions in mind. They think in terms of the
  least favorable one (ambiguity aversion).
• Notion of Model risk.

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Irrational beliefs learning and reaction to
news

• Representativeness
• Prior PA underweighted
• New evidence carries too much weight
• Posterior excessively different from prior
• ? Sample size neglect. People are too easily
  convinced to change their minds.
• ?Extrapolation bias
• Good news can only come from good companies.
• Good news about a company used too hastily to
  conclude that the company is good.
• ? People trade too aggressively
• Conservatism
• Prior PA overweighted
• ?Excessive confidence in the familiar
• ?Belief perseverance
• Anchoring/framing news interpreted differently
  depending on frame
• Availability or saliency bias

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Irrational behavior preferences

• Prospect theory
• Utility of gains differs from disutility of
  losses
• What is the reference point?
• ?People shy away from owning shares because they
  would suffer short-term losses
• ?Trading practices
• Disposition to realize gains and aversion to
  realize losses
• Reference points the price at which you bought
  the share loser stocks vs. winner stocks
• Lower trading volume in bear markets
• narrow framing or mental accounting
• ? The very idea that risk is defined at the level
  of their entire portfolio of activities remains
  foreign to many investors.
• They think of their gains and losses in their
  various activities separately from each other

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Case of delegated portfolio management agency
problem

• Even if manager is rational person, he/she cares
  about his/her compensation
• Clients may not observe skill or effort of
  manager
• For this reason, they may base compensation on
  relative performance,
• This measurement of performance is not incentive
  compatible
• Manager proceeds to maximize expected utility not
  of absolute but of relative return
• Incentives are not aligned
• This is a distortion of decision making but not a
  form of irrationality

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One implementation of rational behavior

• Risk measured by variance (or standard deviation)
  of portfolio return

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Rates of return

• Different from dividend yield
• Holding-period rate of return
• Or
• Translation of holding period rate of return from
  one currency to another
• Excess rate of return RtEUR rtEUR

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Implementation of a special caseRisk measured
by variance or standard deviation

• Variance of your portfolio return
• definition probability weighted squared
  deviations from the expected value
• based on probability distribution
• Please, go and open your statistics textbook, if
  you do not know this very well already

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First application Sharpe ratio

• Definition
• Funds can be ranked by Sharpe ratio
• Used to choose among several funds the one in
  which you are going to put all your money
• Not used to apportion money across several
  investments
• for portfolio construction, see next

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Risk aversion

• Investment houses establish typical investment
  profiles and present them to their clients, to
  gauge their risk aversion
• Aggressive for growth
• Aggressive for income
• Conservative etc..
• Instead, we are going to use a number ? between 2
  and, say, 50
• Maximize portfolio exp. return - 1/2 ? ?
  portfolio variance

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Optimal diversification the ingredients

• Excess expected rate of return for each security
  i (organized into vector)
• Variance of rate of return for each security i or
  standard deviation
• Covariances of rate of return of security i with
  security j (organized into matrix) or
  correlations

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The optimization process
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Optimal diversification

• What is covariance between Ri and Rj?
• Statistical estimate
• Why does covariance come in?
• Correlation defined as the covariance divided by
  the product of the two standard deviations
• ?By definition of correlation, covariance is also
  correlation between Ri and Rj ? standard
  deviation of Ri ? standard deviation of Rj
• Please, go and open your statistics textbook if
  you do not know this very well already

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Properties of covariances and variances

• Covariance can be expanded
• So can variance, since
• Application

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Using Excel to optimize

• Step 1 Set up row or column of portfolio weights
  xi
• Step 2 Obtain portfolio variance
• compute xi ? cov(Ri,Rj) ? xj
• Sum these both ways (over i and j) to get
  portfolio variance
• Step 3 Obtain portfolio expected return
• compute xi ? E(Ri)
• Or, if there is riskless asset, xi ? E(Ri) r
  (use expected excess return)
• Sum these to get portfolio expected return
• Step 4
• Maximize portfolio exp. return - 1/2 ? ?
  portfolio variance for given ?.
• Recall that ? is risk aversion.
• Or maximize portfolio exp. return for given
  portfolio variance (or standard deviation),
• Or minimize portfolio variance for given
  portfolio exp. return ,
• under constraint that portfolio weights sum to 1
  (in the absence of riskless asset) and possibly
  other constraints
• Use Solver. Can have many securities, add
  constraints.

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Risk accounting
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Optimal diversification condition of optimality
(without constraint)

• How can you tell whether a portfolio p is well
  diversified or efficient?
• For each security i, E(Ri) or E(Ri) - r must be
  lined up with cov(Ri,Rp)
• A mini CAPM holdsfor each investor

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Optimal diversification condition of optimality

• If that condition is not satisfied, the
  composition of portfolio p must be changed
• Define
• If ?i gt 0, increase weight of security i
• If ?i lt 0, decrease weight of security i

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Interpretation Risk accounting(simplest form of
Value at Risk)

• Define ? of investment item i with respect to
  portfolio return as just a re-scaled covariance
  with portfolio
• Here, Ri refers to return on security i
• Rp refers to return of portfolio
• ? can be computed as a regression coefficient
• Please, go and open your statistics textbook if
  you are likely to confuse regression coefficient
  and correlation coefficient

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Risk accounting

• Risk accounting
• share of standard deviation measured by means of
  beta of each security with respect to portfolio
  return
• where xi is share of portfolio value invested
  in security i.
• Interpretation of beta relative to investors
  portfolio
• If an investment item has a beta equal to 2 and
  if 1 of the total portfolio value is invested
  there, then that investment accounts for 2 of
  the total risk (standard deviation) of the
  portfolio. (This the basis of Value at Risk
  scheme)
• It is not variance or stdev of investment item
  that counts
• Only systematic risk matters

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Example
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Conclusion risk and return

• Recall if an investment item has a beta equal to
  2 with respect to the portfolio and if 1 of the
  total portfolio value is invested there, then
  that investment accounts for 2 of the total risk
  (standard deviation) of the portfolio
• In a portfolio that is properly constructed, all
  the investment items should plot along a
  (positively sloped) line, so that each bit of
  risk receives its proportionate reward.