Portfolio Selection

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Portfolio Selection
Nakorn Indra-Payoong Maritime College, Burapha
University
Last updated 15.08.04
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Motivation

• Do not leave all of your money with your wife
  spread out to the others
• Transport a basket of eggs from one point to
  another there are several ways to do this but
  different risks are attached.
• Distributing the eggs using different ways is the
  benefit of the diversification of risks ?

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Portfolio theory

• Efficient portfolio
• Given a level of risk, investors prefer higher
  return
• Given a level of return, investors prefer less
  risk
• Markowitz portfolio selection
• Efficient portfolio risk is diversified away,
  Optimal portfolio risk is minimised (there still
  contains a small amount of risk) ..this is almost
  identical

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Optimal portfolio

• Maximise return subject to a given variance

Subject to
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Optimal portfolio

• Minimise variance (risk) subject to a given
  return

Subject to
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Efficient frontier

• Efficient frontier or mean variance efficient
  (MVE)

• 10,000 random portfolios, 30 assets

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Efficient frontier

• Portfolio is said to be efficient if ..
• - no portfolio having the same risk with greater
  return
• - no portfolio having the same return with
  lesser risk
• The efficient frontier is the collection of all
  optimal portfolios

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Balancing risk and return

• To trace out efficient frontier, we introduce
  as risk aversion factor and vary values of
  from 0 to 1

Subject to
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Trace out

• The greater is , the more risk aversion the
  investor has
• If 0, we dont consider risk optimal
  solution will involve a single asset with the
  highest return
• If 1, we dont consider return optimal
  solution will involve a number of assets
• In slide 3 and 4, if we vary values of and
  in the same way we can also get the efficient
  frontiers

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Efficient frontier
Optimal portfolios lie on this curve
High risk /high return
Low risk/low return

• Portfolios below curve are not efficient..why?

• Portfolios above curve are impossible..why?

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Example

• Given the expected returns 0.1, 0.2, 0.15 for 3
  assets
• Risk aversion factor 0.3
• Given the covariance between the returns of
  assets

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Example

• Suppose that
• Expected return of portfolio

• Variance (risk) of portfolio

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Example
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Example

• From the objective function z

• Then

• We want to find the values of that maximize z

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Utility.

• Each investor has a certain utility for money
• Utility determines how much risk he is willing to
  take in order to obtain an expected amount of
  money
• Let x be the amount of money, k be the risk
  aversion factor utility function is

• This function describes relationship between risk
  and return for an investor

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Example
k 10
k 1.0
k 0.1
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Utility

• Assume that the asset return
  is normally distributed with mean and
  variance
• So..the portfolio return is also normally
  distributed with mean and
  variance
• The expected value of utility is ..

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Utility

• Since is an increasing
  function in x, maximizing utility is equivalent
  to maximizing

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Maximise the portfolio utility
Subject to

• The optimal portfolio utility is determined by
  solving for the weighting parameter

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Example

• Two investments risky and risk-free
• We want to find the weights for risky and
  risk- free investments that maximise
  portfolio utility
• Suppose that risk aversion parameter 3

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Example (cont)

• Vary the weights risky and risk-free .

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Example (cont)
maximum utility 0.0910
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Points to notice

• So far..adding more assets, we can get more
  diversification and reduce portfolio risk
• Can we eliminate all portfolio risk?
• Covariance is the key factor for diversify risk
• If average covariance is not close to zero, we
  can never eliminate all risk, even after
  diversifying a lot, we are still left with some
  risk

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Investment risk

• Two basic forms of investment risk specific risk
  and market risk
• Specific risk (or diversifiable risk) derives
  from factors specific to
• - an individual company
• - companies with particular industry or region
• Specific risk is, thus, additional to market risk

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Diversifying specific risk

• As specific risks only affect specific business,
  it is possible to diversify away this form of
  risk
• Diversification is achieved by the development of
  an efficient optimal portfolio of investment
• Efficient portfolio is one where all of specific
  risk is diversified away (minimised)
• Market risks cannot be reduced by
  diversification

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Specific vs. market risk

• A fire at IBMs warehouse affects IBM stock
  market. This risk is unique as it can be
  diversified (specific or diversifiable risk)
• A change in interest rates or oil prices affects
  the world economy. The risk always remains even
  after diversifying risks (market or
  undiversifiable risk)

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Input data

• So far we assume known expected return, variance
  and covariance
• Where do we get these input?
• Historical data ) but may not directly !
• We need to think about this very carefully !!

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Butterfly effect .

• The flapping of a butterflys wings in Beijing
  will cause a tornado in Toronto.
• The same effect as portfolio selection
• The small change to an input will result in a
  large change in the optimal asset weightings
• So.. it is very easy to arrive at a set of
  non-sense asset allocation

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Concepts

• Portfolio optimisation is a rubbish in-rubbish
  out tool.
• Once a set of portfolio allocation is put into
  place, small change in the market will result in
  large changes in the portfolio returns
• Practitioners are losing confidence in portfolio
  theory but we are trying hard to explain

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Concepts

• Over fitting the past data may explain 99 of
  what happened during the past but may only
  forecast 5 of what happens in the future
• Reducing the complexity of the model may reduce
  the explanatory power to 60-70 but forecasting
  ability may increase 40-50

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The differences
Present
Future
Past

• Practitioners forecast
• Practitioners look foreword
• Subjective experience
• Simple, crazy and stupid

• Academics explain
• Academics look backward
• Theory
• Complexity

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Techniques

• There are 3 commonly used techniques for input
  data
• Technique 1 based on the historical observation
• Consider the arithmetic mean as the expected
  return. Then calculate variance and covariance
• This technique is very simple and provides a
  rough approximation but may never be used in the
  real world

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Technique 2 using Beta

• Measurement of Beta is fixed by two definitions
• 1) A rate of return for risk-free investments.
  This rate
• of risk is given a Beta 0
• 2) A rate of return for investments carrying
  market
• risk. This rate of risk is given a Beta 1

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Technique 2 using Beta

• The return of security i composes of two
  components one is independent of the market
  and the other is due to the returns of market
  .

is a measure of the change in for a
given change in
is a random error now we assume 0
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Example

• Suppose the return on risk-free investments (i.e.
  Beta 0) is 6 per annum and the return on
  securities carrying market risk (i.e. Beta 1) is
  10 per annum
• Then the premium for a Beta of 1 is 4 per annum

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Example (cont)

• A security with Beta 0.7 would offer a return
• 6 (0.74) 8.8 per annum
• A security with Beta 2.5 would offer a return
• 6 (2.54) 16 per annum

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Relationship of return to market price

• The higher Beta (and, therefore, the expected
  return), the lower will be the market price of
  the security (people payoff well)
• The lower Beta (and the expected return), the
  higher will be the market price of the security
• In equilibrium, each security has to offer high
  expected return in order to attract investors

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Technique 2

• If we believe that security i is boomed (under
  priced) ..so well invest more in security i in
  our portfolio
• Other investors will do the same and the
  proportion of security i in the market portfolio
  will be increase
• Until proportions of security i and others are in
  an equilibrium

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Technique 3

• Using a multi-index model to capture some of the
  non-market influences
• Expected return of security i

is index k, is the measure of
sensitivity to index k
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Capital asset pricing model

• Now we consider the price rather than expected
  return
• Let be the expected future price of security
  i
• be the current price of security i
• Expected return
• For example

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Setting a current asset price

• From technique 2
• Thus..
• From Beta...

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CAPM

• Capital asset pricing model (CAPM)

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In practice

• We concern about market risk
• Price volatility, cyclicality of demand
• Market equilibrium imbalances
• Long lead time for capacity adjustment

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How company manages the risk

• Visibility risk-based valuation with portfolio
  analysis (input market data, contract)
• Pricing pass on risk, charge for option, min.
  time of contract, sharing risk with shippers
• Portfolio flexibility optimise commitment to
  shippers (contracted vs. spot), optimise
  capacity/cost