Portfolio Selection

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Portfolio Selection

• 1. Overview
• From our previous discussion, we know that
  different securities can exhibit different
  returns and different risk
• In the following, we will briefly discuss a
  strategy how a rational agent can choose out of a
  large number of different risky securities

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Portfolio Selection

• 1. Overview
• In the context of the risk structure of interest
  rates we found that there exists a trade-off
  between the risk of an asset and its interest
  rates if future payments are uncertain (which is
  normally the case).
• The higher an assets risk, the higher are its
  promised interest rates as borrowers require a
  risk premium to be willing to hold this asset

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Portfolio Selection

• 1. Overview
• Since we further also know that borrowers tend to
  care more about the expected return of an asset
  rather than only its interest rate, this logic
  carries right over to expected returns.
• More risky assets have higher expected returns,
  since agents need to be compensated for the
  higher default risk associated with those assets.

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Portfolio Selection

• 1. Overview
• This insight has found its way into the analysis
  of how agents should ideally invest their savings
  and many strategies are, hence, based on measures
  of expected returns and risk.
• The oldest established theory of choosing assets
  optimally is portfolio selection, which was
  developed by Harry Markowitz in 1954 and acts as
  the foundation for many modern investment
  theories.

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Portfolio Selection

• 2. The expected return on an asset
• In the context of asset demand we have already
  developed the tool most commonly used to describe
  an assets return its expected value.
• To briefly review how we find an expected value,
  consider the following example

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Portfolio Selection

• 2. The expected return on an asset
• Assume a bond pays a rate of return of 2 if
  market development is average, 3 if its good an
  -0.5 if its bad.
• The expected return on this bond is a weighted
  average of these three outcomes, where the
  weights indicate how likely each outcome is to
  occur

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Portfolio Selection

• 2. The expected return on an asset
• If we assume, that market conditions are good
  with a 20 chance, that they are average with a
  50 chance and that they are bad with a 30
  chance, the expected rate of return on this asset
  is given by
• E (RET) 0.2 3 0.5 2 0.3
  (-0.5) 1.45
• In the special case, that all scenarios are
  equally likely, the expected return on an assets
  is just the arithmetic mean of all possible
  outcomes.

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Portfolio Selection

• 2. The expected return on an asset
• Two useful properties of the expected value
  operator E
• E(RET1) E(RET2) E(RET1 RET2)
• E(c RET) c E(RET), where cconst.

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Portfolio Selection

• 3. The expected risk of an asset
• Its not a great deal to understand that the risk
  of assets can vary quite significantly and that
  we, hence, need to find a measure how strongly
  actual returns realized on a security deviate
  from its expected return.
• The measures most commonly used in this context
  are an assets variance and its standard
  deviation.

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Portfolio Selection

• 3. The expected risk of an asset
• These two statistics are approximate measures of
  how much actual returns on a security deviate
  from its expected return on average.
• Variance is denoted by s2 and is a weighted
  average of the squared deviations of the actual
  outcomes from the average outcome, where the
  weights, indicate how likely each outcome is to
  occur.
• Standard deviation is simply the square root of
  the variance.

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Portfolio Selection

• 3. The expected risk of an asset
• In the previous example, the assets expected
  rate of return was given by 1.45, while its
  actual returns were given by 2, 3 and -0.5
  with probabilities of 0.2, 0.5 and 0.3,
  respectively.
• This assets variance is, hence, found by
• s2 0.5(2-1.45)2 0.2(3-1.45)2
  0.3(-0.5-1.45)2
• 1.7725
• The assets standard deviation, finally, is equal
  to s 1.33.

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Portfolio Selection

• 4. An asset portfolio
• Rather than picking only single assets at a time,
  agents are typically better off if they spread
  out their wealth over a portfolio, i.e. over a
  number of different assets.
• The aim of this strategy is to hedge against risk
  specific to a particular single asset (Dont put
  all your eggs in the same basket). To see this,
  consider the following example

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Portfolio Selection

• 4. An asset portfolio
• Assume you are given a second bond on top of the
  first in the previous example. An agent could
  hold either one of these assets or a combination
  of both.
• Both of these bonds actual returns in each
  scenario along the respective probabilities of
  each of these scenarios are given by the
  following table.

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Portfolio Selection

• 4. An asset portfolio

Asset A, as a above, has an expected return of
1.55 with a variance of 1.7725 and a standard
deviation of 1.33. Asset B, by contrast has an
expected return of 1.335 with a variance of
1.401025 and a standard deviation of 1.18.
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Portfolio Selection

• 4. An asset portfolio
• If an individual holds a fraction XA of her/his
  wealth in shares of bond A and a fraction XB in
  shares of bond B, the expected return and risk on
  this portfolio is a function of these shares, the
  bonds expected returns and risks.
• While the expected return on such a portfolio is
  found in a straightforward manner, the expected
  risk turns out to be somewhat more tricky.

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Portfolio Selection

• 4. An asset portfolio
• The actual portfolio return for each of the three
  scenarios is given by the fraction of an
  individuals wealth invested in bond A times the
  return on bond A in that scenario plus the
  fraction invested in bond B times the return on
  bond B.
• An example Assume an agent invests 50 of his
  savings in bond A (XA0.5) and 50 in bond B
  (XB0.5).
• If e.g. scenario 1 occurs, her/his portfolio
  return is equal to RET0.5(3) 0.5(-1) 1

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Portfolio Selection

• 4. An asset portfolio
• The expected portfolio return E(RETp) is then
  given by a weighted average of all actual
  portfolio returns, where the weights are once
  more equal to the probability of each scenario i.
• E(RETp) S Pi (XA RET1i XB RET2i)

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Portfolio Selection

• 4. An asset portfolio
• If we hold 50 of our portfolio in Bond A
  (XA0.5) and Bond B (XB0.5), the expected
  portfolio return is, hence, given by
• E(RETp) 0.2 0.5 3 0.5 (-1) 0.5
  0.5 2 0.5 1.75) 0.3 0.5
  (-0.5) 0.5 2.25 1.3925

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Portfolio Selection

• 4. An asset portfolio
• The portfolio variance is found in analogy to the
  single asset case as a weighted average of the
  squared deviations of the actual portfolio
  returns from the expected portfolio return, where
  the weights are equal to the probabilities of
  each outcome.

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Portfolio Selection

• 4. An asset portfolio
• Given the numbers in this table and an average
  portfolio return of 1.3925, the portfolio
  variance is given by
• sp2 0.2 (1 1.3925)2 0.5 (1.88
  1.3925)2 0.3 (0.85 1.3925)2 0.2355062

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Portfolio Selection

• 4. An asset portfolio
• Note, that this number is much smaller than the
  variance of each bond separately, indicating that
  the risk of this portfolio is smaller than the
  risk of each single bond (the standard deviation
  of the portfolio is equal to 0.85).
• For this very reason, an individual is better off
  by spreading out her/his savings across a mix of
  assets rather than holding a single asset alone!

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Portfolio Selection

• 4. An asset portfolio
• What drives this results?
• The actual returns of bond A and B are negatively
  correlated. If the good result for bond A
  (scenario A) occurs, the bad result for bond B
  occurs, et vice versa.
• Holding a mix of A and B allows an individual to
  diverisfy some of the risk specific to A and B
  away. This strategy is known as portfolio
  selection.

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Portfolio Selection

• 4. An asset portfolio
• Note, that there are three possible situations
• The returns of two assets are negatively
  correlated
• The returns of two assets are uncorrelated
• The returns of two assets are positively
  correlated
• While an agent can always gain by diversifying in
  the first two cases, in the third case at least
  she/he is equally well off as if only one asset
  alone were held.

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Portfolio Selection

• 5. The general case
• Somewhat more formally, we can express the
  expected portfolio return over a portfolio of n
  assets, each denoted by j as
• E(RETp) E (S (XjRETj) S E(XjRETj)
• S Xj E (RETj)

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Portfolio Selection

• 5. The general case
• Further, the portfolio variance of an n asset
  portfolio can be found by (j indicating the jth
  asset)
• s2 E S S(xjRETj)-ES(xjRETj)2
• For the two asset - case, after some tedious
  algebra, we can show that
• s2 XA2sA2 XB2 sB2 2XAXBsAB

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Portfolio Selection

• 5. The general case
• Without going too deep into the technical
  details, this expression states, that the risk
  expressed as the variance of a portfolio is a
  function of the fractions held of each asset XA
  and XB, the individual risk / variance sj2 of
  each asset j and finally, the so-called
  covariance sAB between asset A and asset B.

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Portfolio Selection

• 5. The general case
• This covariance term indicates how strongly the
  returns of A and B are moving together.
• If it is positive, a positive return on A occurs
  when a positive return on B occurs and a negative
  return on A occurs when a negative return on B is
  realized.
• If the covariance is negative, positive returns
  on asset A are associated with negative returns
  on asset B and vice versa.

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Portfolio Selection

• 5. The general case
• Note, that this covariance term affects the
  overall portfolio risk.
• If it is positive, the overall portfolio variance
  is larger indicating that there is little room
  for risk reduction through diversification.
• If it is negative, the overall portfolio variance
  is smaller indicating that there is more room for
  risk reduction through diversification.