Topic II The Efficient Frontier

1
Topic II The Efficient Frontier

• Expected return and risk
• The minimum variance frontier
• The efficient frontier
• with riskless lending and borrowing
• with no riskless borrowing
• when the riskless borrowing rate is greater than
  the lending rate
• Maximizing Expected Utility
• Risk Aversion
• Absolute Risk Aversion
• Relative Risk Aversion
• Mean-Variance Analysis

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Expected Return and Risk on a Two Asset Portfolio

• The Minimum Variance Frontier shows the minimum
  risk for a given expected return.
• In order to derive the minimum variance we need
  to examine all feasible combinations of assets in
  expected return and risk space.
• We will examine the two asset portfolio case for
  simplicity.
• The average return and risk on a two asset
  portfolio is

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Example II.1Average Return and Riskon a Two
Asset Portfolio

• Suppose that you have data on the mean, variance
  and covariance of returns
• R1 R2 var(R1) var(R2) cov(R1 R2)
• 10 5 3 2 1
• and you have invested in a portfolio with the
  following weights
• X1 X2
• 0.7 0.3
• Then the average return on this portfolio is
  given by
• 0.710 0.35 8.5
• Then the variance of the return on this portfolio
  is given by
• 0.70.73 0.30.32 20.70.31 2.07

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The Minimum Variance Frontier

• Equations (1) - (3) imply a non-linear
  relationship between Rp and ?p.
• Substitute for X2 from (3) into (1), solve for X1.

• Substitute X1 into (2).
• This relationship will be a hyperbola (in
  general).

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The Minimum Variance Frontier

• Rather than solving algebraically plot (1) and
  the square root of (2) in risk-return space for
  different percentage shares and different
  correlations between return.

Return
MVP
Risk
6
The Minimum Variance Frontier (cont.)

• One point (portfolio) on the minimum frontier
  will have minimum risk.
• We can find what the Xi are that achieves minimum
  risk by differentiating ?p in (2) with respect
  to, say, X1 and solving for X1.

• We will examine the effects on risk as we change
  X1 under four possible correlations between the
  return on asset 1 and the return on asset 2 (1,
  -1, 0, 0.5).

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Expected Return and Risk

• We now have four equations to examine the two
  asset portfolio case.
• The first three give a relationship in risk and
  return.

• The fourth equation allows us to work out the
  portfolio on the minimum variance frontier that
  has the minimum variance of all portfolios.

• Thus we can draw the minimum variance frontier
  for different values of the inputs (the means,
  variances and correlations among returns)

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Example II.2The Minimum Variance Frontier
Correlation Between Returns1

• The risk on a two asset portfolio is

• Equations (1) and (5) imply a linear relationship
  between Rp and ?p.
• We will switch to an Excel file called corpfin.xls

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Example II.3The Minimum Variance Frontier
Correlation Between Returns-1

• The risk on a two asset portfolio is

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Example II.3 (cont.)The Minimum Variance
Frontier Correlation Between Returns-1

• Equations (1) and (6) imply two linear
  relationships between Rp and ?p.
• Note since risk is always positive there is
  always a unique solution.
• Risk is at a minimum using (4) when X1?2/ (?1
  ?2).
• In this case risk is zero!!!
• We will switch to an Excel file called corpfin.xls

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Example II.4The Minimum Variance Frontier
Correlation Between Returns0

• Using (2) The risk on a two asset portfolio is

• Equations (1) and (7) imply a non-linear
  relationship between Rp and ?p.
• We can calculate the X1 that minimizes risk by
  using (4).
• We will switch to an Excel file called corpfin.xls

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Example II.5The Minimum Variance Frontier
Correlation Between Returns0.5

• Equations (1)-(3) imply a non-linear relationship
  between Rp and ?p.
• We can calculate the X1 that minimizes risk by
  using (4).
• In general, if short sales are not allowed, there
  is some value of ?12 such that the risk on a
  portfolio can no longer be reduced below the risk
  on the least risky asset.
• This occurs when ?12?2/?1 (3/60.5 in the
  example used in lectures.)
• In this case only the least risky asset (Asset 2
  in the example) is held.
• Consider extreme cases in one graph.
• The minimum variance frontier is always bounded
  by the triangle.
• We will switch to an Excel file called corpfin.xls

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The Shape of theMinimum Variance Frontier

• When there are many assets the minimum variance
  frontier is still bounded by the triangle.
• Consider the top half of the Minimum Variance
  Frontier.
• We consider three possible shapes (a), (b), and
  (c) below.
• A line connecting two points on the MVF should
  result in more risk for a given expected return.
  Thus the MVF should be a concave curve above the
  minimum variance point. This rules out panels
  (b) and (c).

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The Shape of theMinimum Variance Frontier

• Consider the bottom half of Minimum Variance
  Frontier.
• We consider three possible shapes (a), (b), and
  (c) below.
• The line connecting two points on the MVF should
  result in more risk for a given expected return.
  The MVF should be a convex curve BELOW the
  minimum variance point. This rules out panels
  (b) and (c).
• Thus the shape of the MVF is both panels (a).
  MAIN SLIDE

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The Efficient Frontier

• The Efficient Frontier gives maximum return for a
  given risk.
• It is the top half of the MVF hyperbola (See the
  figure below).
• In terms of return the bottom half of the MVF is
  dominated by the top half.

Return
MVP
Risk
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The Efficient Frontierwhen Short Sales are
Allowed

• Suppose that the efficient frontier is MVP-A in
  the figure below.
• Allowing for short sales extends the efficient
  frontier infinitely beyond A.

Return
A
MAIN SLIDE
MVP
Risk

• We will switch to an Excel file called corpfin.xls

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The Efficient Frontier with Riskless Lending and
Borrowing

• What is the shape of the efficient frontier if an
  investor can lend or borrow at a risk free rate
  of interest, RF?
• We can lend all our investment funds and receive
  an expected return RF at no risk.
• Or we can invest all our investment funds in a
  risky portfolio, A, and receive an expected
  return E(RA) with risk ?A.
• Or we can invest some fraction, X, of our
  investment funds in A and receive expected return
  and risk of

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The Transformation Line

• We derive a relationship between E(Rp) and ?p by
  solving for X in (9) and substituting into (8) to
  get

• This is a straight line in (E(R p), ?p) space and
  is referred to as the transformation line and
  is illustrated in figure on slide 19.
• The slope of this line is called the Sharpe
  Ratio.

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Example II.6 The Sharpe Ratio and Transformation
Line

• Suppose that E(RA)8, ?A4 and RF 5 then the
  
• Sharpe Ratio (8-5)/4 0.75
• And the transformation line is given by
• E(RP) 5 0.75?P

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The Transformation Line
Return
back to slide 17
A
RF
Risk

• At RF we have X0.
• Between RF and A an investor lends and 0ltXlt1.
• At A we have X1.
• After A then an investors borrows and Xgt1 (1-Xlt0).

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The Efficient Frontier with Riskless Lending and
Borrowing
Return
B
A
RF
Risk

• The efficient frontier is obtained by finding the
  tangency point between the transformation line
  and the minimum variance frontier when only risky
  assets are included (on the upper half of the MVF
  hyperbola).
• This occurs at point B, The Optimum Portfolio,
  which dominates portfolio A.
• Along this line an investor holds some percentage
  of funds in portfolio B which can be comprised on
  many risky assets.

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The Efficient Frontier with Riskless Lending and
Borrowing

• Investors with the same beliefs about expected
  returns, risks and correlations will all hold B
  in the figure on the last slide.
• All the investor has to do is choose according to
  preferences along RFB and beyond.
• The ability to determine the optimum portfolio of
  risky assets without having to know anything
  about investors preferences is called the
• Two Fund Separation Theorem.
• If short sales are allowed some of the Xi that
  make up portfolio B may be negative.
• Later in this course (Topic IV) it will be
  demonstrated how an investor can solve for
  optimal Xi mathematically.

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Drawing the Minimum Variance Frontier
Return
E(Rhigh)
MVP
E(Rmvp)
E(Rlow)
Risk

• The minimum variance frontier will be a hyperbola
  (in general).
• The tangency from E(Rmvp) to the MVF is at
  infinity.
• The tangency from E(Rlow) is correct. MAIN SLIDE

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The Efficient Frontierwith no Riskless Borrowing

• The efficient frontier is obtained by finding the
  tangency point between the transformation line
  and the minimum variance portfolio at point A.
• However the transformation line ends at A.
• The rest of the efficient frontier is the the
  minimum variance frontier AB.
• If an investor wants to choose somewhere between
  AB, an investor invests 100 of funds in risky
  assets. MAIN SLIDE

Return
B
A
MVP
RF
Risk
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The Efficient Frontier with Different Borrowing
and Lending Rates

• The efficient frontier is RLABC.
• If an investor wants to choose somewhere between
  RLA, an investor LENDS and invests LESS THAN 100
  of funds in risky portfolio A.
• If an investor wants to choose somewhere between
  AB, a 100 of funds is invested in risky assets.
• If an investor wants to choose somewhere between
  BC, an investor BORROWS and invests MORE THAN
  100 of funds in risky assets. MAIN SLIDE

C
Return
B
A
RB
RL
Risk
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Maximizing Expected Utility

• The material discussed in the course up to this
  point have been concerned with calculating the
  efficient frontier.
• Since all points on the efficient frontier
  represent the maximum expected return give a
  level of risk how does an investor choose among
  them?
• One assumption is that investors care about
  wealth, W, and that this can be mathematically
  represented by a utility function U(W).

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Maximizing Expected Utility (cont.)

• We make standard assumptions about U(W)
• - More is preferred to less (marginal utility is
  positive ?U/?Wgt0)
• - Investors are risk averse (diminishing
  marginal utility ?2U/?W2lt0)
• We assume that since the future is uncertain
  investors choose a point on the efficient
  frontier so as to maximize the Expected Utility
  of future wealth.
• Before considering expected utility we will
  review what we mean by an expected value of a
  random variable.

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Example II.7Expected Returns

• The expected returns is given by ?PiRi where Pi
  is the probability of return Ri.
• The expected returns from Investments A and B are
• 1/3(15) 1/3(10) 1/3(5) 10 and 1/3(20)
  1/3(12) 1/3(4) 12 respectively.

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Expected Utility

• Expected utility from N outcomes is calculated as
  follows

• where P is the probability of an outcome.
• Suppose that the utility function is quadratic
  (more on its properties later)

• Now consider the data in the Table on the
  following slide.

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Example II.8Expected Utility

• The expected utilities of investments A, B and C
  are 36.3, 27.0, 34.4 respectively.
• Example of calculating utility U(WB) U(19)
  419 - 0.1(19)2 39.9
• Example of calculating utility E(U(WB))
  39.91/5 302/5 17.52/5 27.0

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Risk Aversion

• Consider a fair gamble (investment opportunity)
  where there is a 50/50 chance of winning either
  2 or nothing. It costs 1.
• The alternative is to keep the certain 1 and do
  not take the gamble.
• A definition of risk aversion is an investor who
  would not accept a fair gamble.
• This means that U(1) gt 0.5U(2) 0.5U(0)
• or U(1) - U(0) gt U(2) - U(1) implies ?2U/?W2lt0
• The utility from the additional unit increase in
  wealth is less valuable than the last unit
  increase for a risk averse investor.
• Replace the inequality signs with an equality
  sign for a risk neutral investor.
• Reverse the inequality signs for a risk loving
  investor.

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Utility Functions

• Utility function of a risk loving investor is 1.
• Utility function of a risk neutral investor is 2.
• Utility function of a risk averse investor is 3.

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Utility Functions (cont.)

• Utility function of a risk loving investor is 1.
• Utility function of a risk neutral investor is 2.
• Utility function of a risk averse investor is 3.

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Measures of Risk AversionAbsolute Risk Aversion

• One measure of risk aversion is absolute risk
  aversion.
• Algebraically it is defined as

• If an investor decreases the amount invested as
  wealth increases, the investor is said to show
  increasing absolute risk aversion, ARA(W)gt0.
• If an investor held the same amount invested as
  wealth increases, the investor is said to show
  constant absolute risk aversion , ARA(W)0.
• If an investor increases the amount invested as
  wealth increases, the investor is said to show
  decreasing absolute risk aversion , ARA(W)lt0.

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Example II.9Absolute Risk Aversion

• Below we calculate ARA(W) for a commonly assumed
  utility function, namely, log-utility
  (U(W)ln(W))

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Example II.10Absolute Risk Aversion

• Below we calculate ARA(W) for another commonly
  used utility function, namely, quadratic-utility
  (U(W)aW-bW2, a,bgt0)

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Measures of Risk AversionRelative Risk Aversion

• One measure of risk aversion is relative risk
  aversion.
• Algebraically it is defined as

• If an investor decreases the percentage invested
  as wealth increases, the investor is said to show
  increasing relative risk aversion, RRA(W)gt0.
• If an investor held the same percentage invested
  as wealth increases, the investor is said to show
  constant relative risk aversion , RRA(W)0.
• If an investor increases the percentage invested
  as wealth increases, the investor is said to show
  decreasing relative risk aversion , RRA(W)lt0.

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Example II.11 Relative Risk Aversion

• Below we calculate RRA(W) for log-utility
  (U(W)ln(W))

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Example II.11 Relative Risk Aversion

• Below we calculate RRA(W) for quadratic-utility
  (U(W)aW-bW2, a,bgt0)

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Mean-variance Analysisand Quadratic Utility
Functions

• One bad point about assuming quadratic utility is
  that it produces increasing absolute risk
  aversion which is unlikely.
• One good point about assuming quadratic utility
  is that it leads to mean-variance analysis to be
  optimum in expected utility terms (see below).
• The expected value of a quadratic-utility
  function is

• which is in terms of mean and variance. MAIN
  SLIDE