Chapter 6 Portfolio Selection

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Chapter 6 Portfolio Selection

• Business 3059
• Problem Solutions

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NOTE

• Problems 1 through 10 are based on the initial
  given information that
• You manage a risky portfolio
• The risky portfolio has an expected return of 18
  percent and a standard deviation of 28 percent
• The T-bill rate (riskless rate of return
  available) is 8percent.

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Problem 6 - 1
The expected return on the clients portfolio is
the simple weighted average of the two returns as
illustrated above.
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Problem 6 1
The standard deviation on the clients portfolio
is equal to 70 of the standard deviation of the
risky portfolio, assuming that there is no risk
associated with the T-bill investment. Standard
deviation is therefore equal to .7 28 19.6
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Problem 6 2
Investment proportions 30.0 is invested in
T-bills Therefore, 70 is invested in stocks A, B
and C in their relative proportions. Therefore
their weights in the overall portfolio are as
follows .7 27 18.9 in stock
A .7 33 23.1 in stock B .7 40
28.0 in stock C In summary the weights of
each investment in the portfolio are as follows
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Problem 6 3 Reward to Variability Ratio
The reward is measured as the excess returns
expected on the risky portfolio divided by the
standard deviation of that portfolio. This is
known as the Sharpe Ratio. Excess returns are
equal to the expected return minus the risk-free
rate (T-bill rate).
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Problem 6 4 CAL
The slope of the CAL is equal to the Sharpe ratio
(reward-to-variability) ratio.
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Problem 6 5 CAL
The slope of the funds CAL is equal to the funds
Sharpe ratio (reward-to-variability) ratio. The
clients position is dependent upon what
proportion is invested in the fund and what
proportion is invested in T-bills.
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Problem 6 5 Investment Proportions subject to
return objective
Your client wants a 16 return on her portfolio.
We want to find the relative proportions that
must be invested in T-bills and the risky
portfolio that will achieve that objective. Let
wyweight invested in the risky portfolio
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Problem 6 5 Investment Proportions subject
to return objective
Therefore, to get an expected return of 16 on
the overall portfolio, the client must invest 80
of total funds in the risky portfolio and 20 in
T-bills. The relative proportions
invested in each stock is therefore(see next
slide)
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Problem 6 5 Investment Proportions subject
to return objective
Therefore, to get an expected return of 16 on
the overall portfolio, the client must invest 80
of total funds in the risky portfolio and 20 in
T-bills. The relative proportions
invested in each stock is therefore(see next
slide)
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Problem 6 5 Investment Proportions subject
to return objective
Investment proportions 20.0 is invested in
T-bills Therefore, 80 is invested in stocks A, B
and C in their relative proportions. Therefore
their weights in the overall portfolio are as
follows .8 27 21.6 in stock
A .8 33 26.4 in stock B 8 40
32.0 in stock C
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Problem 6 5 Investment Proportions subject
to return objective
The standard deviation on the clients portfolio
is equal to 80 of the standard deviation of the
risky portfolio, assuming that there is no risk
associated with the T-bill investment. Standard
deviation is therefore equal to .8 28 22.4
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Problem 6 6 Location on CAL subject to risk
constraint

• If your client wants a standard deviation of at
  most 18, then the proportion that must be
  invested in the risky portfolio is equal to
  18/28 .629 64.29
• The expected rate of return on the resulting
  portfolio is

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Problem 6 7 Investors Risk Aversion drives
the allocation decision
Therefore the investor will invest 36.44 in the
risky portfolio and 63.56 in T-bills.
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Problem 6 7 Investors Risk Aversion drives
the allocation decision
Expected return on the optimized portfolio where
36.44 is invested in the risky portfolio and
63.56 is invested in T-bills is The
standard deviation of the optimized portfolio is
.3644 2810.20
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Problem 6 8 CAL and CML compared
In question 7 you were told that the passive
portfolio (TSE 300 stock index) yields an
expected rate of return of 13 percent with a
standard deviation of 25 percent. Given the rf
8 percent, this determines the slope of the CML.
The slope of the CML (13 8)/25 .2
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Problem 6 8 CAL and CML compared .
The risky fund allows an investor to achieve a
higher mean for any given standard deviation than
would a passive strategy. Combining the risky
portfolio with the riskfree investment results in
a set of superior portfolio combinations over the
whole range of possible combinations.
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Problem 6 - 10

• The formula for the optimal proportion to invest
  in the passive portfolio is
• The answer is the same as in 9b.
• The fee that you can charge a client is the same
  regardless of the asset allocation mix of your
  clients portfolio. You can charge a fee that
  will equalize the reward-to-volatility ratio of
  your portfolio with that of your competition.

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Problem 6 - 11

• If the riskfree rate is 5, but the borrowing
  rate is 9, then the CML and indifference curves
  are as follows

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Problem 6 - 12

• For y to be less than 1.0 (so that the investor
  is a lender), risk aversion must be large enough
  that

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Problem 6 12

• For y to be greater than 1.0 (so the investor is
  a borrower), risk aversion must be small enough
  that

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Problem 6 12

• For values of risk aversion within this range,
  the investor neither borrows nor lends, but
  instead holds a complete portfolio comprised only
  of the optimal risky portfolio.

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Problem 6 - 13

• The graph of problem 11 is redrawn with E(r)
  11 and s15

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Problem 6 13

• For a lending position,
• Agt (11-5)/(.01225)2.67
• For a borrowing position,
• Alt(11-9)/(.01225) .89
• In between,
• Y1 for .89 lt A gt 2.67

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Problem 6 - 16

• Assuming no change in tastes, that is, an
  unchanged risk aversion coefficient, A, the
  denominator of the equation for the optimal
  investment in the risky portfolio will be higher.
  
• The proportion invested in the risky portfolio
  will depend on the relative change in the
  expected risk premium (the numerator) compared to
  the change in the perceived market risk.
  Investors perceiving higher risk will demand a
  higher risk premium to hold the same portfolio
  they held before.
• If we assume that the risk-free rate is
  unaffected, the increase in the risk premium
  would require a higher expected rate of return in
  the equity market.

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Problem 6 - 17
Standard deviation of the clients overall
portfolio .6 14 8.4 So the correct answer
is C.
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Problem 6 - 18
Reward-to-variability ratio for the equity fund
is Risk premium / Standard deviation 10/14
0.71 Correct answer is A.
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Problem 6 - 19
.6 50,000 .4 (-30,000) 5,000 13,000
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Problem 6 - 22
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Problem 6 - 23
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Problem 6 - 23

• The foregoing graph approximates the points

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Problem 6 - 25

• The reward-to-variability ratio of the optimal
  CAL is

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Problem 6 - 27

• Using only the stock and bond funds to achieve a
  portfolio mean of 14 we must find the
  appropriate proportion in the stock fund (wS),
  and therefore wB1 wS in the bond fund.
  Solving for the weight of stocks

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Problem 6 27

• Since the proportions will be 25 stock and 75
  bonds, the standard deviation of the portfolio
  will be

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Problem 6 - 28
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Problem 6 - 28
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Problem 6 31

• False. If the borrowing and lending rates are
  not identical, then depending on the tastes of
  the individuals (that is, the shape of their
  indifference curves), borrowers and lenders could
  have different optimal risky portfolios.

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Problem 6 32

• False.
• The portfolio standard deviation equals the
  weighted average of the component-asset standard
  deviations only in the special case that all
  assets are perfectly positively correlated.
  Otherwise, as the formula for portfolio standard
  deviation shows, the portfolio standard deviation
  is less than the weighted average of the
  component-asset standard deviations. The
  portfolio variance will be a weighted sum of the
  elements in the covariance matrix, with the
  products of the portfolio proportions as weights.

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Problem 6 36

• Risk reduction benefits from diversification are
  not a linear function of the number of issues in
  the portfolio. Rather, the incremental benefits
  from additional diversification are most
  important when you are least diversified.
  Restricting Hennesey to 10 instead of 20 issues
  would increase the risk of his portfolio by a
  greater amount than would reducing the size of
  the portfolio from 30 to 20 stocks. In our
  example, restricting the number of stocks to 10
  will increase the standard deviation to 23.81.
  The increase in standard deviation of 1.76 from
  giving up 10 of 20 stocks is greater than the
  increase of 1.14 from giving up 30 stocks when
  starting with 50.

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Problem 6 37

• The point is well taken because the committee
  should be concerned with the volatility of the
  entire portfolio. Since Hennessey's portfolio is
  only one of six well-diversified portfolios and
  smaller than the average, the concentration in
  fewer issues could have a minimal effect on the
  diversification of the total fund. Hence,
  unleashing Hennessey to do stock picking may be
  advantageous.

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Problem 6 38

• Since all stocks have the same expected return
  and standard deviation, we know we want to choose
  to add a stock that will result in a portfolio
  with the lowest riskand that would be a stock
  with the lowest correlation with stock A.
• This would be stock D. (Corr(A,D) .45)

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Problem 6 39

• No, at least as long as they are not risk lovers.
  Risk neutral investors will not care which
  portfolio they hold since all portfolios yield
  8.

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Problem 6 40

• No change. The efficient frontier of risky
  assets is horizontal at 8, so the optimal CAL
  runs from the risk-free rate through G. The best
  G is here, again, the one with the lowest
  variance. The optimal complete portfolio will,
  as usual, depend on risk aversion.

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Problem 6 41

• d. Portfolio Y cannot be efficient because it is
  dominated by another portfolio. For example,
  Portfolio X has higher expected return and lower
  standard deviation

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Problem 6 42

• C.
• This is the Evans and Archer study conclusions
  and is universally accepted.
• Today, however, research seems to indicate that
  it may take more investments to achieve full
  diversification of company specific risk than in
  the pastbecause of increasing integration of
  financial markets.

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Problem 6 43

• C.
• Since all stocks have the same expected return
  and standard deviation, we must focus on risk
  reduction in the resultant portfolio.
• Combining B and C together allows you to take
  full advantage of the negatively correlated
  returns (-0.4) between the two.

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Problem 6 44

• We know nothing about expected returns of each
  stock, so we focus exclusively on reducing
  variability.
• Stocks C and A have equal standard deviations,
  but the correlation of stock C with stock B (.1)
  is lower than that of stock A with stock B (.9),
  so a portfolio comprised of C and B will have
  lower total risk than a portfolio comprised of A
  and B.

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Problem 6 45

• We must rearrange the table converting rows to
  columns and the compute the serial correlation
  (use the built-in correlation function in Excel)

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(No Transcript)
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Serial Correlation illustrated

• To compute serial correlation in decade nominal
  returns for large company stocks we set up the
  following two columns and use the correlation
  function of the spread sheet

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Note
Each correlation is based on only seven
observations, so we cannot determine the
statistical significance of the correlation
results (too few observations). Nevertheless, if
you examine the numbers, it does appear there is
a persistent serial correlation in the asset
class sets with perhaps the exception of
large-company stocks.
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Problem 6 - 46
The table for real rates (using the approximation
of subtracting a decades average inflation from
the decades average nominal return) is
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Problem 6 - 46
The positive serial correlation in decade nominal
returns has disappeared and it now appears that
real rates are serially correlated. The decade
time series suggests that real rates of return
are independent from decade to decade.
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Problem 6 - 47

• True
• It will be a weighted average, with the same
  weights as those of the securities in the
  portfolio.

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Problem 6 - 48

• It is equal to (12 10 6) / 3 9.33 (a
  simple arithmetic average since the portfolio is
  equally split, meaning that 33.3 of the
  portfolio is invested in each asset)

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Problem 6 - 49

• Total risk seems implied by the lack of
  specificity in the type of riskas measured by
  the standard deviation or variance of returns.
• Specific risk is a term that holds no meaning in
  the field of finance.

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Problem 6 - 50

• Systematic risk
• By definition, risk that cannot be removed
  through diversification is risk that is common to
  all investmentsand the system or overall
  macroeconomic risks impinge on all securities to
  one extent or another, depending on the stock in
  question.