Investment Management Tutorial

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Investment Management Tutorial

• October 10, 2008
• James Kozyra

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Chapter 5 Problem 16

• You are faced with the probability distribution
  on the stock market index fund given below.
  Suppose the price of a put option on a share of
  the index fund with an exercise price of 110 and
  maturity of one year is 12. The current share
  price is 100 and a cash dividend of 4 per share
  is expected to be paid during the year.

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Chapter 5 Problem 16

• What is the probability distribution of the HPR
  on the put option

Stock
Put Option
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Chapter 5 Problem 16

• What is the probability distribution of the HPR
  on a portfolio consisting of one share of the
  index fund and a put option?
• The cost of one share and a put is 112 (110
  12).

HPR Boom (140 - 112 4) / 112 28.6 HPR
Normal (110 - 112 4) / 112 1.8 HPR
Recession (110 - 112 4) / 112 1.8
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Chapter 5 Problem 16
C) In what sense does buying the put option
constitute a purchase of insurance in this
case? A minimum HPR of 1.8 is guaranteed
regardless of what happens to the stock price.
You are thus protected against a price decline.
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Chapter 8 Problem 14

• Two investment advisors are comparing
  performance. One averaged a 19 rate of return
  and the other a 16 rate of return. However, the
  beta of the first investor was 1.5, whereas the
  that of the second was 1.
• a) Can you tell which investor was a better
  predictor of individual stocks (aside from the
  issue of general movements in the market)?

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Chapter 8 Problem 14

• a) To determine which investor was the better
  predictor we look at their abnormal return, which
  is the ex-post alpha. This means that the
  abnormal return is the difference between the
  actual return and the return predicted by the
  SML.
• Without the parameters of the equation
  (risk-free rate and market rate of return) we
  cannot determine which investor was more accurate.

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Chapter 8 Problem 14

• b) If the t-bill rate were 6 and the market
  return during the period were 14, which investor
  would be the superior stock selector?
• Investor 1 19 6 1.5(14-6)
• 19 18 1
• Investor 2 16 6 1(14-6)
• 16 14 2

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Chapter 8 Problem 14

• c) What if the t-bill rate were 3 and the
  market return were 15?
• Investor 1 19 3 1.5(15-3)
• 19 21 -2
• Investor 2 16 3 1(15-3)
• 16 15 1

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Chapter 8 Problem 14

• First case the second investor has the larger
  abnormal return and appears to be the superior
  stock selector. He appears to have done a better
  job finding underpriced stocks.
• Second case the second investor again is the
  superior stock selector. The first investors
  predictions appear to be worthless.

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Chapter 6 Problem 21

• You manage a risky portfolio with an expected
  rate of return of 18 and a standard deviation of
  28. The t-bill rate is 8. A passive portfolio
  has an expected rate of return of 13 and a
  standard deviation of 25.
• Your client ponders whether to switch the 70
  that he has invested in your risky portfolio to
  the passive portfolio.

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Chapter 6 Problem 21

• a) Explain to your client the disadvantages of
  the shift.
• Current Portfolio
• E(r) (.3 x 8) (.7 x 18) 15
• Std Dev .7 x 28 19.6
• Portfolio after the shift
• E(r) (.3 x 8) (.7 x 13) 11.5
• Std Dev .7 x 25 17.5

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Chapter 6 Problem 21

• a) Therefore, the shift entails a decline in the
  expected return from 14 to 11.5 and a decline
  in the standard deviation from 19.6 to 17.5.
• The disadvantage is that the client could
  achieve an 11.5 expected return in my portfolio,
  with a lower standard deviation.

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Chapter 6 Problem 21

• We first must write the mean of the complete
  portfolio as a function of the proportions
  invested in my portfolio, y
• E(r) 8 y(18-8)
• E(r) 8 10y
• Given the target 11.5 return, the proportion
  that must be invested in the fund is determined
  as follows
• 11.5 8 10y

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Chapter 6 Problem 21

• 11.5 8 10y
• 10y 11.5 8
• 10y (11.5 8) / 10
• y 0.35
• The standard deviation of the portfolio would
  thus be 9.8 (.35 x 28). Achieving the 11.5
  return can be done with a standard deviation of
  9.8 in my portfolio as opposed to 17.5 in the
  passive portfolio.

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Chapter 6 Problem 21

• b) Show your client the maximum fee you could
  charge that would still leave him/her at least as
  well off investing in your fund. (Hint the fee
  will lower the slope of the CAL by reducing E(r)
  net of the fee.
• The fee would reduce the reward-to-variability
  ratio (CAL slope). Clients will be indifferent
  if the slope of the after-fee CAL and the CML are
  equal.

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Chapter 6 Problem 21

• Let f denote the fee
• Slope of the CAL with the fee
• (18 8 f) / 28
• (10 f) / 28
• Slope of the CML (no fee required)
• (13 8) / 25 .20
• We must set the slopes to be equal.

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Chapter 6 Problem 21

• (10 f) / 28 0.20
• (10 f) 28 x 0.20
• (10 f) 5.6
• f 10 5.6
• f 4.4
• Therefore the maximum fee that can be charged to
  make the client indifferent between portfolios is
  4.4 per year.

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Portfolio Selection Problem

• You are considering investing 1,000 in a
  T-bill that pays 0.05 and a risky portfolio, P,
  constructed with two risky securities, X and Y.
  The weights of X and Y in P are 0.6 and 0.4,
  respectively. X has an expected rate of return of
  0.14 and variance of 0.01, and Y has an expected
  rate of return of 0.10 and a variance of 0.0081.

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Portfolio Selection Problem

• If you want to form a portfolio with an
  expected rate of return of 10, what percentages
  of your money must you invest in the t-bill, X,
  and Y respectively if you keep X and Y in the
  same proportions to each other as in portfolio P
  (6040).
• E(r) Portfolio (0.6 x 14) (0.4 x 10)
• E(r) T-bill w x 5

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Portfolio Selection Problem

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Portfolio Selection Problem
Weighting by Asset T-Bill 32.4 X (67.6 x
.6) 40.6 Y (67.6 x .4) 27
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Portfolio Selection Problem

• What would be the dollar value of your
  position in X, Y, and the t-bills, respectively
  if you decide to hold a portfolio that has an
  expected outcome of 1,120
• HPR (1,120 - 1,000) / 1,000
• HPR 12

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Portfolio Selection Problem

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Portfolio Selection Problem
T-Bill 5.4 X (94.6 x .6) 56.8 Y
(94.6 x .4) 37.8
T-Bill (5.4 x 1,000) 54 X (56.8 x
1,000) 568 Y (37.8 x 1,000) 378