MeanVariance Portfolio Theory

1
Mean-Variance Portfolio Theory
2

• This chapter deals with a single investment
  period, e.g.,
• - a zero-coupon bond held to maturity
• - investment in a physical project providing
  payment upon completion.
• Publicly traded stocks are not tied to a single
  period. Still, such investments are often
  analyzed on a single period basis as a
  simplification. Parts 3 and 4 of the text relax
  the single-period assumption.

3

• Implications
• R 1 r (1.20 1 0.20)
• X1 (1r) X0 (12,000 (1.20) 10,000)

4

• (Idealized) Short Sale
• CBA stock now sells for 10/share.
• I borrow 100 shares from my broker (with a 1-year
  stock repayment obligation).
• I sell these shares in the stock market for
  1,000.
• One year later, CBA stock sells for 9/share. I
  buy 100 shares for 900. I give these shares to
  my broker in repayment.
• Because the stock fell in price I make a profit
  of 100.

5

• In the short sale example
• I receive 1,000 initially, my outlay is X0
  -1,000. (A receipt is a negative outlay.)
• One year later
• I pay 900, so I receive X1 -900. (A payment
  is a negative receipt.)
• This means
• R X1/X0 -900/-1000 0.90 total return
• r R 1 -0.10 rate of return

6

• My profit (shorting converts a negative return to
  a profit) is
• r ? X0 (-0.10) ? (-1,000) 100.
• Note. Someone who purchased the CBA stock at the
  first of the year and sold it at the end would
  lose (1,000 - 900) 100. That person would
  have
• R 900/1000 0.90,
• r (900 1,000)/(1,000) - 0.10.
• The rate of return is negative for this person.

7

• It is a bit strange to refer to a rate of return
  associated with the idealized shorting procedure,
  since there is no initial commitment of
  resources. Nevertheless, it is the proper
  notion.
• Summary of (Idealized) Short Selling or Shorting
• You borrow an asset from someone who owns it.
• You sell the borrowed asset to someone, receiving
  an amount X0.
• Later you buy the asset for X1 and use it to
  repay your loan.
• IF X1 lt X0 you make a profit of X0 X1.
• Short selling is profitable if the asset price
  declines.

8

• Observations
• Short selling is risky even dangerous. If X1 gt
  X0 you lose money. There may be no limit on how
  large X1 can be.
• Some financial institutions prohibit short
  selling.
• Many individuals and institutions purposely avoid
  short selling as a policy.
• There is a considerable level of short selling of
  stock market securities.
• Short selling is supplemented by certain
  restrictions and safeguards in practice, e.g.,
  you post a security deposit with the broker when
  you borrow the asset.

9

• Observations (Contd)
• When you short sell, you essentially duplicate
  the role of the issuing corporation. You sell
  the stock to raise immediate capital. If the
  stock pays dividends during the period you
  borrowed it, you too must pay the same dividend
  to the broker you borrowed the stock from.
• Downtick short sell rule on NYSE prevents short
  selling of stocks with declining price

10

• Short Selling Return Analysis
• We receive X0 initially and pay X1 later. Our
  outlay is -X0 and our final receipt is -X1.
• Total return R -X1/-X0 X1/X0. (minus signs
  cancel out)
• The return value R applies algebraically to both
  purchases and to short sales.
• Note -X1 -X0 R -X0 (1 r).

11

• Portfolio Return

Initial Investment Information.
Security
Shares
Price/share
Total
Weight in
Cost
portfolio
Bach, Inc.
100
40
4,000
0.25
Brahms,
400
20
8,000
0.50
Inc.
Mozart, Inc.
200
20
4,000
0.25
Totals
16,000
1.00
12
(No Transcript)
13

• Motivation
• X0 16,000 total initial investment
• X1 4,680 9,040 4,920
• 1.17 (4,000) 1.13 (8,000) 1.23 (4,000)
• 1.17 (1/4) (16,000) 1.13 (1/2) (16,000)
• 1.23 (1/4) (16,000)
• R1 w1 X0 R2 w2 X0 R3 w3 X0

14

• Thus, for total return,
• R X1/X0 R1 w1 R2 w2 R3 w3
• w1 R1 w2 R2 w3 R3.
• For rate of return,
• r R 1 (w1 w2 w3) ?( R 1)
• w1(R1 1) w2(R2 1) w3(R3 -1)
• w1 r1 w2 r2 w3 r3
• Conclusions
• R w1 R1 w2 R2 w3 R3
• r w1 r1 w2 r2 w3 r3

15

• Generalization
• A master asset, or portfolio, has n assets.
• X0 total initial value of portfolio
• X0i initial value of asset i in portfolio
• X0 X01 X02 ? X0n
• (X0i lt 0 if we short sell asset i)
• wi X0i/X0 is the weight/fraction of asset i in
  portfolio
• (wi lt 0 if X0i lt 0 and X0 gt 0)

16

• Ri total return of asset i at end of period ?
• Asset i generates an income of Ri X0i Ri wi X0
  ?
• X1 R1 X01 R2 X02 ? Rn X0n
• R1 w1 X0 R2 w2 X0 ? Rn wn X0
• Thus
• R X1/X0 w1 R1 w2 R2 ? wn Rn
• r R 1 (w1 w2 ? wn) ? (R 1)
• w1(R1 1) w2(R2 1) ? wn(Rn-1)
• w1 r1 w2 r2 ? wn rn

17

• Portfolio Return Summary
• Both the total return R and the rate of return r
  of a portfolio of assets are equal to the
  weighted sum of the corresponding asset returns.
  The weight of each asset is its relative weight
  (in purchase cost) in the portfolio. That is,
• R w1 R1 w2 R2 ? wn Rn
• r w1 r1 w2 r2 ? wn rn

18

• Random Variables (a short review of basics)
• Expectations
• discrete r.v.
• Ex p1 x1 p2 x2 ? pn xn
• continuous r.v.
• ?
• Ex ? x p(x) dx
• - ?

19

• Variance
• Varx E(x - Ex)2
• Standard Deviation
• ?x ? Varx
• Note
• Varx E(x - Ex)2 Ex2 2 x Ex
  Ex2
• Ex2 2 Ex Ex Ex Ex
• Ex2 (Ex)2

20

• Covariance
• covx1,x2 ? E(x1 - Ex1) (x2 - Ex2)
• Note covx,x Varx .
• Common notation
• ?x1,x2 covx1,x2, ?1,2 covx1,x2.
• Observation (Useful for computations)
• covx1,x2 Ex1 x2 - Ex1 Ex2

21

• Random variables x1 and x2 are
• - uncorrelated if ?1,2 0
• - positively correlated if ?1,2 gt 0
• - negatively correlated if ?1,2 lt 0

22
(No Transcript)
23

• Covariance Bound. For any r.v.s x1 and x2,
• ?1,2 ?1 ?2 ? - ?1 ?2 ?1,2 ?1 ?2.
• x1 and x2 are perfectly correlated if ?1,2 ?1
  ?2
• x1 and x2 show perfect negative correlation if
• ?1,2 - ?1 ?2.

24

• Correlation Coefficient.
• For any r.v.s x1 and x2
• ?1,2 ? ?1,2/?1 ?2
• Note
• ?1,2 1 ? -1 ?1,2 1

25

• Thus
• x1 and x2 are perfectly correlated if ?1,2 1
• x1 and x2 show perfect negative correlation
• if ?1,2 - 1.
• Variance of a Sum of RVs. For any r.v.s x1
• and x2,
• Varx1 x2 ?12 2 ?1,2 ?22

26
Random Returns

• A return r can be random. We represent its mean
  and standard deviation by Er and ?
  respectively.
• Luenberger shows a Wheel of Fortune.

27

• The wheel shows the payoff you get when you bet
  1.00. Since each payoff is equally likely, if Q
  is the r.v. denoting the payoff,
• EQ (1/6) (4 1 2 1 3 0) 7/6.
• EQ2 (1/6) (16 1 4 1 9 0) 31/6.
• VarQ 31/6 (7/6)2 ? 3.81.

28

• The total return, R, satisfies R Q/1 Q. Thus
• ER EQ 7/6.
• Since the rate of return r satisfies r R 1,
• Er ER 1 ER 1 1/6.
• Varr VarQ 1 VarQ ? 3.81.
• ? ?3.81 ? 1.95, which is large compared to the
  mean.

29

• You bet on one of the three segments of the wheel
  (like roulette).

30

• Bet 1. If you bet 1.00 on the 3.00 segment and
  the arrow points to that segment you win 3.00
  otherwise you lose 1.00.
• Bet 2. If you bet 1.00 on the 2.00 segment and
  the arrow points at that segment you win 2.00
  otherwise you lose 1.00.
• Bet 3. If you bet 1.00 on the 6.00 segment and
  the arrow points at that segment you win 6.00
  otherwise you lose 1.00.

31

• Is there a clearly worst bet?

32

• Recall
• covx1,x2 Ex1 x2 - Ex1 Ex2
• Note payments on bets 1, 2, and 3 are mutually
  exclusive. This means
• R1 R2 0, R1 R3 0, R2 R3 0.
• Thus
• ?1,2 - ER1 ER2 -(3/2) (2/3) - 1.0
• ?1,3 - ER1 ER3 -(3/2) (1) - 1.5
• ?2,3 - ER2 ER3 -(2/3) (1) - 2/3.
• All the bets are negatively correlated with each
  other.

33

• Mean-Standard Deviation Diagram

34

• Which bet is riskiest (safest)?
• Which bet has the highest return?
• Do you see a tradeoff between choosing one bet
  and another?
• Why do we plot the standard deviation not the
  variance?

35
Portfolio Mean and Variance

• A portfolio has n assets with (random) returns
  r1, r2, ..., rn. Their expected returns are
  Er1,Er2, ...,Ern.
• If the weights of the assets in the portfolio are
  w1, w2, ..., wn respectively, we know the
  portfolio return is
• r w1 r1 w2 r2 ? wn rn
• Note the weights are amounts we can choose, while
  the returns are not. Thus
• Er w1 Er1 w2 Er2 ? wn Ern

36

• To get the expected return of the portfolio, take
  the weighted sum of the individual expected rates
  of returns. We only need to concentrate on the
  individual expected rates of returns to compute
  Er.
• r w1 r1 w2 r2 ? wn rn

37

• Variance of Portfolio Return
• ?2 E(r Er)2
• n n
• ? ? wi wj ?i,j (see text, p. 150
  algebra)
• i1 j1

38

• Remark
• If
• M (?i,j) is the n by n variance-covariance
  matrix,
• W (wj) is the n by 1 column vector,
• WT is the 1 by n row vector,
• then
• ?2 WT M W.
• Recall from matrices that WT M W is a quadratic
  form in the vector W. Because this quadratic
  form is equal to ?2, it is always nonnegative,
  even for negative entries in W. A known result
  is that any quadratic form that is always
  nonnegative is a convex function (of W in this
  case).

39

• Example 6.8. (Two-asset portfolio, n 2)
• Er w1 Er1 ? wn Ern
• ?2 E(r Er)2
• n n
• ? ? wi wj ?i,j
• i1 j1

40

• Example 6.8 (Contd)
• Er1 0.12, Er2 0.15, w1 0.25, w2 0.75
• ?1 0.20, ?2 0.18, ?1,2 0.01 ?2,1
• Er w1 Er1 w2 Er2 w1 0.12 w2 0.15
  0.1425
• ?2 w1 w1 ?12 w1 w2 ?1,2 w2 w1 ?2,1 w2 w2
  ?22
• 0.24475.
• Motivating Question. How would you go about
  finding a BEST choice of w1 and w2?

41

• Diversification
• What can happen if you carry all your eggs in one
  basket?
• What can happen if you put all your savings into
  stock options of the company you work for?
• Diversification is like carrying eggs in several
  baskets. For example, you could invest in your
  company stock option, but also in mutual funds
  and bonds.

42

• An advantage of a mutual fund is, in fact,
  diversification.
• Also, you get professional management and lower
  processing charges than in dealing with a stock
  broker. Stock brokers have a strong incentive to
  encourage their customers to trade often, since
  they get income on every transaction mutual
  funds do not have as strong an incentive to do
  so.

43

• Structure of portfolios (stocks from SP100) as a
  function of risk (two weekly CVaR)

44

• Diversification for Uncorrelated Assets.
• Suppose you have n assets, which are mutually
  uncorrelated. Suppose every asset i has
• Eri m, and wi 1/n. Then
• Er w1 Er1 w2 Er2 ? wn Ern ?
• Er (1/n) m (1/n)m ? (1/n) m m.
• Your expected return is independent of n.

45

• If every asset i has Varri ?2, because the
  assets are uncorrelated, then
• Varr w12 Varr1 ? wn2 Varrn
• (1/n2) ?2 ? (1/n2) ?2 ?2/n

46

• By comparison, if covr1,r2 0.3 ?2 for all
  distinct i and j, Luenberger shows
• Varr 0.7 ?2/n 0.3 ?2.
• We can plot ?2/n and 0.7 ?2/n 0.3 ?2 versus n,
  assuming ? 1, to get some idea of the effect
  of diversification for uncorrelated as compared
  to correlated assets.

47

• Diversification helps less for correlated assets
  than for uncorrelated assets.

48
(No Transcript)
49
(No Transcript)
50

• Types of risk
• unique risk the risk that potentially can be
  eliminated by diversification.
• market risk risk intrinsic to the market, that
  cannot be eliminated by diversification.
• Unique risk stems from the fact that many of the
  perils that surround an individual company are
  peculiar to that company and perhaps its
  immediate competitors.
• Market risk stems from the fact that there are
  other economy-wide perils which threaten all
  businesses. That is why stocks have a tendency
  to move together. And, that is why investors are
  exposed to market uncertainties, no matter how
  many stocks they hold.

51

• Example return and risk calculations
• Covariance 1176 (assuming sds in percent)
• If we put 60 of our portfolio into Georgia
  Pacific and the remaining 40 into Thermo
  Electron, our
• expected return 0.6 ? 15 0.4 ? 21 17.4

52

• Example return and risk calculations (Contd)
• Return variance 0.62 ? 282 2 ? 0.6 ? 0.4 ?
  1176 0.42 ? 422 282.24 564.48 282.24
  1128.96
• standard deviation ? 1128.96 33.6
• If in fact the correlation is 470.4, then
• portfolio variance ? 790, sd ? 28.1 .

53

• Limits to Diversification
• Recall, for a portfolio with n assets, its
  variance is
• ?2 E(r Er)2
• n n
• ? ? wi wj ?i,j
• i1 j1
• This formula, with n2 terms, adds n variances and
  n2 n covariances.

54

• Limits to Diversification (Contd)
• Suppose each weight is 1/n. We invest equally in
  each asset. Then each product of any two
  weights is 1/n2. We conclude
• ?2 1/n2 (sum of all the variances)
• 1/n2 (sum of all the covariances)
• Note, since there are n variances,
• sum of all the variances
• n(sum of all variances/n)
• n ? avg. variance
• sum of all the covariances
• n(n-1)sum of all covariances/n(n-1)
• n(n-1) ? avg. covariance.

55

• Limits to Diversification (Contd)
• ?2 1/n2 (sum of all the variances)
• 1/n2 (sum of all the covariances)
• n/n2 ? (avg. variance)
• n(n-1)/n2 ? avg. covariance
• 1/n ? avg. variance
• (1 1/n) ? avg. covariance
• What happens to the above expression as n becomes
  large?

56

• Limits to Diversification (Contd)
• Variance converges to the average covariance for
  the market
• If the average covariance were 0, it would be
  possible to eliminate all risk by holding a
  sufficient number of securities. Unfortunately,
  common stocks move together, not independently.
  Thus, most of the stocks that the investor can
  actually buy are tied together in a web of
  positive covariances which set the limit to the
  benefits of diversification.
• Major Conclusion. The risk of a well-diversified
  portfolio depends on the market risk of the
  securities included in the portfolio.

57

• Diagram of a Portfolio
• Mean-Standard Deviation Diagram
• Example for the 3 bets of the betting wheel.

58

• The ideas illustrated in the betting wheel mean
  standard deviation diagram provide insight into
  portfolio analysis. Think of each bet as an
  asset, with an expected return and a variance.
  The relative amount of each asset in your
  portfolio (the assets weight) will affect both
  the expected return and portfolio variance. We
  wish to analyze these effects.
•  

59

• Suppose you have two assets. You plot the
  standard deviation and expected return of each.

60

• Suppose these two assets are in a portfolio with
  weights w1 and w2, so that
•  Er w1 Er1 w2 Er2
•  Varr w12 ?12 2 w1 w2 ?1,2 w22 ?22
•  
• Recall w1 w2 1. If we let ? w2, then 1 - ?
  w1. As we vary ? we adjust the relative
  amounts of assets 1 and 2 in our portfolio, and
  change both the expected return and the portfolio
  variance (or standard deviation). In particular,
• return(?) Er (1 - ?) Er1 ? Er2
•  ?(?) ?(1 - ?)2 ?12 2 (1 - ?) ? ?1,2 ?2
  ?22

61

• What does the plot of (?(?),return(?)) look like
  as we vary alpha? Since
•  
• return(?) Er (1 - ?) Er1 ? Er2
•  
• ?(?) ?(1 - ?)2 ?12 2 (1 - ?) ? ?1,2 ?2
  ?22
•  
• we conclude
• (?(?),return(?)) A2 if ? 1,
• (?(?),return(?)) A1 if ? 0.

62
(No Transcript)
63

• As ? varies between 0 and 1, (?(?),return(?))
  traces a curve connecting A1 and A2
•  
• Which point on the curve would you choose?
• Are there better points than just A1 or A2?
•  
• As ? varies outside 0,1 the curve extends to
  include the dashed portions as shown, which
  correspond to short selling. The top dashed
  portion represents short selling of A1 the
  bottom dashed portion represents short selling of
  A2.
•  

64
(No Transcript)
65

• The exact shape of the solid curve depends on
  ?1,2. It can be shown that the curve will always
  be contained in a triangle with vertices A1, A2
  and A. The point A is defined by
•  
• A (0, (Er1 ?2 Er2 ?1)/(?1 ?2 ))
•  

66
(No Transcript)
67

• Indeed
• ?(?) ?(1 - ?)2 ?12 2 ?(1 - ?) ? ?1 ?2 ?2
  ?22
• Using ? 1 we find upper bound
• ?(?) ?(1 - ?)2 ?12 2 (1 - ?) ? ?1 ?2 ?2
  ?22
• ?1 - ?) ?1 ? ?22 (1 - ?) ?1 ? ?2
• Using ? -1 we find lower bound
• ?(?) ?(1 - ?)2 ?12 - 2 (1 - ?) ? ?1 ?2 ?2
  ?22
• ?1 - ?) ?1 -? ?22 (1 - ?) ?1 -? ?2
• Upper and lower bounds are linear
• (1 - ?) ?1 -? ?2 turns to zero when ? ?1/
  (?1 ?2 )

68

• Some Figures with Excel follow
• -1 ? 2, Er1 0.1, Er2 0.3, ?1 0.15,
  ?2 0.20
• - 0.02 ?1,2 0.02 ( ?1,2 ?1 ?2
  0.03)
• When ?2 we have in the portfolio two shares of
  the second stock (long position) and one share of
  the first stock (short position).
• Motivating Questions.
•  
• 1.   Which portfolio mix would be least risky?
•  
• 2.   What happens to the portfolio risk if you
  want a high return?
• Shorting improves portfolio performance for the
  negatively correlated stocks.
•  

69
(No Transcript)
70
(No Transcript)
71
(No Transcript)
72
(No Transcript)
73

• The Mean-Standard Deviation Feasible Set
• Suppose we have n basic assets.
•  
• We plot each as points on the mean-sd diagram.
•  
• Imagine forming portfolios from these n assets,
  using every possible choice of asset weights that
  total to 1.
•  
• The set of points we get that correspond to all
  possible such portfolios is called the mean-sd
  feasible region, feasible set, or feasible
  region.

74

• For two assets, e.g.,

75

• Changing weights for 3 assets

76

• Any two assets define a curved line between them
  as combination portfolios are formed. Given
  three points, P1, P2 and P3, by considering
  portfolios of pairs of points we could sweep out
  the lines shown for these three points. If we
  take a combination of points 2 and 3 we can
  produce point 4, as shown. By moving point 4
  (adjusting the relative weights of points 2 and
  3, we can sweep out the entire solid region
  enclosed by points 1, 2 and 3.
•  

77

• Statement 1. If there are at least three assets
  (not perfectly correlated and with different
  means) then the feasible set will be a solid
  two-dimensional region.
•  
• Statement 2. The feasible region is convex to
  the left. That is, given any two points in the
  region, the straight line connecting them does
  not cross the left boundary of the region. This
  follows from the fact that all portfolios (with
  positive weights) made from two assets lie on or
  to the left of the line connecting them (see the
  next figure).

78
(No Transcript)
79

• The feasible region with n assets (no shorting)

80

• The feasible region with n assets (shorting is
  allowed)

81

• Motivating Questions
•  
•  
• For a given Er, which point in the feasible
  region would you choose?
• For a given variance, which point in the feasible
  region would you choose?

82

• The left boundary of a feasible set is called the
  minimum-variance set. For any point in the
  feasible set, there is another point in the
  minimum-variance set with the same risk and
  smaller variance.
•  
• The point in the minimum-variance set of minimal
  variance is called the minimum-variance point
  (MVP).
•  
• An investor who prefers a point in the
  minimum-variance set is called risk-averse.
•  
• An investor who is not risk-averse is called risk
  preferring, or a risk taker.

83
(No Transcript)
84

• Minimum Variance Set
•  
•  

85

• For a given value of ?, we might as well pick the
  point of highest expected return.
•  
• Nonsatiation everything else being equal,
  investors always want more money.
• The top left portion of the minimum variance set
  is called the efficient frontier.
• The efficient frontier identifies the best
  portfolios.
• We can concentrate on only the portfolios in the
  efficient frontier.

86

• Efficient Frontier

87
Markowitz Model

• A portfolio has n assets with (random) returns
  r1, r2, ..., rn. Their expected returns are
• and the expected return of the portfolio equals
• Variance of portfolio return
•   n n
• ?2 E(r Er)2 ? ? wi wj ?i,j
• i1 j1
• The Markowitz model finds portfolios on the
  efficient frontier of the mean-sd feasible set.

88

• Markowitz Model (Contd)
•  
• We seek a portfolio of minimal risk to achieve
  our desired return (factor ½ for convenience).
• Note the basic model allows short selling.

89

• Markowitz model solving using linear equations
  
•  Lagrangian
•  
• Lagrangian, two dimension case

90

• Markowitz model linear equations (contd)
•   Lagrangian derivatives
• Setting derivatives to zero and using
• This can be solved for

91

• Markowitz model linear equations (contd)
•  
•  Equations for efficient set

92

• Quadratic programming (no short positions)
• w1 ,, wn0
• Note that 1/2 multiplier is not included in the
  performace function

n n

å
å
s
minimize



w
w



i
j
i,j





i1 j1


subject to




¼

w


w


w




1
2
n

¼
w
w


w
1

1
2
n



93

• Quadratic programming (Contd)
• This problem is a special case of a Quadratic
  Programming Program (QPP).
• The QPP is one of the best-solved nonlinear
  programming problems. The financial industry has
  many special-purpose programs to solve the
  Markowitz problem for thousands of assets.
• Excel (Tools,Solver) can be used to solve this
  problem.

94

• Quadratic programming (Excel)
• File Mkwz-5x5.xls, Sheet2, Tools gt Solver
• Do not check linear model,
• assume nonnegative, check use automatic
  scaling.

95

• Summary of Some runs
• r1 0.1, r2 0.3, ?1 0.03, ?2 0.06.

96

• Computational Experience Remark. A significant
  difference between the two formulations is that
  when short selling is allowed, most, if not all,
  of the optimal weights have nonzero values
  positive or negative so essentially all assets
  are used. By contrast, when short selling is not
  allowed, typically many weights are equal to zero.

97

• Example 6.10 (Three uncorrelated assets)
• Each variance is 1, mean returns are 1, 2 and 3
  respectively. Covariances are all 0.

98

•  

99

• The Two-Fund Theorem
•  Equations for efficient set
• Suppose that there are two known solutions

100

• The Two-Fund Theorem (Contd)
• Can be directly verified that combination of the
  portfolios is a new solution
• Two efficient funds (portfolios) can be
  established so that any efficient portfolio can
  be duplicated, in terms of mean and variance, as
  a combination of these two. In other words, all
  investors seeking efficient portfolios need only
  invest in combinations of these two funds.

101

• The Two-Fund Theorem (Contd)
• Assumptions
• - everyone cares only about the mean and
  variance
• - everyone has the same assessments of the
  means, variances, and covariances
• - a single period framework is appropriate

102

• Motivating Questions. Suppose the Markowitz
  model, say P, has n assets. We now expand the
  model to get a larger one, say P, with m assets
  (m gt n), by attaching m-n assets to model P.
• What can you say about the relationship of
  optimal solutions to P and P, and why?
• Which model would be harder to solve?
• Which model would do best?

103

• Inclusion of a Risk-Free Asset
• A risk-free asset has a return that is
  deterministic, that is, ? 0 for this asset.
• Investors can usually borrow or lend.
• Borrowing (negative investing or negative
  lending)
• Examples (lending)
• - Certificate of Deposit
• - US Treasury note or bill (kept until maturity)
• - Money market checking account (almost
  risk-free)

104

• The inclusion of a risk-free asset in the list of
  possible assets for portfolio analysis makes the
  analysis more realistic.
• The inclusion also introduces a mathematical
  degeneracy that simplifies the shape of the
  efficient frontier of the return-sd feasible set.

105

• Suppose (1) rf return of the risk-free asset,
  and (2) ?f 0 standard deviation of return of
  the risk-free asset.
• Also we have a risky asset, with (1) Er
  expected return, and (2) ? gt 0 the standard
  deviation of the risky asset.
• Suppose we form a new portfolio from the
  risk-free asset and the risky asset. Let
• ? weight of risk-free asset in the portfolio,
  ? 1,
• 1 - ? weight of risky asset in the portfolio.

106

• Since ? 1 the weight of the risky asset in our
  portfolio will be nonnegative (buying). The
  weight of the risk-free asset can be either
  negative (borrowing) or positive (lending).
• The covariance of the two assets cannot exceed ?f
  ? ? in absolute value. This means the covariance
  is 0.

107

• For our new portfolio,
• rn ? rf (1-?) r is the return,
• Ern expected return ? rf (1-?) Er
• Varrn Var? rf (1-?)r Var(1-?) r
• 2 ? (1-?) covariance ?2 ?f2
• Var(1-?) r (1-?)2 Varr
  (1-?)2 ?2.
• Thus, st-dev of rn 1-? ? (1-?) ? , since ?
  1.

108

• Summary. For ? 1,
• Ern expected return ? rf (1-?) Er,
• standard deviation of rn ? ?f (1-?) ?.
• Both the mean and standard deviation of the new
  portfolio vary linearly with ?. As ? varies, the
  point representing the new portfolio traces out a
  straight line in the return-sd plane.

s
a
s
a
s
a
(
(r
), Er
)
(
, r
) (1-
) (
, Er,

1.

n
n
f
f
109

• This equation
• (?(rn), Ern) ? (?f, rf) (1-?) (?, Er, ?
  1
• is a half-line through (?f, rf) and (?, Er)
  with its endpoint at (?f, rf) (0, rf).

110

• After Figure 6.13. The feasible set becomes an
  infinite triangle with borrowing and lending of
  risk-free asset allowed.

111

• The half-line idea extends to n risky assets and
  1 risk-free asset. First form the feasible set
  for the risky assets. Then plot (0,rf). Draw
  half-lines beginning at (0,rf) through every
  point in the feasible set. The resulting new
  feasible set is an infinite triangle. Its
  efficient frontier is the top half-line, the
  top edge of the infinite triangle.

112

• If borrowing of the risk-free asset is not
  allowed, instead of half-lines as above we get
  line segments between the point (0,rf) and every
  point in the old feasible set as the new feasible
  set. The new feasible set has a straight-line
  front edge but a rounded top...

113

• The One-Fund Theorem
• Theorem. There is a single fund F of risky
  assets such that every efficient portfolio can be
  constructed as a combination of the fund F and
  the risk-free asset.

114

• The One-Fund Theorem (Contd)
• When risk-free borrowing and lending are
  available, the efficient set consists of a single
  straight line, which is the top of the triangular
  feasible region. This line is tangent to the
  original feasible set of risky assets. There
  will be a point F in the original feasible set
  that is on the line segment defining the overall
  efficient set. Any efficient point (any point on
  the line) can be expressed as a combination of
  this asset and the risk-free asset. We obtain
  different efficient points by changing the
  weighting between these two (including negative
  weights of the risk-free asset to borrow money in
  order to leverage the buying of the risky asset).
  

115

• The One-Fund Theorem (Contd)
• If F (f1,f2) then the points (x,y) on the
  efficient frontier are the ones satisfying
• (x,y) ? (0, rf) (1-?) (f1, f2), ? 1
• This is the equation of a half-line through (0,
  rf) and (f1,f2) with its endpoint at (0, rf).

116

• Solution Method.
• Given a point in the feasible region, we draw a
  line between the risk-free asset and that point.
  Let us denote by ? the angle between that line
  and the horizontal line.
• For any feasible portfolio p , we have
• We have ,
  ,

117

• Differentiation of
  w.r.t.
• and setting derivative to zero gives the
  following system of equations
• where is a constant.
• By denoting we get a system
  of equations w.r.t.

118

• Summary calculation of the efficient frontier
• Suppose there are n risky assets of interest,
  with all the expected risks, and
  variances/covariances known.
• Let M be the n by n variance/covariance matrix.
• Let b be the vector whose entry i is Eri rf.
  
• Solve the equations M V b for V.
• Compute W by dividing each entry in V by the
  total of the entries in V.

119

• Example 6.12. (Three uncorrelated assets). We
  have 3 risky assets, each with variance 1 and
  mean rates of returns of 1, 2, and 3
  respectively. There is a risk-free asset with
  rate rf 0.5.
• The entries in M, V and b are in the following
  table

120

• Example 6.12. (Contd)
• Since the V entries total to 4.5,
• expected return (1/9) 1 (3/9) 2 (5/9) 3
• 22/9 ? 2.4444
• variance of return (1/9)2 ? 1 (1/3)2 ? 1
  (5/9)2 ? 1 35/81 ? 0.432098765 ? sd of return ?
  0.6573.

121

• If F (f1,f2), then the points (x,y) on the
  efficient frontier are the ones satisfying
• (x,y) ? (0, rf) (1-?) (f1, f2), ? 1.
• Thus the efficient frontier consists of those
  points (x,y) satisfying
• (x,y) ? (0, 0.5) (1-?) (0.6753,2.4444), ?
  1

122

123

• If we choose, say ? ½ to define a portfolio,
  say PN, , our expected return is 1.472, and the
  sd of the return is 0.32867. Thus PN (0.32867,
  1.472).
• As compared to fund F, taking half of each of the
  two funds cuts our risk in half while reducing
  our expected return by less than half. Attaching
  even more (less) weight to the risk-free fund
  would give a combined fund with lower (higher)
  return and lower (higher) risk than this one.

124

• Conclusion
• Denote by P any portfolio made up only of the
  risky assets.
• If P has a return of more than PN , it is also
  riskier.
• If P is less risky than PN, its return is also
  lower.
• P can never be both less risky and also have a
  higher return than PN .
• Including a risk-free asset is a very good idea.
• Note shorting (borrowing) the risk-free asset can
  give a portfolio with a higher return (and higher
  risk) than P.