Two fund separation and linear valuation

1
Chapter 4

• Two fund separation and linear valuation

2
Main objective

• we give distributional conditions on the rates of
  return on assets so that individuals will
  optimally choose to hold portfolios on the
  portfolio frontier.
• Discuss the Capital Asset Pricing Model.(CAPM)
• Also discuss the Arbitrage Pricing Theory(APT).

3
4.1 section

• Two (mutual) fund separation phenomenon
• Define given any feasible portfolio, there
  exists a portfolio of two mutual funds such that
  individuals prefer at least as much as the
  original portfolio. The phenomenon is termed two
  (mutual) fund separation.
• E.g. if individuals prefer frontier portfolios,
  they can simply hold a linear combination of two
  frontier portfolios or mutual funds.

4

• This chapter assume that
• 1. all assets are risky.
• 2. asset returns have finite second moments.
• 3. no two asset returns are perfectly
  correlated, which imply that the
  variance-covariance matrix of asset returns
  exists and is positive definite.
• Then portfolio frontier exists, and every
  frontier portfolio is uniquely determined in that
  there is a unique set of portfolio weights
  associated with each frontier portfolio.

5
4.2 section

• The definition of two fund separation
• a vector of asset rate of returns is
  said to exhibit two fund separation if there
  exist two mutual funds and such that for
  any portfolio q there exists a scalar such
  that
• for all concave u(.)

6

• Now termed that the vector of asset rates of
  return, , exhibits two fund separation.
• We first claim that the separating mutual funds
  and must be frontier portfolios.

7

• To see this, for any portfolio q,
• from the definition of two fund separation,
• This is equivalent to

8

• By second degree stochastic dominance.
• Suppose is not a frontier portfolio. The
  there must exist a portfolio that has a
  variance strictly smaller than that of any
  portfolio formed by and .
• This contradicts the hypothesis that and
  are separating portfolios.
• Hence and must be on the portfolio
  frontier.

9
4.3 section

• If there exists two fund separation, any two
  distinct frontier portfolios can be separating
  funds.
• In particular, we can pick any frontier
  portfolio, and its zero covariance
  portfolio, zc(p), to be the separating portfolios.

10

• By Sections 3.16 and 3.17, for any portfolio q,
• Where
• Note that is
  the rate of return of the dominating portfolio.

11
4.4 section

• We show that
• two fund separation
• First proof the necessity of (4.3.3)
• proof. As is the return on the dominating
  portfolio, a0 is a solution of the following
  program

12

• A necessary condition for a0 to be a solution is
  
• Suppose does not hold for
  some q. Denote by and
  the cumulative distribution function of
  by F(.).

13

• As 0 there must exist a real number z
  such that
• By hypothesis that (4.3.3) does not hold for some
  q. we have (4.4.2) can not be zero for all z.
• if the equality of (4.4.2) does not hold for all
  z, that is,

14

• That is,
• Then
• Which contradicts the fact that

15

• Now consider a concave utility function that is
  piecewise linear
• Where gt0

16

• Then
• Which contradicts (4.4.1). Thus (4.3.3) is a
  necessary condition for two fund separation

17
4.5 section

• Now show that the sufficient for two fund
  separation. That is, the separating portfolios
  are p and its zero covariance portfolio. If q is
  a portfolio, then the dominating portfolio is
• To see this, let U(.) be any concave utility
  function.
• Thus (4.3.3) is the sufficient for two fund
  separation

18
4.6 section

• We will say that a vector of asset returns
  exhibits one fund separation if there exists a
  (feasible) portfolio such that every risk
  averse individual prefers to any other
  feasible portfolio.
• That is, when one fund separation, there must
  exist a mutual fund such that for any
  portfolio q, for all concave u(.)

19

• By second degree stochastic dominance, have
• Therefore, write
• Thus, One fund separation

20

• Note that
• 1. one fund separation can be viewed as a
  degenerate case of two fund separation.
• 2. When all assets have the same expected rate of
  return, the portfolio frontier degenerates to a
  single point, the minimum variance portfolio.
• 3.The portfolio frontier is thus trivially
  generated by a single mutual fund.
• 4. A necessary and sufficient condition for one
  fund separation is given by (4.3.3) with

21
4.7 section

• We shall give concrete examples of distributions
  of rates of return that exhibit two fund and one
  fund separation, respectively.
• 1. If rates of return are multivariate normally
  distributed and have nonidentical expectations,
  two fund separation obtains.

22

• By
• As any linear combination of normal random
  variables is itself normal,
• It follows that , , and are
  normal random variables.
• Recall section 3.17
• Have

23

• Consequently,
• We have thus shown that two fund separation
  obtains.
• 2. When returns have identical expectations,
  multivariate normality implies one fund
  separation.

24

• Let denote the return on the minimum
  variance portfolio. For any feasible portfolio q,
  we can always write
• We know
• Hence we know
• Then yields
• This establishes that one fund separation obtains.

25
4.8 section

• In this and the next section, we will show that
  when two fund separation obtains and markets for
  risky assets are in equilibrium, the equilibrium
  relation among asset returns is linear.
• First give a definition of a market portfolio.

26

• Let be individual is initial wealth , and
  let be the proportion of the initial wealth
  invested in the j-th security by individual i.
• The total wealth in the economy is
• Where I is the total number of individuals in the
  economy

27

• In equilibrium, the total wealth is equal
  to the total value of securities.
• For markets to clear, we must have
• Dividing by , we have
• That is, the market portfolio weights are a
  convex combination of the portfolio weights for
  individuals.

28
4.9 section

• We claim that when two fund separation obtains
  and in market equilibrium, the market portfolio
  is a frontier portfolio.
• Next from section 3.16 that if p is a frontier
  (but not mvp) and q is any feasible portfolio, we
  have

29

• Hence, if the market portfolio is not the minimum
  variance portfolio (mvp), we get
• Where
• Since any risky asset is itself a feasible
  portfolio, also have
• For all j1,2,, N . Therefore the equilibrium
  relation among asset returns is linear.

30
4.10 section

• Let us rewrite relation (4.9.3)
• When (i.e. the market
  portfolio is an efficient portfolio, zc(m) is
  inefficient)
• Note that the equilibrium expected rate of return
  on a risky asset depends upon the covariance of
  its rate of return with the rate of return on the
  market portfolio.

31

• Figure

security market line
32

• When 0

security market line
33
4.11 section

• In this section, we shall analyze a special case
  of two fund separation and show that the market
  portfolio is an efficient portfolio.
• Thus a relation like (4.10.1) is valid in
  equilibrium.
• Let us assume that individuals utility functions
  are increasing and strictly concave.
• Also assume that the rates of return on assets
  are multivariate normally distributed.
• We show first that each individual will choose to
  hold an efficient portfolio.

34

• The expected utility of individual is choice is
  
• Where denotes a standard normal random
  variable.
• Defining

35

• Differentiating
• Where
• That is, individual i prefers a higher expected
  rate of return, ceteris paribus.

36

• Also get
• Where
• So a strictly risk averse individual prefers a
  portfolio with a lower standard deviation,
  ceteris paribus.

37

• Now we show that individuals will choose to hold
  efficient portfolios.
• Totally differentiation
  (indifference curves)
• Seeing figure.

38

• Figure
• Thus an individual will choose the point of
  tangency. Then his strictly positively sloped
  indifference curves imply that he will choose an
  efficient portfolio.

39

• We know that all individuals hold efficient
  portfolios. Therefore, the market portfolio is an
  efficient portfolio.
• Furthermore , we have for any feasible portfolio
• This is known as the Zero-Beta Capital Asset
  Pricing Model which was developed by Black(1972)
  and Lintner(1969).

40
4.12 section

• The above analyses, we have assumed that there is
  no riskless asset.
• These next three sections are devoted to the case
  where there is a riskless asset.
• Suppose first the expected rate of return on the
  minimum variance portfolio exceeds the riskless
  rate,
• i.e.,

41

• We know that the rate of return on any feasible
  portfolio q can be expressed as
• With
• We claim that
• two fund
  separation.

42

• We prove the necessity part first.
• Suppose that two fund separation holds.
• Thus two separating portfolios must be frontier
  portfolios and can be chosen to be portfolio e
  and the riskless asset.
• Similar to section 4.4 then show that
• For all portfolios q.

43

• As is nonstochastic and we can always pick q
  to be any risky asset j, it follows that
• It is implied by two fund separation when a
  riskless asset exists.

44

• The sufficiency part follows easily from the
  arguments of section 4.5.
• Thus implies two fund separation
  when a riskless asset exists.
• Similar also show that when gtA/C
• two fund separation
• where

45
4.13 section

• Assume that two fund separation holds.
• When A/C and risky assets are in
  strictly positive supply, the tangent portfolios
  must be the market portfolios of risky assets in
  equilibrium.
• Hence, using(3.19.1), we know
• For any portfolio q in the market equilibrium.
• This is the traditional Capital Asset Pricing
  Model (CAPM) independently derived by
  Lintner(1965), Mossin(1965), and Sharoe(164).

46

• When A/C, the story is a little different.
• We claim that the riskless asset is in strictly
  positive supply and the risky assets are in zero
  net supply in equilibrium.
• Note that the relation(3.19.1) holds independent
  of the relationship between and A/C

47
4.14 section

• In this section we will show that when investors
  have strictly increasing and concave utility
  functions, the risk premium of the market
  portfolio must be strictly positive when the
  risky assets are in strictly positive supply and
  the presence of two fund separation.

48

• We first claim that an investor will never choose
  to hold a portfolio whose expected rate of return
  is strictly lower than the riskless rate when his
  utility function u(.) is strictly increasing and
  concave.
• Let be the random return of a portfolio chosen
  by u(.) such that
• Note that
• This contradicts the hypothesis that the
  individual chooses to hold the portfolio with a
  random rate of return

49

• when the risky assets are in strictly positive
  supply and the presence of two fund separation.
• When A/C , an individual puts all his wealth
  into the riskless asset and holds a
  self-financing portfolio.
• this contradicts that the risky assets are in
  strictly positive supply
• Thus A/C

50

• When gtA/C, then no investor holds a strictly
  positive amount of the market portfolio.
• This is inconsistent with market clearing.
• Thus in equilibrium, it must be the case that
  ltA/C and the risk premium of the market
  portfolio is strictly positive.
• The half line in the standard deviation-expected
  rate of return space composed of efficient
  frontier portfolios and the riskless asset is
  termed the Capital Market Line.

51

• We note that in the above proof when there exists
  a riskless asset, an investor will never choose
  to hold a portfolio having an expected rate of
  return strictly less than the riskless rate we
  only used the fact that an investor can always
  invest all his money in the riskless asset.
• Hence the conclusion holds even when
  short-selling portfolio is not allowed.

52

• When investors choose to hold efficient frontier
  portfolios and the riskless asset is in strictly
  positive supply, we have
• For any feasible portfolio and

53

• When the riskless borrowing is allowed at a rate
  strictly higher than the riskless lending rate,
  we have
• For any feasible portfolio and
• Where and denote the riskless lending rate
  and borrowing rate, respectively.
• These relations is termed the constrained
  borrowing versions of the CAPM

54
4.15 section

• In this and the next section, we will discuss two
  simple situations where we can explicitly write
  down how the risk premium of the market portfolio
  is related to investors optimal portfolio
  decisions.
• First, suppose that there exists a riskless asset
  with a rate of return and that the rates of
  return of risky assets are multivariate normally
  distributed.

55

• Section 1.18 shows that
• Where
• Using the definition of a covariance,
• Under normally distributed, have

Steins lemma
56

• By steins lemma,
• Defining the i-th investors global absolute risk
  aversion
• Dividing by summing across i.

57

• We get
• Where
• Note that is the harmonic mean of
  individuals global absolute risk aversion.
• We can interpret to be the
  aggregate relative risk aversion of the economy
  in equilibrium.

58

• Relation (4.15.4) implies that
• Substituting (4.15.5) into (4.15.4) gives the
  familiar CAPM.

(4.15.4)
(4.15.5)
59
4.16 section

• For example
• Now assume that utility functions are quadratic
• Easy obtain CAPM

60
4.17 section

• In the context of the CAPM, a risky assets beta
  with respect to the market portfolio is a
  sufficient statistic for its contribution to the
  riskiness of an individuals portfolio.
• Risky assets whose payoffs are positively
  correlated with those of the market portfolio
  have positive risk premiums.
• In such event, the higher the asset beta, the
  higher the risk premium.
• The intuition of this relationship can be
  understood as follows. Seeing p103.

61
4.18 section

• Recall from previous analyses, two fund
  separation implies that there exists a linear
  relation among expected returns of assets.
• In equilibrium, the coefficients of the linear
  relation are identified to be related to the
  betas of asset returns with respect to the
  return on the market portfolio.
• As saying that the rate of return on an risky
  asset is generated by a one factor model plus a
  random noise whose conditional expectation given
  the factor is identically zero.
• We will give mult-factors model in following
  section

62
4.19 section

• We consider a sequence of economies with
  increasing numbers of assets.
• In the n-th economy, there are n risky assets and
  a riskless asset.
• The rates of return on risky assets are generated
  by a K-factor model
• Where
• And
• Where are rates of return on portfolios
• Using matrix notation, we can write as

63
4.20 section

• We first note that if
• then there exists an exact linear relation among
  expected rates of return on assets in the n-th
  economy, if one cannot create something out of
  nothing.
• This follows since the returns on risky assets
  are completely spanned by the K factors
  (portfolios) and the riskless asset.

64

• Formally, consider a portfolio of the K factors
  and the riskless asset, , with
• Where is the proportion invested in the
  riskless asset and is the proportion
  invested in the k-th factor.

65

• The rate of return on this portfolio is
• Note that the factor components of the rate of
  return on replicate those of asset j.

66

• We show that must be equal to
• We shall show that if this is not the case,
  something can be created out of nothing.
• Suppose that

67

• We establish a portfolio by investing one dollar
  in and shorting one dollars worth of
  security j. this portfolio costs nothing. Its
  rate of return, however, is
• Which is riskless and strictly positive. Hence
  something has been created out of nothing, or
  there exists a free lunch.

68

• Reversing the above arguments for the case
  we thus have
• From
• Obtain
• That is, there exists an exact linear relation
  among expected asset reurns.

69
4.21 section

• When the are not zeros, the story is a
  little complicated.
• Formally, in economy n, a portfolio of the n
  risky assets and riskless asset is an arbitrage
  portfolio if it costs nothing.
• An arbitrage opportunity (in the limit) is a
  sequence of arbitrage portfolios whose expected
  rates of return are bounded below away from zero,
  while their variances converge to zero.
• That is, it almost a free lunch.

70

• We wish to show that if there is no arbitrage
  opportunity, then a linear relation among
  expected asset returns will hold approximately
  for most of the assets in a large economy.
• We first claim that
• For most of the asset in large economies.

71

• To see this, we fix , however small. Let
  N(n) be the number of assets in the n-th economy
  such that the absolute value of the difference
  between the two sides of (4.21.1) is greater than
• Without loss of generality, assume that

72

• If we can show that there exists such that
  for all n, we are done.
• This follows since there will be at most assets
  that satisfy(4.21.2) for arbitrarily large n and
  can be arbitrarily small.
• We shall proceed by contraposition.

73

• Suppose that there does not exist a finite
  such that for all n.
• Then there must exist a subsequence of
• denoted by,
  such that when
• We construct a sequence of arbitrage portfolios
  as follows.
• First, construct arbitrage portfolios
  that have no factor risk in economy .

74

• The rate of return for the j-th arbitrage
  portfolio is
• where

75

• Next, form a portfolio of these arbitrage
  portfolios with a constant weight, on
  each.
• The resulting portfolio is still an arbitrage
  portfolio with an expected rate of return
• And a variance

76

• Since as , the variance
  of the sequence of arbitrage portfolios converge
  to zero, while their expected rates of return are
  bounded below away from zero by .
• Thus there exists an arbitrage opportunity, a
  contradiction.
• We hence conclude that there must exists a finite
  such that for all n.

77
4.22 section

• Now using the fact proven above that for a given
  , however small, there exist at most
  risky assets such that
• We know
• For all but at most risky assets in any
  economy.
• Thus, for economies with the number of assets
  much larger than , a linear relation among
  expected asset returns holds approximately for
  most of the assets.
• This relation is the Arbitrage Pricing Theory
  (APT) originated by Ross (1976)

78
4.23 section

• For the APT to make predictions in an economy
  with a finite number of assets, would like to
  bound the deviation from the APT linear relation
  for any asset.
• Using equilibrium rather than arbitrage
  arguments, therefore called equilibrium APT
• Suppose that there are N risky assets in the
  economy, indexed by j1,2,N, and a riskless
  asset.
• These risky assets are in strictly positive
  supply.
• The rate of return on the riskless asset is

79

• The rates of return on risky assets are generated
  by a K-factor model
• With
• We also assume that the random variables
• are independent and that are rates of
  return on portfolios.

80

• We assume that agents have utility functions that
  are increasing, strictly concave, and absolute
  risk aversion
• From section 4.20 that, when
• Or equivalently
• Our purpose here is to bound the deviation of
  from the right hand side of (4.23.3), in market
  equilibrium, when is not identically zero.

81
4.24 section

• Consider a portfolio of the K factors and the
  riskless asset having a rate of return
• An arbitrage portfolio is constructed by
  investing one dollar in the above portfolio and
  shorting one dollars worth of risky asset j.
• This arbitrage portfolio has a rate of return
• i.e.

82

• Let be individual is initial wealth, let be
  individual is time-1 random wealth.
• Then is the unique solution to the
  following problem
• The first order necessary condition for
  to be an optimum is

83

• Using , the above relation can be
  written as
• Now we claim that, for all j such that

84

• To see this, note that risky assets are in
  strictly positive supply. Suppose risky asset j
  with is held in a strictly positive
  amount in equilibrium by individual i.
• By the strict concavity of and the assumption
  that is independent of all other random
  variable,
• We know that

85

• Then (4.24.3)holds for j by the fact that the
  above arguments can be applied to all j such that
  . Thus we have that (4.24.3) holds for all
  such js.
• As a consequence of the above analysis, we can
  also conclude that if , then asset j is
  held in a strictly positive amount by all
  individuals.

86
2.25 section

• Now fix risky asset j with . Let
  be the dollar amount invested in
  the risky asset j by individual i.
• Define
• Taylors expansion
• Where all the other random variables are
  independent and is a random variable whose
  value lies between and .

87

• By the assumption that 1, we have
• Therefore
• We claim that

88

• To see this, we first note that since
• Where the second inequality follows from the
  concavity of

89

• Next, we observe that
• by
• Therefore

90

• Now we obtain
• Note also that, since , by the
  conditional Jensens inequality, we have

91

• Finally, by
• We have
• Or equivalently,

92

• Let the total market value of asset j be denoted
  by
• In equilibrium there exists an i such that
  , where I is the number of individuals in the
  economy.
• We thus have
• Where we gives an explicit bound of the deviation
  of from the APT relation.