# Introduction to Portfolio Selection and Capital Market Theory: Static Analysis

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Introduction to Portfolio Selection and Capital Market Theory: Static Analysis BaoheWangbaohewang0592@sina.com

Introduction The investment decision by households as having two parts: The investment decision by households as having two parts: (a) the consumption-saving choice (a) the consumption-saving choice (b) the portfolio-selection choice (b) the portfolio-selection choice In general the two decisions cannot be made independently. In general the two decisions cannot be made independently. However, the consumption-saving allocation has little substantive impact on portfolio theory. However, the consumption-saving allocation has little substantive impact on portfolio theory.

One-period Portfolio Selection The solution to the general problem of choosing the best investment mix is called portfolio-selection theory. The solution to the general problem of choosing the best investment mix is called portfolio-selection theory. There are n different investment opportunities called securities. There are n different investment opportunities called securities. The random variable one-period return per dollar on security j is denoted The random variable one-period return per dollar on security j is denoted

Any linear combination of these securities which has a positive market value is called a portfolio. Any linear combination of these securities which has a positive market value is called a portfolio. denote the utility function. denote the utility function. is the end-of-period value of the investor s wealth measure in dollars. is the end-of-period value of the investor s wealth measure in dollars. is an increasing strictly concave function and twice continuously differentiable. is an increasing strictly concave function and twice continuously differentiable. So the investor s decision is relevant to the subjective joint probability distribution for. So the investor s decision is relevant to the subjective joint probability distribution for.

Assumption 1: Frictionless Markets Assumption 1: Frictionless Markets Assumption 2: Price-Taker Assumption 2: Price-Taker Assumption 3: No-Arbitrage Opportunities Assumption 3: No-Arbitrage Opportunities Assumption 4: No-Institutional Restrictions Assumption 4: No-Institutional Restrictions

Given these assumptions, the portfolio- selection problem can be formally stated as Given these assumptions, the portfolio- selection problem can be formally stated as (2.1) (2.1) Where E is the expectation operator for the subjective joint probability distribution. Where E is the expectation operator for the subjective joint probability distribution.

If is a solution (2.1), then it will satisfy the first-order conditions: If is a solution (2.1), then it will satisfy the first-order conditions: Where is the random variable return per dollar on the optimal portfolio. Where is the random variable return per dollar on the optimal portfolio. With the concavity assumptions on U, if the variance-covariance matrix of the return is nonsingular and an interior solution exists, the the solution is unique. With the concavity assumptions on U, if the variance-covariance matrix of the return is nonsingular and an interior solution exists, the the solution is unique.

Formula (2.1) rules out that any one of the securities is a riskless security. Formula (2.1) rules out that any one of the securities is a riskless security. If a riskless security is added to the menu of available securities then the portfolio selection problem can be stated as: If a riskless security is added to the menu of available securities then the portfolio selection problem can be stated as: (2.4) (2.4)

The first-order conditions can be written as: The first-order conditions can be written as: Where can be rewritten as Where can be rewritten as If it is assumed that the variance- covariance matrix of the returns on the risky securities is nonsingular and an interior solution exits, then the solution is unique. If it is assumed that the variance- covariance matrix of the returns on the risky securities is nonsingular and an interior solution exits, then the solution is unique.

But neither (2.1) nor (2.3) reflect that end of period wealth cannot be negative. But neither (2.1) nor (2.3) reflect that end of period wealth cannot be negative. To rule out bankruptcy, the additional constraint that, with probability one, To rule out bankruptcy, the additional constraint that, with probability one, could be imposed on. could be imposed on. This constraint is too weak, because the probability assessments on are subjective. This constraint is too weak, because the probability assessments on are subjective. An alternative treatment is to forbid borrowing and short-selling securities where, by law,. An alternative treatment is to forbid borrowing and short-selling securities where, by law,.

The optimal demand functions for risky securities,, and the resulting probability distribution for the optimal portfolio will depend on The optimal demand functions for risky securities,, and the resulting probability distribution for the optimal portfolio will depend on (1) the risk preferences of the investor; (1) the risk preferences of the investor; (2) his initial wealth; (2) his initial wealth; (3) the join distribution for the securities returns. (3) the join distribution for the securities returns.

The von Neumann-Morgenstern utility function can only be determined up to a positive affine transformation. The von Neumann-Morgenstern utility function can only be determined up to a positive affine transformation. The Pratt-Arrow absolute risk-aversion function is invariant to any positive affine transformation of. The Pratt-Arrow absolute risk-aversion function is invariant to any positive affine transformation of.

The preference orderings of all choices available to the investor are completely specified by absolute risk – aversion function The preference orderings of all choices available to the investor are completely specified by absolute risk – aversion function The change in absolute risk aversion with respect to a change in wealth is The change in absolute risk aversion with respect to a change in wealth is

is positive, and such investor are call risk averse. is positive, and such investor are call risk averse. An alternative, measure of risk aversion is the relative risk-aversion function defined by An alternative, measure of risk aversion is the relative risk-aversion function defined by Its change with respect to a change in wealth is given by Its change with respect to a change in wealth is given by

The certainty-equivalent end-of-period wealth is defined to be such that The certainty-equivalent end-of-period wealth is defined to be such that is the amount of money such that the investor is indifferent between having this amount of money for certain or the portfolio with random variable outcome. is the amount of money such that the investor is indifferent between having this amount of money for certain or the portfolio with random variable outcome. We can proof follows directly by Jensen s inequality: if is strictly concave We can proof follows directly by Jensen s inequality: if is strictly concave Because U is an increase function, So Because U is an increase function, So

The certainty equivalent can be used to compare the risk aversions of two investor. The certainty equivalent can be used to compare the risk aversions of two investor. If A is more risk averse than B and they hold same portfolio, the certainty equivalent end of period wealth for A is less than or equal to the certainty equivalent end of period wealth for B. If A is more risk averse than B and they hold same portfolio, the certainty equivalent end of period wealth for A is less than or equal to the certainty equivalent end of period wealth for B.

Rothschild and Stiglitz define the meaning of increasing risk for a security so we can compare the riskiness of two securities or portfolios. Rothschild and Stiglitz define the meaning of increasing risk for a security so we can compare the riskiness of two securities or portfolios. If for all concave with strict inequality holding for some concave, we said the first portfolio is less risky than the second portfolio. If for all concave with strict inequality holding for some concave, we said the first portfolio is less risky than the second portfolio.

Its equivalence to the two following definitions: Its equivalence to the two following definitions: (1) is equal in distribution to plus some noise. (1) is equal in distribution to plus some noise. (2) has more weight in its tails than. (2) has more weight in its tails than.

If there exists an increasing strictly concave function such that If there exists an increasing strictly concave function such that, we call this portfolio is an efficient portfolio., we call this portfolio is an efficient portfolio. All portfolios that are not efficient are called inefficient portfolios. All portfolios that are not efficient are called inefficient portfolios.

It follows immediately that every efficient portfolio is a possible optimal portfolio, for each efficient portfolio there exists an increasing concave such that the efficient portfolio is a solution to (2.1) or (2.3). It follows immediately that every efficient portfolio is a possible optimal portfolio, for each efficient portfolio there exists an increasing concave such that the efficient portfolio is a solution to (2.1) or (2.3). Because all risk-averse investors have different utility function, so they will be indifferent between selecting their optimal portfolios. Because all risk-averse investors have different utility function, so they will be indifferent between selecting their optimal portfolios.

Theorem 2.1: If denotes the random variable return per dollar on any feasible portfolio and if is riskier than in the Rothschild and Stiglitz sense, then Theorem 2.1: If denotes the random variable return per dollar on any feasible portfolio and if is riskier than in the Rothschild and Stiglitz sense, then ( is an efficient portfolio) ( is an efficient portfolio) Proof: By hypothesis If then trivially. If then trivially. But is a feasible portfolio and is an efficient portfolio. By contradiction, But is a feasible portfolio and is an efficient portfolio. By contradiction,

Corollary 2.1: If there exists a riskless security with return R, then, with equality holding only if is a riskless security. Corollary 2.1: If there exists a riskless security with return R, then, with equality holding only if is a riskless security. Proof: If is riskless, then by Assumption 3,. If is not riskless, by Theorem 2.1,. Proof: If is riskless, then by Assumption 3,. If is not riskless, by Theorem 2.1,.

Theorem 2.2: The optimal portfolio for a nonsatiated risk-averse investor will be the riskless security if and only if for j=1,2, …..,n. Theorem 2.2: The optimal portfolio for a nonsatiated risk-averse investor will be the riskless security if and only if for j=1,2, …..,n. Proof: If is an optimal solution, then we have By the nonsatiation assumption, so Proof: If is an optimal solution, then we have By the nonsatiation assumption, so If then will satisfy because the property of U, so this solution is unique. If then will satisfy because the property of U, so this solution is unique.

From Corollary 2.1 and Theorem 2.2, if a risk-averse investor chooses a risky portfolio, then the expected return on the portfolio exceeds the riskless rate. From Corollary 2.1 and Theorem 2.2, if a risk-averse investor chooses a risky portfolio, then the expected return on the portfolio exceeds the riskless rate.

Theorem 2.3: Let denote the return on any portfolio p that does not contain security s. If there exists a portfolio p such that, for security s,, where Theorem 2.3: Let denote the return on any portfolio p that does not contain security s. If there exists a portfolio p such that, for security s,, where then the fraction of every efficient portfolio allocated to security s is the same and equal to zero. then the fraction of every efficient portfolio allocated to security s is the same and equal to zero. Proof: Suppose is the return on an efficient portfolio with fraction allocated to security s, be the return on a portfolio with the same fractional holding as except that instead of security s with portfolio P

Hence Hence So So Therefore,for, is riskier than Z in the Rothschild-Stiglitz. This contradicts that is an efficient portfolio. Therefore,for, is riskier than Z in the Rothschild-Stiglitz. This contradicts that is an efficient portfolio. Corollary 2.3: Let denote the set of n securities and denote the same set of securities except that is replace with. If and, then all risk averse investor would prefer to choose. Corollary 2.3: Let denote the set of n securities and denote the same set of securities except that is replace with. If and, then all risk averse investor would prefer to choose.

Theorem 2.3 and its corollary demonstrate that all risk averse investors would prefer any unnecessary and noise to be eliminated. Theorem 2.3 and its corollary demonstrate that all risk averse investors would prefer any unnecessary and noise to be eliminated. The Rothschild-Stiglitz definition of increasing risk is quite useful for studying the properties of optimal portfolios. The Rothschild-Stiglitz definition of increasing risk is quite useful for studying the properties of optimal portfolios. But this rule is not apply to individual securities or inefficient portfolios. But this rule is not apply to individual securities or inefficient portfolios.

2.3 Risk Measures for Securities and Portfolios in The One-Period model In this section, a second definition of increasing risk is introduced. In this section, a second definition of increasing risk is introduced. is the random variable return per dollar on an efficient portfolio K. is the random variable return per dollar on an efficient portfolio K. denote an increasing strictly concave function such that for denote an increasing strictly concave function such that for Random variable Random variable

Definition: The measure of risk of portfolio P relative to efficient portfolio K with random variable return is defined by Definition: The measure of risk of portfolio P relative to efficient portfolio K with random variable return is defined by and portfolio P is said to be riskier than portfolio relative to efficient portfolio K if. and portfolio P is said to be riskier than portfolio relative to efficient portfolio K if.

Theorem 2.4: If is the return on a feasible portfolio and is the return on efficient portfolio K, then. Theorem 2.4: If is the return on a feasible portfolio and is the return on efficient portfolio K, then. Proof: From the definition be the fraction of portfolio P allocated to security j, then be the fraction of portfolio P allocated to security j, then and and

By a similar argument, By a similar argument, Hence, Hence, and and By Corollary 2.1,. Therefore By Corollary 2.1,. Therefore

Hence, the expected excess return on portfolio P, is in direct proportion to its risk and the larger is its risk, the larger is its expected return. Hence, the expected excess return on portfolio P, is in direct proportion to its risk and the larger is its risk, the larger is its expected return. Consider an investor with utility function U and initial wealth who solves the portfolio-selection problem: Consider an investor with utility function U and initial wealth who solves the portfolio-selection problem: The first order condition: The first order condition:

If then the solution is. If then the solution is. However, an optimal portfolio is an efficient portfolio. By Theorem 2.4 However, an optimal portfolio is an efficient portfolio. By Theorem 2.4 So is similar to an excess demand function. Measures the contribution of security j to the Rothsechild-Stiglitz risk of the optimal portfolio. So is similar to an excess demand function. Measures the contribution of security j to the Rothsechild-Stiglitz risk of the optimal portfolio.

By the implicit function theorem, we have: By the implicit function theorem, we have: Therefore, if lies above the risk-return line in the plane, then the investor would prefer to increase his holding in security j. Therefore, if lies above the risk-return line in the plane, then the investor would prefer to increase his holding in security j.

is a natural measure of risk for individual securities. is a natural measure of risk for individual securities. The ordering of securities by their systematic risk relative to a given efficient portfolio will be identical with their ordering relative to any other efficient portfolio. The ordering of securities by their systematic risk relative to a given efficient portfolio will be identical with their ordering relative to any other efficient portfolio.

Lemma 2.1: Lemma 2.1: (i) for efficient portfolio K. (i) for efficient portfolio K. (ii) If then (ii) If then (iii) for efficient portfolio K if and only if for every efficient portfolio L. (iii) for efficient portfolio K if and only if for every efficient portfolio L. Proof: (i) is a continuous monotonic function of and hence and are in one to one correspondence. Proof: (i) is a continuous monotonic function of and hence and are in one to one correspondence.

(ii) (ii) (iii)Because (iii)Because if, then. if, then. Property I: If L and K are efficient portfolios, then for any portfolio p,. Proof : From Theorem 2.4 Proof : From Theorem 2.4

Property 2: If L and K are efficient portfolios, then and. Property 2: If L and K are efficient portfolios, then and. Hence, all efficient portfolios have positive systematic risk, relative to any efficient portfolio. Hence, all efficient portfolios have positive systematic risk, relative to any efficient portfolio. Property 3: if and only if for every efficient portfolio K. Property 3: if and only if for every efficient portfolio K. Property 4: Let p and q denote any two feasible portfolios and let K and L denote any two efficient portfolios. if and only if Property 4: Let p and q denote any two feasible portfolios and let K and L denote any two efficient portfolios. if and only if

Proof: From Property 1, we have Proof: From Property 1, we have Thus the measure provides the same orderings of risk for any reference efficient portfolio. Thus the measure provides the same orderings of risk for any reference efficient portfolio. Property 5: For each efficient portfolio K and any feasible portfolio p, Property 5: For each efficient portfolio K and any feasible portfolio p, where and for every efficient portfolio L. where and for every efficient portfolio L.

Proof: From Theorem 2.4. If portfolio q is constructed by holding one dollar p, dollars riskless security, short selling dollars portfolio K, then Proof: From Theorem 2.4. If portfolio q is constructed by holding one dollar p, dollars riskless security, short selling dollars portfolio K, then so for every efficient portfolio L. so for every efficient portfolio L. But implies But implies for every efficient portfolio L. for every efficient portfolio L. Property 6: If a feasible portfolio p has portfolio weight,then Property 6: If a feasible portfolio p has portfolio weight,then

Hence, the systematic risk of a portfolio is the weighted sum of the systematic risks of its component securities. Hence, the systematic risk of a portfolio is the weighted sum of the systematic risks of its component securities. The Rothschild Stiglitz measure provides only for a partial ordering. The Rothschild Stiglitz measure provides only for a partial ordering. measure provides a complete ordering. measure provides a complete ordering. They can give different rankings. They can give different rankings. The Rothschild Stiglitz definition measure the total risk of a security. It is appropriate definition for identifying optimal portfolios and determining the efficient portfolio set. The Rothschild Stiglitz definition measure the total risk of a security. It is appropriate definition for identifying optimal portfolios and determining the efficient portfolio set.

The measure the systematic risk of a security. The measure the systematic risk of a security. To determine the, the efficient portfolio set must be determined. To determine the, the efficient portfolio set must be determined. The manifest behavioral characteristic shared by all risk averse utility maximization is to diversify. The manifest behavioral characteristic shared by all risk averse utility maximization is to diversify.

The greatest benefits in risk reduction come from adding a security to the portfolio whose realized return tends to be higher when the return on the rest of the portfolio is lower. The greatest benefits in risk reduction come from adding a security to the portfolio whose realized return tends to be higher when the return on the rest of the portfolio is lower. Next to such countercyclical investments in terms of benefit are the noncyclic securities whose returns are orthogonal to the return on the portfolio. Next to such countercyclical investments in terms of benefit are the noncyclic securities whose returns are orthogonal to the return on the portfolio.

Theorem 2.5 : If and denote the returns on portfolio p and q respectively and if, for each possible value of, Theorem 2.5 : If and denote the returns on portfolio p and q respectively and if, for each possible value of, with strict inequality holding over some finite probability measure of,then portfolio p is riskier than portfolio q and. with strict inequality holding over some finite probability measure of,then portfolio p is riskier than portfolio q and. Where, is the realized return on an efficient portfolio. Where, is the realized return on an efficient portfolio.

Proof: Proof: is a strictly increasing function, is a nondecreasing function, so is a strictly increasing function, is a nondecreasing function, so From Theorem 2.4

Theorem 2.6: If and denote the returns on portfolio p and q respectively and if, for each possible value of, Theorem 2.6: If and denote the returns on portfolio p and q respectively and if, for each possible value of,, a constant, then, a constant, then and. and. Proof: By hypothesis Proof: By hypothesis

Theorem 2.7: If, for all possible values of Theorem 2.7: If, for all possible values of (i), then (II), then (III), then (IV), a constant, then

Theorems 2.5, 2.6 and 2.7 demonstrate, the conditional expected return function provides considerable information about a security s risk and equilibrium expected return. Theorems 2.5, 2.6 and 2.7 demonstrate, the conditional expected return function provides considerable information about a security s risk and equilibrium expected return.

2.4 Spanning, Separation, and Mutual-Fund Theorems Definition: A set of M feasible portfolios with random variable returns is said to span the space of portfolios contained in the set if and only if for any portfolio in with return denoted by there exist numbers, such that Definition: A set of M feasible portfolios with random variable returns is said to span the space of portfolios contained in the set if and only if for any portfolio in with return denoted by there exist numbers, such that

A mutual fund is a financial intermediary that holds as its assets a portfolio of securities and issues as liabilities shares against this collection of assets. A mutual fund is a financial intermediary that holds as its assets a portfolio of securities and issues as liabilities shares against this collection of assets. Theorem 2.8 If there exist M mutual funds whose portfolio span the portfolio set, then all investors will be indifferent between selecting their optimal portfolios from and selecting from portfolio combination of just the M mutual funds. Theorem 2.8 If there exist M mutual funds whose portfolio span the portfolio set, then all investors will be indifferent between selecting their optimal portfolios from and selecting from portfolio combination of just the M mutual funds.

Therefore the smallest number of such funds is a particularly important spanning set. Therefore the smallest number of such funds is a particularly important spanning set. When such spanning obtain, the investor s portfolio-selection problem can be separated into two steps. When such spanning obtain, the investor s portfolio-selection problem can be separated into two steps. However, if the smallest funds can be constructed only if the fund managers know the preferences, endowments, and probability beliefs of each investor. However, if the smallest funds can be constructed only if the fund managers know the preferences, endowments, and probability beliefs of each investor.

Theorem 2.9: Necessary conditions for the M feasible portfolios with return to span the portfolio set are (a) that the rank of and (b) that there exist numbers such that the random variable has zero variance. Theorem 2.9: Necessary conditions for the M feasible portfolios with return to span the portfolio set are (a) that the rank of and (b) that there exist numbers such that the random variable has zero variance. Proposition 2.1: If is the return on some security or portfolio and if there are no arbitrage opportunities then Proposition 2.1: If is the return on some security or portfolio and if there are no arbitrage opportunities then

Proof: Let be the return on a portfolio with fraction allocated to security j, Proof: Let be the return on a portfolio with fraction allocated to security j, allocated to the security with return ; and allocated to the riskless security with return R, if is chosen such that,then is riskless security and therefore but can be chosen arbitrarily. So we get the result. allocated to the security with return ; and allocated to the riskless security with return R, if is chosen such that,then is riskless security and therefore but can be chosen arbitrarily. So we get the result.

Hence, as long as there are no arbitrage opportunities, it can be assumed without loss of generality that one of the portfolios in any candidate spanning set is the riskless security. Hence, as long as there are no arbitrage opportunities, it can be assumed without loss of generality that one of the portfolios in any candidate spanning set is the riskless security. Theorem 2.10: A necessary and sufficient condition for to span is that there exist number such that Theorem 2.10: A necessary and sufficient condition for to span is that there exist number such that

Proof: If span, then Proof: If span, then such that. Because such that. Because and substituting, we have and substituting, we have we pick the portfolio weights for and, from which it follows that.But every portfolio in can be written as a portfolio combination of and R. we pick the portfolio weights for and, from which it follows that.But every portfolio in can be written as a portfolio combination of and R.

Corollary 2.10: A necessary and sufficient condition for to be the smallest number of feasible portfolio that span is that the rank of equals the rank of Corollary 2.10: A necessary and sufficient condition for to be the smallest number of feasible portfolio that span is that the rank of equals the rank of Proof: If the rank of, then X Proof: If the rank of, then X are linearly independent. Moreover are linearly independent. Moreover hence, if the rank of then there exist number such that hence, if the rank of then there exist number such that for. Therefore where by Theorem 2.10 span for. Therefore where by Theorem 2.10 span

It follows from Corollary 2.10 that a necessary and sufficient condition for nontrivial spanning of is that some of the risky securities are redundant securities. It follows from Corollary 2.10 that a necessary and sufficient condition for nontrivial spanning of is that some of the risky securities are redundant securities. By Theorem 2.10, if investors agree on a set of portfolios such that By Theorem 2.10, if investors agree on a set of portfolios such that and if they agree on the number,then span even if investors do not agree on the joint distribution of and if they agree on the number,then span even if investors do not agree on the joint distribution of

Proposition 2.2: If is the return on a portfolio contained in, then any portfolio that combines positive amount of with the riskless security is also contained in, where is the set of all efficient portfolios contained in. Proposition 2.2: If is the return on a portfolio contained in, then any portfolio that combines positive amount of with the riskless security is also contained in, where is the set of all efficient portfolios contained in. Proof: Let, because is an efficient portfolio, so Proof: Let, because is an efficient portfolio, so Define where and Define where and, Hence, thus Z is an efficient portfolio., Hence, thus Z is an efficient portfolio.

It follows from Proposition 2.2 that, for every number such that, there exists at least one efficient portfolio with expected return equal to. It follows from Proposition 2.2 that, for every number such that, there exists at least one efficient portfolio with expected return equal to. Theorem 2.11: Let denote the return on m feasible portfolios. If, for security j, there exist number such that Theorem 2.11: Let denote the return on m feasible portfolios. If, for security j, there exist number such that where where for some efficient portfolio K, then for some efficient portfolio K, then

Proof: Let Proof: Let because, thus because, thus by construction, and hence by construction, and hence Therefore the systematic risk of portfolio p, is zero. From Theorem 2.4 Therefore the systematic risk of portfolio p, is zero. From Theorem 2.4 therefore therefore

Hence, if the return on a security can be written in this linear form relative to the portfolios, then its expected excess return is completely determined by the expected excess returns on these portfolios and the weights. Hence, if the return on a security can be written in this linear form relative to the portfolios, then its expected excess return is completely determined by the expected excess returns on these portfolios and the weights. Theorem 1.12: If, for every security j, there exist numbers such that Theorem 1.12: If, for every security j, there exist numbers such that Theorem 1.12 Theorem 1.12 where, then span the set of efficient portfolios. where, then span the set of efficient portfolios.

Proof: Proof: Where Where Construct portfolio Construct portfolio Thus where Thus where Hence, for, is riskier than Z, which contradicts that is and efficient portfolio. So. We get the result. Hence, for, is riskier than Z, which contradicts that is and efficient portfolio. So. We get the result.

Theorem 2.13: Let denote the fraction of efficient portfolio K allocation to security j, span if and only if there exist number for every security j such that Theorem 2.13: Let denote the fraction of efficient portfolio K allocation to security j, span if and only if there exist number for every security j such that where for every efficient portfolio K. where for every efficient portfolio K. Corollary 2.13: (X,R) span if and only if there exist a number for each security j, Corollary 2.13: (X,R) span if and only if there exist a number for each security j, such that such that where where

Proof: By hypothesis, for every efficient portfolio K. If, then from Corollary 2.1 for every efficient portfolio K and R span. Otherwise, from Theorem 2.2, for every efficient portfolio. By Theorem 2.13, Proof: By hypothesis, for every efficient portfolio K. If, then from Corollary 2.1 for every efficient portfolio K and R span. Otherwise, from Theorem 2.2, for every efficient portfolio. By Theorem 2.13, so so Since is contained in, any properties proved for portfolios that span must be properties of portfolio that span. Since is contained in, any properties proved for portfolios that span must be properties of portfolio that span.

From Theorem 2.10, 2.12, 2.13, the essential difference is that to span the efficient portfolio set it is not necessary that linear combinations of the spanning portfolios exactly replicate the return on each available security. From Theorem 2.10, 2.12, 2.13, the essential difference is that to span the efficient portfolio set it is not necessary that linear combinations of the spanning portfolios exactly replicate the return on each available security. All the models that do not restrict the class of admissible utility function, the distribution of individual security returns must be such that All the models that do not restrict the class of admissible utility function, the distribution of individual security returns must be such that

Proposition 2.3: If, for every security j, Proposition 2.3: If, for every security j, with linearly independent with finite variances and if the return on security j, has a finite variance, then the in Theorems 2.12 and 2.13 are given by with linearly independent with finite variances and if the return on security j, has a finite variance, then the in Theorems 2.12 and 2.13 are given by where is the ikth element of. where is the ikth element of. Hence given some knowledge of the joint distribution of a set of portfolio that span Hence given some knowledge of the joint distribution of a set of portfolio that span with, we can determining the and with, we can determining the and

Proposition 2.4: If contain no redundant securities, denotes the fraction of portfolio X allocated to security j, and denotes the fraction of any risk- averse investor s optimal portfolio allocated to security j, then for every such risk-averse investor Proposition 2.4: If contain no redundant securities, denotes the fraction of portfolio X allocated to security j, and denotes the fraction of any risk- averse investor s optimal portfolio allocated to security j, then for every such risk-averse investor

Because every optimal portfolio is an efficient portfolio and the holding of risky securities in every efficient portfolio are proportional to the holding in X. Because every optimal portfolio is an efficient portfolio and the holding of risky securities in every efficient portfolio are proportional to the holding in X. If there exist numbers where If there exist numbers where and,then the portfolio with proportions is called the Optimal Combination of Risky Assets. and,then the portfolio with proportions is called the Optimal Combination of Risky Assets. Proposition 2.5: If span, then is a convex set. Proposition 2.5: If span, then is a convex set.

Proof: Let Proof: Let and,. By substitution, the expression for Z can be rewritten as, where and,. By substitution, the expression for Z can be rewritten as, where.Therefore by Proposition 2.2, Z is an efficient portfolio. It follow by induction that for any integer k and number such that and.Therefore by Proposition 2.2, Z is an efficient portfolio. It follow by induction that for any integer k and number such that and is the return on an efficient portfolio. Hence, is a convex set. is the return on an efficient portfolio. Hence, is a convex set.

Definition: A market portfolio is defined as a portfolio that holds all available securities in proportion to their market values. Definition: A market portfolio is defined as a portfolio that holds all available securities in proportion to their market values. The equilibrium market value of a security for this purpose is defined to be the equilibrium value of the aggregate demand by individuals for the security. The equilibrium market value of a security for this purpose is defined to be the equilibrium value of the aggregate demand by individuals for the security. The market value of a security equals the equilibrium value of the aggregate amount of that security issued by business firms. The market value of a security equals the equilibrium value of the aggregate amount of that security issued by business firms.

We use denote the market value of security j and denote the value of the riskless security, then is the fraction of security j held in a market portfolio. We use denote the market value of security j and denote the value of the riskless security, then is the fraction of security j held in a market portfolio. Theorem 2.14: If is a convex set, and if the securities market is in equilibrium, then a market portfolio is an efficient portfolio. Theorem 2.14: If is a convex set, and if the securities market is in equilibrium, then a market portfolio is an efficient portfolio.

Proof: Let there be K risk averse investor in the economy.Define Proof: Let there be K risk averse investor in the economy.Define to be the return on investor k s optimal portfolio. In equilibrium,, where is the initial wealth of investor K, and. Define to be the return on investor k s optimal portfolio. In equilibrium,, where is the initial wealth of investor K, and. Define. By definition of a market portfolio Multiplying by and summing over j, it follows that. By definition of a market portfolio Multiplying by and summing over j, it follows that

because. Hence, is a convex combination of the returns on K efficient portfolios. Therefore, if is convex, then the market portfolio is contained in. The efficiency of the market portfolio provides a rigorous microeconomic justification for the use of a representative man to derive equilibrium prices in aggregated economic models. The efficiency of the market portfolio provides a rigorous microeconomic justification for the use of a representative man to derive equilibrium prices in aggregated economic models.

Proposition 2.6: In all portfolio models with homogeneous beliefs and risk-averse investors the equilibrium expected return on the market portfolio exceeds the return on the riskless security. Proposition 2.6: In all portfolio models with homogeneous beliefs and risk-averse investors the equilibrium expected return on the market portfolio exceeds the return on the riskless security. Proof: From the proof of Theorem 2.14 and Corollary 2.1., because,. Hence Proof: From the proof of Theorem 2.14 and Corollary 2.1., because,. Hence The market portfolio is the only risky portfolio where the sign of its equilibrium expected excess return can always be predicted. The market portfolio is the only risky portfolio where the sign of its equilibrium expected excess return can always be predicted.

Returning to the special case where is spanned by a single risky portfolio and the riskless security, the market portfolio is efficient. So the risky spanning portfolio can always be chosen to be the market portfolio. Returning to the special case where is spanned by a single risky portfolio and the riskless security, the market portfolio is efficient. So the risky spanning portfolio can always be chosen to be the market portfolio. Theorem 2.15: If span, then the equilibrium expected return on security j can be written as Theorem 2.15: If span, then the equilibrium expected return on security j can be written as where where

This relation, called the Security Market Line, was first derived by Sharpe. This relation, called the Security Market Line, was first derived by Sharpe. In the special case of Theorem 2.15, measure the systematic risk of security j relative to the efficient portfolio. In the special case of Theorem 2.15, measure the systematic risk of security j relative to the efficient portfolio. can be computed from a simple covariance between and. But the sign of can not be determined by the sign of the correlation coefficient between can be computed from a simple covariance between and. But the sign of can not be determined by the sign of the correlation coefficient between and and

Theorem 2.16: If contain no redundant securities, then (a) for each value are unique, (b) there exists a portfolio contained in with return X such that span, and (c) where, Theorem 2.16: If contain no redundant securities, then (a) for each value are unique, (b) there exists a portfolio contained in with return X such that span, and (c) where, Where denote the set of portfolios contained in such that there exists no other portfolio in with the same expected return and a smaller variance. Where denote the set of portfolios contained in such that there exists no other portfolio in with the same expected return and a smaller variance.

Proof: Let denote the ijth element of and denote the ijth element of. So all portfolios in with expect return u, we need solutions the problem Proof: Let denote the ijth element of and denote the ijth element of. So all portfolios in with expect return u, we need solutions the problem If then and If then and Consider the case when. The n first-order conditions are Consider the case when. The n first-order conditions are

Multiplying by and summing, we get Multiplying by and summing, we get By definition of, must be the same for all. Because is nonsingular, the linear equation has unique solution By definition of, must be the same for all. Because is nonsingular, the linear equation has unique solution This prove (a). From this solution we have This prove (a). From this solution we have are the same for every value. are the same for every value. Hence all portfolios in are perfectly correlated. Hence we can pick any Hence all portfolios in are perfectly correlated. Hence we can pick any

portfolio in with and call its return X. Then we have portfolio in with and call its return X. Then we have Hence span which proves (b). Hence span which proves (b). and from Corollary 2.13 and Proposition 2.3 (c) follows directly. and from Corollary 2.13 and Proposition 2.3 (c) follows directly.

From Theorem 2.16, will be equivalent to as a measure of a security s systematic risk provided that the chosen for X is such that. From Theorem 2.16, will be equivalent to as a measure of a security s systematic risk provided that the chosen for X is such that. Theorem 2.17: If span and if X has a finite variance, then is contained in. Theorem 2.17: If span and if X has a finite variance, then is contained in. Proof: Let. Let be the return on any portfolio in such that. By Corollary 2.13 Proof: Let. Let be the return on any portfolio in such that. By Corollary 2.13 where where

Therefore Therefore Thus Thus Hence, is contained in. Hence, is contained in. Theorem 2.18: If have a joint normal probability distribution, then there exists a portfolio with return X such that Theorem 2.18: If have a joint normal probability distribution, then there exists a portfolio with return X such that span. span.

Proof: construct a risky portfolio contained in, and call its return X. Define Proof: construct a risky portfolio contained in, and call its return X. Define by Theorem 2.16 part (c) and by construction. Because by Theorem 2.16 part (c) and by construction. Because are normally distributed, X will be normally distributed. Hence is normal distributed, and because, so they are independent. Therefore are normally distributed, X will be normally distributed. Hence is normal distributed, and because, so they are independent. Therefore, From Corollary 2.13 it follows that span, From Corollary 2.13 it follows that span

Theorem 2.19: If is a symmetric function with respect to all its arguments, then there exists a portfolio with return X such that span. Theorem 2.19: If is a symmetric function with respect to all its arguments, then there exists a portfolio with return X such that span. Proof: By hypothesis Proof: By hypothesis for each set of given values. Therefore every risk averse investor will choose. But this is true for all i. Hence, all investor will hold all risky securities in the same relative proportions. Then span for each set of given values. Therefore every risk averse investor will choose. But this is true for all i. Hence, all investor will hold all risky securities in the same relative proportions. Then span

The APT model developed by Ross provides an important class of linear-factor models that generate spanning without assuming joint normal probability distributions. The APT model developed by Ross provides an important class of linear-factor models that generate spanning without assuming joint normal probability distributions. If we can construct a set of m portfolios with returns such that and are perfectly correlated, then If we can construct a set of m portfolios with returns such that and are perfectly correlated, then will span will span

The APT model is attractive because the equilibrium structure of expected returns and risks of securities can be derived without explicit knowledge of investors preferences or endowments. The APT model is attractive because the equilibrium structure of expected returns and risks of securities can be derived without explicit knowledge of investors preferences or endowments.

For the study of equilibrium pricing, the usual format is to derive equilibrium given the distribution of. For the study of equilibrium pricing, the usual format is to derive equilibrium given the distribution of. Theorem 2.20: If denote a set of linearly independent portfolios that satisfy the hypothesis of Theorem 2.12, and all securities have finite variances, then a necessary condition for equilibrium in the securities market is that Theorem 2.20: If denote a set of linearly independent portfolios that satisfy the hypothesis of Theorem 2.12, and all securities have finite variances, then a necessary condition for equilibrium in the securities market is that the hypothesis of Theorem 2.12 the hypothesis of Theorem 2.12 where is the ikth element of where is the ikth element of

Proof: By linear independence by Theorem 2.12 where. Take expectations, we have Proof: By linear independence by Theorem 2.12 where. Take expectations, we have Noting that Noting that From Proposition 2.3 From Proposition 2.3 Thus Thus We can get We can get

Hence, from Theorem 2.20, a sufficient set of information to determine the equilibrium value of security j is the first and second moments for the join distribution of. Hence, from Theorem 2.20, a sufficient set of information to determine the equilibrium value of security j is the first and second moments for the join distribution of. Corollary 2.20a: If the hypothesized conditions of Theorem 2.20 hold and if the end-of-period value a security is given by Corollary 2.20a: If the hypothesized conditions of Theorem 2.20 hold and if the end-of-period value a security is given byTheorem 2.20 Theorem 2.20 then in equilibrium then in equilibrium This property of formula is called value additivity. This property of formula is called value additivity.

Corollary 2.20b: If the hypothesized conditions of Theorem 2.20 hold and if the end-of-period value of a security is given by, where and then in equilibrium Corollary 2.20b: If the hypothesized conditions of Theorem 2.20 hold and if the end-of-period value of a security is given by, where and then in equilibriumTheorem 2.20 Theorem 2.20 Hence, to value two securities whose end of period values differ only by multiplicative or additive noise, we can simply substitute the expected values of the noise terms. Hence, to value two securities whose end of period values differ only by multiplicative or additive noise, we can simply substitute the expected values of the noise terms.

Theorem 2.20 and its corollaries are central to the theory of optimal investment decisions by business firms. Theorem 2.20 and its corollaries are central to the theory of optimal investment decisions by business firms. Although the optimal investment and financing decisions by a form generally require simultaneous determination, under certain conditions the optimal investment decision can be made independently of the method of financing. Although the optimal investment and financing decisions by a form generally require simultaneous determination, under certain conditions the optimal investment decision can be made independently of the method of financing.

Theorem 2.21: If firm j is financed by q different claims defined by the function Theorem 2.21: If firm j is financed by q different claims defined by the function and if there exists an equilibrium such that the return distribution of the efficient portfolio set remains unchanged from the equilibrium in which firm j was all equity financed, then and if there exists an equilibrium such that the return distribution of the efficient portfolio set remains unchanged from the equilibrium in which firm j was all equity financed, then where is the equilibrium initial value of financial claim k. where is the equilibrium initial value of financial claim k.

Hence, for a given investment policy, the way in which the firm finances its investments changes the return distribution of the efficient portfolio set. Hence, for a given investment policy, the way in which the firm finances its investments changes the return distribution of the efficient portfolio set. Clearly, a sufficient condition for Theorem 2.21 to obtain is that each of the financial claims issued by the firm are redundant securities. Clearly, a sufficient condition for Theorem 2.21 to obtain is that each of the financial claims issued by the firm are redundant securities.

An alternative approach to the development of nontrivial spanning theorems is to derive a class of utility functions for investors. An alternative approach to the development of nontrivial spanning theorems is to derive a class of utility functions for investors. Such that even with arbitrary joint probability distributions for the available securities,investors within the class can generate their optimal portfolios from the spanning portfolios. Such that even with arbitrary joint probability distributions for the available securities,investors within the class can generate their optimal portfolios from the spanning portfolios.

Let denote the set of optimal portfolios selected from by investors with strictly concave von Neumann-Morgenstern utility functions. Let denote the set of optimal portfolios selected from by investors with strictly concave von Neumann-Morgenstern utility functions. Theorem 2.22 There exists a portfolio with return X such that span if and only if, where is the absolute risk-aversion function for investor Theorem 2.22 There exists a portfolio with return X such that span if and only if, where is the absolute risk-aversion function for investor in. in.

Because the b in the statement of Theorem 2.22 does not have a subscript, therefore all investors in must have virtually the same utility function. Because the b in the statement of Theorem 2.22 does not have a subscript, therefore all investors in must have virtually the same utility function. Cass and Stiglitz (1970) conclude: it is requirement that there be any mutual funds, and not the limitation on the number of mutual funds. Cass and Stiglitz (1970) conclude: it is requirement that there be any mutual funds, and not the limitation on the number of mutual funds. This is a negative report on the approach to developing spanning theorems. This is a negative report on the approach to developing spanning theorems.

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