Introduction to Portfolio Selection and Capital Market Theory: Static Analysis

1. Introduction to Portfolio Selection and Capital Market Theory: Static Analysis BaoheWang baohewang0592@sina.com

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

3. One-period Portfolio Selection • 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. • The random variable one-period return per dollar on security j is denoted

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

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

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

7. 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. • 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.

8. 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: (2.4)

9. The first-order conditions can be written 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.

10. 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, could be imposed on . • 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, .

11. 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; (2) his initial wealth; (3) the join distribution for the securities’ returns.

12. 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 .

13. 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

14. is positive, and such investor are call risk averse. • 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

15. 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 . • We can proof follows directly by Jensen’s inequality: if is strictly concave • Because U is an increase function, So

16. 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.

17. 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.

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

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

20. 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.

21. 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) Proof: By hypothesis If then trivially . But is a feasible portfolio and is an efficient portfolio. By contradiction,

22. 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, .

23. 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 If then will satisfy because the property of U, so this solution is unique.

24. 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.

25. 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. 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

26. Hence So 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 .

27. 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. • But this rule is not apply to individual securities or inefficient portfolios.

28. 2.3 Risk Measures for Securities and Portfolios in The One-Period model • In this section, a second definition of increasing risk is introduced. • is the random variable return per dollar on an efficient portfolio K. • denote an increasing strictly concave function such that for • Random variable

29. 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 .

30. 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 and

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

32. 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: • The first order condition:

33. If then the solution is . • 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.

34. 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.

35. 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.

36. Lemma 2.1: (i) for efficient portfolio K. (ii) If then (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.

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

38. Property 2: If L and K are efficient portfolios, then and . • 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 4: Let p and q denote any two feasible portfolios and let K and L denote any two efficient portfolios. if and only if

39. Proof: From Property 1, we have • 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, where and for every efficient portfolio L.

40. 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. But implies for every efficient portfolio L. • Property 6: If a feasible portfolio p has portfolio weight ,then

41. 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. • measure provides a complete ordering. • 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.

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

43. 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.

44. 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 . Where , is the realized return on an efficient portfolio.

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

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

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

48. 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.

49. 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

50. 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.