A Dynamic CAPM with Supply Effect: Theory and Empirical Results

1. A Dynamic CAPM with Supply Effect: Theory and Empirical Results Dr. Cheng-Few Lee Rutgers University Chiung-Min Tsai Central Bank of China (Taiwan) Alice C. Lee San Francisco State University

2. Outline • I. INTRODUCTION • II. DEVELOPMENT OF MULTIPERIOD ASSET PRICING MODEL WITH SUPPLY EFFECT • III. DATA and EMPIRICAL RESULTS • IV. SUMMARY • Appendix A Modeling the Price Process • Appendix B. Identification of the Simultaneous Equation System • Appendix C. Modeling the Dividend Processes • Appendix D Granger Causality Test for Pt and dt

3. I. INTRODUCTION

4. I. Introduction • Black (1976) extends the static CAPM by explicitly allowing for the endogenous supply effect of risky securities to derive the dynamic asset pricing model • We first theoretically extend the Black’s dynamic, simultaneous CAPM to be able to test the existence of the supply effect in the asset pricing determination process

5. I. Introduction • Outstanding Shares • Black (1976) • Lee and Gweon (1986) • Trading Volumes • Campbell, Grossman, and Wang, 1993 • Lo and Wang, 2000

6. II. DEVELOPMENT OF MULTIPERIOD ASSET PRICING MODEL WITH SUPPLY EFFECT • A. The Demand Function for Capital Assets • B. Supply Function of Securities • C. Multiperiod Equilibrium Model • D. Derivation of Simultaneous Equations System • E. Test of Supply Effect

7. A. The Demand Function for Capital Assets (1) , where the terminal wealth Wt+1 =Wt(1+ Rt); Wt is initial wealth; and Rt is the rate of return on the portfolio. The parameters, a, b and h, are assumed to be constants.

8. A. The Demand Function for Capital Assets (2) Xj, t+1 = Pj, t+1 – Pj, t + Dj, t+1 , j = 1, …, N, where Pj, t+1 = (random) price of security j at time t+1, Pj,t = price of security j at time t, Dj, t+1 = (random) dividend or coupon on security at time t+1,

9. A. The Demand Function for Capital Assets (3) xj,t+1= Et Xj, t+1= EtPj, t+1 – Pj, t + E t Dj, t+1 , j = 1, …, N, where EtPj, t+1 = E(Pj, t+1 |Ωt), Et Dj,t+1 = E(Dj, t+1 |Ωt), and EtXj,t+1 = E(Xj,t+1|Ωt); Ωt is given information available at time t.

10. A. The Demand Function for Capital Assets (4) wt+1 = EtW t+1 = Wt + r* ( Wt – q t+1’P t) + qt+1’ xt+1, where Pt= (P1, t, P2, t, P3, t, …, PN, t)’, xt+1= (x 1,t+1, x 2,t+1, x 3,t+1, …, x N, t+1)’ = E tPt+1 – Pt + E tD t+1, qt+1 = (q 1,t+1, q 2,t+1, q 3,t+1, …, q N, t+1)’, qj,t+1 = number of units of security j after reconstruction of his portfolio, and r* = risk-free rate.

11. A. The Demand Function for Capital Assets (5) V(Wt+1 ) = E (Wt+1 – wt+1 ) ( Wt+1 – wt+1 )’ = q t+1’ Sq,t+1, where S = E (Xt+1 – xt+1 ) ( Xt+1 – xt+1 )’ = the covariance matrix of returns of risky securities.

12. A. The Demand Function for Capital Assets (6) (7) Max. (1+ r*) Wt + q t+1’ (xt+1 – r* P t) – (b/2) q t+1’ S q t+1. (8) q t+1 = b-1S-1 (xt+1 – r* P t). (9) where c = Σ (bk)-1.

13. B. Supply Function of Securities (10) Min. EtDi,t+1 Qi, t+1 + (1/2) (ΔQi,t+1’ AiΔQi, t+1), subject to Pi,t ΔQ i, t+1 = 0, (11) ΔQ i, t+1 = Ai-1 (λi Pi, t - EtDi, t+1) where λi is the scalar Lagrangian multiplier.

14. B. Supply Function of Securities (12) ΔQ t+1 = A-1 (B P t - EtDt+1), where , , and .

15. C. Multiperiod Equilibrium Model (9) Qt+1 = cS-1 ( EtPt+1− (1+ r*)Pt+ EtDt+1), (12) ΔQ t+1 = A-1 (B Pt− EtDt+1). (13) cS-1[EtPt+1−Et-1Pt −(1+r*)(Pt − Pt-1) +EtDt+1 − Et-1Dt]=A-1(BPt − EtDt+1) +Vt,

16. C. Multiperiod Equilibrium Model (13’) cS-1[Et-1Pt+1-Et-1Pt -(1+r*)( Et-1Pt - Pt-1) +Et-1Dt+1 - Et-1Dt] = A-1(B Et-1Pt - Et-1Dt+1). (14) [(1+ r*)cS-1 + A-1B] (Pt− Et-1Pt) = cS-1(EtPt+1− Et-1Pt+1) + (cS-1+ A-1) (EtDt+1− Et-1Dt+1)−Vt

17. C. Multiperiod Equilibrium Model (15) Pt− Et-1Pt = (1+ r*)-1(EtPt+1− Et-1Pt+1) + (1+r*)-1(Et Dt+1− Et-1Dt+1) + Ut, where Ut = −c-1SVt.

18. D. Derivation of Simultaneous Equations System (16) − (r*cS-1 + A-1B) (Pt − Pt-1) + (cS-1 + A-1) (EtDt+1 − Et-1Dt+1) = Vt. (17) G pt + H dt = Vt , where G = − (r*cS-1 + A-1B), H = (cS-1+A-1), dt = EtDt+1− Et-1Dt+1, pt = Pt − Pt-1,

19. D. Derivation of Simultaneous Equations System (18) pt = Πdt + Ut, where Π is a n-by-n matrix of the reduced form coefficients, and Ut is a column vector of n reduced form disturbances. Or (19) Π = −G-1 H, and Ut = G-1 Vt.

20. E. Test of Supply Effect (20) , Since Π = − G-1 H in equation (21), Π can be calculated as:

21. E. Test of Supply Effect (21)

22. E. Test of Supply Effect (22)

23. E. Test of Supply Effect • Hypothesis 1: All the off-diagonal elements in the coefficient matrix Π are zero if the supply effect does not exist • Hypothesis 2: All the diagonal elements in the coefficients matrix Π are equal in the magnitude if the supply effect does not exist

24. III. DATA and EMPIRICAL RESULTS A. International Equity Markets –Country Indices 1. Data and descriptive statistics 2. Dynamic CAPM with supply effect B. United States Equity Markets – S&P500 1 Data and descriptive statistics 2 Dynamic CAPM with Supply Side Effect

25. A. International Equity Markets –Country Indices 2. Dynamic CAPM with supply effect Hypothesis 1: pi, t = βidi, t + Σj≠i βjdj, t + εi, t, i, j = 1, …,16. Hypothesis 2 H0: πi,i = πj,j for all i, j=1, …, 16

26. IV. SUMMARY

27. IV. SUMMARY • Focus on firm’s financing decision concerning the supply of risky securities into the CAPM: • by explicitly introducing the firm’s supply of risky securities into the static CAPM and allowing supply of risky securities to be a function of security price • The expected returns are endogenously determined by both demand and supply decisions within the model • Our objectives are to investigate the existence of supply effect in both international equity markets and U.S. stock markets

28. IV. SUMMARY • The test results show that the two null hypotheses of the non-existence of supply effect do not seem to be satisfied jointly in both our data sets. • In other words, this evidence seems to be sufficient to support the existence of supply effect • Thus, imply a violation of the assumption in the one period static CAPM, or • A dynamic asset pricing model may be a better choice in both international equity markets and U.S. domestic stock markets

29. Table 1.World Indices andCountry Indices List I. World Indices

30. Table 1.World Indices andCountry Indices List II. Country Indices

31. Table 2: Summary Statistics of Monthly Return1, 2 Panel A: G7 and World Indices

32. Table 2: Summary Statistics of Monthly Return1, 2 Panel B: Emerging Markets Notes: 1 The monthly returns from Feb. 1988 to March 2004 for international markets. 2 * and ** denote statistical significance at the 5% and 1%, respectively.

33. Table 3: Coefficients for matrix П (all sixteen markets)

34. Table 4

35. Table 4

36. Table 5: Test of Supply Effect on off-Diagonal Elements of Matrix П1, 2 • Notes: • pi, t = βi’di, t • + Σj≠i βj’dj, t + ε’i, t, • i, j = 1, …,16. • Null Hypothesis: • all βj = 0, j=1,…, 16, j ≠i • 2 The first test uses an F distribution with 15 and 172 degrees of freedom, and the second test uses a chi-squared distribution with 15 degrees of freedom.

37. Table 6: Characteristics of Ten Portfolios Notes: 1The first 30 firms with highest payout ratio comprises portfolio one, and so on. 2The price, dividend and earnings of each portfolio are computed by value-weighted of the 30 firms included in the same category. 3The payout ratio for each firm in each year is found by dividing the sum of four quarters’ dividends by the sum of four quarters’ earnings, then, the yearly ratios are then computed from the quarterly data over the 22-year period.

38. Table 7: Summary Statistics of Portfolio Quarterly Returns1 Notes: 1 Quarterly returns from 1981:Q1to 2002:Q4 are calculated. 2 * and ** denote statistical significance at the 5% and 1%, respectively.

39. Table 7: Coefficients for matrix П’ (10 portfolios)* Table 8: Coefficients for matrix П’ (10 portfolios)*

40. Table 9: Test of Supply Effect on off-Diagonal Elements of Matrix П1, 2, 3 Notes: 1 pi, t = βi’di, t + Σj≠i βj’dj, t + ε’i, t, i, j = 1, …,10. Hypothesis: all βj = 0, j=1,…, 10, j ≠i 2 The first test uses an F distribution with 9 and 76 degrees of freedom, and the second uses a chi-squared distribution with 9 degrees of freedom. 3 * and ** denote statistical significance at the 5 and 1 percent level, respectively.

41. Figure 1Comparison of S&P500 and Market portfolio

42. Figure 2Return and Payout Return = 0.04478 – 0.01490×Payout (R2 =0.4412) (0.0030) (0.0059) [14.71]** [-2.62]*

43. Figure 3Payout and Beta Payout = 1.3444 – 0.8683×Beta (R2 =0.4147) (0.3672) (0.3648) [3.662]** -2.381]*

44. 0.050 0.045 0.040 R e t u r n 0.035 0.030 Betaa 0.025 0.500 0.600 0.700 0.800 0.900 1.000 1.100 1.200 1.300 Figures 4Return and Beta Return = 0.0202 + 0.0175×Beta (R2 =0.3359) (0.0088) (0.0087) [2.300] [-2.012]

45. Appendix A Modeling the Price Process In Section B, equation (16) is derived from equation (15) under the assumption that all countries’ index series follow a random walk process. Thus, before further discussion, we should test the order of integration of these price series. Two widely used unit root tests are the Dickey-Fuller (DF) and the augmented Dickey-Fuller (ADF) tests. The former can be represented as: Pt = μ + γ Pt-1 + εt, and the latter can be written as: ∆Pt = μ + γ Pt-1 + δ1 ∆Pt-1 + δ2 ∆Pt-2 +…+δp ∆Pt-p + εt. The results of the tests for each index are summarized in Table A.1 It seems that one cannot reject the hypothesis that the index follows a random walk process. In the ADF test the null hypothesis of unit root in level can not be rejected for all indices whereas the null hypothesis of unit root in the first difference is rejected. This result is consistent with most which conclude that the financial price series follow a random walk process.

46. Appendix A Modeling the Price Process Similarly, in the U.S. stock markets, the Phillips-Perron test is used to check the whether the value-weighted price of market portfolio follows a random walk process. The results of the tests for each index are summarized in Table A.2. It seems that one cannot reject the hypothesis that all indices follow a random walk process since, for example, the null hypothesis of unit root in level cannot be rejected for all indices but are all rejected if one assumes there is a unit root in the first order difference of the price for each portfolio. This result is consistent with most studies concluding the financial price series follow a random walk process.

47. Appendix A Modeling the Price Process Table A.1 Unit root tests for Pt * 5% significant level; ** 1% significant level

48. Appendix A Modeling the Price Process Table A.2Unit root tests for Pt Note: 1. * 5% significant level; ** 1% significant level 2. The process assumed to be random walk without drift. 3. The null hypothesis of zero intercept terms, μ, can not be rejected at 5%, 1% level for all portfolio.

49. g11 g12 …… g1n h11 h12 …… h1n g21 g22 …… g1n h21 h22 …… h2n . . . . gn1 gn2 …… gnn hn1 hn2 …… hnn where A = [G H] = π11 π12 …… π1n 1 0 …… 0 ’ π21 π22 …… π1n 0 1 …… 0 . . . . πn1πn2 …… πnn 0 0 …… 1 W = [Π I n]’ = Appendix B. Identification of the Simultaneous Equation System Note that given G is nonsingular, Π = −G-1 H in equation (19) can be written as (B-1) AW = 0

50. Appendix B. Identification of the Simultaneous Equation System That is, A is the matrix of all structure coefficients in the model with dimension of n times 2n and W is a 2n times n matrix. The first equation in (A. 1) can be expressed as (B-2) A1W = 0, where A1 is the first row of A, i.e., A1= [g11 g12….g1n h11 h12…..h1n].