C RAM 3002 C RAROC Rsk-Adjusted Return on Captal C C RAM Rsk-Adjusted erformance Measure C RAM RAM Bootstrap RAM C RAROC RAM Bootstrap F830.9 A CAM 2 CAM 3 Value at Rsk RAROC Rsk-Adjusted Return on Captal 200 RAROC RAROC RAM Rsk-Adjusted erformance Measure Sharpe
RAM= / 200 Artzner et al. 999 2 3 C Condtonal Value-at-Rsk C RAM 2 3 C C RAM RAM C. C RAM x x X y y Y x y ( f xy, ) f ( xy, ) ξ ψ ( x, ξ) = p ( y) dy () f x, y ξ 0, ( x) mn { R: ( x, ) } ξ = ξ ψ ξ (2) ( ) (, ) C C x = f ( x, y) ( x) f x y p y dy (3) φ ξ C C C 2 3 C C 2 2
3 C RI C.2 RAM = RI ξ (4) C = RI φ (5) RI ~ N ( µ, σ ) c = 2 erf (2 ) 2 d = { 2π exp[ erf (2 )] ( ) } 2 x 2 t erf ( x) = e dt 0 π ξ = µ + c σ (6) C 2 φ = µ + d σ (7) erf 95% ( x ) d > 0 d > c > ( ) c ( > ) > 0 x µ = = µ + c( σ ) + c ( σ / µ ) C ( ) = + d ( σ / µ ) σ σ > j j C C < ( 2 ) > j ( 2 ) > j µ µ C 2 j 2 j C ( ) > C.3 C 4 C 2 C 95% 95% 97.5% 99% n d = k k 2 k d = 0 d = d = n 3
D 48 D = k k (8) = 2 D D ξ C Rockafellar 970 3 T ( x, ) ( ) T f ( x, y) + φ ξ = ξ + ξ (9) RAM RAM 5 2002 4 ~2004 2 3 3 2 2002 48 6 www.huaan.com.cn C - 500 Bootstrap 48 3 56 56 2. Bootstrap Bootstrap Bookstaber & McDonld 987 2 500 250000 250000 250 000 250 000 B-M IID B-M 48 56 45 45 2 000 000 Bootstrap RI ξ φ 4
ˆ j 000 2 000 RIˆ ˆ = RIj ˆ 000 ˆ ξ = ξj ˆ 000 ˆ φ = φj ˆ C j k ˆD ˆ j= j= j= 3 2 000 2 Bootstrap 2 2 ˆ C j ˆ j ˆ k ˆD RIˆ ˆ ξ ˆ φ 2 RIˆ ˆ ξ ˆ φ ˆ C ˆ j j ˆ k ˆD 2 2 RIˆ ˆ ξ ˆ φ ˆ C ˆ j j ˆ k ˆD 3 3 3 0 2.2 Bootstrap 2 48 000 56 48 45 2 Kolmogorov-Smnov 0.65 92.43% 0.99 48 Shapro-Wlk 0.07 36 0.05 75% 0.05 C C C C C C C -C 99%-97.5% 99%-95% 97.5%-95% 99%-97.5% 99%-95% 97.5%-95% 99% 97.5% 95% D 46 72 40 26 36 22 26 32 44 Bootstrap 2 3 2 000 000 000 2 2 5
2 RIˆ ˆ ξ ˆ φ ˆ j ˆ C j RI ˆ 0.00797 0.00007-0.07499 0.00474 0.00042 ˆ ξ 0.0024 0.0006-0.09996 0.03089 0.000 ˆ φ 0.576 0.0002-0.00004 0.02827 0.00074 ˆ j 0.08777 0.00077-0.0453-0.000 0.00484 ˆ C j 0.0733 0.00062-0.03577 0.00264 0.00392 2 2 RIˆ ˆ ξ ˆ φ 48 5 0.366 ˆ j ˆ C j 5 4 5.83% 8.6% 3.9% 4.5% 3 95% C 3 Bootstrap 95% 99%-97.5% 38.004.9255-0.074 0.0460 2 34 42 99%-95% 63.3.8747-0.262-0.0400 0 60 66 97.5%-95% 37.658.547 0.0057-0.6940 8 36 40 99%-97.5% 2.28 2.2625-0.266-0.0457 2 6 26 C 99%-95% 37.238.9378-0.4276-0.270 0 34 40 97.5%-95% 23.26 2.0488 0.0458-0.2868 0 20 28 99% 23.954 2.4609-0.347-0.846 4 8 28 -C 97.5% 30.758.8398 0.0379-0.524 0 28 34 95% 39.76.672 0.539-0.3623 8 38 42 3 C C 99.9% 7.5% 97.5% 7.84% 95% 25.67% 95% C 47.2% 7.8% C 6
C C 50 40 30 20 0 0 3 5 7 9 3 5 7 9 2 23 25 27 29 3 33 35 37 39 4 43 45 47 99% 97.50% 95% C 3 C RAROC C C Bootstrap C C 2 C RAM C []Artzner, Delbaen F, Eber J M, Heath D. Coherent measures of rsk[j]. Mathematcal Fnance, 999(9): 203-228. [2]Bookstaber R M, J B McDonald. A General Dstrbuton for Descrbng Securty rce Returns[J]. Journal of Busness, 987(7): 40-424. [3]Ippolto, Rchard A. On Studes of Mutual Fund erformance, 962-99[J]. Fnancal Analysts Journal, 993(-2): 42-50. [4]Katerna Smons. Rsk-Adjusted erformance of Mutual Funds[J]. New England Economc Revew, 998(9-0): 33-48. [5]Kent Danel, Mark Grnblatt, Sherdan Ttman, Russ Wermers. Measurng Mutual Fund erformance wth Characterstc-Based Benchmarks[J]. The Journal of Fnance, 997(7-8): 035-058. 7
[6]Kevn Dowd. Adjustng for Rsk: An Improved Sharpe Rato[J]. Internatonal Revew of Economcs and Fnance 2000(3): 209-222. [7]Lehmann B, D Modest. Mutual Fund erformance Evaluaton: A Comparson of Benchmarks and Benchmark Comparsons[J]. The Journal of Fnance, 987(2): 233-265. [8]Mark Grnblatt, Sherdan Ttman. ortfolo erformance Evaluaton: Old Issues and New Insghts[J]. The Revew of Fnacal Studes, 989(2): 393-42. [9]Rockafellar R T, S Uryasev. Condtonal Value-at-Rsk for General Loss Dstrbutons[J]. Journal of Bankng and Fnance, 2002(7): 443-47. [0]Zhwu Chen, eter J Knez. ortfolo erformance Measurement: Theory and Applcaton[J]. The Revew of Fnancal Studes, 996(8): 5-555. [][ ]. hlppe Joron. Fnancal Rsk Manager Handbook [M]., 2004: 44-445 [2],. C [J]., 2004(8): 38-40. [3]. [EB]., 2, www.sse.com.cn/sseportal/ps/zhs/yjcb/szyj.shtml RAM to Fund erformance Evaluaton: Usng C and Usng Fang Y, Zhang Yshan (Busness School of Jln Unversty, Jln Changchun 3002, Chna) Abstract: The paper ntroduces C nto RAROC (Rsk-Adjusted Return on Captal) to evaluate mutual fund performance. And t compares the results of RAROC by usng C and usng. It can learn from that, n the condton of normal dstrbuton, both the RAM (Rsk-Adjusted erformance Measure) by usng C and that by usng are suffcent, relant and effectve, and the two measures are equvalent. But the RAM by usng C s more suffcent, more conservatve and more relant than that by usng n the condton of abnormal dstrbuton. What s more, we do emprcal study wth bootstrap and ndcate that the concluson have great value on general RAM. Key words: C RAROC RAM. Bootstrap 2005-0-6 949-8
Roll SML 2 3 Roll 978 β SML SML 4 999 Artzmer 5 998 0 9 6 9