To Rebalance or Not to Rebalance? Edward Qian, PhD, CFA PanAgora Asset Management

Transcription

1 To Rebalance or Not to Rebalance? Edward Qan, PhD, CFA PanAgora Asset anagement

2 To Rebalance or Not to Rebalance It s not THE QUESTION but a very mportant one»to rebalance fxed-weght (FW); Not to Buy and hold (BH)» Passve (nactve) versus actve»effcent market theory versus market neffcency»tradtonal cap-weghted ndces versus alternatve betas»asset allocaton FW polcy versus asset-level BH benchmarks

3 To Rebalance or Not to Rebalance There have been no satsfactory answers»does FW portfolos have hgher returns?» Is dversfcaton return real or magnary?»does FW portfolos have lower rsks?»the effects of mean-revertng or trendng on portfolo rebalancng» FW sells wnners and buy losers (n long-only portfolos)»effects of portfolo rebalancng for long-short portfolos?»should we care more about termnal wealth?

4 To Rebalance or Not to Rebalance Outlne»Drect comparson between FW and BH portfolos»termnal wealth nstead of average returns»expected value and varance of termnal wealth wealth Sharpe rato»long-only portfolos and long-short portfolos»effects of seral correlaton (a hard problem)» Qan, Edward, 04, To Rebalance or Not to Rebalance: A Statstcal Analyss of Termnal Wealth of Fxed-weght and Buy-and-Hold Portfolos, avalable at 3

5 Rebalancng Return A smple experment»two securtes A and B go up and down wth zero cumulatve return A B»Portfolo rebalancng generates postve return Year Rebalance Year A $50 $00 $6.5 (50%) $3.5 B $50 $5 $6.5 (50%) $5 Total $00 $5 (5%) $5 $56.5 (5%) 4

6 Rebalancng Return A more realstc experment»s&p 500 sector portfolos: rebalancng always leads to hgher return» Annual returns from for 0 S&P sectors» 50,000 randomly generated portfolos» Alpha = annual return wth rebalancng mnus return wth buy-and-hold 5

7 Dversfcaton Return Dversfcaton return s not rebalancng return»arthmetc mean»geometrc mean r r r g r g / DR g p N w g 0»Dversfcaton return s not return between two real portfolos N» w g IS NOT the geometrc mean of the buy-and-hold portfolo» Qan, Edward, Dversfcaton Return and Leveraged Portfolos, The Journal of Portfolo anagement, Summer 0, Vol. 38, No. 4: pp

8 Termnal Wealth Notatons» assets/n perods, return of th asset n perod n: r n»expected return ndependent of n return vector: μ E r,,,, n, N n,»covarances ndependent of n covarance matrx: E r r,, j,,, n, N n jn j j,»no seral correlaton between returns of dfferent tme perod Σ»Intal portfolo weghts w w,, wn 7

9 Termnal Wealth Notatons»Expected return of the FW portfolo μ p = w μ + w μ + + w μ =»Volatlty of the FW portfolo w μ = w μ. σ p =,j = w w j σ j = w Σw. 8

10 Termnal Wealth Termnal wealth of $ nvestment»fw portfolo product of perod returns N W FW = + w r + w r N = + w r n. n=»bh portfolo weghted sum of termnal wealth W BH = w + r + r N + + w + r + r N N W BH = w + r n n=. 9

11 Termnal Wealth Expected termnal wealth»fw portfolo E W FW = E + w r + w r N = + μ p N N»BH portfolo E W FW = + w μ E W BH = w E + r + r N + + w E + r + r N E W BH = w + μ N. 0

12 Termnal Wealth Expected termnal wealth»theorem: for long-only portfolos,.e., the expected termnal wealth of the BH portfolo s hgher than that of the FW portfolo E W BH E W FW»Proof by Jensen s nequalty (convex functon) w 0, N w + μ N + w μ w»intuton: don t sell wnners f wnners keep on wnnng = f x = + x N

13 Termnal Wealth Jensen s nequalty»for long-only portfolos,.e.,»what about long-short portfolos? w + μ N w 0, + w Is Jensen's nequalty stll true here? = w μ N μ μ

14 Long-Short Portfolos What about long-short portfolos?»short postons: negatve weghts» Short sellng: borrow shares to sell» Borrow money to buy assets» Invest wth dervatves (futures)»athematcally, we stll have w»portfolo leverage f some weghts are negatve = L = w > 3

15 Long-Short Portfolos Weghts of L/S portfolos»l/s 0/0 portfolo wth securty A and B» A returns 00% and B returns -50% Year Year Rebalance A($0/0%) $40 04% $76 (0%) (Buy) B(-$0/-0%) -$0-4% -$46 (-0%) (Sell) Total $00(40%) $30 00%(08%) $30»Portfolo grows from $00 to $30» Leverage decreases from 40% to 08%» Rebalancng requres buyng the wnner and sell the loser 4

16 Long-Short Portfolos Weghts of L/S portfolos»l/s 0/0 portfolo wth securty A and B» A returns -50% and B returns 00% Year Year Rebalance A($0/0%) $60 300% $4(0%) (Sell) B(-$0/-0%) -$40-00% -$4 (-0%) (Buy) Total $00(40%) $0 00%(500%) $0»Portfolo drops from $00 to $0» Leverage ncreases from 40% to 500%!» Rebalancng requres buyng the wnner and sell the loser 5

17 Long-Short Portfolos Weghts of L/S portfolos»when L/S portfolos have gans (losses), leverage decreases (ncreases)» When a L/S portfolo s postoned correctly,.e., long hgher return assets and short lower return assets, ts leverage decreases.» When a L/S portfolo s postoned wrongly,.e., long lower return assets and short hgher return assets, ts leverage ncreases!»buy-and-hold (passve) and leverage don t mx»fw mght perform better than BH E W BH E W FW 6

18 Long-Short Portfolos Expected termnal wealth»theorem: If w < 0, w 0, =,,. And w =»In addton, μ μ, =,,, μ > μ for some»or μ μ, =,, μ < μ for some,»then N w + μ N + w μ E W BH E W FW 7

19 Return Long-Short Portfolos Applcaton»The result can be extended to cases wth more than one short assets»practcal applcaton: Rsk Party portfolos» Long rsky assets: equty, nterest rates, commodtes, etc.» Leveraged by shortng cash 8% Rsk Party Lne and Tradtonal Fronter 7% Rsk Party Lne 6% 5% 4% 3% Levered Rsk Party 5/75 Rsk Party 60/40 % % 0% 5% 0% 5% 0% Rsk 8

20 Termnal Wealth Expected varance»expected value of termnal wealth E W BH E W FW» Long-only portfolos:» Long-short portfolos E W BH E W FW»But varance s also mportant n any nvestment analyss (rsk/return framework) var W BH and var W FW» What about? 9

21 Termnal Wealth Expected varance»statstcal calculaton var x = E x E x.»fw portfolos var W FW = + μ p + σ N p + μ N p var W FW = N n= C N n + μ p (N n) σ p n»bh portfolos var W BH = w w j + μ + μ j + σ j N w + μ N,j = var W BH = N C N n w w j + μ + μ N n n j σ j. n=,j = 0

22 Termnal Wealth Expected varance specal case»theorem: When and weghts and covarances are non-negatve»then var W BH var W FW»In general, BH long-only portfolos varance of termnal wealth s hgher than that of FW portfolos.

23 Rsk-adjusted Termnal Wealth Wealth-volatlty rato E W std W»Example: 0 securtes wth equal expected return (8%), equal volatlty (0%), equal par-wse correlaton (ρ); ntal weght 0% each

24 Rsk-adjusted Termnal Wealth E W Wealth-volatlty rato FW std W > E W BH FW std W FW»Example: 0 securtes wth equal expected return (8%), equal volatlty (0%), equal par-wse correlaton (ρ=0) 3

25 Rsk-adjusted Termnal Wealth Wealth Sharpe rato SR W = E W + μ 0 N std W»Example: assets one rsk-free wth % return and the other 0% rsk and 8% return; ntal weght 50% each 4

26 Effects of Seral Correlatons Long-only portfolos»ean-revertng gves FW portfolos an edge; trendng or momentum gves BH portfolos an edge»example: assets one rsk-free wth % return and the other 0% rsk and 8% return; ntal weght 50% each E W BH E W FW f ρ < μ μ 0 σ 5

27 Conclusons To rebalance or not to rebalance» Long-only portfolos E W BH E W FW var W BH var W FW» FW tends to have hgher rsk-adjusted termnal wealth» Long-short portfolos E W BH E W FW var W BH? var W FW» Buy-and-hold and leveraged portfolo s not a good combnaton» Seral correlaton» ean-revertng s benefcal to FW long-only portfolos; trendng s benefcal to BH long-only portfolos» For long-short portfolos, tmes seres trendng and cross-sectonal reversal s the best. 6

28 Conclusons Practcal mplcatons To Rebalance!»Investors have fxed-weght asset allocaton portfolos but buyand-hold ndces» The practce s rather nconsstent»captalzaton-weghted ndces are BH and they often underperformed naïve equally-weghted portfolo and other knds of alternatve ndces» Cap-weghted ndces are not dversfed (not passve)» Cap-weghted ndces are not rebalanced»long-short portfolos must be rebalanced regularly 7

29 8 Dversfcaton return»arthmetc mean»geometrc mean Appendx r r r r g / g p N p N p p p g w w g 0 p N N p w w g g