To Rebalance or Not to Rebalance?

Transcription

1 To Rebalance or Not to Rebalance? Edward Qian, PhD, CFA PanAgora Asset Management Northfield Research Conference 04 Stowe, Vermont

2 To Rebalance or Not to Rebalance It is not the question but still»to rebalance fixed-weight (FW); Not to Buy and hold (BH)» Passive (inactive) versus active»efficient market theory versus market inefficiency»traditional cap-weighted indices versus alternative betas»asset allocation FW policy versus asset-level BH benchmarks

3 To Rebalance or Not to Rebalance There have been no satisfactory answers»does FW portfolios have higher returns?» Is diversification return real or imaginary?»does FW portfolios have lower risks?»the effects of mean-reverting or trending on portfolio rebalancing» FW sells winners and buy losers (in long-only portfolios)»effects of portfolio rebalancing for long-short portfolios?»should we care more about terminal wealth?

4 To Rebalance or Not to Rebalance Outline»Direct comparison between FW and BH portfolios»terminal wealth instead of average returns»expected value and variance of terminal wealth wealth Sharpe ratio»long-only portfolios and long-short portfolios»effects of serial correlation (a hard problem)» Qian, Edward, 04, To Rebalance or Not to Rebalance: A Statistical Analysis of Terminal Wealth of Fixed-weight and Buy-and-Hold Portfolios, available at

5 Rebalancing Return A simple experiment»two securities A and B go up and down with zero cumulative return A B»Portfolio rebalancing generates positive 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%)

6 Rebalancing Return A more realistic experiment»s&p 500 sector portfolios: rebalancing always leads to higher return» Annual returns from for 0 S&P sectors» 50,000 randomly generated portfolios» Alpha = annual return with rebalancing minus return with buy-and-hold

7 Terminal Wealth Notations»M assets/n periods, return of ith asset in period n: r in»expected return independent of n return vector: μμ E ( r ) = µ, i =,, M, n =, N in i,»covariances independent of n covariance matrix: E [( r µ )( r µ )] = σ, i, j =,, M, n =, N in i jn j ij,»no serial correlation between returns of different time period Σ»Initial portfolio weights w, ( w, w ) = N

8 Terminal Wealth Notations»Expected return of the FW portfolio μμ pp = ww μμ + ww μμ + + ww μμ = ww ii μμ ii = ww μμ.»volatility of the FW portfolio σσ pp = ww ii ii,jj = ww jj σσ iiii = ww Σww.

9 Terminal Wealth Terminal wealth of $ investment»fw portfolio product of period returns NN WW FW = + ww ii rr ii + ww ii rr iiii = + ww ii rr iiii. nn=»bh portfolio weighted sum of terminal wealth WW BH = ww ( + rr ) ( + rr NN ) + + ww ( + rr ) ( + rr ) NN WW BH = ww ii ( + rr iiii ). nn=

10 Terminal Wealth Expected terminal wealth»fw portfolio E(WW FW ) = E + ww ii rr ii + ww ii rr iiii = + μμ pp NN»BH portfolio NN E(WW FW ) = + ww ii μμ ii E(WW BH ) = ww E[( + rr ) ( + rr NN )] + + ww E[( + rr ) ( + rr )] E(WW BH ) = ww ii ( + μμ ii ) NN.

11 Terminal Wealth Expected terminal wealth»theorem: for long-only portfolios, i.e., the expected terminal wealth of the BH portfolio is higher than that of the FW portfolio E(WW BH ) E(WW FW )»Proof by Jensen s inequality (convex function) NN ww ii ( + μμ ii ) NN + ww ii μμ ii ww ii 0, ww ii =»Intuition: don t sell winners if winners keep on winning f(xx) = ( + xx) NN

12 Terminal Wealth Jensen s inequality»for long-only portfolios, i.e., ww ii ( + μμ ii ) NN»What about long-short portfolios? ww ii 0, ww ii + ww ii μμ ii Is Jensen's inequality still true here? = NN μμ μμ

13 Long-Short Portfolios What about long- short portfolios?»short positions: negative weights» Short selling: borrow shares to sell» Borrow money to buy assets» Invest with derivatives (futures)»mathematically, we still have ww ii =»Portfolio leverage if some weights are negative LL = ww ii >

14 Long-Short Portfolios Weights of L/S portfolios»l/s 0/0 portfolio with security 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»Portfolio grows from $00 to $30» Leverage decreases from 40% to 08%» Rebalancing leads to releveraging and buying the winner and sell the loser

15 Long-Short Portfolios Weights of L/S portfolios»l/s 0/0 portfolio with security 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»Portfolio drops from $00 to $0» Leverage increases from 40% to 500%!» Rebalancing requires deleveraging and buying the winner and sell the loser

16 Long-Short Portfolios Weights of L/S portfolios»when L/S portfolios have gains (losses), leverage decreases (increases)» When a L/S portfolio is positioned correctly, i.e., long higher return assets and short lower return assets, its leverage decreases.» When a L/S portfolio is positioned wrongly, i.e., long lower return assets and short higher return assets, its leverage increases!»buy-and-hold (passive) and leverage don t mix»fw might perform better than BH E(WW BH ) E(WW FW )

17 Long-Short Portfolios Expected terminal wealth»theorem: If ww < 0, ww ii 0, ii =,,. And»In addition, μμ ii μμ, ii =,,, and»then NN ww ii ( + μμ ii ) NN + ww ii μμ ii ww ii μμ pp = ww jj μμ jj jj = = μμ ii E(WW BH ) E(WW FW )

18 Long-Short Portfolios Risk Parity»The result can be extended to cases with more than one short assets»practical application: Risk Parity portfolios» Long risky assets: equity, interest rates, commodities, etc.» Leveraged by shorting cash 8% Risk Parity Line and Traditional Frontier 7% Risk Parity Line Return 6% 5% 4% 3% Levered Risk Parity 5/75 Risk Parity 60/40 % % 0% 5% 0% 5% 0% Risk

19 Terminal Wealth Expected variance»expected value of terminal wealth» Long-only portfolios: E(WW BH ) E(WW FW )» Long-short portfolios E(WW BH ) E(WW FW )»But variance is also important in any investment analysis (risk/return framework)» What about var(ww BH ) aaaaaa var(ww FW )?

20 Terminal Wealth Expected variance»statistical calculation var(xx) = E(xx ) [E(xx)].»FW portfolios var(ww FW ) = + μμ pp + σσ pp NN + μμ pp NN NN var(ww FW ) = C NN nn nn= + μμ pp (NN nn) σσ pp nn»bh portfolios var(ww BH ) = ww ii ww jj ( + μμ ii ) + μμ jj + σσ iiii NN ii,jj = ww ii ( + μμ ii ) NN NN var(ww BH ) = C NN nn nn= ww ii ww jj ( + μμ ii ) + μμ jj NN nn nn σσ iiii. ii,jj =

21 Terminal Wealth Expected variance special case»theorem: When non-negative µ = µ = = µ M and weights and covariances are»then var(ww BH ) var(ww FW )»In general, BH long-only portfolios variance of terminal wealth is higher than that of FW portfolios.

22 Risk-adjusted Terminal Wealth Wealth- volatility ratio EE(WW) std(ww)»example: 0 securities with equal expected return (8%), equal volatility (0%), equal pair-wise correlation (ρ); initial weight 0% each

23 Risk-adjusted Terminal Wealth Wealth- volatility ratio EE(WW FW ) std(ww FW ) > EE(WW BH ) std(ww FW )»Example: 0 securities with equal expected return (8%), equal volatility (0%), equal pair-wise correlation (ρ=0)

24 Risk-adjusted Terminal Wealth Wealth Sharpe ratio SSSS WW = EE(WW) ( + μμ 0) NN std(ww)»example: assets one risk-free with % return and the other 0% risk and 8% return; initial weight 50% each

25 Effects of Serial Correlations Long- only portfolios»mean-reverting gives FW portfolios an edge; trending or momentum gives BH portfolios an edge»example: assets one risk-free with % return and the other 0% risk and 8% return; initial weight 50% each E(WW BH ) E(WW FW ) if ρρ < μμ μμ 0 σσ. ρρ < (SSSS)

26 Conclusions To rebalance or not to rebalance» Long-only portfolios» FW tends to have higher risk-adjusted terminal wealth» Long-short portfolios» Buy-and-hold and leveraged portfolio is not a good combination» Serial correlation E(WW BH ) E(WW FW ) var(ww BH ) var(ww FW ) E(WW BH ) E(WW FW ) var(ww BH )? var(ww FW )» Mean-reverting is beneficial to FW long-only portfolios; trending is beneficial to BH long-only portfolios» For long-short portfolios, times series trending and cross-sectional reversal is the best.

27 Conclusions To rebalance»investors often have fixed-weight asset allocation portfolios but buy-and-hold asset indices»capitalization-weighted indices are BH and they often underperformed naïve equally-weighted portfolio and other kinds of alternative indices» Cap-weighted indices are not diversified; it is passive-aggressively active.» Cap-weighted indices are not rebalanced» To rebalance or not to rebalance? Answer: Rebalance everywhere

28 Appendix Diversification return is not rebalancing return»arithmetic mean µ = M»Geometric mean ( r + + ) r M / [( + r ) ( )] M + g = + r M g µ σ N DR = g p wi g i= i 0»Diversification return is not return between two real portfolios N» w i g i IS NOT the geometric mean of the buy-and-hold portfolio i=» Qian, Edward, Diversification Return and Leveraged Portfolios, The Journal of Portfolio Management, Summer 0, Vol. 38, No. 4: pp. 4-5

29 Diversification return»arithmetic mean»geometric mean ( ) r M r M + + = µ ( ) ( ) [ ] M r M r g / + + = + µ σ g p N i i i i p N i i i p p p g w w g σ σ σ µ σ µ + = = = = = 0 = = = p N i i i N i i i p w w g g σ σ Appendix