Abstract
In this article we establish a comparison between the first two moments of the buy-and-hold portfolio and the constantly rebalanced portfolio, under the assumption that the underlying assets follow a geometric Brownian motion. We prove that over a fixed time interval the buy-and-hold portfolio has the greater expected return, with equality if and only if the underlying assets have the same expected returns. We also show that the buy-and-hold portfolio has the larger variance in at least two cases: (i) when the underlying assets have equal expected returns; (ii) when the portfolio weights are mean-variance optimal (unconstrained). These results are easily extended to the case of time varying expected return and volatility. Finally, we consider a numerical comparison of the risk statistics of the two portfolios (standard deviation and value-at-risk) for different portfolio parameters.
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Notes
Note that to achieve this, the portfolio allocations are constantly changing to reflect the evolving values of the underlying assets.
It follows from Lévy’s theorem, since W t * and (W t *)2−t are both continuous martingales.
The weights add up to one since ∑ i ∑ j w i w j =(∑ i w i )2=1.
This condition is automatically satisfied when C is a diagonal matrix.
For a random variable X, qα(X) is defined by P(X ⩽ qα(X)) = α. For the standard normal, q1%*=−2.326 and q5%*=−1.645.
References
Mulvey, J. and Kim, W.C. (2009) Fixed mix strategy. In: R. Cont (ed.) Encyclopedia of Quantitative Finance, UK: John Wiley and Sons.
Hallerbach, W.G. (2014) Disentangling rebalancing returns. The Journal of Asset Management 15 (Oct): 301–316.
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Appendix
Appendix
Log normal statistics
Proposition 5.1
-
Assume z i ~ N(μ i , σ i 2) are joint Gaussian random variables with covariance cov(z i , z j )=C ij and variance σ i 2=C ii . If
, then
, then
Proof. The moment generating function of the standard normal Z~ N(0, 1) is 
. Hence 
and 
. For the covariance, note that 
and z
i
+z
j
~N(μ
i
+μ
j
, σ
i
2+σ
j
2+2C
ij
).
□
Corollary 5.2
-
Let z i and
as before, and w
i
scalars. Then
as before, and w
i
scalars. Then
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Spinu, F. Buy-and-hold versus constantly rebalanced portfolios: A theoretical comparison. J Asset Manag 16, 79–84 (2015). https://doi.org/10.1057/jam.2015.2
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DOI: https://doi.org/10.1057/jam.2015.2