Optimal Investment Under Transaction Costs: A Threshold Rebalanced Portfolio Approach

Abstract

Abstract-We study how to invest optimally in a financial market having a finite number of assets from a signal processing perspective. Specifically, we investigate how an investor should distribute capital over these assets and when he/she should reallocate the distribution of the funds over these assets to maximize the expected cumulative wealth over any investment period. In particular, we introduce a portfolio selection algorithm that maximizes the expected cumulative wealth in i.i.d. two-asset discrete-time markets where the market levies proportional transaction costs in buying and selling stocks. We achieve this using "threshold rebalanced portfolios", where trading occurs only if the portfolio breaches certain thresholds. Under the assumption that the relative price sequences have log-normal distribution from the Black-Scholes model, we evaluate the expected wealth under proportional transaction costs and find the threshold rebalanced portfolio that achieves the maximal expected cumulative wealth over any investment period. Our derivations can be readily extended to markets having more than two stocks, where these extensions are provided in the paper. As predicted from our derivations, we significantly improve the achieved wealth with respect to the portfolio selection algorithms from the literature on historical data sets under both mild and heavy transaction costs.

Growth optimal investment with threshold rebalancing portfolios under transaction costs

“…The wealth growth of a TRP with (b-E, b+E) for aperiod T with no-crossing can be written as [27] T T (4) t=l t=l !:::,. !…”

“…The proof of the Theorem 3.1 can be found in [27]. Theo Theorem 3.1 and find the optimal TRP, i.e., optimal band E, by using a brute force search.…”

“…Hence, we can explicitly describe the event that the mar ket threshold portfolio (b -E, b + E) does not hit any of the thresholds for T consecutive periods, b T E E�c, as the inter section of the events as [27] bT E E�c == n h2Ih(i) ::::; Ih(i) ::::; '")'1Ih(i)}, (10) i=l where Ih(i) � n�=l Xl ( t ), Ih(i) � n�=l X 2 ( t ) ,"1'1 !:::,. b(1-b+E) d !…”

“…Using the quantitative definitions of the events bT E E�c and bT E E�c, we can express P (E�C) T(T) as [27] P (E�C) T(T) = {OO j"Y17l"1 (b1l"l + (1b)1l"2) P(II1(T) = 1l"1, . .. , �� E [�-81, � -82]) d1l"2d1l"1, (12) wh ich follows from the definition of �� where r;, � In 71"2 • <…”

“…The first probability in (12) can be caIculated as [27] wh ich follows since II1(T) � n;=1 X l (t) and II2 (T) � n;=I X2(t) and we have II1(T) rv InN (TJ-il,mi) and II2(T) rv InN(TJ-i2, Td).…”