# Transaction costs, trading volume, and the liquidity premium

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## Abstract

In a market with one safe and one risky asset, an investor with a long horizon, constant investment opportunities and constant relative risk aversion trades with small proportional transaction costs. We derive explicit formulas for the optimal investment policy, its implied welfare, liquidity premium, and trading volume. At the first order, the liquidity premium equals the spread, times share turnover, times a universal constant. The results are robust to consumption and finite horizons. We exploit the equivalence of the transaction cost market to another frictionless market, with a shadow risky asset, in which investment opportunities are stochastic. The shadow price is also found explicitly.

## Keywords

Transaction costs Long-run Portfolio choice Liquidity premium Trading volume## Mathematics Subject Classification

91G10 91G80## JEL Classification

G11 G12## Notes

### Acknowledgements

For helpful comments, we thank Maxim Bichuch, George Constantinides, Aleš Černý, Mark Davis, Ioannis Karatzas, Ren Liu, Marcel Nutz, Scott Robertson, Johannes Ruf, Mihai Sirbu, Mete Soner, Gordan Žitković, and seminar participants at Ascona, MFO Oberwolfach, Columbia University, Princeton University, University of Oxford, CAU Kiel, London School of Economics, University of Michigan, TU Vienna, and the ICIAM meeting in Vancouver. We are also very grateful to two anonymous referees for numerous—and amazingly detailed—remarks and suggestions.

The first author was partially supported by the Austrian Federal Financing Agency (FWF) and the Christian-Doppler-Gesellschaft (CDG). The second author was partially supported by the ERC (278295), NSF (DMS-0807994, DMS-1109047), SFI (07/MI/008, 07/SK/M1189, 08/SRC/FMC1389), and FP7 (RG-248896). The third author was partially supported by the National Centre of Competence in Research “Financial Valuation and Risk Management” (NCCR FINRISK), Project D1 (Mathematical Methods in Financial Risk Management), of the Swiss National Science Foundation (SNF). The fourth author was partially supported by the Austrian Science Fund (FWF) under grant P19456, the European Research Council (ERC) under grant FA506041, the Vienna Science and Technology Fund (WWTF) under grant MA09-003, and by the Christian-Doppler-Gesellschaft (CDG).

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