2021/05/17 by Andrey Sarantsev, Sarantsev, Andrey
Economics, Econometrics and Finance · Mathematics · #60J22 #91G10 #FOS: Economics and business #FOS: Mathematics #Portfolio Management (q-fin.PM) #Probability (math.PR) #math.PR #msc:60J22 #msc:91G10 #q-fin.PM
paper · pdf · doi:10.48550/arxiv.2105.08139
Merton's problem; stochastic optimization; portfolio theory; wealth process
arxiv created 2021/07/22 · arxiv updated 2021/07/23
Portfolio managers often evaluate performance relative to benchmark, usually taken to be the Standard & Poor 500 stock index fund. This relative portfolio wealth is defined as the absolute portfolio wealth divided by wealth from investing in the benchmark (including reinvested dividends). The classic Merton problem for portfolio optimization considers absolute portfolio wealth. We combine absolute and relative wealth in our new utility function. We also consider the case of multiple benchmarks. To both absolute and relative wealth, we apply power utility functions, possibly with different exponents. We obtain an explicit solution and compare it to the classic Merton solution. We apply our results to the Capital Asset Pricing Model setting.