Abstracts

The objective of many portfolio managers is to beat a specific benchmark. This benchmark is typically chosen to be the stock market. The performance of the fund is then compared with a performance of the benchmark, say a stock index SP500 for a fund that invest in US stocks. From the no arbitrage arguments, it is impossible to beat the stock index for sure, thus such an investment strategy can guarantee success only with a certain probability. The problem how to maximize the probability of beating a specific index by a certain percentage has already been widely studied in the previous literature. Thus we focus our attention to another drawback of such a strategy. The fund manager can still beat the stock index, but in the situation of a market downturn, his strategy can significantly under perform the money market, making the investor worse off in comparison to a conservative strategy of holding the currency.
We can formulate the investment problem in the following way, we want the investment fund $X$ to have at least $\alpha$ units of the stock market $S$ and at least $\beta$ units of the money market $M$ at the end of the monitoring period $T$. Thus the objective is $$ X(T) \geq \max(\alpha S(T), \beta M(T)). $$ There is a region of values $a$ and $b$ such that this objective can be satisfied for sure and we find this set of feasible values in the geometric Brownian motion model. Certainly any values above 1 for both $a$ and $b$ are not feasible, this would create an arbitrage opportunity. However, for an arbitrary choice of $a$ and $b$, one can identify a portfolio that beats both benchmarks with the largest possible probability using the techniques of quantile hedging.
We also find the trading strategy that would deliver the objective portfolio for any $a$ and $b$ in the feasible set. Since both values of $a$ and $b$ must be below 1, there is a possibility that the resulting portfolio would under perform both the stock and the money market. We explicitly compute this probability. As it turns out, this probability of underperformance of both benchmarks converges to zero as time goes to infinity.