著者：Jiro Hodoshima, Tetsuya Misawa, Yoshio Miyahara
Evaluation of performance of stocks is fundamental and practically quite important in finance. Normally performance of stocks is evaluated using the Sharpe ratio and information ratio based on the first two moments of stock returns, showing how much excess returns stocks earned compared to the risk-free rate and a benchmark portfolio return adjusted for their volatilities. However, the Sharpe ratio and information ratio do not satisfy the monotonicity property. A higher return can result in a lower Sharpe ratio. In other words, it is possible the Sharpe ratio of $\mb$ is higher than that of $\mb$ when $\mb \geq \mb a.s.$ where $\mb$ and $\mb$ denote random variables of two financial asset returns. Therefore evaluation by the Sharpe ratio and information ratio is not always appropriate for many investors. In this paper, we present an evaluation of performance of stocks based on a value measure using utility indifference pricing which satisfies desirable properties including monotonicity under which a higher return always results in a higher value measure. We compare evaluation by a value measure based on utility indifference pricing and the Sharpe ratio using selective Japanese stock return data. The two measures are quite similar in most of the comparison results but show different results in the two companies of Softbank and Kao where the Sharpe ratio is higher than the value measure based on utility indifference pricing in Softbank but the opposite is the case in Kao. We show the evaluation by the value measure based on utility indifference pricing is more relevant than that based on the Sharpe ratio, giving values corresponding to those of the risk-averse value measure of the underlying performance of stock returns rather than the first two moments of stock returns. In particular, the evaluation by the value measure based on utility indifference pricing depends on the entire underlying distribution of stock returns instead of the first two moments in the Sharpe ratio so that it is not controlled by a few influential observations of stock return data. We derive the comparison results by assuming the underlying data of stock returns to follow discrete normal mixture distributions.