MUTUAL FUND PERFORMANCE 123 cent of net assets per year to 1.5 per cent ernment bond at a price that would have per year Most funds also charge e an guaranteed a return of slightly less than tial fee of approximately 8.5 per cent for 3 per cent if held to maturity. Using 3 selling costs when shares are purchased; per cent as the estimate of p for the annual rates of return are not net of such period, Boston Fund, shown by point y costs n Figure 1, plus borrowing or lending The average annual rate of return(ao) could have provided any combination of and the standard deviation of annual average return and variability lying rate of return Vi for each fund along line PYZ Incorporate shown in Figure 1; the values are listed shown by point Q, could have provided in Table 1. The relationship predicted by any combination lying along line Pe the theory of capital asset prices is clear- Clearly the for ly present-funds with large average re- latter, since for any level of risk it offered turns typically exhibit greater variability er average return. Indeed, the than those with small average returns. steepness of the line associated with a Moreover, the relationship is approxi- fund provides a useful measure of per mately linear and significant. 2 However, formance--one that incorporates both there are differences in efficiency; a num- risk and average return. We define this ber of funds are actually dominated (i.e., as the reward-to-variability ratio: For some other fund provided both a greater Boston Fund the ratio is equal to the value of At and a smaller value of vi. distance XP on Figure 1 divided by the To analyze the differences, we need a distance XY. The larger the ratio, the single measure of performance; once such better the performance a measure is specified, any persistent An alternative interpretation of the differences can be investigated by testing ratio gives rise to the name---reward-to- alternative measures for predicting per- variability ratio (R/V). The numerator formance shows the difference between the funds An intuitively appealing and theoreti- average annual return and the pure in cally meaningful measure of performance terest rate; it is thus the reward provided is easily derived from the Tobin effect. the investor for bearing risk. The denom- With substitution of the ex post meas- inator measures the standard deviation ures (A and V) for the ex ante measures of the annual rate of return: it show (E and o), the formula described in Sec- amount of risk actually borne. The ratio tion ii becomes is thus the reward per unit of variability. The final column of Table 1 shows the A=P+ alues of the R/v ratio for the thirty- four funds. They vary considerably By investing in fund i and borrowing or from almost 078(the Boston Fund)to lending at the riskless rate p, an investor slightly over 0.43(Incorporated Inves could have attained any point along the tors). Those who view the market as line given by this formula. In 1953 it was nearly perfect and managers as goo possible to purchase a ten-year U.S. gov- diversifiers would argue that the differ 1 The results of statistical tests on these data tangent of the angle made by the line Journal of Finance, nother way, the reciprocal of the slope September, 1965, pp. 416-22. (dA/dv)equals the R/V ratio
26% 18 16 H 10 8 FIG. 1. -Average return and variability, 34 open-end mutual funds, 1954-63