In CFAI volume 3 page 217, how come the risk-adjusted return be Um=E(Rm)-0.005R_A * sigma_m^2
In the following example, thy showed U_A=9.7%-0.02*(15%)^2=5.2%, but clearly this does not hold, 0.02*(15%)^2=0.00045 (very small number)
You are right. They are using a bastard notation and mixing up return in % with variance in %^2 because initially they applied a factor of 0.005 instead of 0.5. I’d just concentrate on the real equation and use decimals:
U_m = E(R_m) - 0.5 R_A sigma_m^2
E(R_m) = 0.097
R_A = 4 (on a scale of 1…10)
sigma_m = 0.15
U_m = 0.097 - 0.5 * 4 * 0.15^2 = 0.097 - 2 * 0.0225 = 0.097 - 0.045 = 0.052 = 5.2%.
If you want to use %ages then:
U_m = E(R_m) - 0.005 R_A sigma_m^2
= 9.7% - 0.005 * 4 * 15% * 15% = 9.7% - 0.02 * 225 %% = 9.7% - 4.5 %% but we already divided the variance terms once with 100 so treat 4.5 %% as 4.5%.
Thanks for the answer!