Michael,

Recall that the unit “$” needs to be treated just like any other algebraic quantity and that 100% = 1,

As I understand things, your formulas’ suggestions applied to your example give:

Arithmetic excess return = [Portfolio return – Benchmark return]*100%/(Initial portfolio value)

= [($210/$200 – 1) – ($205/$200 – 1)]*100%/($200)

= [ (0.05) – (0.025) ]*100%/$200 = 0.0125%/$.

Geometric excess return = [Portfolio return – Benchmark return]*100%/(Ending portfolio value)

= [($210/$200 – 1) – ($205/$200 – 1)]*100%/($205)

= [ (0.05) – (0.025) ]*100%/$250 = 0.012195%/$.

As I previously noted, these values have problematic units. If we did the calculation in pennies instead of dollars, the results would change. But a return calculation should not depend on the units used for the values.

Whereas my formulas’ suggestions applied to your example give:

Arithmetic excess return = [Portfolio return – Benchmark return]*100%

= [($210/$200 – 1) – ($205/$200 – 1)]*100%

= [ (0.05) – (0.025) ]*100% = 2.5%.

Geometric excess return = [Portfolio return – Benchmark return]*100%/(1 + Benchmark return)

= [($210/$200 – 1) – ($205/$200 – 1)]*100%/[ 1 + ($210/$200 – 1) ]

= [ (0.05) – (0.025) ]*100%/[1.025] = 2.439%.

These values of (approximately) 2.5% make more sense to me then your answers which come in (ignoring your units) at around half that.

Andre