# Math for Portfolio Value Net of Fees

I'm looking to do a pretty basic Finance 101-type calculation to compare final portfolio values given varying fees.

Let's say I start with a \$50,000 portfolio. It's invested in a fund that returns 7%/year with an investment horizon of 15 years.

Scenario 1: Low-cost, passively managed fund with an annual expense ratio of 0.05%

Scenario 2: Actively managed mutual fund with an annual expense ratio of 1.00% (assuming it also returns 7%)

I'm trying to figure out:

(A) how much the \$50,000 is worth at the end of 15 years, given the 7% growth rate

(B) what the difference is in value if I go with Scenario 1 vs. Scenario 2, eg, how much money I leave on the table just by paying that 1% for active management

It's easy to plug this into an online calculator or model the annual returns in XLS, but I know there's a more parsimonious mathematical way to do this calculation (something with geometric sums, I seem to remember?)

You can just subtract the fee from your expected return. So for the first fund, your expected return would be 6.95%, anf for the second your expected return would be 6.00%. Then just use normal compounding formulas to calculate an expected value over 15 years (but see below)

Generally, the expected return of actively managed funds needs to be higher than passive funds to overcome the extra costs.

As a side note, you cannot say how much the investment will be worth after 15 years because returns are not guaranteed. You can calculate an expected amount, but the actual amount will be a probability distribution depending on the risk (variance) of those returns.

• Suppose you have a fund that has 1% fees. The first year it returns 10%, and the second year 0%. Average return = 4.8809% (geometric mean). Subtract 1% from that and it's 3.8809%. But if we take the geometric mean of 9% and -1%, we get 3.8797%. So with a highly variable fund, your method slightly overestimates the return. – Acccumulation Jan 24 at 20:44
• @Acccumulation True, but expected return assumes the same return each year, so there would be no difference. – D Stanley Jan 25 at 14:47

The mathematical way is

• \$50,000*(1.0695)^t

• \$50,000*(1.06)^t

Where t is the number of years from the initial investment. Subtract those to find your difference. The ^ symbol means the exponent. I’ll try to see if I can get math Jax working here to make that look nicer. It follows a geometric sequence because you’re multiplying by one plus the rate of return each year. The exponent is how many times you multiply by the growth factor.