困惑expense charge处理，求季度复利基金的Annual Effective Rate

Got it, let's break this down step by step so it makes total sense. I’ve worked through similar expense-adjusted rate problems before, so once we walk through each quarter’s mechanics, it’ll click.

First, let’s lay out the key numbers we’re working with:

• Nominal annual rate: 8%, compounded quarterly → so the quarterly nominal rate is 0.08 / 4 = 0.02 (or 2%)
• Quarterly expense charge: 0.4% of the fund balance after interest is credited

Step 1: Find the quarterly net growth factor #

Each quarter, two things happen in order:

1. Interest gets added to the fund: your balance grows by 2% → multiply by 1 + 0.02 = 1.02
2. The 0.4% expense charge is deducted: you keep 99.6% of that new balance → multiply by 1 - 0.004 = 0.996

Combine these two steps to get the net quarterly growth factor:

1.02 * 0.996 = 1.01592

This means every quarter, your fund balance is multiplied by 1.01592.

Step 2: Calculate the annual effective rate #

Since we’re compounding quarterly, we raise the quarterly net factor to the 4th power (for 4 quarters in a year), then subtract 1 to get the effective annual rate:

(1.01592)^4 - 1 ≈ 1.0651 - 1 = 0.0651

Converting that to a percentage, the annual effective rate compounded quarterly is approximately 6.51%.

Quick sanity check #

If there were no expense charges, the annual effective rate would be (1 + 0.02)^4 - 1 ≈ 8.24%. The 0.4% quarterly expense cuts this down by roughly 1.73 percentage points, which makes sense because the expense is applied to the growing balance each quarter.

内容的提问来源于stack exchange，提问作者Gregggg