Help Calculating Monthly Earnings with two co-existing APYs in same "Savings" Account from T-mobile Money

Question

Background

I have a T-Mobile Money (via BankMobile, a division of Customers Bank) interest checking account.

I'm trying to track earnings per month but I'm a bit confused by their fine print and their tiered APY scheme.

Overall Question

How can I calculate monthly and yearly earnings on excel/gsheets for my account, given my bank's fine print and two APY schemes?

The Fine Print Text

Relevant sections are below; full contact as PDF.

An interest rate of 3.93%, with an Annual Percentage Yield (APY) of 4.00%, will be paid on balances up to and including $3,000.00 in your Account.

An interest rate of 1.00%, with an APY of 1.00%, will be paid on balances above $3,000 in the Account.The APY for this tier will range from 4.00% to 2.79% depending on the balance in the Account (calculation based on a $5,000 average daily balance).

We use the average daily balance method to calculate interest on your Account. This method applies a periodic rate to the average daily balance in the Account for the period. The average daily balance is calculated by adding the principal in the Account for each day of the period and dividing that figure by the number of days in the period. Interest begins to accrue no later than the business day we receive credit for the deposit of noncash items (for example, checks). Interest on your Account will be compounded and credited on a monthly basis at the end of your monthly statement cycle.

The Fine Print Image

While, I have copied and pasted the text of the contact above for search and discovery reasons, I am also adding a screenshot of the above information to show where I am confused, and to highlight the parts of the contact that I would like to reference in my question.

Attempts to Understand

According to this question: Does a savings account's advertised "APY" account for compound interest? The FDIC clearly defines the formula banks must use to compute APY here.

The formula being

APY = 100*[(1 + interest/principal)^(365/Days in term) -1]

And according to this question: https://money.stackexchange.com/a/83879/5306

Most savings accounts compound interest daily and credit earned interest monthly.

Is that what I'm dealing with here? Since my bank is calculating interest daily but applying it at the end of the month, it still means the balance is growing at a monthly compounded interest rate, correct? This seems to confirm the language underlined above in blue.

And does "The APY for this tier" language underlined above in red above contradict the previous sentence?

Attempts to Track Earnings With a Spreadsheet

With a starting balance of $3,000 and 17 days in the first month, I received an interest payout of $5.81

To check this math, let's use the following spreadsheet:

The periodic rate formula: =<Interest Rate>/365

My understanding of the language underlined above in green:

Average Daily Balance=(<days in first month> * <principal>)/<days in first month>

So if we are dealing with monthly compound interest the formula would be:

First month's Interest=Principal * Periodic Rate * Days in the First Month

Which gives us:

$5.49 = $3,000 * (B2/365) * 17

Which is less than what my bank paid me. What am I failing to see?

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According to this online calculator, my interest payout for the first month should have been $5.57

Answer 1

Back in the '70s banks would offer savings products whose "contract rate" would not be an entirely straightforward representation what sort of interest you'd get. As a result Congress passed the Truth in Savings Act to create the APY and require financial institutions to use it in advertising.

A savings account pays money to the depositor according to the contract rate, the term of deposit, and the compounding scheme. When these parameters vary the consumer needs a way to compare (normalize) how much s/he is getting. The APY is just that.

A financial institutions may also offer a "stepped" rate account: You get one rate for the first N days, and another rate for the next N days, and so on according to the number of steps. This may be b/c a financial institution wants to offer a "teaser" rate to entice customers to deposit.

In addition to this a financial institution may offer a "tiered" rate account. It defines a number of tiers. Each tier defines a lower bound, an upper bound, and a contract rate. For instance, [$0-$9,999], [$10,000 - $49,000], [$50,000 - $99,000], [$100,000 - $499,999] and [$500,000 - infinity] with rates of, say, 3.25%, 3.5%, 3.75%, 4%, 4.25%, respectively.

It's theoretically possible for an account to be both stepped and tiered. (I know, I wrote the code for a compliance company.) But practically, it's not done. Oh, and did I mention there's two different ways to calculate a tiered rate APY... Method A and Method B.

Ultimately some money earns simple interest rate and is compounded at a particular rate. (Ever hear of "continuously" compounded interest?)

The Truth in Savings Act requires the financial institution to slice and dice all this complexity into individual pieces where a simple interest rate can be compounded a number of times to compute how much money was earned in each piece. Sum the yield of all those pieces. (Now we're almost done.) Now, use a little algebra to back into the interest rate that would yield the same amount. Walla: that's an APY! (This APY calculation is shown here.)

Note that when we've got piddling little interest rates of 1-2%, there's not a lot of difference. But if you're writing and debugging code to calculate APYs, assume you're in the late '70s with high inflation. High interest rates (15-20%) make the maths ill-conditioned. This will amplify any errors in your code.

Back before I retired, I would test all my code with rates nobody'd seen in 20 years. The Federal Reserve used to have an APY calculator running in DOS you could use to independently validate your code. I see they've got an online version now.

Answer 2

Where did the "17 days" come from?

The bank evidently paid you interest for 18 days.

$3000 * 0.0393 * 18/365 = $5.814 which rounds to your exact payout.