nisiprius wrote:Abe wrote:According to my financial calculator (Texas Instrument BAII Plus) the answer is 7.15%.

14,335 days/365=39.27 years

N=39.27 yrs PV=$10k FV=$150,713.00 IY= 7.15%

That's close enough.

Well, that's what I've been doing for a long time, but you and your BAII didn't answer my question, which is "what number should you plug in there: 365, 365.25, 365.2425, or something else?" Or was the "365" built into your calculator somehow?

There's usually a leap year every 4 years, and will be during our lifetimes, so (365+365+365+366)/4 = 365.25 is close enough.

There is no "official" one right or wrong answer. Various banks and other financial institutions, books, websites, and software packages ( including hand-held calculators and spreadsheets) use their own methods.

- The same goes for rounding of cents in calculating interest. In some software packages it may make a difference whether you're defining the numbers as currency or general numbers. Some round up dollar and cent calculations up or down, and some may truncate to integers. So if you come within a few cents in a month or a few dollars for a short-term loan, or a few hundred dollars over 30 years, whether it's your calculation, your bank's, spreadsheets, hand-helds, or different website calculators, it's close enough.

Similar problems come up in calculating payments for loans. The quoted monthly payment is typically based on an average of 1/12 of the yearly balance paid at the end of the month. For most mortgages and other fixed-payment, closed-end loans like mortgages and car loans that works well because interest is calculated as of the posting date regardless of when the payment is received as long as it's with the grace period. There's no penalty for being a few days late, and no savings for making the payment ahead of time.

But student loans, credit cards , lines-of-credit, and some HELOCs and other loans (more typically at credit unions) calculate the interest on the unpaid balance on a daily basis. There's no one single rule for that, either. Some institutions divide the yearly rate by 365, some use 365.25 and a lot of 'em use 360. For calculating monthly interest, some may also use the actual days in the current month.

Even for fixed-length loans like mortgages the last payment may be slightly larger or smaller than the others. For daily interest loans, not only will the total interest vary, but the actual length of the loan might even be a month or so longer or shorter depending on when the payments were made during each month.

Incidentally, this brings up another problem I see often with the yearly performance history "growth of $10,000" and similar calculations that use the CAGR (the geometric mean) for a given number of years.

While that's accurate for a single lump sum invested at the beginning, that's not how most people invest their money. When you're "dollar cost averaging" by contributing a given amount monthly, the interest is compounded monthly. For the lump sum it can make a substantial difference in the actual rate-of-return. And

*when* you start can make a bigger difference than whether you use 365, 365.25 or 360 days per year, especially for a lump sum if you happened to start the calculation at the beginning of a really bad year or month.

Actually, reasonable question: does your calculator have a "days-to-years" conversion key?

Dates are stored as a serial sequence number in most software packages. (Probably starting with day one of the Gregorian calendar).

In excel and other spreadsheets, you can subtract one date from another to get the number of days. Then you can choose whether to use 360, 365, or 365.25.

- For example, if the beginning date is 01/01/1986 and the end date is 12/31/2015 then in a spreadsheet like Excel:

=begindate -enddate … returns 10956 days

.

=10956/365 … returns 30.01644 years.

=10956/365.25 … returns 29.99589 years.

For the above period the growth of a $10,000 lump sum at an average APY of 7% would be a difference of $76,207.26 compared to $76,101.40.

That's a difference of $105.86, which is 0.139% ... spread out over 30 years give or take

By the way, you don't really need to even know the math formulas. Since you're most likely doing the calculations in a spreadsheet or database package anyway, you can use the standard math functions that are provided with the software.

My posts and those of others in threads

HERE and

HERE discuss and show some examples of how you can use spreadsheet library functions to do the math yourself.

If you're calculating the interest for earnings or interest paid with periodic payments, divide the years by 12. Divide the annual rate by 12 in calculations for payments, FV, PV, etc. With the RATE function solving for monthly periods, multiply the periodic rate by 12.

nisiprius wrote:Doc wrote:Nisi, you have too much time on your hands. Get a job or at least a dog.

Somebody, somewhere, is making money on the difference between a 365.25-day year and a 365-day year. And leveraging it up at 300:1 leverage to make it amount to something. I'm sure of it.

If nobody is, I should be able to open a Boston office and

*tell* investors that that's what I'm doing. "See, I've discovered that in Europe they use a 365.25-day year and in the U.S. it's a 365-day year, so I borrow money in Europe at 1% and lend it in the U.S. at 1%... and it's all done with international postal reply coupons."

I'm reminded of the urban legend-like tales told around the programmer camp-fires way back in the days of the coal-fired computers when I got into the business.

( By the way, even in the 70s through 90s a lot of lenders went by amortization schedule tables for payments and prepayments insteasd of calculating it with the financial softrware functions like PMT, NPER, FV, RATE, etc, since the iterative processes and floating-point math took too much expensive computer processing cycles and too much elapsed time.

The story goes like this:

The federal bank examiners noticed that the banks's lead programmer was living a lavish life style far higher than his salary would support. They

*knew* he must be embezzling from the bank, but could not figure out how.

Finally they called him in and offered him immunity from prosecution if he would just tell them

*how* he did it so they could prevent it from happening at other banks.

He explained that he had written the software algorithms so that every time interest was calculated on every customer's loan or savings account, he rounded the cents down on their account and sent the fractional remainder to his own secret account. Supposedly over a long time with millions of individual daily and monthly calculations it gave him a

*bunch* of money that nobody ever missed.

jimb