To calculate CAGR, how long is "a year?"

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To calculate CAGR, how long is "a year?"

Post by nisiprius » Tue Dec 01, 2015 3:40 pm

I guess I'll ask, since I couldn't quickly find out by Googling. I've seen a bazillion descriptions of how to calculate compound average growth rate, and they all just say you use the time lapse "in years." But how long is a year?

To be specific: suppose an investment is worth $10,000 on 8/31/1976 and $150,713.00 on 11/30/2015. According to Excel, that period of time is 14,335 days.

How many years is that? According to the official accountants' rules, do I divide 14,335
--by 365?
--by 365.25 (accounting for one leap year every four years?)
--365.2425 (including the full Gregorian calendar leap year rules?)
--360 (which bankers use when they're charging you interest?)
--some calculation which actually depends on the actual number of February 29ths there were between the start and end date?

(Cue "A paradox, a paradox, a most ingenious paradox" from "The Pirates of Penzance." Or not.)
Last edited by nisiprius on Tue Dec 01, 2015 5:48 pm, edited 1 time in total.
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Re: To calculate CAGR, how long is "a year?"

Post by VA_Gent » Tue Dec 01, 2015 3:59 pm

The answer: 7.156%
Use this: http://www.dinkytown.net/java/AnnualReturn.html
I'm sure the IRS or SEC has an answer to the year definition.

Edit: Regardless of how a year is defined, the answer is almost identical.
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Re: To calculate CAGR, how long is "a year?"

Post by k66 » Tue Dec 01, 2015 4:14 pm

Personally, I have always favoured 365.25 as being the most mathematically succinct value (rightly or wrongly).

edit: fixed my numerical typo (365 not 364)
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Re: To calculate CAGR, how long is "a year?"

Post by TJSI » Tue Dec 01, 2015 5:24 pm

I think you are mixing metaphors.

When computing CAGR for different investments the number of years is an integer.
Say you had three investments. One increased by some mysterious stochastic process over 5 years. The second was a nice money fund that compounded daily. And the third was a bond which you compounded the interest payment two times a year. To compute the CAGR for comparison purposes of all three investments you would use the same compound interest formula with N=5.

Using the CAGR allows you to compare investments with different internal dynamics.

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Re: To calculate CAGR, how long is "a year?"

Post by nisiprius » Tue Dec 01, 2015 5:48 pm

VA_Gent wrote:The answer: 7.156%
Use this: http://www.dinkytown.net/java/AnnualReturn.html
I'm sure the IRS or SEC has an answer to the year definition.

Edit: Regardless of how a year is defined, the answer is almost identical.
Of course it is. That's why I've never bothered to try to find out the "right" answer.

I screwed up, by the way... the average Gregorian year is 365.2425 days. 365.2422 is the real astronomical value, the sidereal year. I'm going to fix that in my original post.

Trying the reverse engineering approach on Dinkytown's number,
360 days per year -> 7.0502%
365 days per year -> 7.1515%
365.2425 days per year -> 7.1564%, which rounds to 7.156%
365.25 days per year -> 7.1566%, which rounds to 7.157%

So if Dinkytown rounds the same way I do, I conclude that they're using 365.2425 days per year.
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Re: To calculate CAGR, how long is "a year?"

Post by Abe » Tue Dec 01, 2015 6:04 pm

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. :happy
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Re: To calculate CAGR, how long is "a year?"

Post by Doc » Tue Dec 01, 2015 7:33 pm

Nisi, you have too much time on your hands. Get a job or at least a dog. :D
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Re: To calculate CAGR, how long is "a year?"

Post by itstoomuch » Wed Dec 02, 2015 1:39 am

Fiscal, Feb 29. :mrgreen:
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Re: To calculate CAGR, how long is "a year?"

Post by nisiprius » Wed Dec 02, 2015 6:09 am

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. :happy
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?

Actually, reasonable question: does your calculator have a "days-to-years" conversion key?
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Re: To calculate CAGR, how long is "a year?"

Post by nisiprius » Wed Dec 02, 2015 6:13 am

Doc wrote:Nisi, you have too much time on your hands. Get a job or at least a dog. :D
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."
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Re: To calculate CAGR, how long is "a year?"

Post by peppers » Wed Dec 02, 2015 6:29 am

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

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Re: To calculate CAGR, how long is "a year?"

Post by jimb_fromATL » Wed Dec 02, 2015 11:31 am

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. :happy
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. :D
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

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Re: To calculate CAGR, how long is "a year?"

Post by #Cruncher » Wed Dec 02, 2015 4:44 pm

nisiprius in original post wrote:To be specific: suppose an investment is worth $10,000 on 8/31/1976 and $150,713.00 on 11/30/2015. According to Excel, that period of time is 14,335 days. How many years is that? ...
In my opinion one should ignore the total number of days and any attempt to convert that to years. Instead:
  • Compute the fraction of a year from 8/31/1976 to 11/30/1976: 91/366 [ 1 ]
  • Compute the number of years from 11/30/1976 to 11/30/2015: 39
  • Add to get total number of years: 39 + 91/366
The corresponding compounded annual growth rate is 7.156%. This is the same answer (to 11 significant digits) returned by the Excel RATE and YIELD functions:

Code: Select all

7.1562787955% =(150713 / 10000) ^ (1 / (39 + 91/366)) - 1
7.1562787955% =RATE(39 + 91/366, 0, -10000, 150713, 0, 10%)
7.1562787955% =YIELD(DATE(1976, 8, 31), DATE(2015, 11, 30), 0, 10000, 150713, 1, 1) [ 2 ]
Congratulations on getting this right. It's easy not to since the Gregorian rule for number of days in a year has an exception; the exception has an exception; and the exception to the exception has an exception!
  • Rule: 365 days in a year
  • Except 366 if year is divisible by 4
  • Except 365 if year is divisible by 100
  • Except 366 if year is divisible by 400
For example in the 400 years from from 1601 through 2000: 1700, 1800, and 1900 were not leap years, but 2000 was. Thus there were 303 years with 365 days and 97 with 366 or an average of

Code: Select all

365.2425 =(303 * 365 + 97 * 366) / 400
  1. 91 days from 8/31/1976 to 11/30/1976 divided by 366 days from 11/30/1975 to 11/30/1976
  2. Note the last parameter to the YIELD function ("1") tells it to compute the fraction of a period until the first anniversary date using the actual number of days -- the same way I did.

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Re: To calculate CAGR, how long is "a year?"

Post by nisiprius » Wed Dec 02, 2015 5:36 pm

#Cruncher wrote:
nisiprius in original post wrote:To be specific: suppose an investment is worth $10,000 on 8/31/1976 and $150,713.00 on 11/30/2015. According to Excel, that period of time is 14,335 days. How many years is that? ...
In my opinion one should ignore the total number of days and any attempt to convert that to years. Instead:
  • Compute the fraction of a year from 8/31/1976 to 11/30/1976: 91/366 [ 1 ]
  • Compute the number of years from 11/30/1976 to 11/30/2015: 39
  • Add to get total number of years: 39 + 91/366
The corresponding compounded annual growth rate is 7.156%.
I really like that answer. Do you have any idea if that is the "official" answer?
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Re: To calculate CAGR, how long is "a year?"

Post by Coles » Wed Dec 02, 2015 6:19 pm

#Cruncher wrote:In my opinion one should ignore the total number of days and any attempt to convert that to years. Instead:
  • Compute the fraction of a year from 8/31/1976 to 11/30/1976: 91/366 [ 1 ]
  • Compute the number of years from 11/30/1976 to 11/30/2015: 39
  • Add to get total number of years: 39 + 91/366
:?: Is this an equally valid methodology (but resulting in a different answer)?
  • Compute the number of years from 8/31/1976 to 8/31/2015: 39
  • Compute the fraction of a year from 8/31/2015 to 11/30/2015: 91/365
  • Add to get total number of years: 39 + 91/365

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Re: To calculate CAGR, how long is "a year?"

Post by FactualFran » Thu Dec 03, 2015 2:20 pm

Coles wrote:
#Cruncher wrote:In my opinion one should ignore the total number of days and any attempt to convert that to years. Instead:
  • Compute the fraction of a year from 8/31/1976 to 11/30/1976: 91/366 [ 1 ]
  • Compute the number of years from 11/30/1976 to 11/30/2015: 39
  • Add to get total number of years: 39 + 91/366
:?: Is this an equally valid methodology (but resulting in a different answer)?
  • Compute the number of years from 8/31/1976 to 8/31/2015: 39
  • Compute the fraction of a year from 8/31/2015 to 11/30/2015: 91/365
  • Add to get total number of years: 39 + 91/365
Another alternative:
  • Compute fraction of the starting year from 8/31/1976 to 12/31/1976: 122/366
  • Compute the number of calendar years from 12/31/1976 to 12/31/2014: 38
  • Compute the fraction of the ending year from 12/31/2014 to 11/30/2015: 334/365
  • Add to get total number of years: 122/366 + 38 + 334/365
To be consistent with the result of using the XIRR function of spreadsheet software: use a year length of 365 days.

Another post mentioned the YIELD function of spreadsheet software. That function has a parameter that specifies day count convention to use: 30 days per month with 360 days per year US (NASD); 30 days per month with 360 days per year European; actual days per month with either 360, 365, or actual days per year. Given that, I doubt if there is a single "official" answer to how long a year is in CAGR calculations.
Last edited by FactualFran on Thu Dec 03, 2015 5:28 pm, edited 1 time in total.

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Re: To calculate CAGR, how long is "a year?"

Post by nisiprius » Thu Dec 03, 2015 5:10 pm

Random observation: With any of the "date-specific" methods, if we measure from M1/D1/Y to M2/D2/Y+N, we will always get exactly the same result if M1 and M2 are both ≤2/28, and we will always get the same result if M1 and M2 are both ≥3/1, but we will get slightly different results if they are on "different sides" of 3/1.

Another random observation: I assume the SEC specifies how to calculate CAGR, the "average" that's required to be stated in mutual fund literature.
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Re: To calculate CAGR, how long is "a year?"

Post by Leeraar » Thu Dec 03, 2015 5:16 pm

nisiprius wrote:
#Cruncher wrote:
nisiprius in original post wrote:To be specific: suppose an investment is worth $10,000 on 8/31/1976 and $150,713.00 on 11/30/2015. According to Excel, that period of time is 14,335 days. How many years is that? ...
In my opinion one should ignore the total number of days and any attempt to convert that to years. Instead:
  • Compute the fraction of a year from 8/31/1976 to 11/30/1976: 91/366 [ 1 ]
  • Compute the number of years from 11/30/1976 to 11/30/2015: 39
  • Add to get total number of years: 39 + 91/366
The corresponding compounded annual growth rate is 7.156%.
I really like that answer. Do you have any idea if that is the "official" answer?
Yes, to calculate the number of days subtract two dates in Excel, and format the result as a number.

And now, the rest of the story. (Well, maybe a footnote):
There was a bug in the date in Lotus123 that is intentionally carried over into Excel.

https://support.microsoft.com/en-us/kb/180162

Type 60 in an Excel cell, then format it as a date. The result is 29 Feb 1900. Except, 1900 was not a leap year.

For the Macintosh, Day 1 was made to be 1 Jan 1904, which avoids the bug but caused dates to change when you cut from a spreadsheet created on a Mac and pasted into a spreadsheet on a PC (and vice versa).

L.
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Re: To calculate CAGR, how long is "a year?"

Post by Valuethinker » Thu Dec 03, 2015 5:19 pm

nisiprius wrote:I guess I'll ask, since I couldn't quickly find out by Googling. I've seen a bazillion descriptions of how to calculate compound average growth rate, and they all just say you use the time lapse "in years." But how long is a year?

To be specific: suppose an investment is worth $10,000 on 8/31/1976 and $150,713.00 on 11/30/2015. According to Excel, that period of time is 14,335 days.

How many years is that? According to the official accountants' rules, do I divide 14,335
--by 365?
--by 365.25 (accounting for one leap year every four years?)
--365.2425 (including the full Gregorian calendar leap year rules?)
--360 (which bankers use when they're charging you interest?)
--some calculation which actually depends on the actual number of February 29ths there were between the start and end date?

(Cue "A paradox, a paradox, a most ingenious paradox" from "The Pirates of Penzance." Or not.)
Nisi

AFAIK in USD you use the 360 day year for all years.

You use compound interest for periods of more than 1 year, and simple interest for periods up to 1 year.

The rational thing to do would be to use the actual number of days in the year (although that's not the convention for USD). Ie include leap years as 366 days

https://plus.maths.org/content/have-we- ... r-interest

http://www.margill.com/Interest-calcula ... -paper.htm


http://www.math.hawaii.edu/~ramsey/Comp ... erest.html

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Re: To calculate CAGR, how long is "a year?"

Post by nisiprius » Thu Dec 03, 2015 6:20 pm

Not sure I understand VT's post. It also doesn't fit the facts, because the difference in calculating with 360 days and either 365 or 365.25 is pretty big, and the real-world examples of "official" numbers don't seem to fit. Maybe the method is used for calculating loans or bond values, but it doesn't seem to be used for CAGR values of mutual funds.

I think I like FactualFran's method best, and here's why. The "compound average growth rate" should be the annual that would give exactly the same growth as the observed growth. If we take "year" literally to mean calendar year, then 1% annual compounded growth should mean $10000 at daybreak of 1/1/2015, then $10100 at daybreak of 1/1/2016, then $10201 at the beginning of 1/1/2017, then $10303.01 at daybreak of 1/1/2018, and so on. Now, because 2015 has 365 days but 2016 has 366 days, 1% exact-calendar-year growth is actually slightly less per day in 2016 than in 2015. To be precise, it is 0.002726% per day in a 365-day year, but only 0.002719% per day in a 366-day year. Which is weird, but what can you do?

Anyway, for the "exactly 1% per actual calendar year" investment, yes, the growth during a fractional year would be calculated the way FactualFran calculates it. And, working it backwards will give us the rate that would exactly match an observed amount of growth.
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Re: To calculate CAGR, how long is "a year?"

Post by 691175002 » Thu Dec 03, 2015 6:33 pm

If you are going year-end to year-end or month-end to month-end then n is just an integer or multiple of 1/12th.

If you are actually going day to day (very unusual, especially for published performance figures) then almost any day count convention is fine if disclosed. I generally use actual/365.

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Re: To calculate CAGR, how long is "a year?"

Post by #Cruncher » Fri Dec 04, 2015 11:34 am

nisiprius in [url=https://www.bogleheads.org/forum/viewtopic.php?p=2706158#p2706158]this post[/url] wrote:
#Cruncher wrote:... The corresponding compounded annual growth rate is 7.156%.
... Do you have any idea if that is the "official" answer?
No. Mutual funds generally report annual returns over a whole number of years (1, 3, 5, and 10). An exception is the return "since inception". E.g., this web page reports a 9.34% average annual return for VTSMX from inception on 4/27/1992 to 11/30/2015. Perhaps the SEC stipulates an "official" way to annualize the return; but I don't know what it is.
Coles in [url=https://www.bogleheads.org/forum/viewtopic.php?p=2706192#p2706192]this post[/url] wrote:
#Cruncher wrote:In my opinion one should ignore the total number of days and any attempt to convert that to years. Instead:
  • Compute the fraction of a year from 8/31/1976 to 11/30/1976: 91/366 [ 1 ]
  • Compute the number of years from 11/30/1976 to 11/30/2015: 39
  • Add to get total number of years: 39 + 91/366
:?: Is this an equally valid methodology (but resulting in a different answer)?
  • Compute the number of years from 8/31/1976 to 8/31/2015: 39
  • Compute the fraction of a year from 8/31/2015 to 11/30/2015: 91/365
  • Add to get total number of years: 39 + 91/365
FactualFran in [url=https://www.bogleheads.org/forum/viewtopic.php?p=2707058#p2707058]this post[/url] wrote:Another alternative:
  • Compute fraction of the starting year from 8/31/1976 to 12/31/1976: 122/366
  • Compute the number of calendar years from 12/31/1976 to 12/31/2014: 38
  • Compute the fraction of the ending year from 12/31/2014 to 11/30/2015: 334/365
  • Add to get total number of years: 122/366 + 38 + 334/365
These are both valid alternatives. However, for what it's worth, my method is consistent with how the Treasury handles fractional interest periods when it converts yield to price for Notes and Bonds that it issues. For example for the TIPS issued 11/30/2015 that matures 7/15/2025 (column C below), it uses an initial fractional period of 46/184 from 11/30/2015 to the first interest date of 1/15/2016, followed by 19 six month periods from 1/15/2016 until maturity.

For those interested, here is the exact method the Treasury uses illustrated with three cases:

Code: Select all

Row    Col A                  Col B       Col C       Col D        Formulas in Col B
 1  Coupon [C]                3.625%      0.375%      2.000%	
 2  Yield [i]                 3.650%      0.664%      2.115%	
 3  Issued                    10/15/1998  11/30/2015  08/17/2015	
 4  First interest            01/15/1999  01/15/2016  02/15/2016	
 5  Matures                   01/15/2008  07/15/2025  08/15/2025	
 6  Days 1st period [s]       184         184         184         =B4-DATE(YEAR(B4),MONTH(B4)-6,DAY(B4))
 7  Days left 1st period [r]   92          46         182         =B4-B3
 8  Full periods [n]           18          19          19         =2*(YEAR(B5)-YEAR(B4))+(MONTH(B5)-MONTH(B4))/6
 9  PV @ first interest date  101.622184   97.531070  100.014954  =100*(-PV(B2/2,B8,B1/2,1,0)+B1/2)
10  Adjustment divisor          1.009125    1.000830    1.010460  =1+(B2/2)*(B7/B6)
11  Adjustment subtrahend       0.906250    0.140625    0.010870  =100*((B6-B7)/B6)*B1/2
12  Price [P]                  99.797017   97.309562   98.968752  =B9/B10-B11
The formulas on rows 9 through 12 are from Appendix B III. Formulas for conversion of inflation-indexed security yields to equivalent price (page 406) of the 31 CFR Part 356. (Note: Row 9 uses the Excel PV function to produce the same result as the Treasury's formula.) Interestingly they work for nominal Treasury Notes and Bonds as well as TIPS. (Illustrated in column D above.) Surprisingly, the slightly different formulas from Appendix B II. Formulas for conversion of non-indexed security yields to equivalent price (page 403) do not produce the correct result for either nominal Treasury notes/bonds or TIPS.

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