TC wrote:
- quote -

The way I think of it is this: APR is the number which, in combination
with the compounding rules (schedule), is sufficient information to
define the interest that will actually be paid/owed. Neither APR nor
compounding rules by themselves are sufficient information to define
the amount of interest that will be paid. However, the two together
are sufficient.

APR is definitely not the same thing as R_continuous. R_continuous
implies a set of compounding rules (specifically, that compounding is
done continuously). APR implies no compounding rules.

The APR is expressed in annual terms. It is pro-rated across the
compounding interval(s). For example, if the compounding rules
specify that interest is compounded every 30 days, then the interest
paid for a 30-day interval is APR*30/365. If the compounding rules
specify that interest is paid on the 1st of every month, the interest
paid could in theory be APR*28/365 one month and APR*31/365 another
month. In the case of continuously-compounding interest, it's
probably easier not to think in terms of an infinite number of
prorated interest amounts over and infinite number of infinitely
small intervals, although there probably is a mathematically valid
way to do it. :-)

The big gray area in my mind is really the definition of "annual".
As far as I know, there is no fixed mathematical definition of "year".
Some years are 365 days and some are 366. The length of an average
year is a physical phenomenon and can have no precise mathematical
foundation. Maybe the law has solved this problem by making a
fixed definition of "year" for certain purposes and in certain
jurisdictions.

The caveat with all of the above is that I'm neither an expert on
math or financial matters. :-) So, if I have any wrong assumptions,
it would actually be nice to hear about that.

- Logan


======================================= MODERATOR'S COMMENT:
Posters to this thread should relate comments to general financial planning.

On Apr 14, 12:38 pm, "TC" <golemdan...[at]yahoo.com> wrote:

- quote -

Actually, for a given year, the APR is not equivalent to the ln R_(t))/
ln R(t-1) -as you imply- . APR is the effective interest rate, adding
fees and expenses (in the case of loans). The effective interest rate
depends on when the interest accrues, and it is closer but not equal
to the interest rate found by applying logarithms. When you apply the
exponential function to a given amount (and the logarithm to
calculate the implicit APR) you are assuming that the interest accrues
continuously. That's not the case of the APR.

The general formula to calculate the effective interest rate is r =
(1+i/n)^n - 1, where i is the nominal rate, and n is the number of
periods in which the interest accrues. In your example, assuming that
the interest accrues every 12 months, the nominal interest rate of 20%
implies a nominal interest rate of 1.666667 % per month (20%/12), and
the APR would be r = (1+1.666667%)^12 -1 = 21.93%, which differs from
the exp(0.2) = 22.14% that you calculated.

On Apr 14, 6:55 am, "D T W .../\\..." <vze3n...[at]verizon.net> wrote:
- quote -

As wikipedia points out there is another important facter than just
the compounding period. One time fees such as points are taken into
consideration to figure out your equivalent cost of borrowing (APR).
A "no points" loan of 6% is quite different than a "2 points" loan at
6%. APR is a better "apples to apples" comparison than just rate.

It's worth pointing out that APR for mortages is a different beast from
APR/APY for bank deposits. This is where finance *really* diverges from
math.

APR for mortages *is* a standardized number. But it includes so much that
you may not recognize it. APR is for mortgages what APY is for banks (the
rate that you would be charged if you paid your mortgage once per year, e.g.
a $100K mortgage at 6% would result in $6K of interest if you paid at the
end of the year).

But ... The APY attempts to incorporate interest that goes beyond the basic
interest due from the outstanding loan. It incorporates points (which the
IRS considers interest, just paid up front), some loan fees, and mortgage
insurance.

HUD's definition: A measure of the cost of credit, expressed as a yearly
rate. It includes interest as well as other charges. Because all lenders
follow the same rule to ensure the accuracy of the annual percentage rate,
it provides consumers with a good basis for comparing the cost of loans,
including mortgage plans.
http://www.hud.gov/glossary.cfm

If you want the full and gory details (I've never looked through them):
http://www.bankersonline.com/lending...ng091905v.html
"Appendix J of Reg. Z contains the formulas and instructions. You cannot do
these calculations manually, a computer is required." (It has a link to Reg
Z, containing Appendix J: http://www.bankersonline.com/regs/226/j226.html )

Mark Freeland
BnetOnewsX[at]sbcglobal.net

"TC" <golemdanube[at]yahoo.com> wrote in message
news:1176519278.160097.162270[at]y80g2000hsf.googlegroups.com...
- quote -

There is no clear answer.

APR is the annualized rate without compounding, so it depends on the
period of the interest, and the calculation method! If you invest $100 at
20% APR where the interest is credited once per year, you'll have $120 at
the end of the year. If you invest $100 at 20% on a 365/365 calendar (where
interest is calculated daily, based on 1/365 of the APR), then you'll have
(assuming I used the calculator correctly :-) $122.13. Less than continuous
compounding, but close.

Another common calculation is 365/360 compounding. Divide the APR by 360
(12 "standardized" 30-day months), but compound daily. Back when savings
banks' interest was regulated (before the 1980s), they offered APRs of
5.25%. But you would usually get 5.39% yield (365/365 calendar) or 5.47%
(365/360 calendar).

This is why banks are required to advertise annual percentage yield (APY) -
the equivalent of interest credited once per year.

http://www.investopedia.com/articles.../04/102904.asp (APR vs. APY)

Mark Freeland
BnetOnewsX[at]sbcglobal.net

"TC" <golemdanube[at]yahoo.com> wrote:

- quote -

As I understand it, it depends on the compounding rate. Investments are
generally express as APR, but actually mean R_yearly/12 and are
compounded monthly. For example that $100 would go to 101.67, 103.36,
105.08... 121.94.

I think some loans are compounded daily which would be very close to
R_continuous.

http://en.wikipedia.org/wiki/Annual_percentage_rate

--
DTW .../\.../\.../\...

I'm good at math, but not very experienced in finance.

Because I'm good at math, I know that every exponential curve grows
according to the formula X(T) = X(0) * e ^ (T * R), where T is time
and R is the continuously compounded interest rate. The continuously
compounded interest rate is different from the effective rate over one
year. The two are related to each other as R_yearly = e ^ R_continuous
- 1.

Because I'm inexperienced at finance, I don't know whether "APR" means
R_yearly or R_continuous. Every example and definition I've seen on
the web suggests that APR = R_yearly. However, when I look at my
mortgage statement, it seems that the bank is using APR =
R_continuous, which works in their favor.

My question is this: Is there a fixed definition for "APR"? If so,
does it correspond to R_yearly or R_continuous?

To express my question in a different way, if I invest $100.00 at 20%
APR, should I expect to have $120.00 or $122.14 at the end of a year?
(Or is there no clear answer?)


-TC