The annual equivalent rate is the actual rate of return you get on some investment, taking into account the impact of compounding.
When you invest in something that has a stated interest rate (say, a savings account or a CD) the actual rate of return will tend to be slightly higher than the advertised amount. That's because these investments typically compound the interest at various intervals, usually monthly. Compounding means the interest you earn eventually earns its own interest. This little extra bit adds up over time, and the annual rate often turns out higher than the stated rate.
So say you have $100 in a CD that pays a nominal rate of 6%. The interest is computed on a monthly basis and compounded. In the first month, you will earn $0.50 in interest. (If you don't take our word for it: 6% over the 12 months of the year means in any given month you earn 0.5% in interest...as in 6% divided by 12 equals 0.5% per month. With a $100 initial investment, a 0.5% monthly rate would lead to $0.50 in interest earned in the first month. We're not doing that math for you; get a calculator if you don't believe us.)
So now we're going into month number two. There's $100.50 in the CD. You earn another month's interest at 0.5% per month. But wait, now we're earning that interest on $100.50 rather than $100. So the resulting interest payment is slightly higher.
The interest earned in the second month is $0.5025, or 50 cents plus a quarter of cent extra. Assuming the CD isn't provided by the companies in either Office Space or Superman III, that quarter of a penny is added to your total.
Do this same process over the course of a full year, adding up all those fractions of a cent, and your closing balance for the year would be $105.12. That's 12 cents more than the 5% advertised interest rate. That means your annual equivalent rate is 5.12%.
There's an equation for figuring out annual equivalent rate. It goes like this:
(1 + i/n)n - 1
Here, "i" equals the stated interest rate (5% in the example we gave), while "n" equals the number of compounding periods (in the example, the number of months in the year, or 12).
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Finance: How Do You Calculate Rates of R...35 Views
finance - a la shmoop how do you calculate rates of return? well invest a dollar get
more than a dollar back right? well yeah you hope so anyway in in finance land [dollar bill on table]
and Wall Street and any other professional gig. well rates of return
from financial investments are generally stated as annual returns, so calculating
a rate of return revolves around the one year at a time thing. there are a ton of
curveballs that get thrown into these calculations. here's a big one,
dividends. well guess what clueless financial journalists with little to no [dividends defined]
real schooling in finance quote stock market returns all the time. let's say
that shares in random example industries traded at the same price at the
beginning of the 1970s as they did at the end of the decade. prices for random
example industries were totally flat from 1970 to 1980. that's what one of
those journalists might say. and they don't even get fired for making such a [man reports news]
narrow statement .no nothing happened at all. and wrong. had they taken this course
they'd have realized that monster-sized dividends were paid out during that time
period. five six seven eight percent a year, each year. yet the journalists
ignored them when they stated that the stock market was in fact flat for a
decade and maybe shares of that company were also flat for a decade. but it
implied that they got no return from their investment which is absolutely [icons of stock market and a stock deflate]
wrong. did readers get their money back for that bad journalistic work? yeah we
doubt it - well what about zero coupon bonds? that is their bonds that pay no
dividends or interest along the way and they sell at a discount to par. what does
that mean? that is $1,000 par value bond pays you a grand in seven years. well how
do you calculate the annualized rates of return there? well today that bond sells
for six hundred forty two dollars. like you buy it today for six hundred forty
two you get a thousand bucks in seven years. well what's the rate of return on [zero coupon bond rates of return listed]
that bond? hmm. well vanilla bonds like these we're a whole lot easier to
calculate. because like you got the interest rate right there on the thingy.
yeah so the question is really what interest rate will accrue and then
compound for this bond such that in exactly seven years you get a thousand
bucks? well if it compounded at ten percent a year the compounding would
look like this. you see the table right there and whoa we've already passed the
grand way ahead of seven years. so the compound rate must be less than ten
percent right well what if it compounded at five percent a year well then the [compound rate listed]
rates of return would look like this and basically we're just multiplying 1.0
five times a 6.2 and we take that compound totally multiply 1.05 again and
so on and so on. much closer .well here's the formula you'll want to remember.
where f is the face value PV is the present value and n is the number of
periods. well in our example the face values a thousand bucks, the present
value is 642 dollars and the number of periods is the number of years or seven
years. all right well then we just you know put our handy-dandy calculator to [mathematical formula shown]
work and get a yield of well right around here. so here's the key idea rates
of return are an annual thing when quoted among finance professionals. among
fun dance professionals well and maybe a different story. [three stooges pictured]
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