Explanation on how interest accrues???

Question

I've always been confused as to exactly what it means when they say Savings acct with 4.75% interest. I've heard this is an annual return. Can someone break this down for me and explain this in plain terms so I know what the heck this means? lol thanks!

Answer 1

Unless expressed otherwise, "interest" is always the annual rate of return on the sum shown, figured as 'simple interest' rather than some other form of interest such as 'compound interest'. Example: If your savings account has $1,000 in it, and the bank is paying you "4.75 %interest", your savings account will earn $1,000 times 0.0475 (which is 4.75 percent expressed in decimals) the first year, or $47.50. Thus, at the end of the first year you will have $1,047.50 in your account.

Answer 2

Savings account will compound the interest for you, so long as you do not withdraw the money. This is basically how they work:

They will calculate how much interest they owe you based on the amount you have in the account (the principal) and the interest rate they quoted you. The rate is expressed as an annual amount, although they usually break this amount into monthly-sized payments.

For example, if you had $1000 in your savings account and the interest is 4.75%, you will receive $47.50 in interest for that year (approximately $3.95 per month). (1000 x 0.0475 = 47.50) Your principal next year will then be the original $1000 plus your $47.50 in interest, giving you a new principal of $1047.50. Now, for the second year, the bank will pay you interest on $1047.50, not $1000. The interest from the first year is put into your account and goes towards earning more interest. This is sometimes referred to "interest-on-interest" earnings, also called "compound interest".

It is the ability to compound your earnings that will allow you to amass a very large sum of money if you invest the money for a long time. This is how people who start investing for retirement in their 20's are able to put away only 8 - 10% of their annual salary and build a multi-million dollar nest egg for retirement.

Answer 3

Imagine you invest $100 on Jan. 1, 2007 at 5%. If the interest accrues annually, on Jan. 1, 2008, you will have $100 plus 5% of $100 for a total of $105. Then, one year later you will have $105 plus 5% of $105 for a total of $110.25. As you can see, you made $0.25 more the second year than the first because the 5% was applied to a higher balance the second year. As you go on for many years you make more and more. This is the miracle of compounding interest.

To see how much money can grow, get a calculator and multiply $100 times 1.05. Then multiply that total by 1.05. Then again and again. Each amount you see is like keeping $100 in a savings account at 5% for that number of years.

Answer 4

When one speaks of interest, there are essentially two kinds: simple interest and compound interest.

If one invests a sum of money at a simple interest rate, say your 4.75% per annum (i.e. per year), they are paid only that fractional amount on the money they lent to the bank for their use during the contract term. So, $50,000 invested in a bank in either a savings account or a CD at the aforementioned rate would earn $50,000 x 0.0475 = $2,375.00 for the first year it is invested. At the end of the year, the interest is paid to the account, and interest begins to accrue anew on the sum of the original amount plus the interest, unless the money is withdrawn at the end of the contracted term. The formula for the amount at the end of one year using a simple interest scheme is:

A (t) = a (1 + r),

where A (t) is the final amount, a is the initial amount, and r is the simple interest rate.

That brings us to compound interest, where the year is broken down into payment sub-intervals, and the interest is paid in smaller increments. Because interest payments are made many more times during the year, the principal grows at a much faster rate than with simple interest. Hence, one makes a truckload more money by comparison over a long period of time.

When dealing with compound interest, there are two terms to remember: nominal rate and annual percentage yield, or APY for short. The APY is the bottom line figure to look for when searching for an investment vehicle. The nominal rate is the percent rate the bank says they will pay for the year. But that is the rate before compounding has its growth effect on one's money. So, suppose a bank says they will pay 6.00% nominal rate compounded quarterly for a year. That means they will calculate the amount of interest the money in the account has earned every three months, or four times every year. At the end of each compounding sub-interval, or quarter in this case, they also pay the interest earned to the account, then start a new quarter. Because the account now has more money in it from the interest added, it earns even more interest during the next quarter because the rate has not changed between quarters. Here's an example using the 6.00% figure mentioned earlier. Let's say one invests the $50,000 at that rate. Since there are four quarters, then 6.00% / 4 = 1.50% is what they will pay per quarter. At the end of the quarter the bank calculates they owe $750.00 to the account ($50,000 x 0.015 = $750.00), which is paid to the account and a new quarter begins. At the end of the second quarter, they must pay 1.5% interest on $50,750 instead of the original $50,000. That works out to $761.25 instead of the original $750. Now that is added to the account at the end of that quarter and a new one begins yet again. By the end of the fourth quarter, the account is accruing interest at the rate of $784.26 due to the effect of compounding.

$50,000 x 0.015 = $750
($50,000 + $750) x 0.015 = $761.25
($50,000 + $750 + $761.25) x 0.015 = $772.67
($50,000 + $750 + $761.25 + $772.67) x 0.015 = $784.26

Notice that the dollars earned increases with each quarter. When we add up all the dollars of interest earned and divide that total by our original investment, $50,000, we notice another interesting phenomenon. The percentage earned is higher than the rate the bank said they would pay per year:

$750 + $761.25 + $772.67 + $784.26 = $3068.18

$3,068.18 / $50,000 = 0.0613636 or 6.13636%, not the 6% figure. The percent actually earned (6.13636%) is called the APY, and is due to the growth effect compounding has on money.

It turns out there is a handy formula for calculating what amount of money one will have at the end of any number of compounding periods given any interest rate. Using this formula, and knowing how to deal with logarithms, one can figure out all sorts of things, like how much money to invest to gain a certain yield over a defined period, etc. The mathematics of finance is fascinating stuff. Here's the formula:

A (t) = a (1 + r / n)^nt,

where A (t) is the value of the account after t years, a is the initial investment, r is the nominal rate, n is the number of compounding periods per year, and t is the number of years the investment lasts. Obviously, the greater the number of compounding periods in the year the better. Let's plug the figures we dealt with earlier into this formula and see what comes out.

A (t) = $50,000 [1 + (0.06 / 4)]^4*1
A (t) = $50,000 [1 + (0.015)]^4
A (t) = $50,000 (1.0613636)
A (t) = $53,068.18

What do you know? We got exactly the same thing we calculated originally. So, the formula works.

If the interest is compounded continuously, which some financial institutions do, then there is a special formula used for that case:

A (t) = a (e^rt),

where A (t) is the final value, a is the original investment, r is the nominal rate expressed as a decimal, t is the number of years the investment runs, and e is the base of the natural logarithmic system. e is actually defined this way:

e = lim as x ---> ∞ of [1 + (1 / x)]^x

What that means in English is that e is equal to the limit as x goes to infinity of the expression [1 + (1 / x)] raised to the xth power. So, we are raising a sum which is increasingly approaching 1 to a power which is growing infinitely large. e is approximately equal to 2.718. Try it on your calculator and see for yourself

It can be proven using advanced calculus techniques that if (1 / x) is replaced by (r t / x), then lim as x ---> ∞ of [1 + (rt / x)]^x = e^rt. Hence the formula is derived.

This explanation has been pretty longwinded, but I hope I have made this "spooky mechanism," as one of the other answerers calls it, sufficiently straightforward to help you understand the subject a little better. Good luck, and may you have many happy investment returns.

Answer 5

Interest is money you earn on your investment via some spooky mechanism. Lets say we are given a principal value (P) of $100 and an interest rate of 4.75% per year.

After letting our $100 sit with the bank (who loans it to other people) for a year, it will have grown to $104.75. The $4.75 would be your return. Next year, if you don't take out any money, you will have $109.73. This is known as compounding interest- the interest you get makes money too, in an exponential way.

There are a couple of ways interest can be calculated, and it really depends on how your bank does it. Annual interest calculates the amount in your account at a fixed date every year; for example, on Jan 1st, your account will grow by 4.75% and do this yearly. Interest can also be calculated monthly, quarterly, daily, or whatever you want. Continuously calculated interest does it instant by instant and tends to return higher than annual.