## Question about Interest compounded daily

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zrzhu111
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### Question about Interest compounded daily

On Bankrate.com, I saw Barclays offers .9% on saving. It says "compound daily", which means earns interest on previous interest. What's the difference from the APY rate in the bank? Can you please give an example, let's say we deposit \$1000? Thanks.

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### Re: Question about Interest compounded daily

Welcome! Are you asking about the difference between APR and APY? There really is no difference, it's all how the information is presented to you. The details are in the wiki: APR versus APY
wiki wrote:There are two ways of expressing compound interest. Although the results are identical, the manner in which the interest rates are presented has implications to borrowers and lenders. As defined below, Annual Percentage Rate (APR) is always a lower number than Annual Percentage Yield (APY).

* Lenders, such as banks or other institutions, will quote the higher APY to entice customers. For example, bank savings accounts are advertised using APY, as customers are looking for the highest rates.
* Borrowers looking for the lowest rates will be attracted to the lower APR. For example, mortgages and automobile loans use APR to entice borrowers with low rates.
I don't know where your background is. If this is confusing please say so. Here's your \$1,000 example, using 10% interest (to make the numbers large). Since it's paid every period, we call that APR. Changing it to the yearly rate is called APY.

Code: Select all

``````Compounding          Effective Rate (APY)    Future Value
quarterly (4)      10.38% =EFFECT(10%,4)      \$1,103.81  =FV(10.38%,1,0,-1000)
monthly (12)       10.47% =EFFECT(10%,12)     \$1,104.71 = FV(10.47%,1,0,-1000)
daily (365)        10.52% =EFFECT(10%,365)    \$1,105.16 = FV(10.52%,1,0,-1000)``````
You can see that compounding more frequently will give you more money. OTOH, if it's a loan, you will pay more.

Try this yourself. If you don't have Excel, just use LibreOffice Calc (free open source).
To some, the glass is half full. To others, the glass is half empty. To an engineer, it's twice the size it needs to be.

zrzhu111
Posts: 99
Joined: Sun Sep 01, 2013 6:03 pm

### Re: Question about Interest compounded daily

Compounding Effective Rate (APY) Future Value
quarterly (4) 10.38% =EFFECT(10%,4) \$1,103.81 =FV(10.38%,1,0,-1000)
monthly (12) 10.47% =EFFECT(10%,12) \$1,104.71 = FV(10.47%,1,0,-1000)
daily (365) 10.52% =EFFECT(10%,365) \$1,105.16 = FV(10.52%,1,0,-1000)

How do you convert the rate for daily/quarterly/monthly compounding to APY?

dm200
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### Re: Question about Interest compounded daily

Most banks and credit unions today post and compound interest at monthly or quarterly periods. The compounding frequency must be disclosed. With interest rates so low, and that APR must be stated to two (not more and not less) decimal places, the rate and APR are often the same (when rounding the APR to two decimal places).

If I get more time tomorrow, I will do the detailed calculations - but at such a low rate 0f 0.90%, the APR is 0.90 whether compounded daily, monthly, quarterly or annually. At a 1.00% rate, compounded daily is 1.01% - and still just 1.00 for longer compounding periods.

Bottom line - daily compounding is zero to noise level for \$1,000 compared with longer compounding or no compounding (annual).

zrzhu111
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### Re: Question about Interest compounded daily

Thanks. I think I figured it out.

APR=0.01 compounding monthly
=10000*(1+0.01/12)^12=10100.4596088718

APR= 0.009 compounding daily
=10000*(1+0.009/365)^365= 10090.405098137

#Cruncher
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### Re: Question about Interest compounded daily

zrzhu111 wrote:How do you convert the rate for daily/quarterly/monthly compounding to APY?
Read the section in the Wiki, Periodic interest rate, just under the section Lady Geek referenced. It explains the relationship between the "periodic rate", the APR, and the APY. Using the Wiki's formula, to convert from a daily periodic rate to APY would be:
APY = (1 + daily rate) ^ 365 - 1

The Wiki and several other web pages I examined show how to determine the APY given a daily periodic rate. But I didn't find any that show the reverse: how to calculate the daily periodic rate given the APY. To do this we need to alter the above formula to be:
daily rate = (1 + APY) ^ (1 / 365) - 1

Banks usually quote the APY -- not the APR -- for savings accounts, so that's probably what the 0.9% for the Barclays account is. Using the formula the daily rate is then:
daily rate = 1.009 ^ (1 / 365) - 1 = 0.002455%

To check this, we'll convert it back to the APY:
APY = 1.00002455 ^ 365 - 1 = 0.9%

If compounding is done monthly or quarterly instead of daily, use 12 or 4 instead of 365 in the above formulas. Now, you asked for an example ... Assume you deposit \$1,000,000 in the Barclays account. The first day you would earn \$24.55 in interest (1,000,000 X 0.002455%). After 30 days you'd earn \$736.76 (1,000,000 X [1.00002455 ^ 30 - 1]). This is a whopping \$0.26 more than the \$736.50 you'd earn if interest weren't compounded daily (1,000,000 X 0.002455% X 30). After 365 days you'd earn \$9,000 (1,000,000 X [1.00002455 ^ 365 - 1]).

Note: the "^" symbol denotes raising to a power. Most scientific calculators have a [y^x] key that does this.
zrzhu111 wrote:I think I figured it out. ... APR= 0.009 compounding daily =10000*(1+0.009/365)^365= 10090.405098137
That would be correct if the 0.9% were an APR. But I'm pretty sure Barclays is quoting an APY; in which case you'd end up with just \$10,090.00. The corresponding APR is 0.896% using the formula in the Wiki (0.002455% X 365).

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### Re: Question about Interest compounded daily

#Cruncher wrote:The Wiki and several other web pages I examined show how to determine the APY given a daily periodic rate. But I didn't find any that show the reverse: how to calculate the daily periodic rate given the APY. To do this we need to alter the above formula to be:
daily rate = (1 + APY) ^ (1 / 365) - 1
Thanks, I added it to the wiki: Comparing Investments (Periodic interest rate). I revised the equations to be consistent with the standard financial variable notation (N).

The main article: Comparing Investments

If you learn how to setup a cash flow diagram, the 6 financial variables, and sign conventions, you'll be able to solve any financial problem. It's worthwhile learning. There are tutorials at the bottom of the article (under "External links").

Update: Via PM, #Cruncher noted a discrepancy in my interpretation of APR and supplied updated formulas. I revised the wiki to use his formulas.
To some, the glass is half full. To others, the glass is half empty. To an engineer, it's twice the size it needs to be.

zrzhu111
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Joined: Sun Sep 01, 2013 6:03 pm

### Re: Question about Interest compounded daily

Thanks! I know now I was confused with APY and APR.
#Cruncher was right. The rates on Bankrate.com are APY, not APR, so I don't even need to do any conversion to compare the daily compounding and monthly compounding.

The Wizard
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### Re: Question about Interest compounded daily

zrzhu111 wrote:Thanks. I think I figured it out.

APR=0.01 compounding monthly
=10000*(1+0.01/12)^12=10100.4596088718

APR= 0.009 compounding daily
=10000*(1+0.009/365)^365= 10090.405098137
There's even something called CONTINUOUS compounding which uses the exponential function:
10000*exp(0.009) = 10090.4062
It matters more when interest rates get bigger...
Attempted new signature...

dm200
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### Re: Question about Interest compounded daily

In several "past lives" (my day jobs), I actually wrote detailed specifications for programmers doing these calculations for posting interest, calculating and disclosing the APY and APYE (for periodic statements). The APY (Annual Percentage Yield) is the projected yield taking into account the rate and compounding, if any. Those calculations are almost always done with an assumed large dollar balance. The APY must be disclosed as being rounded to two decimal places. The Actual Yield you receive on your deposits may vary slightly (in either direction) because of the timing of your deposits and withdrawals on the account, the method used by the bank or credit union (daily balance vs. average daily balance), rate tiers, tiering method (A or B) an minimum amount to earn interest. The APYE (Annual Percentage Yield Earned) is only used on periodic statements and (rounded to two decimal places) and tells you the actual yield you received on your account. Especially for smaller balances, the rounding of posted interest to the nearest cent can produce some odd-appearing APYE's.

To take an extreme case, suppose a bank or credit union pays an interest rate on a savings account of 1.50% and that there is no minimum balance to earn interest. Suppose also that the interest is paid annually, with no compounding during the year. The bank/credit union would then disclose an interest rate of 1.50% and an APY of 1.50%. Now, suppose you have \$1.00 in the account. At the end of the year, the interest calculation comes up with 1.5 cents, which would be rounded to \$0.02. Your APYE would then be 2.00% on your statement. Suppose instead, you had only \$0.99. The interest calculation would be 1.485 cents, which would be rounded to \$0.01. Your APYE would then be 1.00% on your statement.