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Rate vs. Yield


Date: 01/31/2002 at 22:57:31
From: Jill
Subject: Compounded Interest Different From Bank

What is the interest amount for the period from Sept. 05, 2001 to 
Dec. 04, 2001, at 5% interest, compounded daily, on principal of 
$356,964.15? 

My calculation shows the interest should be $4,477.37. The bank shows 
$4,368.67. I used an Excel worksheet for the calculation.  

Here is the explanation from the bank for their calculation:

The interest for this tax deferred annuity is coumpounded daily. The 
formula to determine the daily interest rate on a flexible annuity 
cannot be calculated as simple interest. The formula to achieve the 
correct interest calculation is as follows:

  1. First, determine the number of days for which the interest is 
     being calculated and the interest rate for the time period over 
     which the calculation is to be performed.

  2. Next, take the number of days and divide by 365. This equals 
     factor A.

  3. Then, express the interest rate factor as "1+interest rate."
     A rate of 5% would have the interest rate factor of 1.0500.  
     Using the scientific calculator, take this number to the exponent 
     of factor A. This will provide the interest rate factor for the 
     number of days indicated.

   4. To determine the interest earned for a given dollar amount, 
      simply multiply the interest rate factor by the dollar amount.

Can you tell me why the difference?


Date: 02/01/2002 at 06:38:38
From: Doctor Mitteldorf
Subject: Re: Compounded Interest Different From Bank

Dear Jill,

   The difference between your calculation and the bank's is this

Bank formula: $356,964.15 (1.05)^(91/365) - $356,964.15 = $4368.67
Your formula: $356,964.15 (1 + 0.05/365)^91 - $356,964.15 = $4,477.37

Both are legitimate ways of compounding interest, and the question 
really is more legal or conventional than mathematical. Let me 
introduce the concepts of "true rate" and "yield." True rate is the 
exponential rate at which your money is growing from moment to moment.  
Yield is the effectiveness of that rate compounded over a full year.  

In your example: If the daily rate is 5%, then you get to divide that 
5% by 365 and say 5%/365 = 0.013699% is the interest gained every day 
- in other words, the money grows by a factor of 1.00013699 for every 
day it's on deposit. If you keep this up for a year, then the money 
will grow by more than 5% because of compounding. In fact, at the end 
of a year you will have (1 + 0.05/365)^365 times as much money, or 
5.1267% more than you started with.  This number 5.1267% is called the 
"yield."

In the bank's calculation, they are saying that 5% is the yield, not 
the rate. You can work the above logic backwards, to ask, "what daily 
rate will give me 5% at the end of the year after all compounding?"  
The answer is that the equivalent daily rate is 4.879%.  

So who's right? The answer depends on how your bank is advertising its 
interest.  If it advertises a "rate" of 5%, you are correct. If it 
advertises a "yield" of 5%, then the bank is correct.

You may be interested in a financial calculation program, Per%Sense, 
which I wrote several years ago and which is still used by lawyers and 
accountants for interest calculations. It is now available free for 
download from the Math Forum Web site at 

   http://mathforum.org/~josh/persense.zip   

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/   


Date: 02/02/2002 at 02:56:28
From: Jill
Subject: Compounded Interest Different From Bank

Thank you for the quick response. It helped greatly. Keep up the 
good work.
    
Associated Topics:
High School Interest

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