The following formula is incorrect for compound interest with the symbols defined this way, can someone tell me why?

Question

The following formula is incorrect for compound interest with the symbols defined this way, can someone tell me why?

 

 

Compound Interest is described by A = P(1+rm)^n, where P is the principal, r is the annual rate, m is the number of compounding periods in 1 year, and A is the amount in the account after n compounding periods.

It is an Exponential function since the exponent is a variable.

Answer 1

Hi Tonya,

 

(Please ignore my comment above.  It seems I saved and posted it before I was finished.)

Your instructor is correct. Now let's figure out why by calculating A, using your formula and the following assumptions:

Your initial account balance (P) = $1,000. The account earns an annual rate of 5% (r = .05), which is compounded quarterly (m = 4). Assume further that you wish to know the account balance after 2 years, so the number of compounding periods (n) = 8.

Using your formula, as written above, the calculation is as follows:

A = P * ((1 + (r * m))^n

A = $1,000 * (1 + (.05 * 4))^8

A = $1,000 * (1 + .2)^8

A = $1,000 * 4.3

A = $4,300

 

Would an account with an initial balance of $1,000 grow to $4,300 after 2 years, given interest of 5% per year?  Not likely, when (ignoring the effect of quarterly compounding for the moment) we know that the account will earn $50 in year 1 ($1,000 * 5%) and $52.50 in year 2 ($1,050 x 5%).  Consequently, the account balance at the end of year 2 (compound annually - as opposed to quarterly - for simplicity) would be $1,102.50 ($1,000 + $50 + $52.50).  $1,102.50 vs. $4,300? Something must be wrong.

 

Can you see where the problem is?  Here's a hint:  As written, your formula multiplies the initial deposit of $1,000 by 1.2 (1 + (.05 * 4) = 1.2) to the 8th power.  This would suggest that the quarterly interest rate - compounded for 8 quarters - is equal to 20%. We know that the quarterly interest rate cannot be higher than the annual rate, let alone 4 times higher.  Is it any wonder the calculated balance of $4,300 is so large?

 

How can you tweak your formula ever so slightly to convert the annual rate of 5% to the correct quarterly rate, which we know cannot be 20%?  

 

Once the formula is revised, the ending balance in the account after two years will be $1,104.49.  

 

I look forward to hearing your thought process and conclusion.

 

Jeff

Answer 2

A=P(1+r/m)^mn where

A is the amount

P is the principal

r is the annual rate

m is the number of compounding periods per year

n is the number of years

My apologies to the others.

 

Explanation for Tonya, and anyone else, of course.

Interest = Principal × r=annual rate × t=time in years

For 1/4 of a year, remember we are compounding quarterly, this means that at the end of 1/4 year we have P+P×r×1/4=P(1+r/4).  Note that any amount of money moved forward in time is simply multiplied by 1+r/4.  For the next 1/4 year we take the prior amount P(1+r/4)×(1+r/4)=P(1+r/4)²

In general after k quarters, the amount is P(1+r/4)^k

We are given n the number of years and so may use 4n as the number of quarters.

Finally we get P(1+r/4)⁴ⁿ.  There are some conventions to be considered when the number of years is not an integer, but that is a story for another time.

Answer 3

Does your book or your notes have a similar equation? Does it look the same or different than this one? The variables in your problem are defined like we normally define them for this kind of equation.

Answer 4

Tonya,

 

The formula I'm used to using is

 

A = P[1 + (r/m)]^nt

 

P = principal amount (the initial amount you borrow or deposit)

r = annual rate of interest (as a decimal)
t = number of years the amount is deposited or borrowed for.
A = amount of money accumulated after n years, including interest.
n = number of times the interest is compounded per year

 

I think that, if the equation you wrote is incorrect, it has to do with the expression r*m instead of the correct (r/m).  

 

I hope this helps.