Finding the Optimal Rate for Equal Interest Earnings on Savings Accounts

[Imparcticle]

I have become completely bewildered by the following problem:
Suppose you are a bank manager determining rates on savings accounts. If the account with interest compounded annually offers 5% interest, what rate should be offered on an account with interest compounded daily in order for the interest earned on equal investments to be the same?
the annual investment, I'm guessing is $1000 because in part "a" of the question, it pointed the amount out, although for a slightly different situation. In part "a", it asked the reader to find the compounded interest for once, twice etc. times during the year.(there are 3 parts to it, btw and the one above is part "c").
The answer is 4.88%.
This is the formula I think you're supposed to use:

Compounded interest is defined by the following formula:
A=P(1+ r/n)^nt , where "A" is the total amount
"r" is the interest rate
"n" is number of times it's paid
"t" is the time
"P" is the principal

Just in case, I have typed up the rest of the problem:

a. How much interest would you earn in one year on an $1000 investment earning 5% interest if the interest is compounded once, twice, four times, twelve times, or 365 times in the year?
b.) If you are making an investment that you will leave in an account for one year which account shuld you choose to get the higheest return?
Account rate compounded
statement savings 5.1% yearly
money market savings 5.05% monthly
super saver 5% daily
c.)Suppose you are a bank manager determining rates on savings accounts. If the account with interest compounded annually offers 5% interest, what rate should be offered on an account with interest compounded daily in order for the interest earned on equal investments to be the same?
THE FOLLOWING ARE THE ANSWERS TO A,B,C:
Answers
A.) $50; $50.63; $50.94; $51.16; $51.26
B.) Money Market Savings
C.) 4.88%

Oh, and this is from a precalculus course I am enrolled in.

 

[Fermat]

Let r1 be the annual interest of 5%. This gives an annual "pay-out factor" of (1+r) on a principle of Po. i.e. over one year,

P = Po(1+r1)

Let r2 be the equivalent interest rate compounded daily. Then the "pay-out factor) is (1+r2/365)^365 on a principle of Po over a period of one year. i.e.

P = Po(1+ r2/365)^365

For the interest rates to be equivalent, the "pay-out factors" must be equal. i.e.

(1+r1) = (1+r2/365)^365

You know r1 is 5%, so you can solve for r2.

 

[Architeuthis Dux]

I can provide an explanation for the optimal rate for equal interest earnings on savings accounts. The key factor in determining the optimal rate is the compounding frequency. Compounding refers to the process of adding earned interest back to the principal amount, which then earns additional interest. The more frequently interest is compounded, the higher the overall return on investment will be.

In this scenario, we are comparing two different compounding frequencies - annually and daily. The annual investment of $1000 with 5% interest compounded once a year will result in a total amount of $1050 after one year. On the other hand, if the same $1000 is invested with a daily compounding frequency, the interest will be added to the principal 365 times in a year, resulting in a total amount of $1050.96.

To find the optimal rate for equal interest earnings, we need to solve for the interest rate in the formula A=P(1+r/n)^nt, where A is the total amount, P is the principal, r is the interest rate, n is the number of times interest is compounded, and t is the time.

By setting the total amount to be equal for both annual and daily compounding frequencies, we can solve for r. Plugging in the values, we get:

Annual compounding: 1050 = 1000(1+0.05/1)^1
Daily compounding: 1050 = 1000(1+r/365)^365

Solving for r in the second equation, we get r = 0.0488 or 4.88%.

Therefore, to earn the same amount of interest on an annual investment of $1000, the daily compounding account should offer an interest rate of 4.88%. This is the optimal rate for equal interest earnings on savings accounts.

In conclusion, it is important for bank managers to consider the compounding frequency when determining interest rates on savings accounts. With the right compounding frequency and interest rate, customers can maximize their earnings on their investments.