Question

Alexander deposited money into his retirement account that is compounded annually at an interest rate of 7%.

Alexander thought the equivalent quarterly interest rate would be 2%. Is Alexander correct? If he is, explain
why. If he is not correct, state what the equivalent quarterly interest rate is and show how you got your
answer. i need to show my work and everything

Answer 1

Tow rates are equivalent if tow initial investments over a the same time, produce the same final value using different interest rates.

For the annually rate we have that:

`V_(0) =(1+ i_(a) ) ^(1)`
Where

`V_(0)` = initial investment.

`i_(a)` = annually interest rate in decimal form.
And the exponent (1) represents the full year.

For the quarterly interest rate we have that:

`V_(0) =(1+ i_(q) ) ^(4)`
Where

`V_(0)` = initial investment.

`i_(q)` = quarterly interest rate in decimal form.
And the exponent (4) the 4 quarters in the full year.

Since the rates are equivalent if tow initial investments over a the same time, produce the same final value, then

`(1+ i_(a) )=(1+ i_(q) ) ^(4)`
Notice that we are not using the initial investment
`V_(0)` since they are the same.

The first thin we are going to to calculate the equivalent quarterly rate of the 7% annually rate is converting 7% to decimal form
7%/100 = 0.07
Now, we can replace the value in our equation to get:

`(1+0.07)=(1+ i_(q) ) ^(4)`

`1.07=(1+ i_(q) ) ^(4)`

`\sqrt[4]{1.07} =1+ i_(q)`

`i_(q) = \sqrt[4]{1.07} -1`

`i_(q) =0.017`
Finally, we multiply the quarterly interest rate in decimal form by 100% to get:
(0.017)(100%) = 1.7%
We can conclude that Alexander is wrong, the equivalent quarterly rate of an annually rate of 7% is 1.7% and not 2%.