question text <u>WITH </u>missing information:
After examining the various personal loan rates available to you, you find that you can borrow funds from a finance company at an APR of <em>12 percent compounded monthly</em> or from a bank at an APR of <em>13 percent compounded annually.</em> Which alternative is more attractive?
If you borrow $100 from a finance company at an APR of 9% percent compounded for year, how much do you need to payoff the loan?
Answer:
The finance company option is better as we are taking the loan we want the lower rate possible.
We need $109 to payoff the loan of $100 at 9% annualy after a whole year.
Explanation:
We solve for the effective rate of 12% compounded monthly
= 1.12682503 = 0.126825 = 12.6825%
As this rate is lower than 13% this option is better
If we take 100 dollars after a year we have to pay:
$100 x (1 + r) = 100 x (1 + 0.09) = 100 x 1.09 = $109
I am very sure that the answer to this question is true.
Answer:
<em>The (minimum) annual interest rate should be at 7.28%</em>
Explanation:
<u>Compound Interest</u>
An investment consisting of a principal P, (or present value) earns interest on each period considering the previous period's amount including the interest earned (no withdrawals). This situation is defined as an investment in compound interest unlike simple interest, where each interest amount is withdrawn and the new principal is P again.
To find the future value (FV) of an investment with an interest annual rate i during n years is

If needed, we can solve the equation for i. Dividing by P:

Taking the nth-root:
![\displaystyle \sqrt[n]{\frac{FV}{P}} =1+i](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Csqrt%5Bn%5D%7B%5Cfrac%7BFV%7D%7BP%7D%7D%20%3D1%2Bi)
Finally:
![\displaystyle i=\sqrt[n]{\frac{FV}{P}} -1](https://tex.z-dn.net/?f=%5Cdisplaystyle%20i%3D%5Csqrt%5Bn%5D%7B%5Cfrac%7BFV%7D%7BP%7D%7D%20-1)
The parents will retire in n=27 years and they currently have P=$360,000 as an initial investment that they want to become into their retirement funds. Let's calculate the needed interest rate:
![\displaystyle i=\sqrt[27]{\frac{2,400,000}{360,000}} -1](https://tex.z-dn.net/?f=%5Cdisplaystyle%20i%3D%5Csqrt%5B27%5D%7B%5Cfrac%7B2%2C400%2C000%7D%7B360%2C000%7D%7D%20-1)


The (minimum) annual interest rate should be at 7.28%
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