Question

Mike is looking for a loan. He is willing to pay no more than an effective rate of 8.000% annually. Which, if any, of the following loans meet Mike’s criteria? Loan X: 7.815% nominal rate, compounded semiannually Loan Y: 7.724% nominal rate, compounded monthly Loan Z: 7.698% nominal rate, compounded weekly a. Y only b. X and Z c. Y and Z d. None of these meet Mike’s criteria.

Answer 1

If Mike is willing to pay no more than an effective rate of 8.000% annually, the loans that meet his criteria are loan X and loan Z. Of those two, the lowest would be loan X.  I hope the answer will help you :)

Answer 2

b. X and Z

Explanation:

Since, the effective rate is,

`r=(1+(i)/(n))^i-1`

Where, i is the nominal rate,

n is the number of compounding periods,

For loan X,

i = 7.815 % = 0.07815,

n = 2, ( 1 year = 2 semiannual )

Thus, the effective rate would be,

`r=(1+(0.07815)/(2))^2-1`

`=0.079676855625`

`=7.9676855625\%\approx 7.968\%`

Since, 7.968 % < 8.000 %,

⇒ Loan X meets Mike's criteria,

For loan Y,

i = 7.724 % = 0.07724,

n = 12 ( 1 year = 12 months ),

Thus, the effective rate would be,

`r = ( 1+(0.07724)/(12))^(12)-1`

`=0.0800339518197`

`=8.00339518197\% \approx 8.003\%`

Since, 8.003 % > 8.000 %,

⇒ Loan Y does not meet his criteria,

For loan Z,

i = 7.698 % = 0.07698,

n = 52 ( 1 year = 52 weeks ),

Thus, the effective rate would be,

`r = (1+(0.07698)/(52))^(52)-1`

`=0.0799589986135`

`=7.99589986135\%`

`\approx 7.996\%`

Since, 7.996 % < 8.000 %,

⇒ Loan Z meets his criteria.

Therefore, Option 'b' is correct.