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

The Loan condition of Loan X and Loan Y will meet the effective rate of 8.00% criteria of Mike.

Computation:

Given,

Effective interest rate =8% $(i_{c})$

Nominal interest rates:  $(r)$

Loan X =7.815%, compounded semiannually $(m=2)$

Loan Y: 7. 724% nominal rate, compounded monthly $(m=12)$

Loan Z: 7. 698% nominal rate, compounded weekly $(m=52)$

The formula of the effective interest rate will be used:

$i_{c}=(1+(\frac{r}{m})^{m}-1)$

For Loan X:

$i_{c}=(1+(\frac{r}{m})^{m}-1)$

$i_{c}=(1+(\frac{0.07815}{2})^{2}-1)$

$i_{c}=0.07958 \:0r\: 7.968\%$

As the effective interest rate of Loan X is lower than the actual effective interest rate. Therefore, loan X meets the criteria of Mike.

For Loan Y:

$i_{c}=(1+(\frac{r}{m})^{m}-1)$

$i_{c}=(1+(\frac{0.07724}{12})^{12}-1)$

$i_{c}=0.08003 \:0r\: 8.003\%$

As the effective interest rate of Loan Y is greater than the actual effective interest rate. Therefore, loan Y will not meet the criteria of Mike.

For Loan Z:

$i_{c}=(1+(\frac{r}{m})^{m}-1)$

$i_{c}=(1+(\frac{0.07698}{52})^{52}-1)$

$i_{c}=0.07996 \:0r\: 7.996\%$

As the effective interest rate of Loan  Z is lower than the actual effective interest rate. Therefore, loan Z meets the criteria of Mike.

Therefore, in reference to the computation of the effective interest rate of individual loans. The correct option is b. X and Z.

To know more about the effective interest rates, refer to the link:

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