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.

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 :)

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parmesanchilliwack parmesanchilliwack · Answer 2 · 2019-06-17T09:02:25+02:00

Answer:

b. X and Z

Step-by-step explanation:

Since, the effective rate is,

[tex]r=(1+\frac{i}{n})^i-1[/tex]

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,

[tex]r=(1+\frac{0.07815}{2})^2-1[/tex]

[tex]=0.079676855625[/tex]

[tex]=7.9676855625\%\approx 7.968\%[/tex]

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,

[tex]r = ( 1+\frac{0.07724}{12})^{12}-1[/tex]

[tex]=0.0800339518197[/tex]

[tex]=8.00339518197\% \approx 8.003\%[/tex]

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,

[tex]r = (1+\frac{0.07698}{52})^{52}-1[/tex]

[tex]=0.0799589986135[/tex]

[tex]=7.99589986135\%[/tex]

[tex]\approx 7.996\%[/tex]

Since, 7.996 % < 8.000 %,

⇒ Loan Z meets his criteria.

Therefore, Option 'b' is correct.