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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

AY only
BX and Z
CY and Z
DNone of these meet Mike’s criteria.

Verified Answer :

Answer:

b. X and Z

Step-by-step explanation:

Since, the effective annual rate is,

\(i_a=(1+\frac{r}{m})^m-1\)

Where r is the nominal rate per period,

m is the number of periods in a year,

For loan X,

r = 7.815 % = 0.07815

m = 2,

Thus, the effective annual rate,

\(i_a=(1+\frac{0.07815}{2})^2-1\)

\(=(1+0.039075)^2-1\)

\(=1.07967685563-1=0.07967685563=7.967685563\% \approx 7.968\%\)

Since, 7.968\% < 8.000 %

Thus, Loan X meets his criteria.

For loan Y,

r = 7.724%= 0.07724

m = 12,

Thus, the effective annual rate,

\(i_a=(1+\frac{0.07724}{12})^{12}-1\)

\(=(1.00643666667)^{12}-1\)

\(=1.08003395186-1=0.08003395186=8.003395186\% \approx 8.003\%\)

Since, 8.003 > 8.000 %

Thus, Loan Y does not meet his criteria.

For loan Z,

r = 7.698% = 0.07698

m = 52,

Thus, the effective annual rate,

\(i_a=(1+\frac{0.07698}{52})^{52}-1\)

\(=(1.00148038462)^{52}-1\)

\(=1.07995899887-1=07995899887=7.995899887\% \approx 7.996\%\)

Since, 7.996 % < 8.000 %

Thus, Loan Z meets his criteria.

Hence, option 'b' is correct.

Answer:

X and Z

Step-by-step explanation:

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