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 loans that meet this criteria are loans X and Z.

Effective annual rate

The effective rate is the rate that a person actually pays on a loan when the rate of compounding is accounted for.

Effective annual rate = (1 + APR / m ) ^m - 1

M = number of compounding

Determining the loans that meet the criteria

Loan X: (1 + 0.07815/2)^2 - 1 = 7.9677%

Loan Y: (1 + 0.07724/12)^12 - 1 = 8%

Loan Z: (1 + 0.07698/ 52)^52 - 1 = 7.9959%

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Answer 2

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

Computation of the effective interest rate for 3 different types of loans

Given,

Effective interest rate =8% $(i_e)$

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:

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

For Loan X:

$\begin{aligned}i_e&=(1+\frac{r}{m})^m-1\\&=(1+\frac{0.07815}{2})^2-1\\&=0.07968\;\text{or}\;7.869\%\end{aligned}$

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:

$\begin{aligned}i_e&=(1+\frac{r}{m})^m-1\\&=(1+\frac{0.07724}{12})^{12}-1\\&=0.08003\;\text{or}\;8.003\%\end{aligned}$

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:

$\begin{aligned}i_e&=(1+\frac{r}{m})^m-1\\&=(1+\frac{0.07698}{52})^{52}-1\\&=0.07996\;\text{or}\;7.996\%\end{aligned}$

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.

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