Question

Melanie is looking for a loan. She is willing to pay no more than an effective rate of 9.955% annually. Which, if any, of the following loans meet Melanie’s criteria? Loan A: 9.265% nominal rate, compounded weekly Loan B: 9.442% nominal rate, compounded monthly Loan C: 9.719% nominal rate, compounded quarterly

Answer 1

Answer:

Loan A and B

Step-by-step explanation:

Since, the effective annual interest rate is,

$i=(1+\frac{r}{n})^n-1$

Where, r is the nominal rate( in decimals) per period,

n is the number of periods,

In Loan A :

r = 9.265% = 0.09265,

n = 52,

Thus, the effective annual interest rate is,

$i_1=(1+\frac{0.09265}{52})^{52}-1$

$=(1+0.00178173077)^{52}-1$

$=1.09698725072-1$

$=0.09698725072\approx 0.9670$

$\implies i_1=9.670\%$

In Loan B :

r = 9.442% = 0.09442,

n = 12,

Thus, the effective annual interest rate is,

$i_2=(1+\frac{0.09442}{12})^{12}-1$

$=(1+0.00786833)^{12}-1$

$=1.09861519498-1$

$=0.09861519498\approx 0.9862$

$\implies i_2=9.862\%$

In Loan C :

r =9.719% = 0.09719,

n = 4,

Thus, the effective annual interest rate is,

$i_3=(1+\frac{0.09719}{4})^{4}-1$

$=(1+0.0242975)^{4}-1$

$=1.10078993749-1$

$=0.10078993749\approx 0.10079$

$\implies i_3=10.079\%$

Since, $i_1$, $i_2$ but $i_3>9.955\%$

Hence, Loan A and B meets his criteria.

Answer 2

The answers are both loan A and loan B :)