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

the answers are both loan A and loan B :)

Answer 2

Loan A and B

Explanation:

Since, the effective annual interest rate is,

`i=(1+(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+(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+(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+(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<9.955\%`,
`i_2<9.955\%` but
`i_3>9.955\%`

Hence, Loan A and B meets his criteria.