Question

Tiffany has taken out a loan with a stated interest rate of 8. 145%. How much greater will Tiffany’s effective interest rate be if the interest is compounded weekly than if it is compounded semiannually? a. 0. 3340 percentage points b. 0. 1659 percentage points c. 0. 1681 percentage points d. 0. 1234 percentage points.

Answer 1

Option c is correct, 0.1689.

Effective Interest rate

We know,  the formula for effective interest rate,

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

where,

r = effective interest rate,

i = stated interest rate,

n = number of compounding periods,

Given to us,

Stated interest rate = 8.145%,

Tiffany's effective interest rate

Tiffany's effective interest rate be if the interest is compounded weekly,

Stated interest rate, i = 8.145%,

number of compounding periods, n = 1 week = 1,

$r_w = (1 + \dfrac{i}{n} )^n - 1\\\\r_w = (1 + \dfrac{0.08145}{1} )^1 - 1\\\\r_w= 0.08145 = 8.145\%$

number of compounding periods, n = 1 year = 52 week = 52,

$r_w = (1 + \dfrac{i}{n} )^n - 1\\\\r_w = (1 + \dfrac{0.08145}{52} )^{52} - 1\\\\r_w= 0.084789 = 8.4789\%$

Tiffany's effective interest rate be if the interest is compounded semiannually,

Stated interest rate, i = 8.145%,

number of compounding periods, n = for half year = 1,

$r_s = (1 + \dfrac{i}{n} )^n - 1\\\\r_s = (1 + \dfrac{0.08145}{1} )^1 - 1\\\\r_s= 0.08145 = 8.145\%$

number of compounding periods, n = 2 semiannually = 2,

$r_s = (1 + \dfrac{i}{n} )^n - 1\\\\ r_s = (1 + \dfrac{0.08145}{2} )^2 - 1\\\\ r_s= 0.0831 = 8.31\%$

Thus, for the first week and 1 semiannually, the effective interest rate will be the same but will definitely make a change, if seen for a year.

Therefore, Tiffany's effective interest rate if interest is compounded weekly rather than compounded semiannually will be  $r_w-r_s=(8.4789-8.31)% = 0.1689%$.

Hence, option c is correct, 0.1689.

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