Which interest rate would you choose if you want to borrow money from a ba \( 19 \% \) p.a. compounded monthly or \( 21 \% \) p.a. effective? Justify your answer with calculations.

To determine which interest rate is more favorable for borrowing money, we need to compare the two rates: \( 19\% \) per annum compounded monthly and \( 21\% \) per annum effective. ### Step 1: Calculate the effective annual rate (EAR) for the \( 19\% \) compounded monthly. The formula for the effective annual rate (EAR) when compounding monthly is given by: \[ \text{EAR} = \left(1 + \frac{r}{n}\right)^{nt} - 1 \] Where: - \( r \) is the nominal interest rate (as a decimal), - \( n \) is the number of compounding periods per year, - \( t \) is the number of years. For \( 19\% \) compounded monthly: - \( r = 0.19 \) - \( n = 12 \) - \( t = 1 \) Substituting these values into the formula: \[ \text{EAR} = \left(1 + \frac{0.19}{12}\right)^{12 \times 1} - 1 \] Calculating \( \frac{0.19}{12} \): \[ \frac{0.19}{12} \approx 0.0158333 \] Now, substituting this back into the formula: \[ \text{EAR} = \left(1 + 0.0158333\right)^{12} - 1 \] Calculating \( (1 + 0.0158333)^{12} \): \[ (1.0158333)^{12} \approx 1.2014 \] Now, subtracting 1: \[ \text{EAR} \approx 1.2014 - 1 = 0.2014 \] Converting to percentage: \[ \text{EAR} \approx 20.14\% \] ### Step 2: Compare the effective annual rates. Now we have: - Effective annual rate for \( 19\% \) compounded monthly: \( 20.14\% \) - Effective annual rate for \( 21\% \) effective: \( 21\% \) ### Conclusion Since \( 20.14\% < 21\% \), it is more favorable to choose the \( 19\% \) per annum compounded monthly rate for borrowing money. Thus, the better option is to borrow at \( 19\% \) p.a. compounded monthly.