enny owes Michelle a sum of money that will amount to R 15 000 at an interest rate of 12% per annum, compounded semi-annually, after 8 years. Michelle in turn owes Jenny a sum of money that will amount to R14250 at a rate of 10% per annum, compounded quarterly, after 5 years. They agree to cancel their respective liabilities by making at present a single payment. Who should make this payment and what should this payment amount to?
Jenny owes Michelle: R 2 791.87
Michelle owes Jenny: R 2 791.87
Jenny owes Michelle: R 328.65
Michelle owes Jenny: R 328.65
None of the above
To determine who should make the payment and how much it should be, we need to calculate the present value of each of their liabilities at the present time.
For Jenny's liability to Michelle:
Principal amount (A) = R15 000
Interest rate (r) = 12% per annum compounded semi-annually
Number of compounding periods per year (n) = 2
Number of years (t) = 8
Using the formula for compound interest:
\(P = A\left(1 + \frac{r}{n}\right)^{nt}\)
\(P = R 15000\left(1 + \frac{0.12}{2}\right)^{2*8}\)
\(P = R 15000\left(1 + 0.06\right)^{16}\)
\(P = R 15000(1.06)^{16}\)
\(P = R 15000*2.010599\)
\(P = R 30159.85\)
So, the present value of Jenny's liability to Michelle is R 30 159.85.
For Michelle's liability to Jenny:
Principal amount (A) = R14 250
Interest rate (r) = 10% per annum compounded quarterly
Number of compounding periods per year (n) = 4
Number of years (t) = 5
Using the same formula for compound interest:
\(P = A\left(1 + \frac{r}{n}\right)^{nt}\)
\(P = R 14250\left(1 + \frac{0.10}{4}\right)^{4*5}\)
\(P = R 14250\left(1 + 0.025\right)^{20}\)
\(P = R 14250(1.025)^{20}\)
\(P = R 14250*1.640904\)
\(P = R 23403.87\)
So, the present value of Michelle's liability to Jenny is R 23 403.87.
Since Michelle's liability to Jenny is less than Jenny's liability to Michelle, Michelle should make a single payment to Jenny in order to cancel out their respective liabilities. The amount of this payment would be the difference between the present values of their liabilities:
R 30 159.85 - R 23 403.87 = R 6 755.98
Therefore, Michelle should make a single payment of R 6 755.98 to Jenny in order to cancel out their respective liabilities.