Problem:
Blooie has the following obligations: 100,000 due after 4 years without interest and 200,000 dues after 3 years with accumulated interest at ($j_1$ = 0.12, $m_1$ = 2). Blooie wants to pay a certain amount after 2 years and then double this amount after another year. What should he pay after 2 years if all amounts are covered by an interest rate of ($j_2$ = 0.08, $m_2$ = 2)?
My solution:
Given:
Loans Payments Creditor
$P_1$ = 100,000 $P_2$ = 200,000 x (in 2 years) $j_2$ = 0.08
j = 0 j = 0.12 2x (in 3 years) $m_2$ = 2
m = 0 m = 2
$t_1$ = 0 $t_2$ = 3
Solve for future values
$$F_1 = 100,000$$ $$F_2 = 200,000(1+ \frac{0.12}{2})^{2(3)}\approx 283,703.82$$
Solve for values of loans at reference time
$$Loan 1 = 100,000(1+ \frac{0.08}{2})^{-2(2)}\approx 85,480.42$$ $$Loan 2 = 283,703.82(1+ \frac{0.08}{2})^{-2(1)}\approx 262,300.13$$
Add values of loans
$$Value of loans = 85,480.42 + 262,300.13 \approx 347,780.55$$
Compute for values of payments at reference time
$$Payment 1 = X$$ $$Payment 2 = 2x(1+ \frac{0.08}{2})^{-2(1)}$$ $$= 2x(0.9246)$$ $$\approx 2x(0.9246)+x$$
Write equation of values
$$Value of loans = Value of payments$$ $$2x(0.9246)+x = 347,780.55$$ $$x \approx 122,062.53$$
Final Answer:
Blooie should pay 122,062.53 after 2 years.