An investor accumulates a fund by making payments at the beginning of each month for 6 years. Her monthly payment is 50 for the first 2 years, 100 for the next 2 years, and 150 for the last 2 years. At the end of the 7th year the fund is worth 10000. The annual effective interest rate is [math]\mathrm{i}[/math], and the monthly effective interest rate is [math]\mathrm{j}[/math].

Which of the following formulas represents the equation of value for this fund accumulation?

• [math]\ddot{s}_{\overline{24} \mid i}(1+i)\left[(1+i)^4+2(1+i)^2+3\right]=200[/math]
• [math]\ddot{s}_{\overline{24} \mid i}(1+j)\left[(1+j)^4+2(1+j)^2+3\right]=200[/math]
• [math]\ddot{s}_{\overline{24} \mid j}(1+i)\left[(1+i)^4+2(1+i)^2+3\right]=200[/math]
• [math]s_{\overline{24} \mid j}(1+i)\left[(1+i)^4+2(1+i)^2+3\right]=200[/math]
• [math]s_{\overline{24} \mid i}(1+j)\left[(1+j)^4+2(1+j)^2+3\right]=200[/math]

References

Hlynka, Myron. "University of Windsor Old Tests 62-392 Theory of Interest". web2.uwindsor.ca. Retrieved November 23, 2023.

Solution: C

Note that 6 years [math]=6 \times 12[/math] months [math]=72[/math] months; 4 years [math]=48[/math] months; 2 years [math]=24[/math] months.

After 6 years, the first two years of payments have grown to [math]50 \ddot{s}_{\overline{24} \mid j}(1+i)^4[/math]. After 6 years, the next two years of payments have grown to [math]100 \ddot{s}_{\overline{24} \mid j}(1+i)^2[/math]. After 6 years, the last two years of payments have grown to [math]150 \ddot{s}_{\overline{24} \mid j}(1+i)^0[/math]. Let [math]T[/math] be the total after 6 years. Then the total after 7 years is [math]T(1+i)[/math]. Thus

[[math]] 10000=(1+i)\left[50 \ddot{s}_{\overline{24} \mid j}(1+i)^4+100 \ddot{s}_{\overline{24} \mid j}(1+i)^2+150 \ddot{s}_{\overline{24} \mid j}(1+i)^0\right] . [[/math]]

Divide by 50 and factor to get

[[math]] 200=\ddot{s}_{\overline{24} \mid j}(1+i)\left[(1+i)^4+2(1+i)^2+3\right] . [[/math]]

References

Hlynka, Myron. "University of Windsor Old Tests 62-392 Theory of Interest". web2.uwindsor.ca. Retrieved November 23, 2023.