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Sagot :
We have to find the present value that can make quarterly payments of $4000 for 7 years, with an nominal interest rate of 6% compounded quarterly.
As it is compounded quarterly, we have 3 subperiods in a year. Then, the parameter m, the number of subperiods, is m=3.
The number of yearly periods is n=7.
The interest rate is r=0.06.
The payment is P=4000.
We have to use the annuity formula to calculate the present value:
[tex]\begin{gathered} PV=P\cdot\frac{1-\frac{1}{(1+\frac{r}{m})^{n\cdot m}}}{\frac{r}{m}} \\ PV=4000\cdot\frac{1-\frac{1}{(1+\frac{0.06}{3})^{7\cdot3}}}{\frac{0.06}{3}} \\ PV=4000\cdot\frac{1-\frac{1}{(1+0.02)^{21}}}{0.02} \\ PV\approx4000\cdot\frac{1-\frac{1}{1.5157}}{0.02} \\ PV\approx4000\cdot\frac{1-0.66}{0.02} \\ PV\approx4000\cdot\frac{0.34}{0.02} \\ PV\approx4000\cdot17 \\ PV\approx68000 \end{gathered}[/tex]Answer: the lump sum is approximately $68,000.
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