Question

Find the lump sum that must be set aside today to make quarterly payments of $4,000 for 7 years, assuming 6% compounded quarterly.

Answer 1

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:

`\begin{gathered} PV=P\cdot(1-(1)/((1+(r)/(m))^(n\cdot m)))/((r)/(m)) \\ PV=4000\cdot(1-(1)/((1+(0.06)/(3))^(7\cdot3)))/((0.06)/(3)) \\ PV=4000\cdot(1-(1)/((1+0.02)^(21)))/(0.02) \\ PV\approx4000\cdot(1-(1)/(1.5157))/(0.02) \\ PV\approx4000\cdot(1-0.66)/(0.02) \\ PV\approx4000\cdot(0.34)/(0.02) \\ PV\approx4000\cdot17 \\ PV\approx68000 \end{gathered}`

Answer: the lump sum is approximately $68,000.