If an annuity is set up for this purpose, how much must be invested now if the annuity earns 6.82%, compounded semiannually?

Question

A company wants to have $30,000 at the beginning of each 6-month period for the next 4

years. If an annuity is set up for this purpose, how much must be invested now if the annuity earns 6.82%, compounded semiannually?

Answer 1

This is an annuity due problem since the $30,000 is required at the beginning of each 6-month period.

There are 2 payments/yr

PV = C*[(1-(1+i/2)^-n)/i/2]*1+i/2

C is cash flow/period, i is annual interest rate, n = no. of payments

PV = 30000[(1-(1+0.0682/2)^-8)/0.0682/2]*(1+0.0682/2)

PV=30000*[(1-(1.0341)^-8)/0.0341]*1.0341

PV = 30000[(1-0.7647) /0.0341]*1.0341

PV=30000[0.2353/0.0341]*1.0341

PV=30000(6.9003)*1.0341

PV = $214068