d Find the value on January 1 2010 of an annuity that will m

d. Find the value on January 1, 2010 of an annuity that will make 24 quarterly payments of $5.0 begin ing on January 1, 2014. Assume that the interest rate is 62% compounded annually and that simple interest is paid for part of a period.

Solution

Quaterlt Installment, PMT = $ 5,000
Interest Rate, i = 6.2% annually or 0.0155
Number of quaterly payments starting from 1 Jan 2014, n = 24
Payment at = Beginning of Annuity

Formula for Present Value of Annuity Due
PV = (PMT/i) *(1+i) *[1-(1+i)^-n]

PV at 1 Jan 2014 = (5000/0.0155)*(1+0.0155)*[1-(1+0.0155)^-24]
PV at 1 Jan 2014 = 322581*(1.0155)*[1-(1.0155)^-24]
PV at 1 Jan 2014 = 327581*[1-0.69312]
PV at 1 Jan 2014 = 327581*0.30868
PV at 1 Jan 2014 = 101116

Now,
Value at 1 Jan 2010 + Simple Interest for 1 Jan 2010 to 1 Jan 2014 = Value at 1 Jan 2014
PV + Interest for 4 Years = 101116
PV + (PV * 6.2% *4) = 101116
PV + 0.2480 PV = 101116
1.2480 PV = 101116
PV = 101116/1.2480
PV = 81023

Value at 1 Jan 2010 = $ 81,023