What amount must be set aside on a boy's seventh birthday, which will provide 24 quarterly payments of ?3,500.00 for school expenses, if the first payment is to be made on his 16th birthday? Assume that money is 10% compounded quarterly?

 

·        ?113,221.63

·        ?64,162.39

·        ?26,376.75

·        ?62,597.45

 

?26,376.75

Step-by-step explanation

In the first instance, one needs to understand that the interest rate of 10% is annual rate, compounded quarterly, hence, the quarterly interest rate is 2.5%(10%/4=2.5%).

The next point is that for us to determine initial amount invested on boy's seventh birthday, we need first of all determine the present value of quarterly payments on his 16th birthday using the formula below:

PV=quarterly payment*(1-(1+r)^-n/r*(1+r)

(It is an annuity due since the first payment is due on 16th birthday and not a quarter thereafter)

PV=present value at 16th birthday which is the unknown

quarterly payment=?3,500.00 

r=quarterly interest rate=2.5%

n=number of quarterly payments required=24

PV=3,500*(1-(1+2.5%)^-24/2.5%*(1+2.5%)

PV=3,500*(1-(1.025)^-24/2.5%*1.025

PV=3,500*(1-0.55287535)/2.5%*1.025

PV=3,500*0.44712465/2.5%*1.025

PV=62,597.45*1.025

PV=64,162.39 

Having computed the present value at his 16th birthday, we discount the amount back to 7th birthday as follows:

PV=FV/(1+r)^n

PV=present value at his 7th birthday which is unknown

FV=Present value at 16th birthday( note that the initial amount of 64,162.39 is present value at 16th birthday while it is future value at 7th birthday)=   64,162.39 

r=quarterly interest=2.5%

n=number of quarters between 7th and 16th birthday i.e (16-7)*4=36 quarters(there are 4 quarters in a year)

PV=64,162.39 /(1+2.5%)^36

PV=26,376.76 (closest to ?26,376.75)