A young couple wants to have a college fund that will pay 35

A young couple wants to have a college fund that will pay $35,000 at the end of each half-year for 8 years. If they can invest at 8%, compounded semiannually, how much do they need to invest at the end of each 6-month period for the next 18 years in order to begin making their college withdrawals 6 months after their last investment? (Round your answer to the nearest cent.) $ Suppose 8 years after beginning the annuity payments, they receive an inheritance of $38,000 that they contribute to the account, and they continue to make their regular payments as found in part (a). How many college withdrawals will they be able to make before the account balance is $0? (Round your answer to the nearest whole number.) withdrawals

Solution

Suppose the couple wants to start receivign 35,000 for 8 years at every 6 months period starting from now; their first installment should come in 6 months from now and last in 8 years from now. So present value of that money would be:
PV= E/r * [1- 1/(1+r)^n ] = 35000/ 0.04 * (1- 1/(1.04)^16) = 35/0.04 * 0.466 = 407,830; Here we considered r= 0.04 and n= 16 because the compounding is done semi-annualy thus 8*2=16 periods and r= 0.08/2=0.04% ;
(Let\'s call this equation A where first 35,000 amount is received in 6 months from now)
As per (a) the couple invests at 8% compounded semi-annualy at the end of each 6 month period ofor 18 years and will make their withdrawl 6 months after this period; Thus, let us say their monthly installment is E;
How much money they will have after 18 years=
E* [ (1+r)^n-1 ] / r = E* {(1.04)^36 - 1 }/ 0.04 = E*77.598;
But the withdrawls will be made after a 6 month period further after the installment stops as we discussed in equation A
Hence: 407830= P* 77.598; Thus P= 5255.676= 5255.68Dollars

Thus a sum of 5255.68$ will need to be paid every 6 months for 18 years by teh couple to have that college fund.

b) 8 years after they begin their annuity payments they receive a sum of 38,000 which they contribute to the account as a lump sum. This amount too undergoes an 8% per annum compounded semi-annually; Thus this goes on for 18-8=10 more years;

So total amount at the end of 18 years will be = Amount accumulated through annutiy+ ammount accrued through this inheritance= 407830+ 38000(1.04)²⁰ = 491092;

This amount is to be distribtued in chunks of 35,000 for every 6 months for an indefinite period (which we need to find out) ;
Thus 491092= 35,000/0.04 * (1- 1/(1.04)^n) ;
14.0312*0.04= (1- 1/(1.04)^n);
1-0.561248 = (1.04)^-n;
0.438752= (0.96153)^n
Thus n= log (0.43872) / log (0.96153)=21.006 ~= 21 more 6-month periods;

Notice that initailly they could receive college fund pay for only 16 periods of 6 months or 8 years and now as the inheritance fund has been added to the account the couple can now receive the same for 21 periods of 6 months or for 10.5 years