Question 1126426: A young couple wants to have a college fund that will pay $40,000 at the end of each half-year for 8 years.
(a) If they can invest at 9%, compounded semiannually, how much do they need to invest at the end of each 6-month period for the next 18 years to begin making their college withdrawals 6 months after their last investment? (Round your answer to the nearest cent.) Answer: 5215.18
I got this one right by using the present value ordinary annuity formula: A = R[(1-(1+i)^-n)/i]
I subbed in the numbers 40000[(1-(1+0.045)^-16)/0.045] = 449360.60
then I plugged in the answer into the S variable in the future value ordinary annuity formula: S = R[(((1+i)^n)-1)/i] and solved for R
449360.60 = R[(((1+0.045)^36)-1)/0.045]
R = 5215.18
(b) Suppose 8 years after beginning the annuity payments, they receive an inheritance of $39,000 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.) Answer: 22 withdrawals
Now for this question I'm not sure how to get to this answer. I know I am supposed to use the ordinary annuity formulas but I just can't arrive at 22.
Keep in mind these answers are provided by the web assign online platform so 22 withdrawals is without a doubt the right answer.
I appreciate any solutions using the formulas I provided! Thanks :)
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
Nice work on getting part A correct.
449360.60 is the future value of the amount of money if they put $5215.18 in the account each 6 months. If there is no inheritance, then this FV stays the same. If we add in the inheritance, then we're adding on the amount of money compounded from the 39,000 dollars.
The value 39000 is compounded for 10 years since "8 years after beginning the annuity payments, they receive an inheritance of $39,000 that they contribute to the account" and because the entire timespan is 18 years. So there's 18-8 = 10 years left over to have this 39000 value compound. Let's find out how much money we'll have in the account if we just compound this one time deposit
A = P*(1+r/n)^(n*t)
A = 39000*(1+0.09/2)^(2*10)
A = 94,056.8469686589
A = 94,056.85
So we'll have $94,056.85 if we compound 39000 dollars for 10 years (semiannual compounding at 9% interest)
note: you do not use an annuity formula since periodic deposits are not made with respect to the inheritance money. It's deposited all at once
Add this onto the FV dollar amount found earlier, which was 449360.60 so we end up with 449360.60+94056.85 = 543,417.45 as the total amount in the account after (a) the 5215.18 semiannual deposits are made for 18 years and (b) the 39000 dollar amount is deposited once and it compounds for 10 years.
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This value of 543,417.45 will then be used to figure out how many time periods n we have until the annuity money dries up
The formula we'll use is
n = log(A)/log(B)
where,
A = 1/(1-(L*r)/P)
B = 1+r
L = starting amount in the annuity
P = annuity payment (money received) per period
r = interest rate per period
I'm breaking up the formula like this because it is an ugly mess if you combine it all into one line. Though to be fair, this could be argued as just as messy.
So,
L = 543417.45 is the amount we start with
P = 40000 is the amount we want each 6 months
r = 0.09/2 = 0.045 is the semiannual interest rate
A = 1/(1-(L*r)/P)
A = 1/(1-(543417.45*0.045)/40000)
A = 2.57297359152973
B = 1+r = 1+0.045 = 1.045
Therefore,
n = log(A)/log(B)
n = log(2.57297359152973)/log(1.045)
n = 21.4704484565457
We wouldn't round to n = 21 because the 0.47 indicates we'd still have money in the account after 21 six-month periods. This amount won't be $40,000 but it's still a positive balance indicating that we haven't hit $0 just yet. Only after 22 six-month periods is when the balance will hit $0.
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Side Notes:
If we use L = 449360.60 in the formula above, ignoring the inheritance, then we end up with...
A = 1/(1-(L*r)/P)
A = 1/(1-(449360.60*0.045)/40000)
A = 2.02237014399224
B = 1+r = 1+0.045 = 1.045
which leads to
n = log(A)/log(B)
n = log(2.02237014399224)/log(1.045)
n = 15.9999998984703
which is very very close to n = 16 as expected. So this helps confirm that we have 16 six-month periods (ie 8 years) if we do not include the inheritance money
If you want to have a formula based on what you wrote, then you would solve the present value ordinary annuity for n like so
A = R[(1-(1+i)^-n)/i]
A*i = R[(1-(1+i)^-n)]
(1+i)^(-n) = 1-A*i/R
1/((1+i)^n) = 1-A*i/R
(1+i)^n = 1/(1-A*i/R)
n = log[ 1/(1-A*i/R) ]/log[ 1+i ]
which is effectively what I wrote above, but I broke down the formula into smaller pieces
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