C. Grab Bag
- Anja and Ted both have just set up investment accounts. Suppose that Anja deposits $1,000 each year for 10 years
and then stops, while Ted deposits nothing for 10 years and then starts and continues depositing $2,000 each year.
How much will each person have at the end of 25 years, assuming they both earn 8.2%? How much total interest will
each person earn? - Eigen’s Valuemart needs to have $20,000,000 in a special account 20 years from now in order to meet an employee
benefi ts commitment. The company plans to make equal quarterly deposits for the next 8 years and then leave the money
invested to grow to the needed future value. If their account will earn 8^3 / 4 %, how much should each quarterly deposit be? - Amy wants to have $1,000,000 in her retirement account in 40 years. She believes her account will earn 9%
compounded monthly.
a. Assuming she keeps making the same monthly deposit for the entire 40 years, how much should she deposit each
month to her account?
b. Assuming she wants to make monthly payments for the next 20 years and then stop her deposits, how much
should she deposit each month?
c. Assuming she wants to make monthly payments for the next 10 years and then stop her deposits, how much
should she deposit each month? - How much will I have in my retirement account when I’m 64, assuming that I invest $375 each month beginning at
age 27 and then stop making investments at age 39. Assume that my account earns 11%.
D. Additional Exercises
- Suppose that you deposit $120 each month into an account earning 8% for 5 years. You continue your deposits at this
same pace, but the interest rate is increased to 10% for the next 15 years. How much in total will you have at the end
of 20 years?
196 Chapter 4 Annuities
4.7 Future Values with Irregular Payments: The Bucket
Approach (Optional)
The previous section’s approach works well in some circumstances, but in others it falls
short. It will handle annuities whose payments stop, but it isn’t adequate to handle “annui-
ties” whose payments change, or accounts that begin with some amount of money already
in them. We can address those sorts of problems with another technique, which we will
refer to as the bucket approach.“Annuities” That Don’t Start from Scratch
Suppose you have $5,000 in an account that earns an effective rate of 4.35% for 8 years.
This account will grow to $7,029.26 at the end of the 20-year term. Now, suppose that
instead of one account with $5,000, you opened up two accounts, each with $2,500. Would
you have earned more by doing this? Less? The same?
A deposit of $2,500 at 4.35% for 8 years grows to $3,514.63. Each of your two accounts
would have grown to this. In total you would have $7,029.26, the same as if you had kept
the entire amount in a single account. As we noted back when we were trying to find
a future value annuity formula, it makes no difference whether you kept everything in