Business Application: A Trick for Calculating the Value of Annuities: In several of the previous
exercises, we have indicated that an infinite series 1/11 1 r2 1 1/11 1 r2 2 1 1/11 1 r2 3 1 csums
to 1/r. This can be (and has been, in some of the B-parts of exercises) used to calculate the value of an
annuity that pays x per year starting next year and continuing every year eternally as x/r.
A. Knowing this information, we can use a trick to calculate the value of annuities that do not go on
forever. For this example, consider an annuity that pays $10,000 per year for 10 years beginning next
year, and assume r 5 0.1.
a. First, calculate the value of an annuity that begins paying $10,000 next year and then every
year thereafter (without end).
b. Next, suppose you are given such an annuity in 10 years; that is, suppose you know that the
first payment will come 11 years from now. What is the consumption value of such an annuity
today?
c. Now consider this: Think of the 10-year annuity as the difference between an infinitely lasting
annuity that starts making payments next year and an infinitely lasting annuity that starts
11 years from now. What is the 10-year annuity worth when you think of it in these terms?
d. Calculate the value of the same 10-year annuity without using the trick mentioned in part (c).
Do you get the same answer?
B. Now consider more generally an annuity that pays x every year beginning next year for a period of
n years when the interest rate is r. Denote the value of such an annuity as y1x,n,r2.
a. Derive the general formula for valuing such an annuity by using the trick described in part A.
b. Apply the formula to the following example: You are about to retire and have $2,500,000 in
your retirement fund. You can take it all out as a lump sum, or you can choose to take an annuity that will pay you (and your heirs if you pass away) $x per year (starting next year) for
the next 30 years. What is the least x has to be in order for you to choose the annuity over the
lump sum payment assuming an interest rate of 6%?
c. Apply the formula to another example: You can think of banks as accepting annuities when
they give you a mortgage. Suppose you determine you would be able to pay at most $10,000
per year in mortgage payments. Assuming an interest rate of 10%, what is the most the bank
will lend you on a 30-year mortgage (where the mortgage payments are made annually beginning one year from now)?
d. How does your answer change when the interest rate is 5%?
e. Can this explain how people in the late 1990s and early 2000s were able to finance increased
current consumption as interest rates fell?