PV PMT a _n (^) | (^) i
$557,893 PMT a _ 240 | (^) .00375
$557,893 PMT(158.065436810)
PMT $3,529.51
So Charlie’s account will support a monthly income of $3,529.51 for 20 years.
So far our examples have all been ordinary annuities. There are, however, some cases
where we need the present value of an annuity due, as this example shows:
Example 4.4.11 The New York State Lotto Jackpot is advertised to be $52 million.
However, this jackpot is not paid out all at once as a single lump sum. Rather, it is paid
out in equal annual installments for 26 years (beginning immediately). The advertised
$52 million is the total of all the payments. When you buy a ticket, you can choose
to have the jackpot paid to you as a lump sum should you win, but in that case you
receive the present value of the payments. If the interest rate used is 6%, what is the
value of the jackpot as a lump sum?
The payments do represent an annuity, and since they begin immediately they represent
an annuity due. A lump sum received up front instead of the payments clearly would be
their present value. The $52,000,000 is not the present value, but instead is the total of all
the payments. Since the jackpot annuity is 26 equal payments totaling $52 million, each
payment would be $52,000,000/26 $2,000,000.
PV PMT a _n (^) | (^) i (1 i)
PV $2,000,000 a 26 | (^) .06 (1.06)
PV ($2,000,000)(13.003166187)(1.06)
PV $27,566,712.32
So winners who choose to take their winnings entirely up front will receive just a bit over half
of the advertised jackpot. The practice of advertising the total payments of an annuity as the
prize is very commonly used not only for lotteries but for other sweepstakes as well.
EXERCISES 4.4
A. Using Future Values to Find Present Values
- To buy their new townhouse, Marc and Jun borrowed $116,509 from International National Mortgage Company for
30 years. The interest rate was 7.2% compounded monthly.
a. Suppose that instead of making monthly payments on the loan, the entire balance together with all the interest it
accumulates will be paid all at once at the end of the 30-year term. How much would that amount be?
b. Suppose that Marc and Jun made monthly payments into a sinking fund earning 7.2% compounded monthly to
accumulate the total from part (a). How much would the sinking fund payment be?
c. Parts (a) and (b) are completely hypothetical; in reality, the mortgage company will require Marc and Jun to pay
their loan off with monthly payments. How much will each of their monthly mortgage payments be?
B. Using Tables for Present Value CalculationsQuestions 2 to 5 relate to the scenario below and the table given on the next page.In the United States, the most common terms for a mortgage loan are 15 years or 30 years. In working with his clients, Jeff,
a real estate agent, fi nds it convenient to have a table of annuity factors for monthly loan payments for these two terms at a
Copyright © 2008, The McGraw-Hill Companies, Inc.selection of interest rates typical in the current market. He carries a small card with the table shown below in his briefcase:
Exercises 4.4 177