30 Part One Value
bre44380_ch02_019-045.indd 30 09/02/15 03:42 PM
◗ FIGURE 2.8
Calculations showing
the year-by-year present
value of the installment
payments.0 1 2345 Year$5,000Present value
(year 0)$5,000/1.07^5 = $3,565$5,000 $5,000 $5,000 $5,000$5,000/1.07 = $4,673
$5,000/1.07^2 = $4,367
$5,000/1.07^3 = $4,081
$5,000/1.07^4 = $3,814Total = PV = $20,501● ● ● ● ●In May 2013, an 84-year-old woman invested $10 in five Powerball lottery tickets and won a
record $590.5 million. We suspect that she received unsolicited congratulations, good wishes,
and requests for money from dozens of more or less worthy charities, relations, and newly
devoted friends. In response, she could fairly point out that the prize wasn’t really worth
$590.5 million. That sum was to be paid in 30 equal annual installments of $19.683 million
each. Assuming that the first payment occurred at the end of one year, what was the present
value of the prize? The interest rate at the time was about 3.6%.
These payments constitute a 30-year annuity. To value this annuity, we simply multiply
$19.683 million by the 30-year annuity factor:PV = 19.683 × 30-year annuity factor= 19.683 ×
[__^1
r- __^1
r(1 + r)^30
]
At an interest rate of 3.6%, the annuity factor is
[____^1
.036- ___^1
.036(1.036)^30
]
= 18.1638
Therefore, the present value of the cash payments is $19.683 × 18.1638 = $357.5 million,
much below the well-trumpeted prize, but still not a bad day’s haul.
For winners with big spending plans, lottery operators generally make arrangements so
that they may take an equivalent lump sum. In our example the winners could either take the
$590.5 million spread over 30 years or receive $357.5 million up front. Both arrangements
had the same present value.EXAMPLE 2.3^ ●^ Winning Big at the Lottery
● ● ● ● ●