Traceyc --
Each of the options needs be discounted or divided by the discount
rate. A dollar received a year from now is really worth about $0.91
because the discount rate (or interest rate) will make it less
valuable.
Your question relates to Net Present Values (NPV) over time for
options A, B, C. I've set up a spreadsheet here to show each of the
options:
http://www.mooneyevents.com/annuity.xls
Often these questions are set up so money received today is "Year 0,"
which is what I've done here. On Option C, the payment is received
after 3 years -- it appears in Column E because those calculations
represent the numbers at the end of Year 3.
THE DIFFICULTIES OF PERPETUITIES
================================
A perpetuity is an income stream received forever. Excel doesn't
handle them well, but they're easy to calculate, as you can see from
the concise definition here:
Prof. Sid Systema (Ferris State)
"Basic Mathematics of Finance"
http://www.sytsma.com/cism700/mathfin.html
PERPETUITIES
"A perpetuity is an annuity that continues forever, that is every year
from now on this investment pays the same dollar amount. An example of
a perpetuity is preferred stock which yields a constant dollar
dividend infinitely. The following equation can be used to determine
the present value of a perpetuity:
PV = pp/i
where
PV=the present value of the perpetuity
pp=the constant dollar amount provided by the perpetuity
i=the annual interest or discount rate"
OPTION D
---------------
Prof. Systema's definition misses one thing relevant to this problem:
in Option D the $950 received next year has a "constant" dollar amount
of $863.64 -- because it's received a year late ($950/1.1).
So the PV = $863.63/(0.10) = $8635.30
OPTION E
---------------
Similarly for Option E, the real value of $1,100 after 3 years is
$1,100/ (1.1 * 1.1 * 1.1) = $826.45.
So the PV = $826.45/(0.10) = $8,264.50
THE WINNER!!
============
See column O in the spreadsheet, but Option C has an NPV of $9,519 --
barely outstripping the option of taking the cash today. Note that a
small upwards change in the interest rate (discount rate) would make
taking the cash today (Option A) the superior choice.
NOTES ON DETAIL :
* the spreadsheet NPVs in column O may differ by pennies here due to
rounding differences.
* note that payments are made at the END of each year (one of the
reasons that we use year 0 for a payment made today). There can
sometimes be drastic differences in NPV when payments are received at
the start of a year -- or even during the year.
* Prof. Systema's definition is also missing another typical
assumption for perpetuities, which is the growth rate (g). There is
no growth in the payments here but adjusted for that a perpeuity's NPV
is:
PV = pp / (i-g)
If there's any confusion about this Google Answer, don't hesitate to
ask before rating the answer.
Best regards,
Omnivorous-GA |
