Hi deacon1!!
1 Mrs. Optimistic wishes to retire upon accumulation of $1,000,000.How
many years must she work if she is able to save and invest $17,460 per
year at 10 percent interest compounded annually? - did not match any
documents:
This is a Future value of an annuity case (ordinary), an invest of an
equal amount of money at the end of each year for a specified number
of years and allow it to grow at certain rate.
The formula for this calculation is:
FV(n) = A * [SUM(i=1 to n-1) (1 + r)^î]
= A * [(1+r)^n - 1 / r]
where
A is the annuity.
r is the annual compound interest rate.
n is the number of years for which the annuity will last.
FV(n) is the future value of the annuity at the end of the nth year.
In this case:
A = $17,460 ; r = 0.1 ; n = ?
Oops!! we don´t know n, what can we do?
Note that the [(1+r)^n - 1 / r] part of the equation is independent
of the annuity, for this reason it is used for build tables that show
"SUM OF AN ANNUITY OF $1 FOR n PERIODS" or in other words how many
times the annuity growths at certain rate in n periods. These tables
are the "Future Value of $1 per year (Ordinary Annuity of $1)" tables.
You can see this table here:
"Interest Tables":
At this page there are several tables, look the Future Value of $1 per
year (Ordinary Annuity of $1) one.
http://www.cob.sfasu.edu/rgriffith/interestables.html
To use this table you must look the factor for n periods at r interest
rate in the table, then
Future value = Annuity X Table factor,
then
Table factor = Future value / Annuity
In this case
Table factor = $1,000,000 / $17,460 = 57.2738
Now go to the table and, in the 10% column (this is our rate), search
for the first factor greater than 57.2738 , this period row is the n
that are we looking for, in this case:
n = 20
Mrs. Optimistic must work for at least 20 years.
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2.Mr. Joe Steam may elect to take a lump-sum of $50000.00 from his
insurance policy or an annuity of $5650 annually as long as he
lives.How long must Mr. Steam anticipate lin\ving for the annuity to
be preferable to the lump-sum if his opportunity rate is 8 percent?
This is a Present Value of an Annuity case, a situation where you want
to find the value in today's dollars of a series of future equal
payments that are due annually when payments are made each period at
the given interest rate.
The formula for this is:
PV(n) = A * [SUM(i=1 to n) 1/(1 + r)^î]
where
A is the annuity withdrawn at the end of each year.
i is the annual interest or discount rate.
PV is the present value of the future annuity.
n is the number of years for which the annuity will last.
Again with the [SUM(i=1 to n) 1/(1 + r)^î] part of the equation we can
make a table for Present Value of $1 per year or Present Value of
Annuity of $1.
In the "Interest Tables" page you have one of such tables, where
To use this table you must look the factor for n periods at r interest
rate in the table, then:
Present value = Annuity X Table factor,
then:
Table factor = Present value / Annuity
Again we know all except n, but we can estimate the factor because we
know the Present value and the annuity:
Table Factor = $50,000 / $5,650 = 8.8496
Again we must go to the table and, in the 8% column, search for the
first factor greater than 8.8496 , this period row is the n that are
we looking for, in this case:
n = 16
Mr. Joe Steam must receive the annuity for 16 years for the annuity to
be preferable to the lump-sum.
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For references see the following pages:
"Basic Mathematics of Finance ":
http://www.sytsma.com/cism700/mathfin.html
"Time value of money formulas":
http://people.hofstra.edu/faculty/A_Sinan_Cebenoyan/fin101%20formula%20sheet.pdf
or
http://highered.mcgraw-hill.com/sites/dl/free/0070898669/65177/keyequations.pdf
I hope this helps you. If you need further assistance or find that
something is unclear, feel free to request for a clarification. Please
don't consider the answer finished until you feel completely satisfied
with it.
Best regards.
livioflores-ga |
