Hi everyone we are continuing forward now with accounting and the time value of
money and up until now we've been looking at how to solve for the present
value or the future value amounts but there are times when we actually may
know the present value and the future value and maybe we want to find the
interest rate that we would need or the number of periods we need to save so we
can also use this time value of money information to be able to solve for
those amounts. So let's just look at some examples on how we can do that.
Let's first look at solving for the number of periods so your 'n' in the
equation there. So the village of Somonauk wants to accumulate $70,000
$70,000 for the construction of a veteran's monument in the town square.
At the beginning of the current year the village deposited $47,811
$47,811 dollars in a memorial fund that earns 10 percent
interest compounded annually. How many years will it take to accumulate $70,000
$70,000 in the memorial fund? So as you can see we know lots of
information but we don't know how many years it's going to take so we're
looking for our 'n' as you can see in the timeline here. There's a couple of
methods we can actually use to find this amount. The first method is called the
future value approach. With this approach in the box there you see the
information in red, we're going to set the future value equal to the present
value times the future value factor at a certain number of periods and ten
percent interest. Again we're looking for that little 'n,' we're looking for the
number of periods so the unknown amount is that future value factor. So we have
$70,000 is equal to $47,811
$47,811 times our unknown or our X- in this case the future value factor.
So we're going to divide both sides by the $47,811
$47,811 to get the future value factor on
side by itself just like we're solving an algebraic expression for our X our
unknown. We find the future value factor to be 1.46410.
So if we look for that number in our chart we know it's at 10% interest
so if we go down the 10% column we find that 1.46410
1.46410 at the four period mark so we know that the number of periods it will
take to accumulate $70,000 by first depositing $47,811
$47,811 at 10% it will take us four years to accumulate
$70,000. We can also use what's called the present value approach.
This is saying the present value equal to the future value times the present
value factor at a certain number of periods and 10% so again that's our
unknown that present value factor amount. So we want to get that on one side of
the equation by itself so dividing both sides by the $70,000
gets us a present value factor of 0.68301. We know it's
10% interest so we go to our 10% column in our table and we
scan down until we find that factor 0.68301 happens
to be again at the four periods. So no matter which method or approach you use
the present value approach or the future value approach you should still get the
same answer in this case it's four periods. We can also use these two
different approaches to solve for interest rates. Here we have Advanced
Design Incorporated. They need $1,409,870
$1,409,870 for basic research five years from now. The company
currently has $800,000 to invest for that purpose.
At what rate of interest must it invest the $800,000 to fund basic research projects
of $1,409,870 five years
from now? So again as you can see on our timeline the the piece that we're
missing here is the interest rate. So what interest rate are we going to need
to be able to invest $800,000 today for the next
over the next five years and end up with a little over 1.4 million dollars? Well
again we can use the future value and the present value approaches so let's
look at the future value approach first firstly. Again we start with setting
the future value equal to the present value times the future value factor at
five periods and we don't know the interest rate. Again that piece is our
unknown. So we have our $1,409,870
$1,409,870 set equal to $800,000 times the
future value factor which is what we're solving for. Get that by itself on one
side of the equation and you find that the future value factor is
1.76234 and we know the number of periods is five. So if we start
with the number of periods are five and then we scan over until we find
1.76234 we find that the interest rate we will need to
get to end up with a little over 1.4 million dollars investing
8 million dollars today for five years we would need to get 12% on our
money. Now this was using the future value approach. So remember we can also
use the present value approach to solve for that 12% and it should
come out to be the same answer it should not change our answer so let's take a
look at it. Here we start with the present value now equal to the future
value times the present value factor and if we do that we find that $800,000
$800,000 equals $1,409,870
$1,409,870 X our unknown that present value factor. If we solve for that present
value factor we come up with 0.56743 and again we know
that's at five periods so now we scan over across our paper there across our
time value of money tables and we find that number again in the 12% column so
we are able to see that 12% is our rate to invest $800,000 today over the next
five years we will need to be achieving a 12% rate to be able to come up with a
little over 1.4 million dollars five years from now.
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