1. Title Slide
Let's now see how discounting the future value to find the present value
works when we split one year into different periods, or in other words,
for different frequencies, as well as for different number of years.
Let's imagine that you need $1500 in four years time. You want to
know how much you need to put in the bank today.
In other words, you want to discount the future value of $1500 to
find its present value. The bank pays an annual or nominal interest rate
of 6% per year. However, it does compound its interest on a six
month or semi annual basis. So the frequency f is going to be
2. Because we're working backwards from four years in the future to find
the present value, and because the bank pays semi annual compounded interest,
the number of periods between the future value and the present value is
four multiplied by two, which of course is eight.
So if we have a future value of $1500 four years from today,
and if interest rates were 6% and we're using a discounting frequency of
two, the present value is 1184.11. Another way to say this is that
if you had 1184.11 today, invested it and received 6% interest compounded
semi annually, you'd have 1500 in four years time.
Now from an investing point of view, you wouldn't really mind which finding
you had, the 1184.11 today or the 1500 in four years time,
because they both have the same value. Here in the second example,
we've used the different inputs. Now let's open up Excel again and see
if we can get the same results as these last two examples.
When we enter the formulas, we're going to have to be really careful
about getting our brackets in the right position in our formulas.
So here I've just scrolled down to the next section in our Excel
workbook called Present Values Using Discounting, and I just want to walk
through the last two examples that we've seen to make sure that we
can get the same numbers in Excel. Now in the first example that
we walked through, we had a future value of 1500 four years from
today, nominal rates at 6%, and we're assuming semi annual discounting,
so the opposite of compounding. And we want to know what the present
value is. So let's start by just putting in our input.
So we've got a future value of 1500, we've got nominal rates of
6%, and we've got semi annual discounting, so the frequency is two.
And we want to discount back four years
to get the present value. So we can see the
present value formula in cell D109 there just to help us remember what
it is, because it is quite hard to remember where to put all the
brackets and everything, isn't it? So the present value equals the future
value times one divided by, open bracket, one plus the nominal rate of
6% divided by the discounting frequency of two. And then we're going to
close that, and we're gonna raise that all to the power of N4 times
F2. And we get, when we do that, 1184.11. Now we can also
use the present value formula in Excel just to confirm that number.
So let's start by putting a minus sign out the front of the
PV function to make sure our answer is a positive number.
We want the rate per period. Well, it's a 6% nominal,
and it's a frequency of two, so we divide that by f. The
number of periods we're discounting back is the number of years,
four times two, the number of discounting periods in a year.
We're not making or receiving any payments along the way, so PMT is
zero, and the future value is 1500. And as always, we're assuming that
all the action, the discounting, or the compounding happens at the end of
the period, so type is zero. And we get the same result,
1184.11. So here's the second example that we've just seen. Let's see if
we can get the same result. What is the present value if the future
value three years from today is 1250? And we're going to assume nominal
rates are 10%, so the nominal rate is 10%. And we're using monthly
discounting, so the frequency is 12. And remember, our future value was
three years from today. So what's the present value? Well, we start by
taking the future value, and we multiply that by one divided by open
bracket one plus the nominal rate of 10%, divided by the discounting frequency,
which is 12. We're going to close that bracket and raise that all
to the power of n times f, which I can pick up there,
and we can just close that bracket. So the present value is $927.17. So
let's just make sure that we can get that using the PV function.
And so the rate per period 10% divided by 12, the number of
periods three times 12. We're not making or receiving any payments, so PMT
is zero, and the future value is 1250. And the type is zero.
And we can see when we do that, we get the same result,
927.17. Well done.
2. Let's practice!