We now come to the final section where we will apply
some of the concepts that we have learned. Solving
for Interest rates and Growth rates. Let's take
a recent example. Say you have $100 today, which
you deposit $100 at time 0 and this $100 grows to 121
in 2 years, what is the interest rate given that this growth
took place in 2 years? You can apply the simple formulas
that you have learned. So one way of doing this is
to say that. 100 multiplied by 1 plus the interest rate, which is
unknown so let's say that is r, to the power of 2, because we have
two years, is equal to 121. You solve for r and essentially
what you would get is r= 0.1. You could
also have just seen the fact that the return here
is 10% because a 100 would grow to 110, and then
this would further compound to 121. Using a calculator,
what you could do is the following. Present value
equal to 100, future value equal to 121, and you need to
hit the plus minus and make this negative. When the
calculator does compounding, the future value and
present value have to have opposite signs because
the way calculator looks at it, 100 is going in one direction
perhaps into a project, and 121 is coming out since
the direction of cash flows different therefore,
the sign that we give has to be different for the present
value and future value. The number of periods is
2, the interest rate is what we compute and the payment
in this case is 0, because there is no cash flow in
between. When you compute the interest rate you
will get 10% just as what you get when you use the formula.
Obviously what I've done here is a straightforward
example, but you can imagine that the numbers can
be a whole lot more complicated and clearly it makes
sense to solve problems like this using the calculator.
The population of a small town is 100,000.
on 1 January 2000. On 31st December 2001, the population
is 121,000. What is the growth rate? Notice that we have growth
over two years from 1 January 2012 to 31st December
2001 is two years. So like this problem, the growth
is over two years. Initially, the population is 100,000
and then after two years is 121,000
So even though you're talking about population
growth rate rather than interest rate, the calculation is exactly
the same. You can see that again, the interest rate
is going to be 10%. The growth rate is going to be 10%,
which is the same as 0.1. Given that the calculation
is exactly the same we can apply the calculator
method here, plugging in a 100,000 for the
present value, - 120 1000 for the future value, n=2,
payment=0, and compute the interest rate. The interest
rate here would really be the growth rate which will
be 10. The larger point is that anytime you have
compounding and compound growth, we can use the calculator
function. Let's look at another example. You invest 900 today so the present value
is -900, receive a 100 coupon payment at the end
of every year for five years. This means that the
payment is equal to 100, n is equal to 5, because the
payment is being received for 5 years in addition
you receive a 1000 at the end of your 5 that
means that the future value is equal to 1000. The question
is what is the interest rate? So the I is what you
are supposed to compute. If you plug in the numbers,
you should get 12.83. Notice, I had a different sign
for the present value and future value. The way we
can think of this as an investment is a negative number,
so money going in, the payments, the coupon payments
in this case are received so that's plus, and the
future value is also received so that's a plus. We
could also get the same answer by using a positive
number for present value, but then we would have
to use negative numbers for payment and future value.
Solving for the Number of Periods. You invest 2,500.
How many years will it take to triple the amount?
Given that the interest rate is 6% per annum compounded
annually. Using the calculator method, all we need
to do is the following; present value is 2500, future
value needs to be the opposite sign. The number needs
to triple so future value is 7500 (minus). We are told that
the interest rate is 6%, payment is equal to 0 because
there is no cash flow in between and n is what we compute.
If you plug these numbers, what you should get is
18.85 years. Using the formula, you do the following;
2,500 * 1.06, thats 1 + (the interest rate) raised to the power of N, is
equal to 7500, and then use simple algebra. 2500 comes
down over here. So this becomes 3, you have 1.06 ^N
is equal to 3. Take the log of both sides. So log of
1.06 ^N is equal to log of 3. Using your log maths, hopefully,
you remember that this exponent comes down over
here. So N times log of 1.06 is equal to log of 3, so
N is equal to log of 3 / log of 1.06. On your calculator,
you will see the Ln button on the left, so you can just
plug in ln3 / ln1.06, and here again, you should get
18.85. Next, solving for the size of Annuity Payments. Read
this problem and try to solve it before you listen
to my explanation. The first thing you should do
is figure out how much is the effective loan. Given that Freddie
is paying 15%. Means that the loan amount is 0.85
into 42,000. This is the amount that he's paying over time.
So the numbers you plug in are present value = 35,700, N, 12 periods
because we have monthly compounding. So 1 period is 1 month.
The interest rate per month is 8%, for the year, divided
by 12. The future value is 0 and you compute your payment
and here is the answer that you should get. Present
Value and Future Value Equivalents. This is a straightforward
point which should be obvious given what we have
talked about so far, but in any case we will go over
the concept. Let's say you have a ordinary annuity
with A equal to 10, r= 5%, N= 5 and if you compute the
present value you will get 43.29. A lump sum can be
considered equivalent to an annuity. So a lump sum
amount of 43.29 at time 0 is the same or is equivalent
to this annuity. So if somebody were to give you a
choice between 43.29 today versus a annuity
that looks like this, you would be indifferent assuming
a rate of 5%. So this annuity and this lump sum, are
equivalent. Also an annuity can be considered equivalent
to a future value. If you take this annuity and
compute the future value over here, you will get
55.26, what that means is between an annuity
shown here and a value of 55.26 at time 5, you would be indifferent.
So that means that 43.29 and 55.26 so this money
at the end of five years 43.29 at time 0. These two numbers are also
equivalent given a rate of 5% so a lump-sum can be
considered equivalent to a future value. The Cash
Flow Additivity Principal. This is a complicated
sounding phrase for a straightforward concept.
Amounts of money indexed at the same point in time
are additives. In other words, if you have 100
at the end of one year, 100 at the end of two years,
and 100 at the end of three years, you cannot
simply add these three numbers. What you need to
do is take each of these numbers and bring them back
to a particular point in time. Let's say that we find
the present value at time 0 for each of these cash
flows. These present values that are all indexed
to time 0 can be added. So that is the additivity
principal. Amount of money index at the same point
in time, here that point in time, this time zero, are additives.
So that brings us to the end of this reading. We talked
about the interpretations of interest rates, and
we talked about how a investor can think of interest
rate as a sum of multiple components. We then talked
about future value, the concept of compounding then
we went over the present value concept and the fact
that we can discount a future value to come up with
present value. And these three concepts can be linked
using a very simple relationship future value at
time N is equal to present value you into one plus
the interest rate to the power of N, and this can be rearranged
and expressed in terms of present value. We then
did several problems, then we solved for rates, number
of periods, size of annuity payments, and so on. My suggestion
now is that you review the learning objectives in
the curriculum and make sure that you understand
them, the examples and practice problems in the curriculum
are fairly difficult. if you have trouble with them,
that's okay because most of these examples in practice
problems are a little bit harder than what you are
likely to see on the exam but still it would be useful
if you tried to do them and go over the solutions for
this particular reading. I think it would be useful
to get as many practice questions from other sources,
for example, IFT will be releasing some questions.
So the more questions you do on the subject the better,
because if you practiced more you will understand
the concepts of whole lot better. So that is it. Good luck
with your studies!