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Alright, let's, let's go to the main material in the lecture.
So, I start the course for the discussion of return calculations, and so the primary
thing that we are going to look at in terms of financial data or asset returns
over fund. And so, it is worthwhile to spend a bit of
time to define what we mean by asset returns.
So that's the subject of this topic. And for many of you, you know again.
A lot of this is material that you would see in a financial economics course, the
beginning of finance course, talk about present value calculations, time value of
money and so on. So we'll start with the time value of
money. And we'll start with the, the notion of
future value. So we'll say, we have an investment.
We have V dollars today. So we have $100 today.
We're gonna invest it for N years. And we're gonna get a simple rate of
interest R per year. Okay?
And so [inaudible] think of this. Deposit your money in a money market
account. The money market account pays an interest
rate, one percent per year, you know? How much will your money be worth after
five years. And so that's a typical future value
computation. And interest is assumed to compound once
per year. So the interest rate would be an annual
interest rate and the bank is going to pay you interest each year.
And you want to know how much your money will be worth at the end of n years.
So future value, so how much money you have at the end is equal to how much you
start with times one plus the interest rate raised to the power n.
So that's the, the standard future value computation.
In, in these kinds of problems I like to draw a timeline, that gives you what's
happening to your money at different points in time.
So you get a time zero, time one, time two, up to time N.
We start with B dollars today. And so what happens after one year, so V
dollars grows to V dollars, so you keep your $100 and then the bank pays you an
interest rate of r %. So after one year.
Your. Initial investment, grows to your initial
investment plus the initial investment the interest rate.
So that's the, the interest that you earn. So this is the future value after one time
period. So what happens after two time periods?
Now we have a, an initial investment of V one+R.
This grows, to, an amount. After two years, you're gonna get V
one+R one+R. So that's your, what you started with.
And then you get your principal + your interest.
And so that gives you your V one +R^2. So after two years, this is your future
value. And then you just continue on that same
compounding. And after N years, you get your future
value, which is your initial investment. And then your one+ the interest rate
raised to the power of N, okay? So that's, future value.
So as a practical example, say if you start with $1000 in a checking account,
okay a simple annual interest rate of three%.
By today's standards this is hot, right. So right now banks are paying at best, you
know, one percent per year Which is, historically, extremely low.
So what is your future value after one, five, and ten years?
So after one year, you know, you get $30 in interest.
After five years, then you get an additional $159.
And after ten years, you get an, an additional $343.
So again, this is the power of compounding.
You know, as, as your interest, as your money grows, the, the payment of interest
on interest, you know, leads to this sort of exponential increase in your wealth.
Now this future value function. That we wrote.
So this future value function. Future value.
Is equal to initial value. One Plus interest rate raise to power N.
This is an expression that contains, one, two, three, four different variables,
right? And so given my initial investment, given
the interest rate, given the number of compounding periods, I can figure out what
my future value is. The same thing, if I tell you what the
future value is, I tell you what my initial investment is, I tell you what the
compounding period is, I can solve for what the interest rate is on that
investment. So.
Taking our future value formula, if we solve for the initial value.
In terms of future value of the interest rate.
And the number of compounding periods we get.
Present value. So what is the?
If I know I get this amount in'n' years and I know the interest rate is this, this
percent per year, for this many years this is how much money I started with.
So that's the present value. So when you think of the time line, future
value is out here, that's the money that you receive'n' years from today, the
present value is what is the, what is the future value worth today, so that's what
the present value calculation says. Then you have.
We can take the future value formula, this thing here, and we can solve for the
interest rate. And so if I tell you, you know, in ten
years from now you're going to get $5,000. You start with a thousand dollars and, you
know, it's a ten year investment. What's the compound annual return on that
investment? That's just solving the future value
formula for the interest rate. So the interest rate, which is future
value divided by present value raised to the power of 1/(n-1), that's the average
annual interest rate that get every year that compounds such that your initial
investment grows to the future value. And then we can also take this future
value formula and solve it for the investment horizon.
That is, if I tell you the interest rate, the present value and the future value, I
can solve for N in this relationship. We can think of this as the investment
horizon. And so, how would you do that?
Then, we would just take the logorithm of both sides of this.
Or you take V, [inaudible], future value divided by V.
That's one + R to the power N. Take the logorithm, solve for N.
And then we can solve for N this way. Okay.
So one of the things I want to do with this investment horizon, or sort of an
interesting, Example. Consider the following questions: how long
does it take for your money to double? Right?
So you have an investment. Start with a $100.
And you know the interest rate is three percent per year.
How long will it take you for your money to double?
So, we can use this relationship here to, to solve that question.
And this gives rise to a rule of thumb that it's often called the rule of 70.
So ask the question, how long does it take for your money to double?
Okay. So you take my initial investment'B' and I
want... And so we know that this is equal to,
sorry, the future value want it to be'2B' is equal to'D' times'1+R' raised to the
power'n'. So, our initial investment is'B' and we
want our future value to be twice our initial investment, that's doubling our
money, right. So notice that we take a look this,
that'B' is cancelled. And so I can look at this and say the log
rhythm of two is equal to n times the log rhythm of one plus r.
And so then I can solve this and say n is equal to the logarithm of two divided by
the logarithm of one plus r. So this, an expression, an exact
expression that tells me how many years it will take, for my, for my money to double
if the interest rate is equal to R percent per year, okay?
Now, the logorithm of two is approximately 0.7.
And one of the things we'll see from calculus, if you do a first order taylor
series approximation, of logarithmic function.
If R is, if the rate of return is, close to zero.
So if it's like one percent or two%, then the logarithm of one plus R, is
approximately equal to R. So, we can approximate this exact result
by, 0.7 divided by R. And so this gives the, the rule f thumb
that's called the Rule of 70. So for example.
If the interest rate is zero point one. Then N is equal to 0.7 divided 0.01, which
is 70. So if the interest rate is one%, it takes
70 years for your money to double at compounding at one percent per year.
So all this money you're putting in your savings accounts right now that's getting
paid one%, it'll take 70 years for that money to double.
So again that's just a an, an illustration of, you know, working in computational
finance. You have a formula, you know you do some,
some simple computations to, you know, to get a result.
Alright now very often in. Investment situations compounding interest
can happen more than once per year. So if compounding happens M times per
year. For example, M could be two.
So you could have semi-annual compounding. Or you could have monthly compounding so
that's, M would be twelve times per year. You could have daily compounding, so M is
365. So, if you have compounding M times per
year, what's your future value formula? Well, what you do is you take your annual
interest rate, you divide it by the number of compounding periods.
This becomes what's called the periodic interest rate.
So, if M is two, this would be your semiannual rate.
If M was twelve, that would be your monthly rate, and then your future value
is your initial investment times one plus the periodic rate raised to the power M
times N. So this is the number of years, the number
of compounding periods per year. Now we have something called continuous
compounding so, think if we're going from compounding from daily to hourly to
minutely to every second to every millisecond, so as M essentially goes to
infinity so we're compounding instantaneously, then we can take the
limit of this as M goes to infinity and it turns out that this is equal to V x E to
the RN. So continuous compounding involves the
exponential function and E to the one is, 2.7.
So, what this compounding give us? So here's an example.
Say we have an annual interest rate of ten%.
We start with $1,000. And then we look at the future value.
After one year with different amounts of compounding.
So if you compound annually you get, ten percent interest.
So you get an extra $100. If you compound four times per year, then
you're gonna get an extra $3.81. Okay, so that's what the four times
compounding gives you. If you compound weekly, you get an extra
$5.00. If you compound daily, you don't get a
whole much more than what you get from compounding weekly.
And if you compound continuously, it's a, effectively what you get if you compound
daily. So, increasing your compounding frequency,
leads you to get, slightly higher, interest.
I mean, for historical reasons, thi-, this notion of paying interest more often than
once a year, was one way that banks can get around certain kinds of regulations
that capped interest rates. So, it used to be regulations for savings
and loans that said they can only pay interest rates of five percent per year.
But the regulation didn't specify how often you can compound.
So, if you, you can have an annual interest rate of five%.
But if you compound continuously, you effectively offer a higher interest rate.
So that's an example of, of getting around a regulation, by, being a bit clever.
Now, because we have compounding periods that could be different from a year, there
is a concept known as the effective annual rate.
So we saw that when we compounded. Say daily.
We got a $105.16 of interest, right, and we compounded once per year and we only
got $100 in interest. So effectively we have a higher annualized
interest rate when we compound daily, then when we just compound once per year.
The effective annual rate is, what is the annual interest rate that gives us this
future value? Okay.
And so our interest rate is essentially 10.5 percent instead of ten%.
So how do we solve this? So, to calculate the annual rate that
gives you the same future value with compounding M times per year to
compounding with one time per year, you solve this equation.
So, here, our initial investment, our periodic rate.
This gives us our future value with compounding M per year.
This is future value compounding once per year at this interest rate, which would be
the effective annual rate. And then we, take this equation and we
want to solve for RA. So just do some simple manipulations.
We get that the effective annual interest rate is equal to one plus the periodic
rate raised to the power m minus one. Okay.
And so we can see in our example, or, and similarly if we have continuous
compounding do the same computation. We have, this is the future value with
continuous compounding. This is the future value compounding once
per year; and then we want to solve for the interest rate.
That's going to be E to the interest rate minus one.
So let's do some comparisons or computations.
Suppose we have, a situation where we look at a comput-, effective annual rate with
semi annual compounding, okay? So we have an investment with a simple
annual interest rate of ten%. But interest gets compounded twice per
year. What's the effective annual rate on that
investment? So you, you saw one plus the annual rate.
Is equal to one plus the. Semiannual rate squared.
Solve for the, the annualized rate. You get one plus.
This would be five percent squared. Minus one.
And so our effective annual rate. Is in fact 10.25%.
So it's effectively higher. Than just getting, simple interest.
So if we do a comparison, compounding once per year.
Ten percent interest compounding quarterly we get an extra.38%.
Weekly. You get an extra.5%.
And then, [inaudible] continuously, an extra.52%.
So you can see, you know, effectively, we get slightly higher interest when we
compound more often. So this is one of the reasons why when you
take out a credit card, they don't charge you an annual interest rate.
Well, they quote you an annual interest rate, but they compound interest daily.
So your effective interest rate is higher than the annual rate that they quote you
on your interest, on your credit card. And how much higher is it?
You do this calculation.