Speaker 0: Hi, everyone.
Speaker 0: My name is Simil.
Speaker 0: And in this short video, I will walk you through
Speaker 0: an example of the importance of geometric series in the
Speaker 0: context of economics, and in particular in the context of
Speaker 0: investment decisions.
Speaker 0: So an investment projects typically involves incurring a large cost
Speaker 0: today in anticipation of a certain stream of payoffs in
Speaker 0: the future.
Speaker 0: For example, an investment project as well, where a bomb
Speaker 0: went build the factory today in order to produce and
Speaker 0: sell out for a particular amount of time, for example,
Speaker 0: 30 years, so long as the factory is in operation.
Speaker 0: A firm should implement an investment project.
Speaker 0: All if the payoffs from the project exceed the cost.
Speaker 0: However, one of the issues that occurs in the context
Speaker 0: of investment projects is that that the costs and payoffs,
Speaker 0: in fact, the crew at different points in time.
Speaker 0: And this creates a difficulty because of what is known
Speaker 0: as the time value of money.
Speaker 0: The time value of money is the principal loosely speaking,
Speaker 0: that because if interest rates went down today is not
Speaker 0: the same in terms of value as £1.
Speaker 0: One year from now, for example, if the interest rates
Speaker 0: are equal to five percent.
Speaker 0: And if I have one thing today, then through my
Speaker 0: access to a bank which base interest rates, I can't
Speaker 0: convert this one pound today into one point five pounds
Speaker 0: one year from now.
Speaker 0: By depositing this one pound in a bank.
Speaker 0: We say in this context that the future value of
Speaker 0: one pound today in one year is exactly one pound
Speaker 0: multiplied by one, plus the interest rate.
Speaker 0: But similarly, if I actually hold my money in the
Speaker 0: deposit for two years, I can convert my one pump
Speaker 0: even more money to pyrates into the future.
Speaker 0: And we say equivalently that the future value of £1
Speaker 0: today in two years is this one bond multiplied by
Speaker 0: one, plus the interest rate squared and so forth.
Speaker 0: Now, this arguments underlying the time value of money also
Speaker 0: works backwards.
Speaker 0: For example, if the interest rate is five percent and
Speaker 0: I am to receive £1 one year into the future,
Speaker 0: then this is very much like receiving one divided by
Speaker 0: one point five or approximately ninety five pence today, because
Speaker 0: if I had ninety five pence today, I can deposit
Speaker 0: them in a bank at the interest rate, the five
Speaker 0: percent, and convert it into account.
Speaker 0: One year from now.
Speaker 0: In this context, we say that the present value of
Speaker 0: £1 received one year from now is Wangan divided by
Speaker 0: one.
Speaker 0: Plus, the interest rate, the present value of £1 received
Speaker 0: two years from now is £1 divided by one plus
Speaker 0: R squared.
Speaker 0: And so far.
Speaker 0: When a firm decides whether to implement an investment project
Speaker 0: or not, it should compare all the payoffs and costs
Speaker 0: from the project in present value terms.
Speaker 0: And one January to do this is to use the
Speaker 0: what's known as the net present value calculation.
Speaker 0: So the net present value of a project is the
Speaker 0: sum of for the present value of all payoffs from
Speaker 0: the project, minus the present value of all costs from
Speaker 0: the project.
Speaker 0: If the net present value of the project exceeds zero,
Speaker 0: then the project is worthwhile implementing.
Speaker 0: But if it is below zero, then the project should
Speaker 0: not be implemented and the firm is better off depositing
Speaker 0: their money in a bank account instead.
Speaker 0: Let's consider a numerical example which will clarify the importance
Speaker 0: of geometric series in this discussion, so suppose that there
Speaker 0: is a project, a form Kesem project, and the cost
Speaker 0: of implementing this project is one point five million pounds,
Speaker 0: which has to be incurred today, for example, in order
Speaker 0: to build the factory.
Speaker 0: If implemented, the project will pay off one hundred thousand
Speaker 0: pounds each year for 20 years with the first payoff.
Speaker 0: A great one year into the future after the cost
Speaker 0: is incurred, perhaps because it takes one year to build
Speaker 0: the factory and then you start producing and selling out,
Speaker 0: but only from one year into the future.
Speaker 0: So an interesting question is, is this project worthwhile?
Speaker 0: Should it be implemented by the four?
Speaker 0: Well, in order to answer this question, we should calculate
Speaker 0: the net present value of the project.
Speaker 0: Well, given that the interest rate is denoted by the
Speaker 0: net present value of this project in thousands of pounds,
Speaker 0: this minus fifteen hundred thousand pounds, this is the negative
Speaker 0: of the cost incurred today.
Speaker 0: And then one year into the future, where receive pay
Speaker 0: of one hundred thousand.
Speaker 0: But because it's one year into the future, we have
Speaker 0: to divide it by one.
Speaker 0: Plus, sorry to convert the present values that three years
Speaker 0: into the future, we receive another 100000.
Speaker 0: But in order to discount them in present value terms,
Speaker 0: we divide by one plus our square and so forth,
Speaker 0: all the way until twenty one years into the future.
Speaker 0: We receive 20 years into the future, receive £100, and
Speaker 0: we discounted by one.
Speaker 0: Plus are the power of twenty.
Speaker 0: Now, the usefulness of geometric series in the context of
Speaker 0: this calculation comes from recognising that the payoffs shown on
Speaker 0: the slides, in fact, form a geometric series with first
Speaker 0: element.
Speaker 0: Why can't rates over one plus are a common ratio
Speaker 0: of the elements?
Speaker 0: One over one, plus the interest rate and with 20
Speaker 0: terms.
Speaker 0: If we know the formula for the sum of the
Speaker 0: first 20 terms of a geometric.
Speaker 0: Sequence.
Speaker 0: We can therefore rewrite this some as simply one over
Speaker 0: one, plus the interest rate multiplied by one minus Y,
Speaker 0: plus the interest rate to the power of twenty oh,
Speaker 0: divided by one minus one over one, plus the interest
Speaker 0: rate.
Speaker 0: And this is the formula for the sum of geometric
Speaker 0: series with 20 elements and the common ratio one over
Speaker 0: one plus hour in the first element, one hundred over
Speaker 0: one.
Speaker 0: So therefore this formula signifies that whether the project is
Speaker 0: worthwhile undertaking will ultimately depend not only on what are
Speaker 0: the payoffs and the costs of the project, but also
Speaker 0: on the interest rate.
Speaker 0: For example, suppose that the interest rate is five percent.
Speaker 0: If we plug in zero point zero five into our
Speaker 0: formula for the net present value, we are going to
Speaker 0: find that the net present value of the project would
Speaker 0: be negative.
Speaker 0: Two hundred and fifty four thousand pounds.
Speaker 0: And since the net present value is negative, it implies
Speaker 0: that the project should not be undertaken because it is
Speaker 0: better to deposit your money in a bank account, instead
Speaker 0: yielding the interest rate of five percent.
Speaker 0: And over the duration of 20 years, this is going
Speaker 0: to hold your fire profits.
Speaker 0: Then investing in the project.
Speaker 0: However, if the interest rate lasts only one percent, then
Speaker 0: by blackening one percent inside this formula, we are going
Speaker 0: to be able to calculate that the net present value
Speaker 0: is, in fact, positive and large, positive and equal to
Speaker 0: three hundred and four thousand pounds.
Speaker 0: Therefore, in this case, if the interest rate to slow
Speaker 0: the project should be undertaken because it yields better returns,
Speaker 0: then the next best option, which is to deposit your
Speaker 0: money in a bank.
Speaker 0: This brief example demonstrates the usefulness of geometric series in
Speaker 0: the context of economics and also demonstrates one important, you
Speaker 0: know, concept in economics, which is about the relationship between
Speaker 0: interest rates and investment.
Speaker 0: As we know often in macroeconomics, we assume that higher
Speaker 0: interest rates discourage investment.
Speaker 0: And the underlying logic behind this, the underlying argument behind
Speaker 0: this statement is exactly one based on net present value.
Speaker 0: As we have seen here, when the interest rates are
Speaker 0: low, the project was worthwhile undertaking and therefore investment should
Speaker 0: be undertaken.
Speaker 0: But atai interest rates, investment should not be undertaken.