Brave Assignment.pdf

T HE A PPLICATIONS OF S ERIES AND THE C ONCEPT OF T IME VALUE OF M ONEY IN B USINESS AND F INANCE March 30, 2019

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Executive Summary

A short statement that gives the important facts, conclusions and suggestions of a report, usually printed at the beginning of a report.

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Introduction

Mathematics is the key to our understanding of the dynamics around us, e.g. how fast cars move, how much profit a business project rakes in. In trying to simplify real world situations, we make assumptions about complex systems and develop theories to predict the future. In business, private firms and individuals operate for profit and thus have to ensure that the money they plough in to purchase machines, build factories, and hire workers yield a positive rate of return. Because firms and investors aim to make profit, the decision-making can be difficult given that high returns are invariably accompanied by high risk: there is an inverse relationship between rate of return and risk. It is the responsibility of economists and financial analysts to make predictions about the economy, the business environment, and, at times, business earnings. There are numerous factors that influence the profit of a business—namely, the price, cost of production, exchange rates, inflation, and the rate of interest. In this essay, I explore two important uses of series in business and finance. First, we delve into the concept of time value of money to prove that immediate money is more valuable than future money. Second, I try to explain the fundamentals of geometric series and the importance of finite and infinite sums which will then be used to show how geometric series are applied in finance. In the last section, we explain what are perpetuities, annuities, and net present value, and we finally review the shortcomings of our model.

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Time Value of Money

Let’s take an example: In 2019, James decides to borrow £100 from Jones and promises to pay back next year. In Jones’s perspective, is the amount that he will receive in 2020 worth more or less than the original £100 that he borrowed? In simple terms, if he had access to that particular £100 today, he could use it to purchase a panoply of items to satisfy his needs and wants. Alternatively, that £100 could have been used to deposit in a bank to earn interest rate of, say, 3%. This option would give Jones 100 × (1 + 0.03) = £103 which is more than what James will repay him in 2020. We say that Jones has incurred an opportunity cost by lending to James without demanding interest, because he could have stuffed that £100 at the bank instead. Every decision to spend money today is accompanied by the choice to save that money for interest. As a result, we can conclude that the present value of money is worth more than the future value of the same amount. Mathematically, this relationship is represented by the following equation:

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PV =

FV , (1 + r ) n

(1)

where PV is the present value, FV is the future value, the rate of interest is r, and n is the number of years from the present time. Thus, in an economy where the ongoing rate of interest is 5%, the = £78.35. It is now quite obvious that the value of £100 five years from now is worth PV = (1+100 0.05)5 rate of interest underpins the present value of money. For a higher rate of interest, lenders demand to be paid more for the use of their money—in effect, the return from saving is higher and a higher FV is required to maintain the same PV.

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Geometric Series

Before we move on to the applications of the time value of money, we first need to develop the necessary mathematical tools to extend our understanding. When a business receives a cash inflow more than once in the future, the computations of the present value becomes more complex as we have to take into account more than one value. Luckily, with some assumptions, we can use series to help make life easier. To understand what are series, I define sequences as lists of numbers that are in order. The following are examples of sequences: 1, 3, 5, 7, 9, ... 1 1 1 1 1, , , , , ... 3 9 27 81 1, 2, 4, 8, 16 A sequence can either be an arithmetic or geometric sequence. An arithmetic sequence involves a constant difference between the terms, while a geometric is one where two connected terms are separated by a multiple. For example, in the second sequence above, each term is the previous term multiplied by a third, so it is a geometric sequence. Furthermore, we can distinguish sequences by their limit: those that goes on forever are called infinite sequences like the first two sequences above, and those that end after a certain number of terms are called finite sequences like the last one above. Now, what are series? Series are the sum of all the terms in a sequence. From the above sequences, we will have the following series: 1 + 3 + 5 + 7 + 9 + ... 1 1 1 1 1+ + + + + ... 3 9 27 81 1 + 2 + 4 + 8 + 16 We can find the value of both finite and infinite series (only possible in a convergence series where the next term is always smaller like the second series above) using the following formulas: Sn =

a (1 − r n ) 1−r

(2)

a , 1−r

(3)

S∞ =

where a is the first term, n in the number of terms in a series, and r is the common ratio (the multiple between a term and the next). We can now apply S∞ to the second sequence above to obtain the following: S∞ =

1 1− 2

1 3

= 1.5.

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Annuities, Perpetuities and Business Cash Flows

The concept of time value of money and series can be applied in analyzing the present value of financial assets that generate constant streams of income either finitely or infinitely. Annuities are financial assets that pays out a fixed stream of payments to an individual while perpetuities do the same forever. These financial assets are mostly used by retirees but can also be used by parents to allocate a certain amount of income to their children over a period of time. In any case, as long as it is not for illegal purposes, annuities and perpetuities can be devised to suit the owner’s needs in a number of situations. Take the following example: Earlier this year, hedge fund billionaire David Winton and his wife donated £100 million to his alma mater to fund PhD programs and other projects in perpetuity. We can only but guess how he funds the donation since the University of Cambridge does not divulge the details. For the sake of simplicity, let’s assume that he decided to donate a fixed amount each year in perpetuity but all those amounts sum up to £100 million in present value. And we want to find out how much he donates each year in this perpetuity. Assuming that the rate of interest i is 2% forever, let’s do the maths to find out how much he donates each year: PV = a +

= S∞

a a a + + ... + 1 + 0.02 (1 + 0.02)2 (1 + 0.02)n a a = = = £100 million 1 1−r 1 − 1+ i

1 1 = = 0.98039... (1 + i ) (1 + 0.02)   1 a = £100 million × 1 − = £1.96 million. (1 + 0.02) Common ratio r =

Resolving the equation involving S∞ we can work out the amount that David Harding is supposed to donate every year perpetually, i.e. a = £1.96 million. The other application of the concept of present value in business is the valuation of projects that involves more than multiple cash flows over a given period of time. For example, consider this problem: should a business invest £1 million in a car production factory that generates a positive cash flow of £300,000 per annum for 5 years? For business leaders to decide whether to invest in a certain project, they normally employ finance professionals, consultants or even investment bankers to aid them in their decision-making. (That’s probably the reason why investment bankers and consultants are paid so much.) Without going into too much detail, one of the tools that these professionals employ is the Net Present Value method to ascertain the true value of a project. The net present value aggregates all the negative and positive cash flows over the lifetime of the project: if it is positive, it means the project generates a positive cash flow, and investors should undertake the project. The net present value can be calculated using the following formula: NPV = C0 +

C1 C2 Cn + , + ... + 2 (1 + i ) (1 + i ) (1 + i ) n

(4)

where we define C0 as cash flow at year 01 while Cn is the cash flow received at year n, and the interest rate is i. If we apply the NPV formula to the problem formulated above, we can work out if it is worthwhile to invest in the project. Let’s first assume in our base case that the interest rate i is 5%, our calculations will yield the following: 1 usually,

this is taken to be the amount of money invested at the beginning of the project

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NPV = −1, 000, 000 +

300, 000 300, 000 300, 000 + ... + = £298, 843. + 2 (1 + 0.05) (1 + 0.05) (1 + 0.05)5

Given the positive net present value, the business should invest money in constructing the factory in a heartbeat. In principle, the NPV rule states that if the net present value of a project is positive, business/investor(s) should accept the project; on the other hand, if it is negative, the project should be rejected since the cash flow generated over the lifetime of the project will not cover the initial investment. Note that if we were to calculate the above NPV with a higher interest rate, say, of 20%, the net present value becomes negative 2 . This is telling since it shows that interest rate is a key determinant when it comes to business investment: the lower the interest rate, the more likely will businesses invest since they can borrow less costly. Furthermore, in the real world the cost of capital, another term for interest rate, can fluctuate from time to time which will, in turn, affect the project NPV. Thus, financial analysts may want to work out the value of the internal rate of return IRR—which is simply the rate of interest that renders the NPV = 0. A higher IRR is usually interpreted as favorable, primarily because investors could accept higher interest rate and the project would still have a positive NPV. In other words, it outlines the profitability threshold of a project. To illustrate this idea, we first have to express IRR mathematically: NPV = C0 +

Cn C2 C1 + ... + = 0. + (1 + IRR) (1 + IRR)2 (1 + IRR)n

(5)

In our example of car production factory, we can either use Microsoft Excel or a calculator by trial and error to find the value of IRR, which turns out to be 15.24%. The interpretation of this value is that to finance this projects, firms can accept an interest rate as high as 15.24% and the project will still break even. More importantly, if there is an alternative project with lower IRR compared with the current one, it is unlikely to be as attractive, because when the cost of capital is higher it is less likely to yield a positive net present value. Therefore, we see that IRR is one of the key parameters for financial analysts in determining whether a project is good or bad. If financial analysis entails only the calculations NPV and IRR, finance professionals are unlikely to earn as much as they do now. For one it is because the real world is much more complex than the little model delineated above. In short, our model fails to incorporate the effects of inflation, fluctuations in interest rate and exchange rates, and the growth of cash flows. Assuming away inflation will result in an overestimation of the NPV, because the effect of inflation is to eat away the value of money as time goes by. While the fluctuations of interest rates and exchange rates are less harmful in the sense that they may, in fact, increase the NPV if, for example, the rate of interest decreases, the effect is to make our NPV model much less reliable. Lastly, in the real world, it is rare for any type of business to be earning a constant amount of profit over a period of time, so a more complex model of NPV will account for an increase or decrease in cash flow with a growth multiplier.

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Concluding Remarks

The goal of this essay is to inform readers of the importance of the applications of mathematics in finance by exploring first how geometric series are put to use. I defined the concept of present value to illustrate that immediate money is more valuable than future money. Then, geometric sum formulae is applied to solve financial problems. In the last section, I define the NPV and explain its significance in business and its shortcomings in modeling the real world. Although the concepts are a simple conception of the real world, these are still valuable tools to help simplify financial problems. And like the case of present value, it explains what is generally believed to be true in mathematical terms. But for a more realistic model, we perhaps need to dig deep into the mathematics at the expense of simplicity and understandability. 2 NPV

= -1,000,000 +

300,000 (1+0.20)

+

300,000 (1+0.20)2

+ ... +

300,000 (1+0.20)5

= -£102,816.

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References https://www.cam.ac.uk/news/cambridge-university-secures-unprecedented-ps100-milliongift-to-support-students

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