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# Essential Mathematics of Money by Tim Hill

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Essential Mathematics of Money: A Self-Teaching Guide by Tim Hill Copyright © 2018 by Questing Vole Press. All rights reserved. Editor: Kevin Debenjak Proofreader: Diane Yee Compositor: Kim Frees Cover: Questing Vole Press Self-Teaching Guides by Tim Hill Essential Trigonometry Essential Algebra Essential Geometry Essential Precalculus Essential Advanced Precalculus Essential Permutations & Combinations Essential Mathematics of Money Essential Excel Loan & Mortgage Formulas For a complete list of titles, go to questingvolepress.com. Contents 1. Interest 2. Present Value 3. Annual Percentage Rate 4. Mortgages 5. Annuities 6. Stocks 7. Personal Savings 8. Student Loans 9. Kelly Strategy 10. Mathematics Review 1. Interest Mathematical prerequisites. The material in this book requires a command of high-school mathematics, especially exponents, logarithms, geometric series, sigma notation, and some statistics. In a few places, the tools of elementary calculus—derivatives, Taylor series, L’Hôpital’s rule, and iteration—are used to prove a result, solve an equation, or derive an optimal value. If you like, you can skip or skim the calculus-based parts without loss of continuity. Chapter 10 provides a mathematical refresher and a short list of key mathematical facts. Keep in mind that mathematical concepts aren’t confined to the context in which they’re introduced. You can apply the methods in this book to a wide variety of problems. As illustrated by various examples, the same calculations that determine mortgage payments can also help you decide how to pay down credit-card debt, and the optimal strategy for investing in speculative business ventures also applies to casino betting. When you borrow money for a period of time, interest will be charged by the lender (think of interest as the cost of using someone else’s money). Conversely, when you lend money you will receive interest. Interest is payment above repayment of the amount borrowed, at a particular rate. This interest is either simple or compound, and the dramatic difference between the two is best illustrated with an example. Suppose that you invest $100 for 100 years at an annual interest rate of 5%. With simple interest, $5 is added each year, or $5 × 100 = $500 in total, so the end total is $100 + $500 = $600. With compound interest, let S and E denote the amounts at the start and end of a given year, respectively. During the year, 5% of S is added, so E = S(1 + 5/100). This addition happens every year, so after 100 years the total amount that you have is $100(1 + 5/100)100 = $13,150.13. Note that interest is distinct from any fees that a lender might charge a borrower in addition to interest. Unless otherwise specified, interest calculations in this book Use compound interest Ignore fees and other transaction costs Use natural (base e) logarithms, denoted by log(x), loge(x), or ln(x) Example 1.1 Suppose that you borrow an amount C on which you’re charged an annual rate of interest of r%. If you make no repayments, then how much will you owe after n years? Solution After n years, you’ll owe C(1 + r/100)n Example 1.2 If the annual interest rate is 12%, and you borrow $1000 for six months, then how much should you repay? Solution It’s tempting to think that by borrowing for half a year the interest charge is 6% (half of 12%), so the repayment should be $1060. This answer is incorrect, however. Let x% be the interest rate for six months. After six months you owe $1000(1 + x/100), so after another six months, one year in total, you would owe $1000(1 + x/100)2. Because the annual rate is 12%, we have leading to The interest rate for six months is actually 5.83%, so you should repay $1058.30. Example 1.3 If the annual rate is 12%, and you borrow $1000 for one month, then how much should you repay? Solution The appropriate interest rate is that value x such that giving 1 + x/100 = 1.121/12 ≈ 1.0094888. The interest rate for one month is 0.94888%, so you should repay $1009.49. Example 1.4 What is the annual interest rate if a bank advertises (a) a quarterly interest rate of 2%, (b) 1% per month? Solution (a) After a full year of four quarters, with 2% interest charged each time, a loan of size C becomes C(1.02)4 ≈ 1.0824C. The true annual interest rate is 8.24%, not the 8% that the bank suggests. (b) A rate of 1% per month means that the debt is C(1.01)12 ≈ 1.1268C at the year’s end—a true annual rate of 12.68%, not 12%. No matter how an interest rate is presented or quoted, you can use a simple two- step process to calculate how much is owed at any specified future time. 1. If necessary, convert the rate to the true annual rate by using the method shown in the preceding example. 2. With r% as the true annual rate, calculate the amount owed when borrowing C for t years (t can be a whole number or a fraction) as We say that interest is compounded continuously. Example 1.5 If the true annual interest rate is 10%, then how much should you repay if you borrow $1000 for (a) six months, (b) 40 months? Solution (a) Six months is half a year, so you should repay $1000(1.10)1/2 = $1048.81. (b) 40 months is three and one-third years, so you should repay $1000(1.10)10/3 = $1373.96. Example 1.6 A loan is offered at 15% interest compounded quarterly. Show that the true annual rate is about 15.865%. Find the true annual rate, to three decimal places, if the nominal rate is 15% compounded (a) monthly, (b) weekly, (c) daily, (d) hourly, (e) more frequently (continuously). Solution If you borrow an amount C on these terms, then after one year of four quarters you’ll owe C(1 + 0.15/4)4 = C ∙ 1.158650..., so the true rate is about 15.865%. (a) Monthly, (1 + 0.15/12)12 = 1.160754..., so the rate is 16.075%. (b) Weekly, (1 + 0.15/52)52 = 1.161583..., so the rate is 16.158%. (c) Daily, (1 + 0.15/365)365 = 1.161798..., so the rate is 16.180%. (d) Hourly, (1 + 0.15/8760)8760 = 1.161832..., so the rate is 16.183%. (e) Continuously, exp(0.15) = e0.15 = 1.161834...; the limit is also a rate of 16.183%. Example 1.7 You invest $10,000 in property, which increases in value at the compound rate of 5% annually, but you are charged 2% of the total value at the end of each year. What will your investment be worth at the end of five years? Solution If, at the beginning of any year, your investment is worth C, then it increases to 1.05C, but the 2% charge then turns this amount into 1.05C × 0.98 = 1.029C, so at the end of 5 years, you’ll have 1.0295C, or about $11,536.57. Note that a 5% increase, with a 2% decrease, is not the same as a 3% increase. Example 1.8 You borrow an amount C at r% annual interest. If you make no repayments, then at what time will your debt double? Solution Your debt will double at that time n when C(1 + r/100)n = 2C; that is, when n log(1 + r/100) = log(2) = 0.69314718.... Thus where x = r/100. When |x| < 1, we have the Taylor series log(1 + x) = x − x2/2 + x3/3 − ⋯, so when x (> 0) is small, the right side of the equation above is just under x and consequently n is just over 0.693/x = 69.3/r. The preceding example justifies a handy method for estimating the doubling time of a debt. The Rule of 72 At an annual interest rate of r%, it will take approximately 72/r years for a debt to double, if no repayments are made. Naturally, the same argument shows that if you invest a sum of money that earns annual interest of r%, then it will take