Lecture Business mathematics - Chapter 5: Financial mathematics. The main topics covered in this chapter include: arithmetic and geometric sequences and series; simple interest, compound interest and annual percentage rates; depreciation; net present value and internal rate of return; annuities, debt repayments, sinking funds;... Please refer to this chapter for details!
BUSINESS MATHEMATICS CHAPTER 5: FINANCIAL MATHEMATICS Lecturer: Dr Trinh Thi Huong (Hường) Department of Mathematics and Statistics Email: trinhthihuong@tmu.edu.vn CONTENT 5.1 Arithmetic and Geometric Sequences and Series 5.2 Simple Interest, Compound Interest and Annual Percentage Rates 5.3 Depreciation 5.4 Net Present Value and Internal Rate of Return 5.5 Annuities, Debt Repayments, Sinking Funds 5.6 The Relationship between Interest Rates and the Price of Bonds 5.7 Excel for Financial Mathematics 5.1 ARITHMETIC AND GEOMETRIC SEQUENCES AND SERIES A sequence is a list of numbers which follow a definite pattern or rule Arithmetic sequence with d common difference: after the first, is obtained by adding a constant, d, to the previous term A geometric sequence with r common ratio: after the first, is obtained by multiplying the previous term by a constant r ARITHMETIC SERIES (OR ARITHMETIC PROGRESSION, AP) • • The value of the nth term: Tn = a + (n − 1)d The sum of the first n terms, 𝑆𝑛 : 𝑛 𝑆𝑛 = [2𝑎 + 𝑛 − 𝑑] GEOMETRIC SERIES (OR GEOMETRIC PROGRESSION, GP) 5.2 SIMPLE INTEREST, COMPOUND INTEREST AND ANNUAL PERCENTAGE RATES Simple interest (Lãi đơn) Compound interest (Lãi kép) In the modern business environment, the interest on money borrowed (lent or invested) is usually compounded Compound interest pays interest on the principal plus on any interest accumulated in previous years The total value, 𝑃𝑡 , of a principal, 𝑃𝑜 , when interest is compounded at i% per annum is 𝑃𝑡 = 𝑃0 + 𝑖 𝑡 Present value at compound interest 5.5 ANNUITIES, DEBT REPAYMENTS, SINKING FUNDS Trái phiếu đồng niên Debt Repayments: Lịch trình trả nợ Sinking Funds: Quỹ tích lũy, quỹ tốn nợ Annuities: 5.5.1 COMPOUND INTEREST FOR FIXED DEPOSITS AT REGULAR INTERVALS OF TIME Finally the value 𝑉𝑡 of the investment at the end of t years is equal to initial investment 𝑃𝑜 plus the annual investments 𝐴0 all compounded annually: 𝑡 −𝑖 + 𝑖 𝑉𝑡 = 𝑃0 + 𝑖 𝑡 + 𝐴0 𝑖 5.5.2 ANNUITIES: TRÁI PHIẾU ĐỒNG NIÊN An annuity is a series of equal deposits (or withdrawals) made at equal intervals of time The total amount of the annuity at the end of t years is 1+𝑖 𝑡−1 𝑉𝐴𝑁𝑈,𝑡 = 𝐴0 𝑖 5.5.3 DEBT REPAYMENTS: LỊCH TRÌNH TRẢ NỢ A loan is said to be amortised (khấu hao) if both principal and interest are to be repaid by a series of equal payments made at equal intervals of time, assuming a fixed rate of interest throughout For such a repayment scheme, the value of the loan (L) and interest rate are usually known but the amount to be repaid at each interval must be calculated Example: Mortgage (Thế chấp) 5.5.3 DEBT REPAYMENTS: LỊCH TRÌNH TRẢ NỢ SINKING FUNDS: QUỸ TÍCH LŨY, QUỸ THANH TỐN NỢ A sinking fund is created by putting aside a fixed sum each year for the purpose of paying debts, replacing equipment, etc In other words, an annuity is set up to repay the debt If a fixed sum, 𝐴0 , is set aside at the start of each year and interest is compounded annually at i%, the fund will grow year by year as follows 5.6 THE RELATIONSHIP BETWEEN INTEREST RATES AND THE PRICE OF BONDS Bond: Trái phiếu A bond is a cash investment made to the government, usually in units of £1000 for an agreed number of years In return, the government pays the investor a fixed sum at the end of each year; in addition, the government repays the original value (face value) of the bond to the investor with the final payment To make bonds attractive to investors, the size of the fixed annual payments (sometimes referred to as the coupon) must be based on the prevailing rate of interest (i) at the time of purchase The fixed annual payment is calculated as follows: Annual payment = i × (price of bond) 5.7 EXCEL FOR FINANCIAL MATHEMATICS ... Sinking Funds 5.6 The Relationship between Interest Rates and the Price of Bonds 5.7 Excel for Financial Mathematics 5.1 ARITHMETIC AND GEOMETRIC SEQUENCES AND SERIES A sequence is a list of numbers... value, A, each year Straight-line depreciation: is the converse of simple interest with equal amounts being subtracted from the original asset value each year Reducing-balance depreciation: is... annual payment is calculated as follows: Annual payment = i × (price of bond) 5.7 EXCEL FOR FINANCIAL MATHEMATICS