Lecture Business mathematics - Chapter 5: Financial mathematics

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!

B USINESS M ATHEMATICS

CHAPTER 5: FINANCIAL

MATHEMATICS

Lecturer: Dr Trinh Thi Huong (Hường)

Department of Mathematics and

Statistics

Email: trinhthihuong@tmu.edu.vn

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:

𝑆𝑛 = 𝑛

2 [2𝑎 + 𝑛 − 1 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 1 + 𝑖 𝑡 Present value at compound interest

WHEN INTEREST IS COMPOUNDED

SEVERAL TIMES PER YEAR

 Interest may be compounded several times per year (namely conversion period or interest period): daily, weekly, monthly, quarterly, semi-annually or

continuously

 The number of conversion periods per year: m

 The interest rate applied at each conversion is 𝑖

𝑚

 The number of year: t

 The value of the investment at the end of n conversion

periods is

CONTINUOUS COMPOUNDING

 With continuous compounding, the future value is given by the formula

𝑃𝑡 = 𝑃0𝑒𝑖𝑡

ANNUAL PERCENTAGE RATE (APR)

-LÃI SUẤT PHẦN TRĂM HÀNG NĂM

 The nominal rate is compounded m times per

5.3 DEPRECIATION (OR AMORTIZATION,

KHẤU HAO)

 Depreciation is an allowance made for the wear and tear

of equipment during the production process It involves

the deduction of money from the original asset 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 the converse of compound interest with larger amounts being subtracted from the original asset value each year The formula for reducing-balance depreciation is

𝐴𝑡 = 𝐴0 1 − 𝑖 𝑡where 𝐴𝑡 is the value of the asset after t years taking account of depreciation 𝐴0 is original value of the asset, i: depreciation rate and t: number of years.

5.4 NET PRESENT VALUE AND INTERNAL

RATE OF RETURN

 Net Present Value: Giá trị hiện tại ròng

 The present value of a sum due to be paid in t

years’ time is:

𝑃𝑜 = 𝑃𝑡

1 + 𝑖 𝑡

 The net present value is the present value of

several future sums discounted back to the

present

 NPV = present value of cash inflows − the

present value of cash outflows

 A decision rule is: NPV > 0 Invest in the project;

NPV <0 Dont invest in the project

 The IRR is the interest rate for which the NPV is

zero A project is viable if the prevailing interest

rate is less than the IRR, but not profitable if the prevailing interest rate is greater than the IRR.

 The decision rule is: Accept the project if the IRR

is greater than the market rate of interest

INTERNAL RATE OF RETURN (LÃI SUẤTHOÀN VỐN NỘI BỘ)

5.5 ANNUITIES, DEBT REPAYMENTS,

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:

𝑉𝑡 = 𝑃0 1 + 𝑖 𝑡 + 𝐴0 1 + 𝑖

𝑡 − 𝑖 𝑖

5.5.2 ANNUITIES: T RÁI 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

𝑉𝐴𝑁𝑈,𝑡 = 𝐴0 1 + 𝑖

𝑡 − 1 𝑖

5.5.3 DEBT REPAYMENTS: L ỊCH 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: L ỊCH TRÌNH TRẢ NỢ

S INKING F UNDS : Q UỸ TÍCH LŨY , QUỸ

THANH TOÁN NỢ

 A sinking fund is created by putting aside a fixedsum 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 setaside 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