Chapter3 Advanced mathematics

Chapter 3 Mathematics of finance Chapter 3 Mathematics of finance Nguyen Thi Minh Tam ntmtam vnuagmail com November 7, 2020 1 3 1 Percentage 2 3 2 Compound interest 3 3 3 Geometric series 4 3 4 Investment appraisal 3 1 Percentage Example 1 a) A firm’s annual sales rise from 50 000 to 55 000 from one year to the next Express the rise as a percentage of the original b) The government imposes a 15% tax on the price of a good How much does the consumer pay for a good priced by a firm at 1360? c) I.

Chapter 3: Mathematics of finance

Nguyen Thi Minh Tam

ntmtam.vnua@gmail.com

November 7, 2020

1 3.1 Percentage

2 3.2 Compound interest

3 3.3 Geometric series

4 3.4 Investment appraisal

3.1 Percentage

Example 1

a) A firm’s annual sales rise from 50 000 to 55 000 from one year

to the next Express the rise as a percentage of the original.b) The government imposes a 15% tax on the price of a good.How much does the consumer pay for a good priced by a firm

If the percentage rise is r %, then the scale factor is

1 + r100

If the percentage decrease is r %, then the scale factor is

1 − r100The final value = the scale factor × the original

The original = the final value

the scale factor

Example 2.

a) The value of a good rises by 13% in a year If it was worth

$6.5 million at the beginning of the year, find its value at theend of the year

b) The GNP of a country has increased by 63% over the pastfive years and is now $124 billion What was the GNP fiveyears ago?

Example 3

a) Current monthly output from a factory is 25 000 In a

recession, this is expected to fall by 65% Estimate the newlevel of output

b) After a 15% reduction in a sale, the price of a good is $39.95.What was the price before the sale began?

When finding index numbers, a base year is chosen and thevalue of 100 is allocated to that year.

index number = scale factor from base year × 100

Example 4 The following table shows the values of householdspending (in billions of dollars) during a five-year period Calculatethe index numbers when Year 1 is taken as the base year and give

a brief interpretation

Example 5 The following table shows the index numbers of theoutput of a particular firm for two consecutive years Use theseindex numbers to find the percentage change in output froma) Y2Q1 to Y2Q4

b) Y1Q1 to Y2Q4

c) Y1Q1 to Y2Q1

It is possible to create sensible index numbers to measure thevariation of a bundle of goods over time.

Laspeyre index: An index number for groups of data which areweighted by the quantities used in the base year

Paasche index: An index number for groups of data which areweighted by the quantities used in the current year

Example 6 Suppose that a firm buys three product The

following table shows the number of each type bought in Year 1together with the unit prices of each item in Year 1 and Year 2

The total purchase cost in Year 1 is

If the quantities bought in Year 2 are those shown in thefollowing table

then the current weighted index in Year 2 is

17 × 10 + 38 × 23 + 12 × 5

850 ×100 = 129.9 ←−Paasche index

The annual rate of inflation is the average percentage change

in a given selection of these goods and services, over theprevious year

Inflation distorts economic magnitudes, making them lookbigger than they really are Economists deal with this bydistinguishing between nominal and real data

- Nominal data are the original, raw data.

- Real data are the values that have been adjusted to take inflation into account.

Example 7 Table 3.11 shows the average annual salary (inthousands of dollars) of employees in a small firm, together withthe annual rate of inflation for that year Adjust these salaries tothe prices prevailing at the end of Year 2 and so give the realvalues of the employees’ salaries at constant ‘Year 2 prices’.Comment on the rise in earnings during this period.

3.2 Compound interest

Simple interest

Simple interest is the interest that accrues on a given sum in aset time period

It is not reinvested along with the original capital

The amount of interest earned on a given investment eachtime period will be the same if the total amount of capitalinvested and interest rates do not change

Example 8 An investor puts $20,000 into a deposit account andhas the annual interest paid directly into a separate current

account and then spends it The deposit account pays 8.5%interest How much interest is earned in the fifth year?

Compound interest

Compound interest is the interest which is added to the initialinvestment every time it accrues

The interest added in one time period will itself earn interest

in the following time period

Example 9 If $600 is invested for 3 years at 8% interest

compounded annually at the end of each year, what will the finalvalue of the investment be?

Compound interest formula

Consider an investment at compound interest where:

P is theprincipal (the initial sum invested),

S is the future value (the final value of the investment),

r is the interest rate per time period (as a decimal fraction)

n is the number of time periods

The compound interest formula is

S = P(1 + r )n

Example 10 Find the value, in 10 years’ time, of $1000 invested

at 8% interest compounded annually

Example 11 A principal of $25 000 is invested at 12% interestcompounded annually After how many years will the investmentfirst exceed $250 000?

Example 12 A principal, $30, is invested at 6% interest for twoyears Determine the future value if the interest is compoundeda) annually

b) semi-annually

c) quarterly

d) monthly

d) weekly

The future value rises as the frequency of compounding rises.The type of compounding in which the interest is added onwith increasing frequency is calledcontinuous compounding.The future value, S , of a principal, P, compounded

continuously for t years at an annual rate of r is

S = Pert

Example 13 Determine the rate of interest required for a

principal of $1000 to produce a future value of $4000 after 10years compounded continuously

APR, AER

Annual percentage rate(APR) is the annual rate of interestpaid for a loan, taking into account the compounding over avariety of time periods

The phrase ‘annual equivalent rate’(AER) is frequently usedwhen applied to savings

Example 14 Determine the annual percentage rate of interest ifthe nominal rate is 12% compounded quarterly

3.3 Geometric series

Geometric progression, Geometric series

A geometric progressionis a sequence of the form

a, ar , ar2, , arn, where a and r are real numbers

r is called geometric ratio

A sum of the consecutive terms of a geometric progression iscalled a geometric series

Example 15 Decide which of the following sequences aregeometric progressions For those sequences that are of this type,write down their geometric ratios.

Example 16 A person saves $100 in a bank account at thebeginning of each month The bank offers a return of 12%compounded monthly.

a) Determine the total amount saved after 12 months

b) After how many months does the amount saved first exceed

$2000?

Example 17 A person requests an immediate bank overdraft of

$2000 The bank generously agrees to this but insists that itshould be repaid by 12 monthly instalments and charges 1%interest every month on the outstanding debt Determine themonthly repayment

Example 18 It is estimated that world reserves of oil currentlystand at 2625 billion units Oil is currently extracted at an annualrate of 45.5 billion units and this is set to increase by 2.6% a year.After how many years will oil reserves run out?

P = S (1 + r )−n (discrete compounding)

P = Se−rt (continuous compounding)

In this case,

- the principal, P, is called the present value ,

- the rate of interest is sometimes referred to as the discount rate

Example 19 Find the present value of $1000 in four years’ time ifthe discount rate is 10% compounded

a) annually

b) semi-annually

c) continuously

The net present value

The net present value(NPV) of an investment project is thedifference between the present value of the revenue and theoriginal cost

A project is considered worthwhile when the NPV is positive

If a decision is to be made between two different projects,then the one with the higher NPV is the preferred choice

The internal rate of return

The internal rate of return(IRR) is the annual rate which,when applied to the initial outlay, yields the same return asthe project after the same number of years

The investment is considered worthwhile if the IRR exceedsthe market rate

Example 20 An investment project requires an initial outlay of

$8000 and will produce a return of $17 000 at the end of fiveyears Use the

a) net present value

b) internal rate of return

methods to decide whether this is worthwhile if the capital could

be invested elsewhere at 15% compounded annually

Example 21 A firm needs to choose between two projects, A and

B Project A involves an initial outlay of $13 500 and yields $18

000 in two years’ time Project B requires an outlay of $9000 andyields $13 000 after two years Which of these projects would youadvise the firm to invest in if the annual market rate of interest is7%?

Consider the case of a sequence of payments over time Thesimplest cash flow of this type is an annuity, which is a sequence ofregular equal payments.

Example 22 Find the present value of an annuity that yields anincome of $2000 at the end of each month for 10 years, assumingthat the interest rate is 6% compounded monthly

Example 23 A firm has a choice of spending $10 000 today onone of two projects The revenue obtained from these projects islisted in Table 3.28 Assuming that the discount rate is 15%compounded annually, which of these two projects would youadvise the company to invest in?

Exercise 3.1, 3.1*: 8, 12-15 (page211-212), 4,8 (page 213,214)Exercise 3.2, 3.2*: 7, 9, 12-15 (page 227), 1-3 (page 228)Exercise 3.3, 3.3*: 2, 5, 7 (page 238-239), 2, 8 (page 239- 240)Exercise 3.4, 3.4*: 1, 2, 4, 9, 11 (page 253-254), 8 (page 255)