TOPIC TWO:

Chapter 3: Financial Mathematics

By Diana Beal and

Michelle Goyen

Cash flows, interest and time value

• Cash flows are funds flowing either into or out of a business, that is, cash receipts or cash expenses

• Interest is the price charged for the temporary use of money

• Interest performs an allocative role in economies, by sharing out scarce funds to users

• For borrowers, interest can be thought of as a penalty for wanting to consume before income is earned.

• For lenders, on the other hand, interest is compensation for putting off consumption until a later time

• The time value of money is the concept that a dollar is worth more the sooner it is received

• Money now is worth more than the same amount received in the future

– If you have money now, you can invest

it and receive a larger sum in the future

• When cash sums are received at different times– their values need to brought to a

common date– you cannot add money amounts at

different dates because it is like adding apples and cats

• The common date can be either today or some future date

Simple interest

• Simple interest is interest calculated on the original sum borrowed

• is applied to many sorts of bank deposits, such as savings accounts and term deposits, as well as many types of loans

• The interest rate and the loan period need to be in the same units– 6% per annum = 0.06/12 per month– The simple interest for 7 months

= 0.06/12 x 7

• To calculate how much would be repaid on a simple interest loan

• To calculate how much money is equivalent to a future sum on a simple interest loan

Simple interest application

• The calculation of values relating to commercial bills provides examples of simple-interest problems

• The yield is the amount producedor the return based on the initial investment

• A discount rate (in relation to pricing a security) is the percentage difference between the face value and the purchase price of the security

• Some of the most important conventions or rules for solving commercial bill problems are:– Interest rates are quoted on a nominal

annual basis and pro-rated for the actual period of the loan

– Pro-rating takes place on the basis of the exact number of days/365

– In a leap year, 29 February is charged for if the loan is outstanding over this day, but the year is assumed still to have 365 days

Compound interest

• Compound interest is interest calculated on the actual outstanding amount each period

• the outstanding amount can include:– the whole or part of the initial principal – plus unpaid interest

• Compound interest is applied to many different types of loans such as personal and credit-card finance, mortgages and overdrafts

• Calculating future value:

• A time line is a diagrammatic representation of cash flows, either received or paid or both

P FV1 FV2

0 1 2

• Calculating present value:

• Discount rate is the interest factor that is used to calculate a present value (PV) from a future value (FV).

• The nominal interest rate is the advertised or quoted annual rate that does not reflect the impact of multiple compounding periods during the year

• The effective interest rate reflects the impact of multiple compounding during the year

• Calculating the annual effective rate:

• Using the correct rate:

• general rule is that the timing of the cash flows and the compounding period need to be equal– if they are, you can use the nominal

rate– if not, use the effective rate

• When the rate is quoted as annual nominal and compounding and cash flows are more frequent than annual– divide the annual nominal rate by the

number of compounding periods

• When the rate is quoted as annual nominal, cash flows are annual and compounding more frequent than annual– use equation 3.7 for the annual

effective rate and use it on the annual cash flows

Future and present values of equal sums

• An annuity is a series of equal cash flows which are evenly spaced over time

• An ordinary annuity is a series of equal cash flows which are evenly spaced over time and are received at the end of each period

9 9 9 9 9

0 1 2 3 4 5

• Present value of an ordinary annuity:

• Future value of an ordinary annuity:

• An annuity due has the first payment made at the inception of the annuity

9 9 9 9 9

0 1 2 3 4 5

• Present value of an annuity due:

• Future value of an annuity due:

• A deferred annuity has the first payment made after the end of the first equal-length period

• for a two-period deferral:

9 9 9

0 1 2 3 4 5

• Present value of an annuity due:

• Future value of an annuity due:

• is calculated the same way as the future value of an ordinary annuity

• A perpetuity is an ordinary annuity that has cash flows that continue forever

9 9 9 ……. 9 ……… 9

0 1 2 3 … ... ∞

• Present value of a perpetuity:

Present and future value of unequal sums

• A stream of unequal amounts is not an annuity so we need to use a combination of calculations

• Any problem with different amounts must be solved using the single-sum formulae for either the FV or PV

• We can use the annuity equations for those parts of the stream that form a series of equal cash flows

– make sure you have the value of the annuity is at time zero or FVn

• Once all cash flows are in FV or PV terms, we can add like to like and thus calculate the total FV or PV

Solving cash flow problems

• Four key steps to follow:1. Draw a time line, showing when the

cash flows occur2. Identify which required information is

unknown3. Determine if you are looking for PV or

FV4. Write down the equations, insert the

values and solve