Ch2_time Value of Money

THE TIME VALUE OF MONEY

CHAPTER 2

Chapter outline

• Introduction• Interest rates• Future value and compounding of lump sums• Compounding interest more frequently than

annually • Nominal and effective interest rates• Present value and discounting• More on present and future values• Valuing annuities• Perpetuities• Amortising a loan• Sinking funds• Conclusion

Learning outcomesBy the end of this chapter, you should be

able to:• use various computation tools to analyse the role of

time value in finance• calculate, interpret and explain the future value

and present value of single amounts or lump sums, and investigate the relationship between them

• calculate, interpret and explain the future value and present value of annuities (ordinary annuities, annuities due and ordinary deferred annuities)

• calculate, interpret and explain the present value of a perpetuity

Learning outcomes (cont.)

By the end of this chapter, you should be able to:

• calculate and interpret the present value and future value of a mixed stream of cash flows

• determine deposits needed to accumulate a future sum, calculate installments to amortise a loan and calculate an interest rate or growth rate

• calculate the present value, future value, interest rate and time period using discounting and compounding principles.

Introduction • The value of an investment depends on:

size timing of cash flows

• The larger the cash inflows, and the sooner the receipt of these cash flows, the more valuable the investment

• Time value of money: Cash flows to be received in the near future

are more valuable than ones to be received in the distant future

Interest rates

• Simple interest Interest earned on the principal amount only

– interest earned is not reinvested • Compound interest

All interest earned is reinvested together with principal amount – interest is earned on original principal as well as on interest that has been reinvested

Sibusiso receives a R1 000 bonus. He invests the R1 000 in a savings account that offers a simple interest rate of 10% p.a. for a period of five years. How much money will Sibusiso have after five years?

Example 4.1 Simple interest

Initial principal InterestYear 1: 10% of R1 000 = R100Year 2: 10% of R1 000 = R100Year 3: 10% of R1 000 = R100Year 4: 10% of R1 000 = R100Year 5: 10% of R1 000 = R100 Initial principal: R1 000Total interest earned over this period: R 500Final amount after five years: R1 500

Sibusiso invests R1 000 in a savings account offering interest at 10% p.a. but the interest earned will be re-invested. How much money will Sibusiso have after five years?

Example 4.1 Compound interest

Initial PreviousPrincipal Interest Principal New amount

Year 1: 10% of R1 000,00 = R100,00 R1 000,00 R1 100,00Year 2: 10% of R1 100,00 = R110,00 R1 100,00 R1 210,00Year 3: 10% of R1 210,00 = R121,00 R1 210,00 R1 331,00Year 4: 10% of R1 331,00 = R133,10 R1 331,00 R1 464,10Year 5: 10% of R1 464,10 = R146,41 R1 464,10 R1 610,51 Initial principal: R1 000,00Final amount after five years: R1 610,51Total interest earned over this period: R 610,51

Future value and compounding of a lump sum

• Future value (FV): determine accumulated value of all cash flows at end of a project (Tn)

• Present value (PV): discounts all cash flows to start/beginning of a project (time zero, T0) FVn = PV0 × (1+i)n

Example 4.2Sibusiso invests his money for a period of one year at an interest rate of 10%. What will the FV of his investment be at the end of the year?

Example 4.3Sibusiso invests his money for two years at 10%. Calculate the FV of his investment after two years.

Compounding interest more frequently than annually

• Interest often computed more frequently than once a year

• Terminology for different frequencies of compounding: Nominal annual rate compounding annually

(NACA) Nominal annual rate compounding semi-

annually (NACSA) Nominal annual rate compounding quarterly

(NACQ) Nominal annual rate compounding monthly

(NACM)

Semi-annual, quarterly and monthly compounding• Formula when interest is compounded

more than once per period: FVn = PV0 ×

• FV increases when frequency of compounding the interest payments is increased

mn

mi1

Sibusiso invests his money for a period of two years at an interest rate of 10%, compounded semi-annually. What will the FV of his investment be at the end of this period?

Example 4.6

Continuous compounding • Interest sometimes computed

continuously FVn = PV0 × e i × n

Let’s reconsider Sibusiso’s deposit of R1 000 in the savings account for a period of two years at an annual interest rate of 10%, but let’s now assume that the interest is compounded continuously. Calculate the FV. Using the formula: FV2 = PV0 × ei × n = R1 000 × 2,7183 0,10 × 2

= R1 221,40

Nominal and effective interest rates• Nominal (stated) interest rate:

contractual annual percentage rate of interest charged by a lender or promised by a borrower

• Effective (true) annual rate: annual rate of interest actually paid or earned. Effective annual rate includes effects of

compounding frequency; nominal rate does not

EAR =

1mi1

m

Example 4.10

• What is the effective annual rate of interest if an annual nominal rate of 8% is compounded quarterly?

Using the formula:EAR = = = (1,02)4 − 1 = 0,0824 = 8,24%

1mi1

m

14

8%14

Present value and discounting• Present value (PV)

Amount of money invested today at given interest rate for specified period to equal future amount

• Alternatively: PV is amount today that is equivalent to future payment that has been discounted by appropriate interest rate

• Since money has time value: PV of future amount is worth less longer you have to wait to receive it

• Process of finding PVs: discounting

Present value and discounting

• PV Amount of money that would have to be invested today at given interest

rate over specified period to equal future amount

PV0 =

nn

i1

FV

Example 4.11

Fikile wishes to find the current value (PV) of an amount of R1 700 that will be received eight years from now, assuming that the annual interest rate is 8%. Using the formula:PV = FV × (1 + i)-n

= R1 700 × (1,08)-8

= R1 700 × 0,5402 = R918,46

More on present and future values• Determining an interest rate

Sometimes necessary to calculate the return on investment

• Calculating the number of periods Sometimes necessary to work out the

number of time periods for an initial investment to accumulate to a given FV

Valuing annuities

• Annuity Series of equal payments (cash outflows) or

receipts (cash inflows) occurring over specified time period

Consists of constant payments made at regular intervals (monthly, quarterly, annually, etc.)

• Two types of annuities Ordinary annuity (or annuity in arrears):

payments or receipts occur at the end of each period

Annuity due (or annuity in advance): payments occur at the start of each period

Example 4.14

You deposit R2 000 at the end of each of the next five years in an account that pays 10% interest p.a. What will the FV of your account be after five years?

Example 4.14

FVA = PMT ×

ii 11 n

FV of an annuity due

• Annuity due: each payment occurring is moved ahead one period in order to convert the cash flow stream into an ordinary annuity Achieved by multiplying PMT by (1+i) Resulting cash flows of R5 300 (5 000 ×

1,06) at the end of each period are similar to the cash flows of R5 000 at the beginning of each period

Example 4.19

What amount will accumulate if you deposit R5 000 at the beginning of each year for the next five years in an account with an interest of 6% compounded annually?

PV of an ordinary annuity

• PVA: current value of a stream of expected or promised future payments that have been discounted to a single equivalent value today

PVA = PMT ×

ii n11

Example 4.20

Suppose you need an investment that will pay R2 000 at the end of every year for the next five years at an annual interest rate of 10%. How much should you invest today?

Example 4.22

What amount must you invest today at 6% interest compounded annually so that you can withdraw R5 000 at the beginning of each year for the next five years?

Mixed stream of cash flows

• Annuity based on equal payments over a number of periods

• During capital budgeting cash flows from initial investment usually not in form of annuity

• Unequal cash flows will probably be generated over project lifetime In some cases, positive as well as negative

cash flows may occur Mixed cash flow stream Not possible to use annuity formulae and

calculator solutions

Example 4.24

As a reward for taking care of your uncle he makes the following payments into your account over a period of five years: Year Cash flow 1 R5 000 2 R5 000 3 R6 000 4 R6 000 5 R1 000 Assuming an interest rate of 10% per annum, calculate the PV of the cash flows.

Example 4.24

The calculation can also be computed with a financial calculator by using the cash flow function (Cfi): Input Function 0 Cfi 5 000 Cfi 5 000 Cfi 6 000 Cfi 6 000 Cfi 1 000 Cfi 10 i NPV = R17 904,58

Example 4.24

Perpetuities

• Perpetuity: annuity in which periodic payments begin on a fixed date and continue indefinitely (perpetual annuity)

• Three types of perpetuities: Ordinary perpetuity: payments made at the

end of the stated periods Perpetuity due: payments made at the

beginning of the stated periods Growing perpetuity: periodic payments grow

at a given rate (g)

Perpetuities

• Formula used to calculate the PV of an ordinary perpetuity (PV∞):

  PV∞ =  • Formula used to calculate the PV of a

growing perpetuity:  PV∞ =

i

PMT

gi

PMT

Conclusion• A lump sum refers to a single payment or

receipt of cash at a specific point in time. A distinction was made between initial cash flows (occurring at time zero, i.e. now) and future cash flows that occur somewhere in future.

• An annuity can be defined as a stream of equal, periodic cash flows over a specified period of time, in equally spaced time intervals. These payments are usually annual, but can occur at other intervals, such as monthly (e.g. bond payments). Annuity formulae allow complex problems to be resolved in a systematic manner.

Conclusion (cont.)

• A perpetuity is a perpetual stream of constant or constantly growing cash flows.

• A mixed cash stream consists of non-constant cash flows, where different cash flows occur every period.

• The FVs and PVs of lump sum amounts, annuities and mixed cash flows can be calculated by making use of formulae, financial tables or a financial calculator.