1. CHAPTER 4
THE TIME VALUE OF MONEY

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

3. Learning outcomes By 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

4. 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.

5. 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

6. Interest rates Simple interest Compound 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

7. Example 4.1 Simple interest
Sibusiso receives a R1 000 bonus. He invests the R 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?
Initial principal Interest
Year 1: 10% of R1 000 = R100
Year 2: 10% of R1 000 = R100
Year 3: 10% of R1 000 = R100
Year 4: 10% of R1 000 = R100
Year 5: 10% of R1 000 = R100
Initial principal: R1 000
Total interest earned over this period: R 500
Final amount after five years: R1 500

8. Example 4.1 Compound interest
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?
Initial Previous
Principal Interest Principal New amount
Year 1: 10% of R1 000,00 = R100,00 R1 000,00 R1 100,00
Year 2: 10% of R1 100,00 = R110,00 R1 100,00 R1 210,00
Year 3: 10% of R1 210,00 = R121,00 R1 210,00 R1 331,00
Year 4: 10% of R1 331,00 = R133,10 R1 331,00 R1 464,10
Year 5: 10% of R1 464,10 = R146,41 R1 464,10 R1 610,51
Initial principal: R1 000,00
Final amount after five years: R1 610,51
Total interest earned over this period: R 610,51

9. 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

10. Example 4.2
Sibusiso 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?

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

12. 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)

13. 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

14. Example 4.6
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?

15. Continuous compounding
Interest sometimes computed continuously
FVn = PV0 × e i × n

16. 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 =

17. 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%

18. 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

19. Present value and discountingPV
Amount of money that would have to be invested today at given interest rate over specified period to equal future amount
PV0 =

20. 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

21. 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

22. Valuing annuities Annuity Two types of annuities
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

23. 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?

24. Example 4.14
FVA = PMT ×

25. 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

26. 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?

27. 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 ×

28. 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?

29. Example 4.22
What amount must you invest today at 6% interest compounded annually so that you can withdraw R at the beginning of each year for the next five years?

30. 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

31. Example 4.24

32. Example 4.24

33. Example 4.24

34. 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)

35. Perpetuities
Formula used to calculate the PV of an ordinary perpetuity (PV∞):
PV∞ =   
Formula used to calculate the PV of a growing perpetuity:

36. 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.

37. 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.

38. Conclusion (cont.)
Loan amortisation refers to the determination of equal loan payments (the extinction of a debt by means of equal periodic payments over a period of time). Therefore, amortisation is a schedule showing the repayment details for a loan, including the amount of each payment that is apportioned to interest and to capital (the principal debt).
Sinking funds are used to accumulate money over time, by depositing periodic payments in a fund. Examples of sinking funds are to make provision for the replacement of assets, or to make provision for a loan that needs to be repaid.