1. Time Value of Money LECTURER: ISAAC OFOEDA

2. Chapter Objectives Understand what gives money its time value. Explain the methods of calculating present and future values. Highlight the use of present value technique (discounting) in financial decisions. Introduce the concept of internal rate of return.

3. Time Preference for Money Time preference for money is an individual’s preference for possession of a given amount of money now, rather than the same amount at some future time. Four reasons may be attributed to the individual’s time preference for money: – risk – preference for consumption – investment opportunities – inflation

4. Required Rate of Return The time preference for money is generally expressed by an interest rate. This rate will be positive even in the absence of any risk. It may be therefore called the risk-free rate. An investor requires compensation for assuming risk, which is called risk premium. The investor’s required rate of return is: Risk-free rate + Risk premium.

5. Time Value Adjustment Two most common methods of adjusting cash flows for time value of money: – Compounding—the process of calculating future values of cash flows and – Discounting—the process of calculating present values of cash flows.

6. Future Value Compounding is the process of finding the future values of cash flows by applying the concept of compound interest. Compound interest is the interest that is received on the original amount (principal) as well as on any interest earned but not withdrawn during earlier periods. Simple interest is the interest that is calculated only on the original amount (principal), and thus, no compounding of interest takes place.

7. Future Value Lump sum payment The general form of equation for calculating the future value of a lump sum after n periods may, therefore, be written as follows: The term (1 + i) n is the compound value factor (CVF) of a lump sum of Re 1, and it always has a value greater than 1 for positive i, indicating that CVF increases as i and n increase.

8. Example If you deposited GHȼ55,650 in a bank, which was paying a 12 per cent rate of interest on a ten-year time deposit, how much would the deposit grow at the end of ten years?

9. Future Value of an Annuity Annuity is a fixed payment (or receipt) each year for a specified number of years. If you rent a flat and promise to make a series of payments over an agreed period, you have created an annuity. The term within brackets is the compound value factor for an annuity of Re 1, which we shall refer as CVFA.

10. Future Value of an Annuity Annuity is a fixed payment (or receipt) each year for a specified number of years. If you rent a flat and promise to make a series of payments over an agreed period, you have created an annuity This is used in the case of ordinary annuity (payment in arrears) The term within brackets is the compound value factor for an annuity of Re 1, which we shall refer as CVFA

11. Future Value of an Annuity Suppose that a firm deposits GHȼ 5,000 at the end of each year for four years at 6 per cent rate of interest. How much would this annuity accumulate at the end of the fourth year? We first find CVFA which is 4.3746. If we multiply 4.375 by GHȼ5,000, we obtain a compound value of GHȼ 21,875:

12. Future Value of an Annuity If the payments are made at the beginning of each period, then we have an annuity due. Rental lease payments, life insurance premiums, and lottery payoffs (if you are lucky enough to win one) are examples of annuities due.

13. Example A customer has instructed his bankers to transfer GHȼ400 from his current account to his savings account at the end of each year for 4 years. Interest rate on savings is 10% per annum. What is the size of the savings account at the end of the 4 th year. Assuming the customer’s instruction to the bank was that the first transfer should be made now, followed by the same amount of GHȼ400 for three more years. What is the size of the account after 4 years.

14. Future Value of an Uneven Periodic Sum Investments made by of a firm do not frequently yield constant periodic cash flows (annuity). In most instances the firm receives a stream of uneven cash flows. Thus the future value factors for an annuity cannot be used. The procedure is to calculate the future value of each cash flow and aggregate all present values.

15. Example Dosky expects to receive GHȼ300, GHȼ500, GHȼ1000 and GHȼ1500 in year1,2,3 and 4 respectively. What is the value of these cash flows at the end of the fourth year?

16. Application of the “f” Rule The above discussion assumes that payments are made once a year. However, in real life situation, payments may be made daily, weekly, monthly, quarterly or even semi- annually. In this case, we apply the f rule. That is, wherever we see r we divide it by f and wherever we see n we multiply by f.

17. Sinking Fund Sinking fund is a fund, which is created out of fixed payments each period to accumulate to a future sum after a specified period. For example, companies generally create sinking funds to retire bonds (debentures) on maturity. The annuity formulas may be used in computing the periodic payments for both ordinary annuity or annuity due.

18. Example Cleanborn Ltd has bought an asset with a life span of 4 years. At the end of the 4 years, replacement of the asset will cost GH¢12,000. In this direction, the company has decided to provide for the future commitment by setting up a sinking fund account into which equal annual investment will be made at the end of each year. Interest rate on the investment will be 12% per annum.

19. Example Cont’d Required: a. Calculate the annual instalments b. Draw up the sinking fund schedule to show the growth fund c. Assuming the first payments will be made now and 12 months thereafter, what are the annual payments? d. Draw up the sinking fund schedule under (c)

20. Solution-----Arrears Finding the periodic payment Sinking Fund Schedule YearBal b/fPaymentsInterestBal c/f 102510.00 2 301.25321.2 3 2510.0638.58469.7 4 2510.01016.411996.1

21. Solution-----Advance Finding the periodic payment Sinking Fund Schedule YearBal b/fPaymentsInterestBal c/f 102241.8269.02510.8 2 2241.8570.35323.0 35321.22241.8907.68470.6 48469.72241.81285.411996.9

22. Changes in Interest Rates If the interest rates changes during the period of an investment, the compounding formula must be amended as follows;

23. Present Value Present value of a future cash flow (inflow or outflow) is the amount of current cash that is of equivalent value to the decision-maker. Discounting is the process of determining present value of a series of future cash flows. The interest rate used for discounting cash flows is also called the discount rate.

24. Present Value of a Single Cash Flow The following general formula can be employed to calculate the present value of a lump sum to be received after some future periods: The term in parentheses is the discount factor or present value factor (PVF), and it is always less than 1.0 for positive i, indicating that a future amount has a smaller present value.

25. Example Suppose that an investor wants to find out the present value of GH¢50,000 to be received after 15 years. Her interest rate is 9 per cent. First, we will find out the present value factor, which is 0.275. Multiplying 0.275 by GH¢50,000, we obtain GH¢ 13,750 as the present value:

26. Present Value of an Uneven Periodic Sum Investments made by of a firm do not frequently yield constant periodic cash flows (annuity). In most instances the firm receives a stream of uneven cash flows. Thus the present value factors for an annuity cannot be used. The procedure is to calculate the present value of each cash flow and aggregate all present values.

27. Present Value of an Uneven Periodic Sum Your company invested in a project which is expected to generate GH¢50,000 in year1, GH¢70,000 in year2 and GH¢85,000 in year3. If the cost of capital for the company is 15%, what will be the value of the future cash flows today.

28. Present Value of an Annuity The computation of the present value of an annuity can be written in the following general form: The term within parentheses is the present value factor of an annuity of GH¢ 1, which we would call PVFA, and it is a sum of single- payment present value factors.

29. Present Value of an Annuity If the payments are made at the beginning of each period, then we have an annuity due.

30. Capital Recovery and Loan Amortisation Capital recovery is the annuity of an investment made today for a specified period of time at a given rate of interest. Capital recovery factor helps in the preparation of a loan amortisation (loan repayment) schedule. The reciprocal of the present value annuity factor is called the capital recovery factor (CRF).

31. Example You have applied to your bankers for a loan of GH¢30,000 to complete your dream house for deductions to be made over 3 years equal annual instalments. Your bankers, however, maintained that your 40% annual salary which amounts to GH¢12,000 cannot meet both the principal and interest payment. It is the bank’s policy to maintain a debt service ratio of 40%. Interest rate charged by the bank is 18% per annum.

32. Example Required: a. Calculate the size of the loan you qualify for. b. Prepare amortization table to show how the loan will be liquidated

33. Solution Value of the loan Annual Payment = GH¢12,000

34. Solution

35. Example Mr Zidi borrowed GH¢20,000 to renovate his house. He will pay it back in equal annual payments, which begins today and every 12 months for 4 years. Interest rate on the loan is 8% per annum. a. Calculate the annual loan payments b. Show the amortization table

36. Solution Finding annual payment

37. Solution

38. Present Value of Perpetuity Perpetuity is an annuity that occurs indefinitely. Perpetuities are not very common in financial decision-making:

39. Present Value of Growing Annuities The present value of a constantly growing annuity is given below: Present value of a constantly growing perpetuity is given by a simple formula as follows:

40. Finding Interest Rates Your brother is planning to retire in 18 years time. He currently has GHC250,000, and he would like to have GHC1,000,000 when he retires. You are required to compute the annual rate of interest he would have to earn on his GHC250,000 in order to reach his goal, assuming he saves no more money.

41. Finding Number of Periods At 9% interest, how long does it take to double your money? To quadruple it?

42. Multi-Period Compounding If compounding is done more than once a year, the actual annualised rate of interest would be higher than the nominal interest rate and it is called the effective interest rate.