1. Introduction To Finance The time value of money [Chapter 3] 1

2. Learning Objectives Understand the meaning of “the time value of money”. Understand the relationship between present and future value. Describe how interest rate can be used to adjust the value of cash flows – both forward and backward – to single point in time. Calculate both the future value and present value of cash flows. 2

3. Learning Objectives cont. Distinguish between an “ordinary annuity” and an “annuity due”. Use interest factor tables to calculate present and future values. Use interest factor tables to find unknown interest rates or growth rates. Build an “amortisation schedule” for an instalment-style loan. 3

4. Introduction Given the choice between the receipt of GH100 today and the same amount in a year’s time, a rational person would prefer the GH 100 today. The following reasons would support this choice: The money could be invested to worth more than GH100 in a year’s time. Inflation would mean that the GH100 would buy less in a years time than it would today, that is, inflation devalues disposable income. 4

5. Introduction Cont’ There is some element of risk or uncertainty, however small attached to the receipt of GH 100 at a future date. For instance, the debtor may become bankrupt, die or if is a gift, the person may change his mind. A sudden investment opportunity may present its self requiring immediate cash, which may be lost out. You already recognize that there is TIME VALUE TO MONEY!! Why is TIME such an important element in your decision?. 5

6. The Time Value of Money The value of money (its purchasing power) tends to be different at different times due to inflation/deflation This means that People who lend out their money face the risk of losing value due to inflation and so they should be given a compensation for inflation risk Thus, the nominal value of cash flow should be adjusted accordingly to obtain real value of cash flows over time. 6

7. The Time Value of Money Cont’ Inflationary Pressure  Value of money declines over time Compensation of deferred consumption  There is always an opportunity cost in making a decision to spend money now or later and this relates to time value of money.  Interest rate should include a premium for inflation risk  TIME allows you the opportunity to postpone consumption and earn INTEREST. Adjustment of Cash Flows  Nominal Value of cash flows should be adjusted in order to maintain its original purchasing power 7

8. The Interest Rate Interest rate is a cost of capital. The price paid for a sum of money borrowed. Thus, investors are paid interest to compensate them for the opportunity cost of forgoing possible returns from alternative investments. Due to future uncertainties, a risk premium should be added to the real risk-free interest rate to compensate for certain risks. 8

9. Determination of Market Interest Rate The nominal or quoted interest rate on a security is determined thus: k* = Real risk-free interest rate IP = Inflation premium DRP = Default risk premium LP = Liquidity premium MRP = Maturity risk premiums NIR = Nominal Interest Rate (or quoted interest rate) 9 k* IPDRP LP MRP NIR

10. Calculation of Interest Methods Simple Interest Interest paid (earned) on only the principal borrowed (lent). Compound Interest Interest paid (earned) on any previous interest earned and on the principal borrowed (lent). 10

11. Simple Interest The interest on a single amount is computed using following formula: Where SI = simple interest in cedis. P 0 = principal or original amount borrowed or lent i = interest rate per time period n = number of time periods 11

12. Simple Interest cont. The Future Amount is the sum of the principal and simple interest. The future amount is calculated using the following formula: 12 Principal Simple Interest Future Value

13. Simple Interest cont. The present value of a future receipt or payment is calculated using the following formula: 13

14. Simple Interest Illustration If you borrowed 10 million cedis from bank X, and the bank charges you 10% interest per annum and principal and interest are due at the end of two years. What is the amount due to the bank? Calculate ◦ The simple interest ◦ The future value of the amount in the account at the end of 2 years. 14

15. Solution Simple Interest At the end of the 1st year, you owe: 10m + interest on 10 m 10 + 10x10%x1 = 11m: At the end of 2nd year, you owe amount at end of 1st year, plus 2 nd year interest. 2nd year interest is calculated on the original 10 m: 10% of 10 x 1 = 1m Total Amount due Bank = 11m + 1m = 12m Short cut: amount due = principal + principal x interest rate x time. 15

16. Compound Interest Interest is paid on the principal and all previously accumulated interest. Compound interest means that interest is earned on interest. The result: the income you earn would be higher than it would be with simple interest. This is because you earn interest on both the original investment and the interest earned in previous years. Suppose in the previous case interest is to be compounded at the end of every year. Thus at the end of the 1st year, interest is paid on the 10 m, giving 11m. At the end of the 2nd year, interest will be paid on the entire 11 m outstanding 11m + 10% of 11m = 12.1m 16

17. Compound Interest Finding the value at some future time of amount(s) paid or received using a given interest rate Compounding Finding the current value of amount(s) to be paid or received in the future using a given discount rate. Discounting 17

18. Compounding This may be continued for any number of years Mathematically: an amount today of value, PV, earning interest at r% p.a. for n years will grow to a future value (FV) given by: FV = PV x (1 + r ) n Reworking the previous cases> for, n = 2; FV = 10x(1 +10%) 2 = 12.4 For, n = 3; FV = 10x(1 + 10%) 3 = 13.2. PV is called present value. It is the amount today. FV is called future value. It is the amount in the future. Example: Calculate the amount due at the end of a 10 year period. 18

19. Compounding FV = PV x (1 + r ) n. (1 + r ) n is called the compound factor. It may be read from interest tables. We say that is present value, PV, is compounded at r% for n years to obtain the future value, FV. Other Examples:  How much will GHS100m become after 5 years if the interest rate is 24%?  Which is more GHS10m invested at 12% for 6 years, or 8m invested at 12% for 8 years? 19

20. Compounding More Than Once A Year Suppose your saving account pays you interest twice a year. You have 1 million in the account and the interest rate is 24% p.a. In general, interest rates are quoted per annum. Six months hence, you will receive 120,000 in interest, half the annual amount or 12% In another six months, you will receive another 120,000, and every six months thereafter 12% in interest. We say that interest is compounded every six months or twice a year or semi-annually. To find the FV in such situations, where interest in compounded m times a year, we use the expression FV = PV(1 + r/m) nm 20

21. Discounting Suppose 100m will be received one year from now. We may want to know how much today will be equivalent to the 100. We reason as follows: if I can invest PV and earn r% p.a., then 1 year from now PV(1 + r) should equal 100. Thus, PV(1 + r) = 100. IF r = 12%; then PV = 100/(1.12) = 1/(1.12) x 100 = We say that the 100 in the future has been discounted to obtain its present value. That is, FV is discounted to obtain PV. 1/(1 + r) n is called the discount factor. In general, from FV = PV(1 + r ) n, we deduce PV = FV/(1 + r) n, FV = [(1/(1 + r ) n ] x PV, where 1/(1 + r ) n is called the discount factor. 21

22. Questions  How long will it take for GHC300 to grow to GHC1,500 at the interest rate specified? a. 4% b. 8% c. 16%  Would you rather receive GHC1,000 a year for 10 years or GHC800 a year for 15 years if a. the interest rate is 7 percent? b. the interest rate is 18 percent? c. Why do your answers to (a) and (b) differ?  You will require GHC1,200 in 5 years. If you earn 6 percent interest on your funds, how much will you need to invest today in order to reach your savings goal?  At what rate will an expected value of GHC1000 in 25 years offer, if an investment value of 129 is offered today? 22

23. Patterns of Cash Flow Cash flows that are compounded or discounted may be one of the following patterns: 23 A one-off payment or receipt Single Amount Ordinary Annuity Annuity due Stream of equal cash flows Different amounts paid or received Stream of mixed cash flows

24. Single Amount Future (Compounded) Value Present (Discounted) Value 24

25. Effective Annual Interest Rate A nominal annual interest rate, r% used to compound interest once in a year will not give the same future value when it is used to compound interest several times in a year. An annual rate of interest that compounds interest once in a year to give the same future value as using the nominal annual rate of interest to compound interest several times in a year is the effective annual rate of interest 25

26. Effective Annual Interest Rate The actual rate of interest earned or paid after adjusting the nominal rate for factors such as the number of compounding periods per year. Definition Formula 26

27. Multiple Compounding Specified number of compounding periods Continuous compounding 27

28. Annuities, Sinking Fund and Amortization In some business transactions, instead of making a single payment at the beginning or end of the period, a series of uniform payments may be made at regular intervals of time. If these payments are made into a fund that pays interest, at r% per annum, they will accumulate to a certain capital in the future. The creation of the accumulated capital is the Sinking fund, and the equal payments made are the annuities. Annuities is are fixed amounts paid at regular intervals for a pre- determined length of times. At times, we may also wish to find the present value of these equal payments (Amortization) 28

29. Car Loan Payments Mortgage Payments Retirement Savings Funeral Insurance Educational Insurance Examples of Annuities

30. Annuities Forms of Annuity Ordinary Annuity Payments or receipts occur at the end of each period(In arrears) Annuity Due Payments or receipts occur at the beginning of each period (In advance) 30

31. Ordinary Annuity Future (Compounded) Value Present (Discounted) Value 31

32. Annuity Due Future (Compounded) Value Present (Discounted) Value 32

33. Amortising a Loan Determine the regular equal instalment payments required to discharge fully all liabilities by the end of the term. It helps in determining:  which part of the monthly payment is used to pay interest (interest element) and  the part used to reduce the principal borrowed (capital element). Usefulness:  Determine Interest Expense  Determine Interest Expense – Interest expenses may reduce taxable income of the firm.  Calculate Debt Outstanding  Calculate Debt Outstanding – The quantity of outstanding debt may be used in financing the day-to-day activities of the firm. 33

34. Prepare an amortisation schedule to show repayments of interests and principal: 1. Calculate the payment per period (Use the P.V of an annuity, either ordinary or annuity due). 2.Determine the interest in Period t. (Loan Balance at t – 1) x (i% / m) principal payment 3. Compute principal payment in Period t. (Payment - Interest from Step 2) 4. Determine ending balance in Period t. principal payment (Balance - principal payment from Step 3) 5.Start again at Step 2 and repeat. Steps to Amortizing a Loan

35. $10,000 Julie Miller is borrowing $10,000 at a compound annual interest rate of 12%. Amortize the loan if annual payments are made for 5 years. Step 1:Payment PV 0 PV 0 = R (PVIFA i%,n ) $10,000 $10,000 = R (PVIFA 12%,5 ) $10,000 $10,000 = R (3.605) R$10,000$2,774 R = $10,000 / 3.605 = $2,774 Amortizing a Loan Example

36. End of Year PaymentInterestPrincipalEnding Balance 0———$10,000 1$2,774$1,200$1,5748,426 22,7741,0111,7636,663 32,7748001,9744,689 42,7745632,2112,478 52,7752972,4780 $13,871$3,871$10,000 [Last Payment Slightly Higher Due to Rounding] Amortizing a Loan Example

37. A fund established by a government agency or business organisation for the purpose of reducing debt. Definition Sinking fund was first used in Great Britain in the 18th century to reduce national debt. Historical Context Sinking fund is used by organizations by setting aside money over time to retire indebtedness or to finance the purchase of assets. Modern Context 37 Sinking Fund

38. Perpetuity An ordinary annuity whose payments or receipts continue for ever. Definition Present Value 38

39. Mixed Flows Future Value Present Value 39