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# Time value of money

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### Time value of money

1. 1. Time Value of Money
2. 2. THE INTEREST RATE Which would you prefer – Rs10,000 today or Rs10,000 in 5 years? Obviously, Rs10,000 today. You already recognize that there is TIME VALUE TO MONEY!!
3. 3. WHY TIME? Why is TIME such an important element in your decision? TIME allows you the opportunity to postpone consumption and earn INTEREST A rupee today represents a greater real purchasing power than a rupee a year hence Receiving a rupee a year hence is uncertain so risk is involved
4. 4. TIME VALUE ADJUSTMENTTwo 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.
5. 5. TYPES OF INTEREST  Simple Interest Interest paid (earned) on only the original amount, or principal borrowed (lent).  Compound Interest Interest paid (earned) on any previous interest earned, as well as on the principal borrowed (lent).
6. 6. SIMPLE INTEREST FORMULA Formula SI = P0(i)(n) SI:Simple Interest P0: Deposit today (t=0) i: Interest Rate per Period n: Number of Time Periods
7. 7. SIMPLE INTEREST EXAMPLE  Assume that you deposit Rs1,000 in an account earning 7% simple interest for 2 years. What is the accumulated interest at the end of the 2nd year?  SI = P0(i)(n) = Rs1,000(.07)(2) = Rs140
8. 8. SIMPLE INTEREST (FV)  What is the Future Value (FV) of the deposit? FV = P0 + SI = Rs1,000 + Rs140 = Rs 1,140  Future Value is the value at some future time of a present amount of money, or a series of payments, evaluated at a given interest rate.
9. 9. SIMPLE INTEREST (PV)  What is the Present Value (PV) of the previous problem? The Present Value is simply the Rs 1,000 you originally deposited. That is the value today!  Present Value is the current value of a future amount of money, or a series of payments, evaluated at a given interest rate.
10. 10. FUTURE VALUE SINGLE DEPOSIT (GRAPHIC) Assume that you deposit Rs 1,000 at a compound interest rate of 7% for 2 years. 0 1 2 7%Rs 1,000 FV2
11. 11. Future Value Single Deposit (Formula)FV1 = P0 (1+i)1 = Rs 1,000 (1.07) = Rs 1,070FV2 = FV1 (1+i)1 = P0 (1+i)(1+i) = Rs1,000(1.07)(1.07) = P0 (1+i)2 = Rs1,000(1.07)2 = Rs1,144.90 You earned an EXTRA Rs 4.90 in Year 2 with compound over simple interest.
12. 12. GENERAL FUTURE VALUE FORMULA FV1 = P0(1+i)1 FV2 = P0(1+i)2 etc.General Future Value Formula: FVn = P0 (1+i)nor FVn = P0 (FVIFi,n)
13. 13. PROBLEM Reena wants to know how large her deposit of Rs10,000 today will become at a compound annualinterest rate of 10% for 5 years. 0 1 2 3 4 5 10% Rs10,000 FV5
14. 14. SOLUTION Calculation based on general formula: FVn = P0 (1+i)n FV5 = Rs10,000 (1+ 0.10) 5 = Rs 16,105.10
15. 15. DOUBLE YOUR MONEY!!! Quick! How long does it take to double Rs 5,000 at a compound rate of 12% per year (approx.)? We will use the ―Rule-of-72‖.
16. 16.  Doubling Period = 72 / Interest Rate 6 years For accuracy use the “Rule-of-69”.Doubling Period =0.35 +(69 / Interest Rate) 6.1 years
17. 17. PRESENT VALUE SINGLE DEPOSIT (GRAPHIC)Assume that you need Rs 1,000 in 2 years.Let’s examine the process to determine howmuch you need to deposit today at a discountrate of 7% compounded annually. 0 1 2 7% Rs 1,000 PV0 PV1
18. 18. PRESENT VALUE SINGLE DEPOSIT (FORMULA)PV0 = FV2 / (1+i)2 = Rs 1,000 / (1.07)2 = FV2/ (1+i)2 = Rs 873.44 0 1 2 7% Rs 1,000 PV0
19. 19. GENERAL PRESENT VALUE FORMULA PV0 = FV1 / (1+i)1 PV0 = FV2 / (1+i)2 etc.General Present Value Formula: PV0 = FVn / (1+i)nor PV0 = FVn (PVIFi,n)
20. 20. PROBLEMReena wants to know how large of a depositto make so that the money will grow to Rs10,000 in 5 years at a discount rate of 10%. 0 1 2 3 4 5 10% Rs 10,000 PV0
21. 21. PROBLEM SOLUTION Calculation based on general formula: PV0 = FVn / (1+i)n PV0 = Rs 10,000 / (1+ 0.10)5 = Rs 6,209.21
22. 22. TYPES OF ANNUITIES An Annuity represents a series of equal payments (or receipts) occurring over a specified number of equidistant periods. Ordinary Annuity: Payments or receipts occur at the end of each period. Annuity Due: Payments or receipts occur at the beginning of each period.
23. 23. EXAMPLES OF ANNUITIES  Student Loan Payments  Car Loan Payments  Insurance Premiums  Retirement Savings
24. 24. PARTS OF AN ANNUITY(Ordinary Annuity)End of End of End ofPeriod 1 Period 2 Period 3 0 1 2 3 Rs 100 Rs 100 Rs 100 Today Equal Cash Flows Each 1 Period Apart
25. 25. PARTS OF AN ANNUITY (Annuity Due) Beginning of Beginning of Beginning of Period 1 Period 2 Period 3 0 1 2 3 Rs 100 Rs 100 Rs 100 Today Equal Cash Flows Each 1 Period Apart
26. 26. FUTURE VALUE OF ORDINARY ANNUITY -- FVA Cash flows occur at the end of the period 0 1 2 n n+1 i% . . . A A AA = PeriodicCash Flow FVAnFVAn = A(1+i)n-1 + A(1+i)n-2 + ... + A(1+i)1 + A(1+i)0
27. 27. EXAMPLE OF AN ORDINARY ANNUITY -- FVA Cash flows occur at the end of the period0 1 2 3 4 7% Rs1,000 Rs1,000 Rs1,000
28. 28. EXAMPLE OF AN ORDINARY ANNUITY -- FVA Cash flows occur at the end of the period0 1 2 3 4 7% Rs1,000 Rs1,000 Rs1,000 Rs1,070 Rs1,145 FVA3 = 1,000(1.07)2 +1,000(1.07)1 + 1,000(1.07)0 Rs3,215 = FVA3 = 1,145 + 1,070 + 1,000 = Rs 3,215
29. 29. General Formula for CalculatingFuture Value of an Ordinary Annuity n 1 n 2FVAn A(1 i ) A(1 i ) ... A n (1 i) 1 A i
30. 30. FUTURE VALUE OF ANNUITY DUE - - FVAD Cash flows occur at the beginning of the period0 1 2 3 n-1 n . . . i%R R R R R FVADn = R(1+i)n + R(1+i)n-1 + FVADn ... + R(1+i)2 + R(1+i)1 = FVAn (1+i)
31. 31. EXAMPLE OF AN ANNUITY DUE -- FVAD Cash flows occur at the beginning of the period 0 1 2 3 4 7%1,000 1,000 1,000 1,070 Rs1,145 Rs1,225 FVAD3 = 1,000(1.07)3 + Rs 3,440 = 1,000(1.07)2 + 1,000(1.07)1 FVAD3 = 1,225 + 1,145 + 1,070 = Rs 3,440
32. 32. PRESENT VALUE OF ORDINARY ANNUITY -- PVA Cash flows occur at the end of the period 0 1 2 n n+1 i% . . . R R R R = Periodic Cash FlowPVAn PVAn = R/(1+i)1 + R/(1+i)2 + ... + R/(1+i)n
33. 33. EXAMPLE OF AN ORDINARY ANNUITY -- PVA Cash flows occur at the end of the period0 1 2 3 4 7% Rs1,000 Rs1,000 Rs1,000
34. 34. EXAMPLE OF AN ORDINARY ANNUITY -- PVA Cash flows occur at the end of the period 0 1 2 3 4 7% Rs1,000 Rs1,000 Rs1,000 934.58 873.44 816.30Rs 2,624.32 = PVA3 PVA3 = 1,000/(1.07)1 + 1,000/(1.07)2 + 1,000/(1.07)3 = 934.58 + 873.44 + 816.30 = 2,624.32
35. 35. General Formula for CalculatingPresent Value of an OrdinaryAnnuity A A APVAn 2 ... n (1 i ) (1 i ) (1 i ) n (1 i ) 1 A n i (1 i )
36. 36. PRESENT VALUE OF ANNUITY DUE -- PVAD Cash flows occur at the beginning of the period 0 1 2 n-1 n i% . . . R R R R R: PeriodicPVADn Cash FlowPVADn = R/(1+i)0 + R/(1+i)1 + ... + R/(1+i)n-1 = PVAn (1+i)
37. 37. EXAMPLE OF AN ANNUITY DUE -- PVAD Cash flows occur at the beginning of the period 0 1 2 3 4 7%1,000.00 1,000 1,000934.58873.442,808.02 = PVADn PVADn = 1,000/(1.07)0 + 1,000/(1.07)1 + 1,000/(1.07)2 = Rs 2,808.02
38. 38. MIXED FLOWS EXAMPLEReena will receive the set of cash flowsbelow. What is the Present Value at adiscount rate of 10%? 0 1 2 3 4 5 10% 600 600 400 400 100 PV0
39. 39. SOLUTION 0 1 2 3 4 5 10% 600 600 400 400 100545.45495.87300.53273.2162.09Rs 1677.15 = PV0 of the Mixed Flow
40. 40. SHORTER COMPOUNDING ANDDISCOUNTING PERIODS General Formula: FVn = PV0(1 + [i/m])mn Or PVn = FV0 /(1+[i/m] )mn n: Number of Years m: Compounding Periods per Year i: Annual Interest Rate FVn,m: FV at the end of Year n PV0: PV of the Cash Flow today
41. 41. EXAMPLE Reena has Rs1,000 to invest for 1 year at an annual interest rate of 12%.Annual FV = 1,000(1+ [.12/1])(1)(1) = 1,120Semi FV = 1,000(1+ [.12/2])(2)(1) = 1,123.6
42. 42. EFFECTIVE VS. NOMINAL RATE OF INTERESTRs. 1000 Rs.1123.6So,Rs. 1000 grows @ 12.36% annuallyEffective Rate of Interest m r = 1 + i/m -112% is the nominal rate but 12.36% is the effective rate.
43. 43. PROBLEM Basket Wonders (BW) has a Rs1,000 CD at the bank. The interest rate is 6% compounded quarterly for 1 year. What is the Effective Annual Interest Rate (EAR)?EAR = ( 1 + 6% / 4 )4 - 1 = 1.0614 - 1 = = 0.0614 or 6.14%!
44. 44. PERPETUITY A perpetuity is an annuity with an infinite number of cash flows. The present value of cash flows occurring in the distant future is very close to zero.  At 10% interest, the PV of Rs 100 cash flow occurring 50 years from today is Rs 0.85!
45. 45. PRESENT VALUE OF A PERPETUITY A A APVAn 2 ... n (1 i ) (1 i ) (1 i ) When n= PVperpetuity = [A/(1+i)] [1-1/(1+i)] = A(1/i) = A/i
46. 46. PRESENT VALUE OF A PERPETUITY What is the present value of a perpetuity of Rs270 per year if the interest rate is 12% per year? PV A Rs270 Rs 2250 perpetuity i 0.12
47. 47. STEPS TO AMORTIZING A LOAN1. Calculate the payment per period.2. Determine the interest in Period t. Loan balance at (t-1) x (i%)3. Compute principal payment in Period t. (Payment - interest from Step 2)4. Determine ending balance in Period t. (Balance - principal payment from Step 3)5. Start again at Step 2 and repeat.
48. 48. AMORTIZING A LOAN EXAMPLE Reena is borrowing Rs10,000 at a compoundannual interest rate of 12%. Amortize the loan if annual payments are made for 5 years.Step 1: Payment PV0 = A(PVIFA i%,n) Rs10,000 = A(PVIFA 12%,5) Rs10,000 = A(3.605) A = Rs10,000 / 3.605 = Rs2,774
49. 49. AMORTIZING A LOAN EXAMPLEEnd of Payment Interest Principal Ending Year Balance 0 1 2 3 4 5
50. 50. AMORTIZING A LOAN EXAMPLEEnd of Payment Interest Principal Ending Year Balance 0 --- --- --- Rs10,000 1 Rs2,774 Rs1,200 Rs1,574 8,426 2 3 4 5 [Last Payment Slightly Higher Due to Rounding]
51. 51. AMORTIZING A LOAN EXAMPLEEnd of Payment Interest Principal Ending Year Balance 0 --- --- --- Rs10,000 1 Rs2,774 Rs1,200 Rs1,574 8,426 2 2,774 1,011 1,763 6,663 3 2,774 800 1,974 4,689 4 2,774 563 2,211 2,478 5 2,775 297 2,478 0 Rs13,871 Rs3,871 Rs10,000 [Last Payment Slightly Higher Due to Rounding]
52. 52. USEFULNESS OF AMORTIZATION1. Determine Interest Expense -- Interest expenses may reduce taxable income of the firm.2. Calculate Debt Outstanding -- The quantity of outstanding debt may be used in financing the day-to - day activities of the firm.
53. 53. EXERCISE Ashish recently obtained a Rs.50,000 loan. The loan carries an 8% annual interest. Amortize the loan if annual payments are made for 5 years.
54. 54. SOLUTION 50000 5 0.08 12523TIME PAYMENT INTERESTPRINCIPAL AMOUNT OUTSTANDING 0 50000 1 12523 4000 8523 41477 2 12523 3318 9205 32272 3 12523 2582 9941 22331 4 12523 1786 10737 11594 5 12522 928 11594 0
55. 55. EXERCISE  Compute the present value of the following future cash inflows, assuming a required rate of 10%: Rs. 100 a year for years 1 through 3, and Rs. 200 a year from years 6 through 15. ANS: 1011.75
56. 56. SOLUTION Cash flows occur at the end of the period 0 1 2 3 6 7 15 . . . . . . i% 100 100 100 200 200 200 th Till 5 year 248.70 1228.9 763.051011.75