1. Financial Management [FIN501] Suman Paul Suman Paul Chowdhury spc@bracu.ac.bd Suman Paul Suman Paul Chowdhury spc@bracu.ac.bd

2. Time Value of Money Besley & Bringham, 14ed Time Value of Money Besley & Bringham, 14ed

3. $10,000 today Obviously, $10,000 today. TIME VALUE TO MONEY You already recognize that there is TIME VALUE TO MONEY!! $10,000 today $10,000 in 5 years Which one would you prefer -- $10,000 today or $10,000 in 5 years?

4. Why TIME? TIME INTEREST TIME allows you the opportunity to postpone consumption and earn INTEREST……explain!! TIME Why is TIME such an important element in your decision?

5. Types of Interest Compound Interest Compound Interest Interest paid (earned) on any previous interest earned, as well as on the principal borrowed (lent). Simple Interest Simple Interest Interest paid (earned) on only the original amount, or principal, borrowed (lent).

6. Simple Interest Formula Formula FormulaSI = P 0 (i)(n) SI:Simple interest P 0 :Deposit today (t=0) i:Interest rate per period n:Number of time periods

7. Simple Interest Example $140 SI = P 0 (i)(n) = $1,000(.07)(2) = $140 Assume that you deposit $1,000 in an account earning 7% simple interest for 2 years. What is the accumulated interest at the end of the 2nd year?

8. FV $1,140 FV = P 0 + SI = $1,000 + $140 = $1,140 Future Value 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. Future Value FV What is the Future Value (FV) of the deposit?

9. The Present Value is simply the $1,000 you originally deposited. That is the value today! Present Value Present Value is the current value of a future amount of money, or a series of payments, evaluated at a given interest rate. Present Value PV What is the Present Value (PV) of the previous problem?

10. Why Compound Interest? Future Value (U.S. Dollars)

11. General Future Value Formula Future Value General Future Value Formula: FV n FV n = P 0 (1+i) n FV 1 FV 1 = P 0 (1+i) 1 FV 2 FV 2 = P 0 (1+i) 2 FV n FVIF or FV n = P 0 (FVIF i,n ) -- See Table A-1

12. Example $10,000 5 years Julie Miller wants to know how large her deposit of $10,000 today will become at a compound annual interest rate of 10% for 5 years. 5 0 1 2 3 4 5 $10,000 FV 5 10%

13. Solution FV 5 FVIF $16,110 Calculation based on Table I: FV 5 = $10,000 (FVIF 10%, 5 ) = $10,000 (1.611) = $16,110 [Due to Rounding] FV n FV 5 $16,105.10 u Calculation based on general formula: FV n = P 0 (1+i) n FV 5 = $10,000 (1+ 0.10) 5 = $16,105.10

14. Present Value Formula Present Value General Present Value Formula: PV 0 FV n PV 0 = FV n / (1+i) n PV 0 FV 1 PV 0 = FV 1 / (1+i) 1 PV 0 FV 2 PV 0 = FV 2 / (1+i) 2 PV 0 FV n PVIF or PV 0 = FV n (PVIF i,n ) See Table A-3 for PVIF -- See Table A-3 for PVIF i,n

15. Example $10,000 5 years Julie Miller wants to know how large of a deposit to make so that the money will grow to $10,000 in 5 years at a discount rate of 10%. 5 0 1 2 3 4 5 $10,000 PV 0 10%

16. Solution PV 0 FV n PV 0 $10,000 $6,209.21 Calculation based on general formula: PV 0 = FV n / (1+i) n PV 0 = $10,000 / (1+ 0.10) 5 = $6,209.21 PV 0 $10,000PVIF $10,000 $6,210.00 Calculation based on Table I: PV 0 = $10,000 (PVIF 10%, 5 ) = $10,000 (.621) = $6,210.00 [Due to Rounding]

17. Self test questions Investment decision If your opportunity cost is 20%, which is better, receipt of $5,000 today or $10,368 on four years? Why? Solving for ‘r’ The current value of a security is $78.35, which is expected to be valued $100 after 5 years. What is the rate of return? Solving for ‘n’ A security will provide a return of 10% per annum. It will cost @68.30, and you are going to receive $100 at maturity. How long the security will take to mature?

18. Terminologies (contd..) Compounding: the process of determining the value of a cash flow or series of cash flows at some time in the future when compound interest is applied. Discounting: The process of determining the present value of cash flow or a series of cash flows received (paid) in the future, the reverse of compounding. Compounded interest: Interest earned on interest that is reinvested. Opportunity cost rate: The rate of return on the best alternative investment of equal risk.

19. Annuities Annuity Ordinary Annuity: Payments or receipts occur at the end of each period. Due Annuity Due: Payments or receipts occur at the beginning of each period. u An Annuity u An Annuity represents a series of equal payments (or receipts) occurring over a specified number of equidistant periods.

20. Examples of Annuities Student Loan Payments Car Loan Payments Insurance Premiums Mortgage Payments Retirement Savings

21. Parts of an Annuity 0 1 2 3 $100 $100 $100 (Ordinary Annuity) End End of Period 1 End End of Period 2 Today Equal Equal Cash Flows Each 1 Period Apart End End of Period 3

22. Parts of an Annuity 0 1 2 3 $100 $100 $100 (Annuity Due) Beginning Beginning of Period 1 Beginning Beginning of Period 2 Today Equal Equal Cash Flows Each 1 Period Apart Beginning Beginning of Period 3

23. Overview of an Ordinary Annuity -- FVA FVA n FVA n = R(1+i) n-1 + R(1+i) n-2 +... + R(1+i) 1 + R(1+i) 0 R R R n 0 1 2 n n+1 FVA n R = Periodic Cash Flow Cash flows occur at the end of the period i%...

24. Example of an Ordinary Annuity -- FVA FVA 3 FVA 3 = $1,000(1.07) 2 + $1,000(1.07) 1 + $1,000(1.07) 0 $3,215 = $1,145 + $1,070 + $1,000 = $3,215 $1,000 $1,000 $1,000 3 0 1 2 3 4 $3,215 = FVA 3 7% $1,070 $1,145 Cash flows occur at the end of the period FVA n FVA n = PMT (FVIFA i%,n ) [See Table A-2]

25. Overview View of an Annuity Due -- FVAD FVAD n FVAD n = R(1+i) n + R(1+i) n-1 +... + R(1+i) 2 + R(1+i) 1 R R R R R n-1 0 1 2 3 n-1 n FVAD n i%... Cash flows occur at the beginning of the period

26. Example of an Annuity Due -- FVAD FVAD 3 FVAD 3 = $1,000(1.07) 3 + $1,000(1.07) 2 + $1,000(1.07) 1 $3,440 = $1,225 + $1,145 + $1,070 = $3,440 $1,000 $1,000 $1,000 $1,070 3 0 1 2 3 4 $3,440 = FVAD 3 7% $1,225 $1,145 Cash flows occur at the beginning of the period FVAD n FVAD n = PMT (FVIFA i%,n )(1+i) [See Table A-2]

27. Overview of an Ordinary Annuity -- PVA PVA n PVA n = R/(1+i) 1 + R/(1+i) 2 +... + R/(1+i) n R R R n 0 1 2 n n+1 PVA n R = Periodic Cash Flow i%... Cash flows occur at the end of the period

28. Example of an Ordinary Annuity -- PVA PVA 3 PVA 3 = $1,000/(1.07) 1 + $1,000/(1.07) 2 + $1,000/(1.07) 3 $2,624.32 = $934.58 + $873.44 + $816.30 = $2,624.32 $1,000 $1,000 $1,000 3 0 1 2 3 4 $2,624.32 = PVA 3 7% $934.58 $873.44 $816.30 Cash flows occur at the end of the period PVA n PVA n = PMT (PVIFA i%,n ) [See Table A-4]

29. Overview of an Annuity Due -- PVAD PVAD n PVAD n = R/(1+i) 0 + R/(1+i) 1 +... + R/(1+i) n-1 R R R R n-1 0 1 2 n-1 n PVAD n R: Periodic Cash Flow i%... Cash flows occur at the beginning of the period

30. Example of an Annuity Due -- PVAD PVAD n $2,808.02 PVAD n = $1,000/(1.07) 0 + $1,000/(1.07) 1 + $1,000/(1.07) 2 = $2,808.02 $1,000.00 $1,000 $1,000 3 0 1 2 3 4 $2,808.02 PVAD n $2,808.02 = PVAD n 7% $ 934.58 $ 873.44 Cash flows occur at the beginning of the period PVAD n PVAD n = PMT (PVIFA i%,n ) (1+i) [See Table A-4]

31. Steps to Solve Time Value of Money Problems 1. Read problem thoroughly 2. Create a time line 3. Put cash flows and arrows on time line 4. Determine if it is a PV or FV problem 5. Determine if solution involves a single cash flow, annuity stream(s), or mixed flow 6. Solve the problem

32. Mixed Flows Example Present Value 10% Julie Miller will receive the set of cash flows below. What is the Present Value at a discount rate of 10%. 5 0 1 2 3 4 5 $600 $600 $400 $400 $100 $600 $600 $400 $400 $100 PV 0 10%

33. Solution 5 0 1 2 3 4 5 $600 $600 $400 $400 $100 $600 $600 $400 $400 $100 10% $545.45$495.87$300.53$273.21 $ 62.09 $1677.15 = PV 0 of the Mixed Flow

34. Frequency of Compounding General Formula: PV 0 FV n = PV 0 (1 + [i/m]) mn n: Number of Years m: Compounding Periods per Yeari: Annual Interest RateFV n,m : FV at the end of Year n PV 0 PV 0 : PV of the Cash Flow today

35. Impact of Frequency $1,000 Julie Miller has $1,000 to invest for 2 Years at an annual interest rate of 12%. 1,000 1,254.40 Annual FV 2 = 1,000(1+ [.12/1]) (1)(2) = 1,254.40 1,000 1,262.48 Semi FV 2 = 1,000(1+ [.12/2]) (2)(2) = 1,262.48

36. Impact of Frequency 1,000 1,266.77 Qrtly FV 2 = 1,000(1+ [.12/4]) (4)(2) = 1,266.77 1,000 1,269.73 Monthly FV 2 = 1,000(1+ [.12/12]) (12)(2) = 1,269.73 1,000 1,271.20 Daily FV 2 = 1,000(1+[.12/365]) (365)(2) = 1,271.20

37. Effective Annual Interest Rate The actual rate of interest earned (paid) after adjusting the nominal rate for factors such as the number of compounding periods per year. (1 + [ i / m ] ) m - 1

38. BWs Effective Annual Interest Rate EAR Basket Wonders (BW) has a $1,000 CD at the bank. The interest rate is 6% compounded quarterly for 1 year. What is the Effective Annual Interest Rate (EAR)? EAR 6.14%! EAR= ( 1 + 6% / 4 ) 4 - 1 = 1.0614 - 1 =.0614 or 6.14%!

39. Steps to Amortizing a Loan 1.Calculate the payment per period. 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) principal payment 4.Determine ending balance in Period t. (Balance - principal payment from Step 3) 5.Start again at Step 2 and repeat.

40. Amortizing a Loan Example $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 = PMT (PVIFA i%,n ) $10,000 $10,000 = PMT (PVIFA 12%,5 ) $10,000 $10,000 = PMT (3.605) PMT$10,000$2,774 PMT = $10,000 / 3.605 = $2,774

41. Amortizing a Loan Example [Last Payment Slightly Higher Due to Rounding]