Lecture 05 - Time Value of Money 2g.pptx

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1. FIN 1302-Basic Finance 03 Credit Points Level I Semester II – 2023 26/09/2016 1 Ms.Udeshka Pathirana Lecturer (Probationary) Department of Finance Faculty of Management and Finance University of Colombo

2. Time Value of Money (Part 02) 2 2020

3. Learning Outcomes At the end of this lesson you will be able to; Differentiate between the usage Interest factor formulas and Annuity formulas as well as between Future value and Present value formulas clearly Relate Time Value of Money concepts in to practice through solving real case study questions Reflection on improving own strengths and Weaknesses in understanding time value of money concepts 26/09/2016 3

4. Annuity An Annuity is a series of equal/ fixed payments or receipts occurring over a specified number of periods. Example: A constant sum of Rs.10/= is deposited in a savings account for 4 years at a 6% rate of interest. Equal installment loans by house financing companies Agreed sum of house rental payments Lease payments 8/22/2017 4

5. Annuity Types There are 4 types; Ordinary Annuity - payments or receipts occur at the end of each period. Annuity Due - payments or receipts occur at the beginning of the each period. Deferred Annuity - An annuity which the payment or receipts are deferred for a specified future period. Perpetuity : An Annuity that has infinite life time. 5 8/22/2017

6. 01. Ordinary Annuity FV of an Ordinary Annuity 6 8/22/2017

7. Future value of an Ordinary Annuity…… Example 01 A constant sum of Rs. 1 is deposited in a savings account at the end of each year for 4 years at 10% compounding interest. - Calculate the future value at the end of the 4th year . 7 8/22/2017

8. Future value of an Ordinary Annuity…… 0 1 2 3 4 [ This implies Rs 1/= deposited at the end of first year will grow for three years, Rs 1/= at the end of second year for two years, Rs 1/= at the end of third year for one year and Rs 1/= and at the end of fourth year will not yield any interest. ] 1.0000 1.1000 1.2100 1.3310 1(1.1 )^1 1(1.1 )^2 1(1.1)^3 4.641 8 8/22/2017

9. Formula for Future value of an Ordinary Annuity…… Developing the formula: FV A = A (1+i) ^ n-1 + A (1+i) ^ n-2 + A(1+i) ^ n-3 + A (1+i)^ n-4 FV A = A [(1+i) ^ 3 + (1+i) ^ 2 + (1+i) ^ 1 + 1] Extend the equation for n periods; 8/22/2017 9

10. Formula for Future value of an Ordinary Annuity…… Where, A = Constant Periodic payment / Annuity i = Interest rate per period n = Number of periods As per example 1: 10 = 4.641 8/22/2017

11. Interest Tables for Future value of an Ordinary Annuity …… FVIFA : Future value interest factor of an annuity at i % for n periods. 4.641 = 1 * 4.6410 ; (P= Rs.1/=, i= 10%, n= 4) Example 02 Suppose that you have agreed to deposit Rs . 100 at the end of the next three years at 9% interest rate. Calculate the future value compounded using the formula as well as annuity table? FV A = A(FVIFA i,n ) 11 8/22/2017

12. Future value of an Ordinary Annuity…… Example 03 Assume you are going to open up a savings account in a bank. You plan to make ten annual deposits of Rs.5,000/= at the end of each year. The bank will pay you 16% interest. Calculate how much you will have in the account at the end of the 10th year? Future Value of Annuity; 5000(1+0.16)^9 + 5000(1+0.16)^8 + 5000(1+0.16)^7 + 5000(1+0.16)^6 + 5000(1+0.16)^5 + 5000(1+0.16)^4 + 5000(1+0.16)^3 + 5000(1+0.16)^2 + 5000(1+0.16)1 + 5000 = Rs. 106,607.35/= 8/22/2017 12

13. Future value of an Ordinary Annuity…… Formula: FV A = 5000 * [ (1+0.16)^10]-1 0.16 = Rs. 106,607.35/= 8/22/2017 13

14. Future value of an Ordinary Annuity…… Year Deposit ( Rs ) No: of years until maturity Future value factor (FVIF) Future Value 1 5,000 9 FVIF 9,16% 3.8030 (21.321-17.519) 19,014.81 2 5,000 8 FVIF 8,16% 3.2784 16,392.07 3 5,000 7 FVIF 7,16% 2.8262 14,131.10 4 5,000 6 FVIF 6,16% 2.4364 12,181.98 5 5,000 5 FVIF 5,16% 2.1003 10,501.71 6 5,000 4 FVIF 4,16% 1.8106 9,053.20 7 5,000 3 FVIF 3,16% 1.5609 7,804.48 8 5,000 2 FVIF 2,16% 1.3456 6,728.00 9 5,000 1 FVIF 1,16% 1.1600 5,800.00 10 5,000 FVIF 0,16% 1.0000 5,000.00 21.3215 106,607.35 14 8/22/2017

15. Ordinary Annuity PV of an Ordinary Annuity 8/22/2017 15

16. Present Value of an Ordinary Annuity This concentrates on how much it is worth today of a sum of money to be received or paid constantly at the end of certain future time intervals. It requires to find out the present value of each amount in every time interval and to aggregate all the present values to get total PV of the annuity. Example 04 Jaliya has an opportunity to receive Rs. 1 at the end of each year for 4 years under a 10% compounding rate of return. Find out the PV of this annuity? 16 8/22/2017

17. Present Value of an Ordinary Annuity… Y2 Rs. 1 Y3 Rs. 1 Y4 Rs. 1 Y1 Rs. 1 Y0 0.909 0.826 0.751 0.683 3.169 17 8/22/2017 1(1.1 )^-1 1(1.1 )^-2 1(1.1 )^-3 1(1.1 )^-4

18. Formula for Present Value of an Ordinary Annuity… PV A = 1* [1- (1.1) ^ -4] 0.1 = Rs. 3.1699/=   18 8/22/2017

19. Interest Tables for Present Value of An Ordinary Annuity… PVIFA : Present value interest factor of an annuity at i % for n periods. Using tables; PV A = A * (PVIFA n,i ) = 1*PVIFA 4,10% = 1 * 3.1699 = Rs. 3.1699 8/22/2017 19

20. Present Value of an Ordinary Annuity… Example 05 You have planned to purchase a house of Rs.500,000 on a mortgage. The mortgage should be settled by 10 equal installments which should be paid at the end of each year. The interest rate relevant to the mortgage is 18%. Calculate the installment value. 20 8/22/2017

21. 02. Annuity Due FV of an Annuity Due 8/22/2017 21

22. Annuity Due… Ex: When you buy a fridge on an installment basis, the dealer requires you to make an immediate payment and to make other subsequent installments periodically. 8/22/2017 22

23. Future Value of an Annuity Due… Example 06 Suppose you deposit a constant sum of Rs. 1/= in a savings account at the beginning of each year for 4 years to earn 10% interest. How much will be at the end of 4 years? FV A = 1*(1.1^4)+ 1*(1.1^3)+1*(1.1^2)+1*(1.1^1) = Rs 5.1051 *** Future Value of an annuity due is more than of an annuity( ordinary Annuity). Because it earns extra interest for one more year. Hence ; FV of an Annuity Due= FV of an Ordinary Annuity * (1+i) 8/22/2017 23

24. Formula for Future Value of an Annuity Due… Where, A = Constant Periodic payment / Annuity i = Interest rate per period n = Number of periods FV A = 1* [(1.1)^4]-1 *(1.1) 0.1 = Rs 5.1051/= * (1+i) 8/22/2017 24

25. Interest tables for Future Value of an Annuity Due… FV A = A * FVIFA n,i * (1+i) = 1 *[4.6410 * (1.1)] = Rs 5.1051/= (n= 4, i= 10%) 8/22/2017 25 FV A = A * FVIFA * (1+i)

26. Future Value of an Annuity Due… Example 07 Assume that you are going to open a savings account in a bank. On agreement, you have to make ten payments of Rs.5,000/= at the beginning of each year. The bank pays you 16% interest. You can withdraw the deposit at the end of the 10th year. Calculate the amount you can withdraw at the end of year 10? 8/22/2017 26

27. Future Value of an Annuity Due… 27 Year Deposit No: of years until maturity Future Value Factor Future Value 1 5,000 10 F VIF 16%,10 4.4114 (25.733-21.321) 22,057.18 2 5,000 9 FVIF 16%,9 3.8030 19,014.81 3 5,000 8 FVIF 16%,8 3.2784 16,392.07 4 5,000 7 FVIF 16%,7 2.8262 14,131.10 5 5,000 6 FVIF 16%,6 2.4364 12,181.98 6 5,000 5 FVIF 16%,5 2.1003 10,501.71 7 5,000 4 FVIF 16%,4 1.8106 9,053.20 8 5,000 3 FVIF 16%,3 1.5609 7,804.48 9 5,000 2 FVIF 16%,2 1.3456 6,728.00 10 5,000 1 FVIF 16%,1 1.1600 5,800.00 24.7329 123,664.52 8/22/2017

28. Annuity Due… PV of an Annuity Due 8/22/2017 28

29. Present Value of an Annuity Due… Example 08 Let us consider a 4 year annuity of Rs. 1/= each year , the interest rate being 10%. What is the present value of this annuity if each payment is made at the beginning of the year? 29 8/22/2017

30. Formula for Present Value of an Annuity Due…   8/22/2017 30

31. Interest Tables for Present Value of an Annuity Due… 8/22/2017 31 PV A = A * PVIFA * (1+i)

32. Present value of Annuity Due… Example 09 An investment made today offers you Rs. 5,000 per year at the beginning of each year for the next five years on 15% interest. How much this series of cash flow worth in today’s term? 32 8/22/2017

33. 03. Deferred Annuity A deferred annuity is a type of annuity contract that delays payments of income, installments or a lump sum until the investor elects to receive them or an annuity in which payback does not start until a specified time in the future, such as after a certain number of years from the annuity contract date, or at a certain age of the annuitant or the beneficiary. 8/22/2017 33

34. Deferred Annuity… Example 10 Maumi will receive the following payments at the end of the next three years; Rs.2000, Rs.3500, Rs.4500. Then from the end of the 4th through the end of the 10th year, he will receive an annuity of Rs.5000 per year. At a discount rate of 9%, what is the present value of all future benefits? 8/22/2017 34

35. Deferred Annuity… Step 01- Find the PV of first three payments. PV of Lump sum Step 02- Find the PV of the deferred annuity. The value of the annuity at the beginning of the 4th year is; 8/22/2017 35

36. Deferred Annuity… Step 03-The value found in step 02, discount back for 3 years to get the PV of the deferred annuity. Step 04- Finally find the total present value of all future payments. 8/22/2017 36

37. 04. Perpetuity Perpetuity is an annuity that occurs indefinitely. Ex: Irredeemable Preference shares (Preference shares without a maturity); the company is expected to pay preference dividend perpetually. Example 11 : Eagle Ltd issued preferred stock with an annual dividend of Rs. 5.50 forever. Thilan requires a 15% rate of return on this type of investment. How much would the Thilan be willing to pay for this security now? P = A / i = 5.5 / 0.15 = Rs.36.67 P = A / i 37 8/22/2017

38. Sinking Fund A fund which is created out of fixed payments each period to accumulate to a future sum after a specified period. Ex: Companies generally make sinking funds to retire bonds/debentures or loan on maturity. Rather than repay a payment in installments (amortization method), a borrower may choose to repay it by means of one lump – sum payment at the end of a specified period of time. In many such cases the borrower will accumulate a fund which will be sufficient to exactly repay the loan at the end of the specified period of time. 38 8/22/2017

39. Sinking Fund… The sinking fund factor is useful in determining the annual amount to be put in a fund to repay bonds or debentures at the end of a specific period. The factor used to calculate the annuity for a given future sum is called the sinking fund factor. Formula; A = FV * i (1+ i )^n-1 39 8/22/2017

40. Sinking Fund… The general approach to SF problems is to answer two questions: What is the interest paid to the lender? What is the SF deposit? Example 11 Suppose we want to accumulate Rs.21,873/= at the end of 4 years from now. How much should we deposit each year at an interest rate of 6% so that it grows to Rs.21,873/= at the end of fourth year? 8/22/2017 40

41. Capital Recovery If we make an investment today for a given period of time, at a specified rate of interest, we may like to know the annual income. Capital recovery is the annuity of an investment made today, for a specified period of time, at a given rate of interest. It utilizes the capital recovery factor in computations. Formula; A = P * 1 PVAF n,i 41 8/22/2017

42. Capital Recovery Example 13 Suppose you plan to invest Rs 10,000/= today for a period of 4 years. If the interest rate is 10%, how much income per year should you receive to recover your investment? 8/22/2017 42

43. Loan Amortization… Loan amortization is the process of scheduling out a fixed-rate loan into equal payments. A portion of each installment covers interest and the remaining portion goes toward the loan principal. 8/22/2017 43

44. Loan Amortization… Example 14 Asanka has borrowed a 3 year loan of Rs. 1 Mn at a 9% from Bank of Ceylon. If Bank requires three equal end of year payments; (I) Calculate the annual installment Asanka is supposed to pay? (II) Prepare the loan amortization schedule for Asanka . Answer I) Annual Installment Rs.1,000,000 = A * (PVAF 3, 9%) Rs . 1,000,000 = A * 2.5313 A = Rs 395,100/= (Rounded off) 395,054.7573 8/22/2017 44

45. Loan Amortization… II) Loan amortization schedule End of Year Payment Interest Principal Repayment Outstanding Balance 1000,000 1 395,100 90,000 305,100 694,900 2 395,100 62,541 332,559 362,341 3 395,100 32,610 362,490 Nil 45 8/22/2017

46. Additional Questions 1. Tao invested Rs . 5000 in a fund at the beginning of every three months for five years. The fund was earning an interest rate of 5% compounded monthly. a. What was the total amount invested? b. What was the accumulated value of the investment? c. What was the interest earned over the period? 2. Alexandria inherited money that was invested in an account which provided her Rs . 4500 at the beginning of every month for 30 years. If the interest rate on the savings account was 4% compounded semi-annually. What was the amount of the inheritance? Round your answer to the nearest hundred dollars. 8/22/2017 46

47. Additional Questions 3. Mr.Emerson wishes to purchase an annuity contract that will pay him, Rs.7000 a year for the rest of his life. The Sri Lanka Insurance Corporation figures that his life expectancy is 20 years. The company imputes a compound annual interest rate of 6% in its annuity contracts. How much will Emerson has to pay for the annuity? How much would he have to pay if the interest were 8%? 4. Your late Uncle will entitles you to receive Rs.1000 at the end of every other year for the next two decades. The first cash flow is two years from now. AT a 10% compound annual interest rate, what is the present value of this unusual cash-flow pattern? 47 8/22/2017

48. Additional Questions 5. You borrow Rs . 10,000 at 14% compound annual interest for four years. The loan is repayable in four equal annual installments payable at the end of each year. a. What is the annual payment that will completely amortize the loan over four years? (to the nearest rupee) b. Of each equal payment, what is the amount of interest? The amount of loan principal? (Hint: In early years, the payment is composed largely of interest, whereas at the end it is mainly principal) 6. Mr. Akash recently obtained a 10-year, Rs . 50,000 loan. The loan carries an 8% compound annual interest rate and calls for annual installement payments of Rs.7451.47 at the end of each of the next 10 years. How much (in Rs .) of the first year’s payment is principal? How much total interest will be paid over the life of the loan? Thank You 8/22/2017 48

49. Any Questions?? 26/09/2016 49 “Some people want it to happen, some wish it would happen, others make it happen.”   -Michael Jordan