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### Lecture 06

1. 1. Lesson 6 Chapter 3 The time value of money Unit 1 Core concepts in financial management After reading this lesson you will be able to: - Calculate the future value of a stream of equal cash flows (an annuity). Calculate the present value of a stream of equal future cash flows. Understand the concept of Sinking Fund Factor. Understand the concept of Capital Recovery Factor. Understand the concept of Perpetuity. ANNUITY Till now we talked about the future value of single payment made at the time zero (PV0). Now we will speak about annuities. An annuity is an equal annual series of cash flows. Annuities may be equal annual deposits, equal annual withdrawals, equal annual payments, or equal annual receipts. The key is equal, annual cash flows. Note that the cash flows occur at the end of the year. This makes the cash flow an ordinary annuity. If the cash flows were at the beginning of the year, they would be an annuity due. Annuity = Equal Annual Series of Cash Flows Assume annual deposits of Rs 100 deposited at end of year earning 5% interest for three years
2. 2. Year 1: Rs100 deposited at end of year = Rs100.00 Year 2: Rs100 x .05 = Rs5.00 + Rs100 + Rs100 = Rs205.00 Year 3: Rs205 x .05 = Rs10.25 + Rs205 + Rs100 = Rs315.25 Translating a series of cash flows into a present value is similar to translating a single amount to the present; we discount each cash flow to the present using the appropriate discount rate and number of discount periods. Translating a series of cash flows into a future value is also similar to translating a single sum: simply add up the future values of each cash flow. Again, there are tables for working with annuities. Future Value of Annuity Factors is the table to be used in calculating annuities due. Basically, this table works the same way as Table 1. Just look up the appropriate number of periods, locate the appropriate interest, take the factor found and multiply it by the amount of the annuity. We use table A-2 for FVIFA For instance, on the three-year, 5% interest annuity of Rs100 per year. Going down three years, out to 5%, the factor of 3.152 is found. Multiply that by the annuity of Rs100 yields a future value of Rs315.20. another example of calculating the future value of an annuity is illustrated. You deposit Rs 300 each year for 15 years at 6%. How much will you have at the end of that time? = 300 X 23.276 = 6982.8 The following exercise should aid in using tables to solve annuity problems. Use table A- 2. FVIFA
3. 3. 1.You deposit Rs 2,000 in recurring account each year for 5 years. If interest on this recurring account is 4%, how much will you have at the end of those 5 years? Rs 10,000 Rs 10,832.60 Rs 8,903.60 2.If you deposit Rs 4,500 each year into an account paying 8% interest, how much will you have at the end of 3 years? Rs 13,500 Rs 14,608.80 Rs 11,596.95 To find the present value of an annuity, use Table A-4. Find the appropriate factor and multiply it times the amount of the annuity to find the present value of the annuity. For instance Find the present value of a 4-year, Rs 3,000 per year annuity at 6%. Using the present value of annuity table, going down the left column for 4 yrs and to 6% the corresponding factor is 3.465 =3000 X 3.465 = 10395 Rs FUTURE VALUE OF ANNUITY
4. 4. Annuity as discussed just now is the term used to describe a series of periodic flows of equal amounts. These flows can be either receipts or payments. For example, if you are required to pay Rs. 200 per annum as life insurance premium for the next 20 years, you can classify this stream of payments as an annuity. If the equal amounts of cash flow occur at the end of each period over the specified time horizon, then this stream of cash flows is defined as a regular annuity or deferred annuity. When cash flows occur at the beginning of each period the annuity is known as an annuity due. The future value of a regular annuity for a period of n years at a rate of interest ‘k’ is given by the formula: FVAn = A (1 +K)n-1 + A ( 1+ K)n-2 + A( 1 + k)n-3 + ..+ A Which reduces to  (1 + K ) n − 1 FVAn = A    K  Where A = amount deposited/ invested at the end of every year for n years. K = rate of interest (expressed in decimals) N = time horizon FVAn = accumulation at the end of n years.  (1 + K ) n − 1 The expression   is called the future value  K  Interest factor for Annuity (FVIFA, hereafter) and it represents the accumulation of Re. 1 invested or paid at the end of every year for a period of n years at the rate of interest ‘k’. As in the case of the future value of a single flow, this expression has also been evaluated for different combinations of ‘k’ and ‘n’ and tabulated in table A.2 at the
5. 5. end of this book. So, given the annuity payment, we have to just multiply it with the appropriate FVIFA value and determine the accumulation.
6. 6. Example Under the recurring deposit scheme of the Vijaya Bank, a fixed sum is deposited every month on or before the due date opted for 12 to 120 months according to the convenience and needs of the investor. The period of deposit however should be in multiples of 3 months only. The rate of interest applied is 9% p.a. for periods from 12 to 24 months and 10% p.a. for periods form 24 to 120 months and is compounded at quarterly intervals. Based on the above information the maturity value of a monthly installment of Rs. 12 months can be calculated as below: Amount of deposit = Rs. 5 per month Rate of interest = 9% p.a. compounded quarterly Effective rate of interest per annum 4  0.09  = 1 + − 1 = 0.0931  4  0.0931 Rate of interest per month = =n 0.78% 12 Alternative method Rate of interest per month = (r + 1 )1/m - 1 = (1 + 0.0931)12 - 1 = 1.0074 - 1 = .0074 = .74% Maturity value cab be calculated using the formula
7. 7.  (1 + k ) n − 1 FVAn = A    k   (1 + 0.0078)12 − 1 =5    0.0078  = 5 x 12.53 = Rs. 62.65 If the payments are made at the beginning of every year, then the value of such an annuity called annuity due is found by modifying the formula for annuity regular as follows. FVAn (due) = A (1 + k) FVIFAK,n Example Under the Jeevan Mitra plan offered by Life insurance Corporation of India, if a person is insured for Rs. 10,000 and if he survives the full term, then the maturity benefits will be the basic sum of Rs. 10,000 assured plus bonus which accrues on the basic sum assured. The minimum and maximum age to propose for a policy is 18 and 50 years respectively. Let us take two examples, one of a person aged 20 and another a person who is 40 years old to illustrate this scheme. The person aged 20, enters the plan for a policy of Rs. 10,000. The term of policy is 25 years and the annual premium is Rs. 41.65. The person aged 40, also proposes for the policy of Rs. 10,000 and for 25 years and the annual premium he has to pay comes to Rs. 57. What is the rate of returns enjoyed by these two persons?
8. 8. Solution: Rate of Return enjoyed by the person of 20 years of age Premium = Rs. 41.65 per annum Term of policy = 25 years Maturity value = Rs. 1,000 + bonus which can be neglected as it is a Fixed amount and does not vary with the term of policy. We know that the premium amount when multiplied by FVIFA factor will give us the value at maturity. i.e., P X (1 X k)* FVIFA (k,n) = MV Where P = Annual premium n = Term of policy in years k = Rate of return MV = Maturity Value Therefore 41.65 x (1 + k) FVIFA (k, 25) = 10,000 (1 + k) FVIFA (k, 25) =240.01 From the table A.2 at the end of the book, we can find that (1 + 0.14) FVIFA (14,25) = 207.33 i.e., (1.14) FVIFA (15,25) = 1.15 X 212.793 = 244.71 By Interpolation
9. 9. 240.01 − 207.33 K = 14% + (15% - 14%) x 244.71 − 207.33 32.68 = 14% + 1% X 37.38 = 14% + 0.87% = 14.87% Rate of return enjoyed by the person aged 40 Premium = Rs. 57 per annum Term of policy = 25 years Maturity value = Rs. 10,000 Therefore 57 X ( 1 + k) FVIFA (k,25) = 10,000 (1 + k) FVIFA (k, 25) = 175.87 i.e., (1.13) (155.62) = 175.87 i.e., k. = 13% (appr.) Here we find that the rate of return enjoyed by the 20-year-old person is greater than that of the 40-year-old person by about 2% in spite of the latter paying a higher amount of annual premium for the same period of 25 years and for the same maturity value of Rs. 10,000. This is due to the coverage for the greater risk in the case of the 40 year old person. Now that we are familiar with the computation of future value, we will get into the mechanics of computation of present value. SINKING FUND FACTOR Here is the equation
10. 10.  (1 + K ) n − 1 FVA = A    k  We can rewrite it as  K  A = FVA  − 1  (1 + K ) n   K  The expression  − 1 is called the sinking Fund factor. It represents the amount  (1 + K ) n  that has to be invested at the end of every year for a period of “n” years at the rate of interest “k”, in order to accumulate Re. 1 at the end of the period. PRESENT VALUE OF AN ANNUITY The present value of an annuity ‘A’ receivable at the end of every year for a period of n years at a rate of interest K is equal to A A A A PVAn = + + + ... (1 + K ) (1 + K ) 2 (1 + K ) 3 (1 + K ) Which reduces to  (1 + K )n − 1 PVAn =A X  n   k ((1 + k )  The expression  (1 + K )n − 1  n   k ((1 + k ) 
11. 11. is called the PVIFA (Present Value Interest Factor Annuity ) and it represents the present value of regular annuity of Rs. 1 for the given values of k and n. The values of PVIFA (k, n) for different combinations of ‘k’ and ‘n’ are given in Appendix A.4 given at the end of the book. It must be noted that these values can be used in any present value problem only if the following conditions are satisfied: (a) The cash flows are equal; and (b) The cash flows occur at the end of every year. ALWAYS REMEMBER It must also be noted that PVIFA (k, n) is not the inverse of FVIFA (k, n,) although PVIF (k, n) is the inverse of FVIF (k, n). Example The Swarna Kailash Yojana at rural and semi-urban branches of SBI is a scheme open to all individuals /firms. A lump sum deposit is remitted and the principal is received with interest at the rate of 12% p.a. in 12 or 24 monthly installments. The interest is compounded at quarterly intervals. The amount of initial deposit to receive a monthly installment of Rs. 100 for 12 months can be calculated as below: Firstly, the effective rate of interest per annum has to be calculated. m  k r = 1 +  −1  m 4  k = 1 +  − 1 =12.55%  m After calculating the effective rate of interest per annum, the effective rate of interest per month has to be calculated which is nothing but
12. 12. 0.1255 = 0.01046 12 The initial deposit can now be calculated as below:  (1 + k )n − 1 PVAn = A  n   k (1 + k )   (1 + 0.01046)12 − 1  =100  12   0.01046(1 + 0.01046)   0.133  =100    0.01185  =100 x11.22 = Rs.1122 Example The annuity deposit scheme of SBI provides for fixed monthly income for suitable periods of the depositor’s choice. An initial deposit has to be made for a minimum period of 36 months. After the first month of the deposit, the depositor receives monthly installments depending on the number of months he has chosen as annuity period. The rate of interest is 11% p.a., which is compounded at quarterly intervals. If an initial deposit of Rs. 4,549 is made for an annuity period of 60 months, the value of the monthly annuity can be calculated as below: Firstly, the effective rate of interest per annum has to be calculated. m  k r = 1 +  −1  m
13. 13. 4  0.11 = 1 + − 1 = 11.46%  4   After calculating the effective rate of interest per annum, the effective rate of interest per month has to be calculated which is nothing but 0.1146 = 0.00955 12 The monthly annuity can now be calculated as  (1 + K ) n − 1 PVAn = A  n   k (1 + k )   (1 + 0.00955)60 − 1  4549 = A  60   0.00955(1.00955)  0.7688 4549 = A X 0.0169 A = Rs. 100 Capital Recovery Factor Manipulating the relationship between PVAn, A, K & n we get an equation:  k (1 + k ) n  A = PVAn    (1 + k ) − 1 n
14. 14.  k (1 + k ) n    is known as the capital recovery factor.  (1 + k ) − 1 n KEEP IN MIND Inverse of FVIFA factor is Sinking Fund Factor Inverse of PVIFA factor is Capital Recovery Factor Example A loan of Rs. 1,00,000 is to be repaid in five equal annual installments. If the loan carries a rate of interest of 14% p.a. the amount of each installment can be calculated as below: If R is defined as the equated annual installment, we are given that R X PVIFA (14.5) = Rs. 1,00,000 Rs.1,00,000 There fore R = PVIFA(14.5) Rs.1,00,000 = 3,433 = Rs. 29,129 Notes: 1. We have introduced in this example the application of the inverse of the PVIFA factor, which is called the capital recovery factor. The application of the capital recovery factor helps in answering questions like:
15. 15. What should be the amount that must be paid annually to liquidate a loan over a specified period at a given rate of interest? How much can be withdrawn periodically for a certain length of time, if a given amount is invested today? 2. In this example, the amount of Rs. 29,129 represents the sum of the principal and interest components. To get an idea of the break-up of each installment between the principal and interest components, the loan- repayment schedule is given below. Year Equated annual Interest Capital content Loan outstanding installment content of (b) of (B) after payment (Rs.) (Rs.) (Rs.) (Rs.) (A) (B) (C) [(D) = (B – C)] (E) 0 - - - 1,00,000 1 29,129 14,000 15,129 84,871 2 29,129 11,882 17,247 67,624 3 29,129 9,467 19,662 47,962 4 29,129 6,715 22,414 25,548 5 29,129 3,577 25,552 -- The interest content of each installment is obtained by multiplying interest rate with the loan outstanding at the end of the immediately preceding year. As it can be observed form this schedule the interest component declines over a period of time whereas the capital component increases. The loan outstanding at the end of the penultimate year must be equal to the capital content of the last installment but in practice there will be a marginal difference on account of rounding off errors. 3. The equated annual installment method is usually adopted for fixing the loan repayment schedule in a hire- purchase transaction. But the financial institutions in India
16. 16. like IDBI, IFCI and ICICI do not follow this scheme of equal periodic amortization. Instead, they stipulate that the loan must be repaid in equal installments. According to this scheme, the principal component of each payment remains constant and the total debt-serving burden (consisting of principal repayment and interest payment) declines over time. Perpetuities Perpetuity is a cash flow without a fixed time horizon. For example if someone were promised that they would receive a cash flow of Rs400 per year until they died, that would be perpetuity. To find the present value of perpetuity, simply take the annual return in Rs and divide it by the appropriate discount rate. Suppose you will receive a fixed payment every period (month, year, etc.) forever. This is an example of perpetuity. Perpetuities PV of Perpetuity Formula PV = C R C = cash payment R = interest rate Example If someone were promised a cash flow of Rs 400 per year until they died and they could earn 6% on other investments of similar quality, in present value terms the perpetuity would be worth Rs 6,666.67. (Rs 400 / .06 = Rs 6,666.67) Tell me the answer You want to create an endowment to fund a football scholarship, which pays Rs 15,000 per year, forever, how much money must be set-aside today in the rate of interest is 5%?
17. 17. By now you should be an expert in using the following two tables: A-1 The Compound Sum of one rupee FVIF A-3 The Present Value of one rupee PVIF A-2 The Compound Value of an annuity of one rupee FVIFA A-4 The Present Value of an annuity of one rupee PVIFA We will practice questions on time value of money in the following session.