SLM_(1)

Dec. 15 151 days Banker’s Discount = Interest on ` 5,460 at 5% p.a. for 151 days. 151 5 i.e., B.D. = 5,460 × 365 × 100 = ` 112.94 CU IDOL SELF LEARNING MATERIAL (SLM)

22 Business Mathematics and Statistics Example 11: A sewing machine is offered for ` 300 cash or 8 monthly instalments of ` 40 each, the payment starting at the end of the first month. What is the rate of interest charged? Solution: At the end of each month the following amounts are outstanding: ` 300 at the end of the 1st month ` 260 at the end of the 2nd month ` 220 at the end of the 3rd month ` 180 at the end of the 4th month ` 140 at the end of the 5th month ` 100 at the end of the 6th month ` 60 at the end of the 7th month ` 20 at the end of the 8th month Now ` 20 being the difference between ` 320 and ` 300 is the interest that is charged on: 300 + 260 + 220 + 180 + 140 + 100 + 60 + 40 + 20 for one month. The sum of the above arithmetic series is: n S = 2 [ 2a + (n – 1) d] a = 20, d = 40, n = 8 8 = 2 [40+(8–1)40] = 4 (40 + 280)

= 4×320 [Ans: Rate of interest is 18.75%] = 1280 r1 Now 20 = 1280 × 100 12 150 = 8 = 18¾ = 18.75 CU IDOL SELF LEARNING MATERIAL (SLM)

Simple and Compound Interest 23 2.6 Compound Interest There is much difference between Simple Interest considered earlier and the Compound Interest. The main difference is that in the case of Compound Interest the interest earned at the end of the year is added to the Principal and therefore the Principal becomes a variable quantity and in this manner interest grows. That is, the interest that is left to accumulate earns interest on itself, and the interest is added periodically to the Principal. For example, the calculation of Compound Interest on ` 8,000 for 3 years at 5% per annum is as follows: Principal at the beginning =8000=P  = (8000) (0.05) Interest at the end of the Interest at the  1st year end of the 3rd year  =400 Amount at the end of the 1st year Amount at the = 8000 + 8000 (0.05) end of the 3rd  Interest at the end of the year 2nd year  = 8000 + 400 Amount at the end of the  2nd year  = (8400) (0.05)   =420  = 8400 + 420   = 8820

 = (8820) (0.05)  = P + Pi    =441  = P (1 + i)  = 8,820 + 441   = (P + Pi) i    = (P + Pi) + (P + Pi) i  = (P + Pi) (1 + i)  = P (1 + i)2   = P (1 + i)2 × i  P=(P1=(+P1i(+)12i+×)3Pi i)2 + ? The amount at the end of 3 years is ` 9,261 In brief, Amount at the end of the 1st year = P (1 + i) Amount at the end of the 2nd year = P (1 + i)2 CU IDOL SELF LEARNING MATERIAL (SLM)

24 Business Mathematics and Statistics Amount at the end of the 3rd year = P (1 + i)3 ...................................................................... ...................................................................... Amount at the end of the nth year = P (1 + i)n That is, A = P (1 + i)n Or A = P (1 + r/100)n By logarithmic Calculation, A = Antilog [log P + n log (1 + i)] Further, A/P = (1 + i)n By logarithmic calculation it follows that log A – log P = n log (1 + i) log A log P ? =n log (1 + i) Illustrative Example: Example 1 : Find the time that is necessary for a certain sum of money to double itself, the rate of interest per ` 1/- per annum being i. Solution: Putting A = 2P in the equation log A log P , we get log (1 + i)

log 2P log P log 2 + log P log P log (1 + i) = log (1 + i) log 2 = log (1 + i) For instance, if i = 0.05 log 2 log 2 Then n = log (1 + .05) = log (1.05) 0.3010 = 0.0212 CU IDOL SELF LEARNING MATERIAL (SLM)

Simple and Compound Interest 25 = Antilog [log 3010 – log 0212] = Antilog [3.4786 – 2.3263] = Antilog [1.1523] = 14.20 Money doubles itself in 14.2 years, taking the rate of interest as 0.05 for ` 1/- per annum. The following table shows the different time periods required for money to double itself at different rates of interest. ri n 3 .03 23.44 3.50 .035 20.15 4 .04 17.68 4½ .045 15.74 5 .05 14.20 5½ .055 12.95 6 .06 11.89 6½ .065 11.01 7 .07 10.25 7½ .075 9.58 8 .08 9.01 8½ .085 8.49 9 .09 8.04

9½ .095 7.64 10 .1 7.27 Example 2: Find the compound interest earned from ` 16,000 for 3 year at 12% per annum? Solution: Here, P = ` 16,000 Rate of interest, r = 12% = 0.125 No. of periods, n = 3 Interest compounded annually ? i = r Amount A = P (1 + i)n ? A= 16,000 (1 + 0.125)3 = 16,000 (1·423828) = ` 22,781.25 CU IDOL SELF LEARNING MATERIAL (SLM)

26 Business Mathematics and Statistics Now Compound Interest I = A – P = 22,781.25 – 16,000 = ` 6,781.25 ? Interest paid is ` 6,781.25. Example 3: If ` 1,750 is invested at 9% per annum interest for 10 years and interest is compounded half-year, find the amount and interest. Solution: P = ` 1,750 and r = 9% = 0.09 Interest is calculated half-yearly. r ? i = 2 = 0.045 and n = 10 × 2 = 20 Amount A = P (1 + i)n ? A = 1,750 (1.045)20 Let x = (1.045)20 log x = 20 log 1.045 = 20 × 0.0191 = 0.3820 x = antilog (0.03820) = 2.41 Now, A = 1,750 × 2.41 = 4,217.50 Interest, I =S–P = 4,217.50 – 1,750 = ` 2,467.50 ? The amount is ` 4,217.50, and interest is ` 2,467.50

Example 4: To what amount ` 10,000 accumulate in 6 year, if invested at 8% compounded quarterly? [(1.02)24 = 1.6084]. Solution: P = ` 10,000, and r = 8% = 0.08 Rate of interest as per conversion period r 0.08 i = 4 = 4 = 0.02 and n = 6 × 4.24 Amount A = P (1 + i)n ? A = 10,000 (1 + 0.02)24 = 10,000 (1.02)24 = 10,000 × 1.6084 = ` 16,084 ? The amount is ` 16,084. CU IDOL SELF LEARNING MATERIAL (SLM)

Simple and Compound Interest 27 Example 5: Find the rate of interest, if the sum of money will double itself in 10 years by investing at compound interest. Solution: Let P be the principal. Let the rate of interest be r. Interest is compounded annually, r = i and n = 10 Given that the amount will be doubled in 10 years. i.e., A = 2 P But A = P (1 + i)n ? 2P = P (1 + i)10 ? 2 = (1 + i)10 Taking log on both sides log 2 = 10 log (1 + i) log 2 0.0310 log (1 + i) = 10 10 = 0.0301 Now, 1 + i = antilog (0.0301) 1 + i = 1.072 i = 0.072 = 7.2% ? The required rate of interest is 7.2%. Example 6: The amounts for a certain sum with compound interest at a certain rate in two years and in three years are ` 8,820 and ` 9,261 respectively. Find the rate and sum.

Solution: Let P be the principal amount and i be the rate of interest. … (1) Amount A = P (1 + i)n … (2) At the end of two years, A = ` 8,820 ? 8,820 = P (1 + i )2 At the end of three years, A = ` 9,261 ? 9,261 = P (1 + i)3 CU IDOL SELF LEARNING MATERIAL (SLM)

28 Business Mathematics and Statistics Dividing (2) by (1), we get 9,261 P(1i)3 = 8,820 P(1i)2 Ÿ 1.05 = 1 + i Ÿ i = 0.05 = 5% Now, substituting i = 0.05 in equation (1) we get, 8,820 = P (1.05)2 Ÿ 8,820 = P × 1.1025 × P = ` 8,000 ? Required rate is 5% and principal is ` 8,000. Example 7: A certain sum of money is invested at 4% p.a. compound annually. The interest for 2nd year is ` 25. Find the interest for 3rd year. Solution: Let P be the principal. A = P (1 + i)n Amount at the end of 1st year = P (1.04) Amount at the end of 2nd year = P (1.04)2 Interest for 2nd year = Amount at the end of 2nd year – Amount at the end of 1st. Given interest for second year is ` 25.

25 = P (1.04)2 – P (1.04) P (1.04) (1.04 – 1) = 25 P (1.04) (0.04) = 25 … (1) Amount at the end of 3rd year = P (1.04)3 Interest for 3rd year = Amount at the end of 3rd year – Amount at the end of 2nd year ? Interest for 3rd year = P (1.04)3 – P (1.04)2 = P (1.04)2 [1.04 – 1] = P (1.04)2 (0.04) = P (1.04) (1.04) (0.04) = 25 × 1.04 [using (1)] = ` 26. ? Interest for third year is ` 26. CU IDOL SELF LEARNING MATERIAL (SLM)

Simple and Compound Interest 29 Example 8: Lal deposited an amount of ` 50,000 in two different Banks A and B, dividing the amount into two investments. Bank A calculates interest at a rate of 7% per annum and Bank B calculates interest at the rate of 6% per annum convertible semi-annually. At the end of 3 years, he received ` 10,632.35 as the return on his investment. What amount he has deposited in both Banks? Solution: Let ` X and ` Y be the amounts deposited in Bank A and Bank B respectively. Total investment is ` 50,000 ? X+Y=50,000 … (1) Compound interest is given by … (2) I =A – P where A = P (1 + i)n Interest earned from Bank A: I =X (1.07)3 – X 1 ? I1 =1.225043 X – X ? I1 =0.225043 X Interest earned from Bank B: I =(1.03)6 Y – Y 2 I2 =1.194052 Y – Y I2 =0.194052 Y Total interest earned I =I1+I2 ? I =0.225043 X + 0.194052 Y It is given that total interest received is ` 10,632.35 ? 0.225043 X + 0.194052 Y = 10632.35

Solving (1) and (2) we get, X =` 30,000 and Y = ` 20,000. ? ` 30,000 and ` 20,000 are the sums deposited in Bank A and Bank B respectively. CU IDOL SELF LEARNING MATERIAL (SLM)

30 Business Mathematics and Statistics 2.7 Summary Borrowers take loans from money lenders and repay the amount borrowed at a certain rate of interest per annum, for the specific period. This is called ‘Amount of Repayment’, consisting of principle plus simple interest. If interest is charged on interest, then such an interest is called Compound Interest. Pnr S.I. = 100 Pnr  Pnr  Amount = P – S.I. = P + = P1  100  100  Amount A – [Compound Interest + Principal] =  r n P 1   100  r n ? C.I. = P 1 P  100 2.8 Key Words/Abbreviations S.I. = Simple Interest, C.I. = Compound Interest

r i Time period = n, (r = interest); Percentage of Interest = 100 P.W. = Present Worth; T.D. = True Discount; B.D. = Banker’s Discount; A.D.D. = Average Due Date. 2.9 Learning Activity After thoroughly studying and solving the examples, 1 to 25, fill in the blanks in the table given in Example 6. ............................................................................................................................................................ ............................................................................................................................................................ CU IDOL SELF LEARNING MATERIAL (SLM)

Simple and Compound Interest 31 2.10 Unit End Questions (MCQ and Descriptive) A. Descriptive Type: Short Answer Type Questions 1. In what time will ` 1.250 amount to ` 1,400 at 6% per annum? [Ans: 2 years] 2. The present worth of a bill due at the end of 4 years is ` 575; and if the bills were due at the end of 2½ years its present worth would be ` 620. Find the rate per cent. [Ans: 6%] 3. If the discount on ` 161 due 2½ years hence be ` 21, at what rate percent is the interest claculated? [Ans: 6%] 4. What sum will amount to ` 1,000 in 50 years at 4% p.a. compound interest, the interest being paid half-yearly? [Ans: ` 138] 5. Find the difference between simple and compound interest on ` 1,500 for 5 years at 4½ years at 4½ p.a. [Ans. 31.50] 6. A sum of money at simple interest amount to ` 2,800 in two years and ` 3,250 in five years. Find the sum and the rate of interest. [Ans: 1,500, 6%] 7. The amount of ` 3,000 at 3% p.a. compound interest for 10 years is equal to the amount of ` 2,000 at 6% p.a. compound interest for a certain period. Find this period. [Ans: 12 years] 8. A sum of money put out at Compound Interest amounts in 2 years to ` 672 and in 3 years to ` 714. Find the rate of interest. [Ans: 61/4%]

9. A man buys a house on condition that he shall pay ` 8,820 now and equal sums at the end of one year and two years. What would be the cash value of the house, if Compound Interest payable yearly be calculated at 5% per annum? [Ans: ` 25,220] 10. A simple interest on a certain sum of money for 3 years at 4% is ` 303.60 p. Find the Compound Interest on the sum for the same period and at the same rate. (Fractions of a paisa to be ignored). [Ans: 315.90] 11. The Compound Interest on a certain sum of money for 2 years is ` 920.25 and the simple interest is ` 900. Find the sum and the rate per cent. [Ans: ` 10,000, 4½] CU IDOL SELF LEARNING MATERIAL (SLM)

32 Business Mathematics and Statistics 12. A certain sum of money put out at Compound Interest at 5 percent yields ` 20 more in two years than an equal sum put out at simple interest at the same rate. Find the sum of money. [Ans: ` 8,000] 13. Divide the sum of ` 3,903 between a brother and a sister who are 18 years and 16 years old respectively, in such a way that their shares invested at 4% Compound Interest should be equal when they attain the age of 21 years. [Ans: ` 1,875, ` 2,028] 14. A sum of ` 5,150 lent for 2 years at 6% Compound Interest (compounded annually). If the borrower prefers to pay the money back by two equal annual payments, one at the end of the first year and the other at the end of the second, how much should be pay at the end of each year? [Ans: ` 2,809] 15. A money lender borrows a certain sum of money at 3% per annum simple interest and invests the same at 5 per annum Compound Interest (compounded annually). After 3 years he makes a profit of ` 1,082. Find the amount he borrowed. [Ans: ` 16,000] 16. Find the Simple Interest on ` 75,500 at 81/ % per annum for 2.5 years.[Ans: ` 15,722.87] 3 17. Find the Simple Interest on ` 8,450 at 12% per annum for 100 days. [Ans: 277.80] 18. Find the Simple Interest on ` 1,25,000 at 18% per annum for the period from 11th April 2008 to 23rd May 2008. [Ans: ` 2,589.04] 19. At what rate percent per annum will a sum of money double itself in 6½ years? [Ans: ` 15.38%] 20. A sum of money was invested at Simple Interest at a certain rate for 3 years. Had it been invested at a rate 2% higher than the present rate it would have given the investor ` 360

more. What is the sum of money invested? [Ans: ` 6,000] 21. ` 800 amounts to ` 920 in 3 years at Simple Interest. Had the interest rate been increased by 3% then what would be the amount? [Ans: ` 992] 22. If a sum of money doubles itself in 8 years at Simple Interest, then what is the rate of interest per annum? [Ans: 12.5] 23. If the difference between Compound Interest and Simple Interest on a certain sum of money at 10% per annum for 2 years is ` 631, then what is the sum? [Ans: ` 6,100] CU IDOL SELF LEARNING MATERIAL (SLM)

Simple and Compound Interest 33 24. Find the Compound Interest on ` 2,800 at 16% per annum for 9 months compounded quarterly. [Ans: ` 349.60] 25. A sum of money at Compound Interest amounts to thrice itself in 3 years. In how many years will it be 9 times itself. [Ans: 6 years] 26. Fill in the blanks in the following table.  nr   Pnr Ex. No. Principal (P) r (%) n (Yrs) P1 +    100  100 Amount Simple Interest 1. 18.250 63 — 21,957.031 3,707.031 4 2. 29,540 81 5 3 43,694.58 — 3 4 6½ — 32,449.21 3. 44,375 11¼ 4. 73,295 — 11 3 1,31,427.09 58,132.096 4 5. 1,23,984 81 125 days — 3,538.36 3 [Ans: (1) 3¼, (2) 14,154.58, (3) 76,824.21, (4) 6¾%, (5) 2,398.36] B. Multiple Choice/Objective Type Questions 1. If the rate of interest is 8%, money doubles itself in ________.

(a) 51/ years (b) 61/ years 2 3 (c) 71/ years (d) 9.01 years 2 (e) None of these 2. If the S.I. on ` 4,600 is 4 years is ` 1,150, then the rate of S.I. is ________. (a) 91/ % (b) 51/ % 2 3 (c) 61/ % (d) 7.2% 4 (d) None of these CU IDOL SELF LEARNING MATERIAL (SLM)

34 Business Mathematics and Statistics 3. If the difference between C.I. and S.I. for 2 years at 10% per annum is ` 631, then the principle is ________. (a) 4,100 (b) 5,300 (c) 6,100 (d) 7,400 (e) None of these 4. The C.I. on ` 2,800 for 9 months compounded quarterly is ` 349.6, then the rate of interest per annum is ________. (b) 14% (a) 111/ % 2 (c) 15% (d) 16% (e) None of these 5. A steel cupboard worth ` 10,000 is purchased on instalment basis for ` 200 down and of monthly instalments of ` 1,000 each. The rate of interest charged is ________. (a) 23% p.a. (b) 24% p.a. (c) 25% p.a. (d) 26% p.a. (e) None of these Ans wers: (1) (d); (2) (c); (3) (c); (4) (d); (5) (b); 2.11 References

References of this unit have been given at the end of the book. ˆˆˆ CU IDOL SELF LEARNING MATERIAL (SLM)

Annuities 35 UNIT 3 ANNUITIES Structure 3.0 Learning Objectives 3.1 Introduction 3.2 Immediate Annuity 3.3 Present Worth 3.4 Perpetuity 3.5 Annuity Due 3.6 Present Value of an Annuity Due 3.7 Deferred Annuity 3.8 Present Value of a Deferred Annuity

3.9 Endowment Fund 3.10 Annuity in the Case of which Payments are made other than Annually 3.11 Sinking Fund 3.12 Repayment of Loan by Instalments 3.13 Summary 3.14 Key Words/Abbreviations 3.15 Learning Activity 3.16 Unit End Questions (MCQ and Descriptive) 3.17 References

CU IDOL SELF LEARNING MATERIAL (SLM) 36 Business Mathematics and Statistics 3.0 Learning Objectives After studying this unit, you will be able to:  Explain the meaning of various types of annuities and the importance of each.   Elaborate the methods of calculation by noting down the various formulae.   Work out the examples given in the exercise. 3.1 Introduction An annuity is a fixed sum paid regularly at equal intervals of time. In other words, an annuity stands for a series of equal payments regularly paid at equal intervals of time. The payments may be made yearly, half-yearly, quarterly or monthly depending upon the conditions of agreement. The person who receives an annuity is known as ‘annuitant’. When an annuity is payable unconditionally for a certain specified period, then it is said to be ‘Annuity Certain’. There are three types of ‘Annuity Certain’ viz :

1. Immediate Annuity 2. Annuity Due 3. Deferred Annuity Immediate Annuity: According to this type, the annuity payment falls due at the end of the first interval. Therefore payment is made regularly at the end of each of the successive intervals of time. Annuity Due: In the case of this type the annuity payment falls due at the beginning of the first interval. Hence regular payment is made at the beginning of each of the successive intervals of time. Deferred Annuity: This type of annuity is also known as ‘Reversion’. According to this, the annuity begins after the lapse of a certain period. Therefore the annuity that is deferred for n years commences only after n years, the first payment being made at the end of n + 1 years. We note the following:  A “Life Annuity” is the type in the case of which, annuity is payable only during the lifetime of a person or of the surviors of a number of persons.

CU IDOL SELF LEARNING MATERIAL (SLM)

Annuities 37  An ‘Annuity Contingent” is the type where annuity is payable till some contingency, that is, the happening of some event such as marriage of a girl, commencement of education of a child, or the death of the annuitant.   An annuity is said to be a “Perpetuity” when it is to continue for ever. It is said to be “Deferred Perpetuity” if it does not commence at once. The annuity that is left unpaid for a certain number of years is said to be ‘Forborne’ for that number of years. Application of Annuities Annuities can be purchased from insurance providers, banks, mutual fund companies, stockbrokers, and other financial institutions. They come in several different forms, including immediate and deferred annuities, and fixed and variable annuities. Each form has different properties and involves different costs. Although the money placed in an annuity is first subject to taxation at the same rate as ordinary income, it is then invested and allowed to grow tax-deferred until it is withdrawn. Distribution is flexible and can take the form of a lump sum, a systematic payout over a specified period, or a guaranteed income spread over the remainder of a person’s life. In most cases annuities are a long-term investment vehicle, since the costs involved make it necessary to hold an annuity for a number of years in order to reap financial benefits. Because of their flexibility, annuities can be a good choice for small business owners in planning for their own retirement or in providing an extra reward or incentive for valued employees. An annuity is an interest-bearing financial contract that combines the tax-deferred savings and investment properties of retirement accounts with the guaranteed-income aspects of insurance. Annuities can be described as the flip side of life insurance. Life insurance is designed to provide financial protection against dying too soon. Annuities provide a hedge against outliving your retirement savings. While life insurance plans are designed to create principal, an annuity is designed to liquidate principal that has been created, usually in the form of regular payments over a number of years.

Annuities can be purchased from insurance providers, banks, mutual fund companies, stockbrokers, and other financial institutions. They come in several different forms, including immediate and deferred annuities, and fixed and variable annuities. Each form has different properties and involves different costs. Although the money placed in an annuity is first subject to taxation at the same rate as ordinary income, it is then invested and allowed to grow tax-deferred until it is CU IDOL SELF LEARNING MATERIAL (SLM)

38 Business Mathematics and Statistics withdrawn. Distribution is flexible and can take the form of a lump sum, a systematic payout over a specified period, or a guaranteed income spread over the remainder of a person’s life. In most cases annuities are a long-term investment vehicle, since the costs involved make it necessary to hold an annuity for a number of years in order to reap financial benefits. Because of their flexibility, annuities can be a good choice for small business owners in planning for their own retirement or in providing an extra reward or incentive for valued employees. Types of Annuities There are several different types of annuities available, each with different properties and costs that should be taken into consideration as business owners put together their retirement investment portfolio. The two basic forms that annuities take are immediate and deferred. An immediate annuity, as the name suggests, begins providing payouts at once. Payouts may continue either for a specific period or for life, depending on the contract terms. Immediate annuities — which are generally purchased with a one-time deposit, with a minimum of around $10,000 — are not very common. They tend to appeal to people who wish to roll over a lump-sum amount from a pension or inheritance and begin drawing income from it. The immediate annuity would be preferable to a regular bank account because the principal grows more quickly through investment and because the amount and duration of payouts are guaranteed by contract. Immediate annuities are also known by the name income annuities. What is important to remember when considering an immediate annuity is that “at the end of the day, you’ve got to remember what you’re buying is insurance, not an investment vehicle like a stock or mutual fund,” explains Rob Nestor in an article by Murray Coleman in Investor’s Business Daily. Deferred annuities delay payouts until a specific future date. The principal amount is invested and allowed to grow tax-deferred over time. More common than immediate annuities, deferred annuities appeal to people who want a tax-deferred investment vehicle in order to save for retirement.

There are also two basic types of deferred annuity: fixed and variable. Fixed annuities provide a guaranteed interest rate over a certain period, usually between one and five years. In this way, fixed annuities are comparable to certificates of deposit (CDs) and bonds, with the main benefit that the sponsor guarantees the return of the principal. Fixed annuities generally offer a slightly higher interest rate than CDs and bonds, while the risk is also slightly higher. In addition, like other types of annuities, the principal is allowed to grow tax-deferred until it is withdrawn. CU IDOL SELF LEARNING MATERIAL (SLM)

Annuities 39 The more popular of the deferred annuity types is the variable annuity which offers an interest rate that changes based on the value of the underlying investment. Purchasers of variable annuities can usually choose from a range of stock, bond, and money market funds for investment purposes in order to diversify their portfolios and manage risk. Some of these funds are created and managed specifically for the annuity, while others are similar to those that may be purchased directly from mutual fund companies. The minimum investment usually ranges from $500 to $5,000, depending on the sponsor, and the investments (or subaccounts) usually feature varying levels of risk, from aggressive growth to conservative fixed income. In most cases, the annuity principal can be transferred from one investment to another without being subject to taxation. Variable annuities are subject to market fluctuations, however, and investors also must accept a slight risk of losing their principal if the sponsor company encounters financial difficulties. Features of Annuities Variable annuities have a number of features that differentiate them from common retirement accounts, such as 401(k)s and IRAs, and from common equity investments, such as mutual funds. One of the main points of differentiation involves tax deferral. Unlike 401(k)s or IRAs, variable annuities are funded with after-tax money — meaning that contributions are subject to taxation at the same rate as ordinary income prior to being placed in the annuity. In contrast, individuals are allowed to make contributions to the other types of retirement accounts using pre-tax dollars. That is why financial specialists usually instruct people to first maximize their contributions to 401(k) plans and IRAs before considering annuities. On the plus side, there is no limit on the amount that an individual may contribute to a variable annuity, while contributions to the other types of accounts are limited by the federal government. Unlike the dividends and capital gains that accrue to mutual funds, however, which are taxable in the year they are received, the money invested in annuities is allowed to grow tax free until it is withdrawn. Another feature that differentiates variable annuities from other types of financial products is the death benefit. Most annuity contracts include a clause guaranteeing that the investor’s heirs will receive either the full amount of principal invested or the current market value of the

contract, whichever is greater, in the event that the investor dies before receiving full distribution of the assets. However, any earnings are taxable for the heirs. CU IDOL SELF LEARNING MATERIAL (SLM)

40 Business Mathematics and Statistics Another benefit of variable annuities is that they offer greater withdrawal flexibility than other retirement accounts. Investors are able to customize the distribution of their assets in a number of ways, ranging from a lump-sum payment to a guaranteed lifetime income. Some limitations, however, do apply. For example, the federal government imposes a 10 percent penalty on withdrawals taken by anyone before they reach the age of 59 1/2 years. But contributors to variable annuities are not required to begin taking distributions until age 85, whereas contributors to IRAs and 401(k)s are required to begin taking distributions by age 70 1/2. Costs Associated with Annuities In exchange for the various features offered by annuities, investors must pay a number of costs. Many of the costs are due to the insurance aspects of annuities, although they vary among different sponsors. One common type of cost associated with annuities is the insurance cost, which averages 1.25 per cent and pays for the guaranteed death benefit in addition to the insurance agent’s commission. There are also usually management fees, averaging 1 percent, which compensate the sponsor for taking care of the investments and generating reports. Many annuities also charge modest administrative or contract fees. One of the more problematic costs of annuities, in the eyes of their critics, is the surrender charge for early removal of the principal. In most cases, this fee begins at around 7 percent but then phases out over time. However, the surrender fee is charged in addition to the 10 percent government penalty for early withdrawal if the investor is under age 59 1/2. All of the costs associated with variable annuities detract somewhat from their attractiveness as a financial product when one compares them to mutual funds. The costs also mean that there are no quick profits associated with annuities; instead, they must be held as a long-term investment. In fact, it can take as long as 17 years for the benefits of tax deferral to outpace the administrative expenses of an annuity. For investors who wish to put money away for an extended period, a variable annuity may be a very good investment vehicle.

Distribution Options On the positive side, investors in annuities have a number of options for receiving the distribution of their funds. The three most common forms of distribution — all of which have various costs deducted — are lump sum, lifetime income, and systematic payout. Some investors CU IDOL SELF LEARNING MATERIAL (SLM)

Annuities 41 who have contributed to a variable annuity over many years may elect to take a lump-sum withdrawal. The main drawback to this approach is that all the taxes are due immediately. Other investors may decide upon a systematic payout of the accumulated assets over a specified time period. In this approach, the investor can determine the amount of payments as well as the intervals at which payments will be received. Finally, some investors choose the option of receiving a guaranteed lifetime income. This option is the most expensive for the investor, and does not provide any money for heirs, but the sponsor of the annuity must continue to make payouts even if the investor outlives his or her assets. A similar distribution arrangement is joint-and-last-survivor, which is an annuity that keeps providing income as long as one person in a couple is alive. Annuities are rather complex financial products, and as such they have become the subject of considerable debate among experts in financial planning. As mentioned earlier, many experts claim that the special features of annuities are not great enough to make up for their cost as compared to other investment options. As a result, financial advisors commonly suggest that individuals maximize their contributions to IRAs, 401(k)s, or other pre-tax retirement accounts before considering annuities (investors should avoid placing annuities into IRAs or other tax- sheltered accounts because the tax shelter then becomes redundant and the investor pays large annuity fees for nothing). Some experts also prefer mutual funds tied to a stock market index to annuities, because such funds typically cost less and often provide a more favorable tax situation. Contributions to annuities are taxed at the same rate as ordinary income, for instance, while long- term capital gains from stock investments are taxed at a special, lower rate — usually 20 percent. Still other financial advisors note that, given the costs involved, annuities require a very long-term financial commitment in order to provide benefits. It may not be possible for some individuals to tie up funds for the 17 to 20 years it takes to benefit from the purchase of an annuity. Despite the drawbacks, however, annuities can be beneficial for individuals in a number of different situations. For example, annuities provide an extra source of income and an added margin of safety for individuals who have contributed the limit allowed under other retirement savings options. In addition, some kinds of annuities can be valuable for individuals who want to protect their assets from creditors in the event of bankruptcy. An annuity can provide a good shelter for a retirement nest

egg for someone in a risky profession, such as medicine. Annuities are also recommended for people who plan to spend the principal during their lifetime rather than leaving it CU IDOL SELF LEARNING MATERIAL (SLM)

42 Business Mathematics and Statistics for their heirs. Finally, annuities may be more beneficial for individuals who expect that their tax bracket will be 28 per cent or lower at the time they begin making withdrawals. Annuities may also hold a great deal of appeal for small businesses. For example, annuities can be used as a retirement savings plan on top of a 401(k). They can be structured in various ways to reward employees for meeting company goals. In addition, annuities can provide a nice counterpart to life insurance, since the longer the investor lives, the better an annuity will turn out to be as an investment. Finally, some annuities allow investors to take out loans against the principal without paying penalties for early withdrawal. Overall, some financial experts claim that annuities are actually worth more than comparable investments because of such features as the death benefit, guaranteed lifetime income, and investment services. A small business owner considering setting up an annuity should consider all options, look carefully at both the costs and the returns, and be prepared to put money away for many years. It is also important to shop around for the best possible product and sponsor before committing funds. “Before investing in an annuity, make sure the insurance company that will sponsor the contract is financially healthy,” counseled Mel Poteshman in Los Angeles Business Journal. Also, find out from the sponsoring company the interest rates that have been paid out over the last five to ten years and how interest rate changes are calculated. This will give you an idea of the annuity’s overall performance and help you identify the annuity that provides the best long-range financial security. 3.2 Immediate Annuity To find the amount of an immediate annuity certain at the end of n years. Let A be the annuity, r the rate per ` l for one year, n the number; of years and M the amount. The first payment is made at the end of the first year. The amount of this sum A in the remaining

(n – 1) years is ARn–l, where R = I + r/100; r% being the effective rate of interest per annum. At the end of the second year another sum A is due and the amount of this sum in the remaining (n – 2) years is ARn–2; and proceeding in this manner; we get M = ARn–1 + ARn–2 + …AR2 + AR + A = A (1 + R + R2 + … + Rn–1) CU IDOL SELF LEARNING MATERIAL (SLM)

Annuities 43 (R n 1) … (1) ?M= A R1 r Putting i = 100 , we get … (2) r ? R= I + 100 = l + i [(1 i)n 1] M= A 111 A = = i [(1+i)” –1] Example 1:How much will an annuity of ` 100 amount to in 20 years at 4½% interest? Answer to the nearest paisa. Given that log 1.045 = .0191163, log 24.117 = 1.3823260 Solution: Using formula (1) and substituting A = 100 and R = 1 + 4½/100 = 1 + .045 = 1.045. We get, M = A (R n 1) z 1 [(1.045)20 1] = 100 1.045 1 20000 = 19 (1.045)20 – 1]

Now, (1.045)20 3. Antilog [20 log (1.045)] 4. Antilog [20 (.0191163] 5. Antilog [.382326] 6. 2.4117 20000  1.4117 M= 9 3. 3137.11 Amount is ` 3137.11 CU IDOL SELF LEARNING MATERIAL (SLM)

44 Business Mathematics and Statistics Example 2: How much will an annuity of ` 500 amount to in 20 years at 3½% interest? Answer to the nearest paisa. Given that log 103.5 = 2.0149403 and log 19.89784 = 1.2988060 [I.C.W.A.] Solution: Using formula (1) (R n 1) M= A R1 [(1.035)20 1] = 500 1.035 1 [(1.035)20 1] = 500 .035 1 Now, (1.035)20 z Antilog [20 log (l.035)] z Antilog [20 (.0149403)] z Antilog (0.2988060) z 1.989784 500(1.989784 1)

?M= 0.035 500  989.784 = 35 100 × 141.397714 14,139.7714 6. Amount is ` 14,139.77 3.3 Present Worth To find the present worth (value) of an immediate annuity of ` A, payable yearly for n years, at r% per annum compound interest. Suppose P1 is the present value of the first payment (say A) that is made at the end of the first year, then CU IDOL SELF LEARNING MATERIAL (SLM)

Annuities 45 A = P1R P1 = A/R Putting V = 1/R, we get P1=AV Similarly, if P2 is the present value of the second payment of ` A that is made at the end of two years, then A = P2R2 P2 = A/R2 = AV2 Proceeding in this manner, we get P3 = A/R3 = AV3 ......................... ......................... Pn = A/Rn = AVn The present value of the annuity is P = P1 + P2 + …+Pn \\emdash A/R + A/R2 + ...... + Pn

\\emdash A/R + A/R2 + ................. + A/Rn \\emdash AV + AV2 + ....... + AVn \\emdash AV(1 + V + V2 + ...... + Vn–1)  1 Vn  ? P = AV   … (3)   … (4) 1V A1Rn Also P =   R 1R1   1 Rn  = A   R1  A  R n 1 ?P= n    R R1 CU IDOL SELF LEARNING MATERIAL (SLM)