Document Sample

Annuities: Math 360 Chapter 2 Lecture 2 Annuities Introduction • An annuity is a series of periodic payments • For us, the payments are contingent only on the passage of time, not on certain events (i.e. all annuities are annuity certain) • We will need a simple, but very useful, result from algebra: Geometric Series 1 x x .... x X 2 n and so (1 x) X 1 x 1 x x .... x 2 n n 1 x x .... x x 1 x x .... x 2 2 n 1 x x 2 .... x n x x 2 .... x x n n 1 n 1 1 x n 1 1 x X k 0 x n k 1 x Example 2.1 • The federal gov’t sends Smith a family allowance payment of 30 every month for Smith’s child. Smith deposits the payments in a bank account on the last day of each month. The account earns interest at the annual rate of 9% compounded monthly and payable on the last day of each month, on the minimum monthly balance. If the first payment is deposited on May 31, 1998, what is the account balance on December 31, 2009, including the payment just made? Ex 2.1 Solution 140 total deposits first deposit 1 imonthly i .09 .0075 12 12 value at time of last deposit 30(1.0075)139 TotalValue 30 30(1.0075) 30(1.0075) ... 30(1.0075) 2 139 1 1.0075140 1 1.0075 7385.91 30 Level Payment Annuities • Number of payments in series of payments is called the term of the annuity • Time between the successive payments is called payment period, or frequency • A series of payments whose value is found at the time of the final payment is known as an accumulated annuity immediate More Notation 1 (1 i ) n (1 i ) 1 n sn|i k 0 (1 i ) n 1 k 1 (1 i ) i or (1 i ) 1 i sn|i n Example 2.2 • What level amount must be deposited on May 1 and Nov 1 each year from 1998 to 2005, inclusive, to accumulate to 7000 on November 1, 2005 if the nominal annual rate of interest compounded semi-annually is 9% ? Ex 2.2 Solution 16 total deposits i ( 2) 0.09 X level amount deposited / year EoV : X (1.045) (1.045) .. (1.045) 1 15 14 X s16|0.045 22.719337 X 7000 X 308.11 Example 2.3 • Suppose that in Example 2.1, Smith’s child is born in April 1998 and the first payment is received in May (and deposited at the end of May.) The payments continue and the deposits are made at the end of each month until (and including the month of) the child’s 16th birthday. The payments stop after the 16th birthday, but the balance continues to accumulate with interest until the end of the month of the child’s 21st birthday. What is the balance in the account at that time? Ex 2.3 solution X (1.0075) 30 s192|.0075 12792.31 60 Some Arithmetic Value@ time _ n growth from n n k (1 i ) n 1 sn|i (1 i ) 1 i k k i (1 i ) n k (1 i ) k (1 i ) n k 1 (1 i ) k 1 i i i sn k |i sk |i or sn k |i sn|i sk |i (1 i ) n Example 2.4 • Suppose that in Example 2.1, the nominal annual interest rate earned on the account changes to 7.5 %, still compounded monthly, as of January 1 2004. What is the accumulated value of the account on December 31 2009? Ex 2.4 Solution Value 30 s68|.0075 (1.00625) s72|.00625 72 Example 2.5 • Suppose 10 monthly payments of 50 each are followed by 14 monthly payments of 75 each. If interest is at an effective monthly rate of 1%, what is the accumulated value of the series at the time of the final payment? • Answer: Value 50s24|.01 25s14|.01 1722.36 Present Value of an Annuity Immediate • Make a lump sum payment X now to receive periodic payments C , starting one period from today. If the market bears a constant interest of i, then Present Value of this Annuity Immediate is calculated as C C C X ... 1 i (1 i ) 2 (1 i ) n 1 n n C k 1 k C 1 .. n 1 C 1 1 n C Can|i i Loan Repayment – Ex 2.7 • Brown has bought a new car and requires a loan of 12000 to pay for it. The car dealer offers Brown two alternatives on the loan: • A.) Monthly payments for 3 years, starting one month after purchase, with an annual interest rate of 12% compounded monthly, or • B.) Monthly payments for 4 years, also starting one month after purchase, with annual interest rate 15% compounded monthly. • Find Brown’s monthly payment and the total amount paid over the course of the repayment period under each of the two options Ex 2.17 Solution a.)12000 P a36|.01 P 398.57 1 1 b.)12000 P2 a48|.0125 P2 333.97 TotalValue( A) 36 P 14348.52 1 TotalValue( B) 48 P2 16030.56 Present Value of an Annuity Some Time Before Payments Begin • Example 2.8: • Suppose that in Example 2.7 Brown can repay the loan, still with 36 payments under option (a) or 48 payments under option (b), with the first payment made 9 months after the car is purchased in either case. Assuming interest accrues from the time of the car purchase, find the payments required under options (a) and (b). • This is known as a deferred annuity. Ex 2.8 Solution 1 ' 9 12000 P 1.01 1.01 10 44 .. 1.01 P 1 ' 1 8 a36|.01 so 12000 P ' 8 431.60 1.01 a36|.01 1 12000 P ' 8 368.86 1.0125 a48|.0125 2 Duality of Total Value an Present Value for Annuities sn|i 1 i an|i n an|i sn|i n Perpetuities • What happens as the term approaches infinity? 1 n 1 a|i lim an|i lim n n i i Ex. 2.10 • A perpetuity immediate pays X per year. Brian receives the first n payments, Colleen receives the next n payments, and Jeff receives the remaining payments. Brian’s share of the present value of the original perpetuity is 40%, and Jeff’s share is K. Calculate K. Ex. 2.10 Solution 1 n PV ( Brian) X an|i X i 0.4 X a|i X 0.4 1 n 0.4 i X X PV (Colleen) X an|i 0.4 0.6 0.24 n i i X PV PV ( Brian) PV ( Brian) PV ( Jeff ) i X X K PV ( Jeff ) (1 0.4 0.24) 0.36 i i Annuity Immediate n 1 (1 i ) 1 n S _ 1 (1 i ) ... (1 i ) n|i i 1 v n a _ v v ... v 2 n n|i i S _ Payment Accumulation at Final Time n|i a _ Present Value one period before n|i first payment Annuity Due S n|i (1 i ) ... (1 i ) n (1 i ) S n|i _ _ value 1 period after final payment a n|i (1 i )a _ Present Value at time of first payment _ n|i 0 1 2 ... n -1 n _ _ _ a_ a n|i S n|i S n|i n|i Level Payment Annuities-Some Generalizations • What happens when compounding interest period and the annuity payment period do not coincide ? Ex 2.11 • Smith deposits quarterly of 1000 each over 16 years. Find balance after last payment. a) Interest quoted at 9% nominal annual rate compounded monthly b) Interest quoted at effective annual rate 10% Compounded interest paid for fraction of years. Ex 2.11 Soln 64 total deposit. Suppose effective quarter- rate is j. (1 j ) 1 64 _ value = 1000 S 64| j =1000 j a) 0.09 3 1 j (1 ) (1 0.0075) 3 12 so j 0.02266917. b) 1 j (1 0.01)1/ 4 so j 0.02411369 .

DOCUMENT INFO

Shared By:

Categories:

Tags:

Stats:

views: | 14 |

posted: | 5/18/2012 |

language: | |

pages: | 29 |

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.