9 Arithmetic and Geometric Sequence


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1 AAU  Busness Mathematcs I Lecture #5, Aprl 4, Arthmetc and Geometrc Sequence Fnte sequence: 1, 5, 9, 13, 17 Fnte seres: Infnte sequence: 1,, 4, 8, 16,... Infnte seres: When Gauss was a boy, the teacher ran out of stuff to teach and asked them, n the remanng tme, to compute the sum of all the numbers from 1 to 40. Gauss thought that 1+40 s 41. And +39 s also 41. And ths s true for all the smlar pars, of whch there are 0. So... the answer s 80. One can wonder what would have happened had the teacher asked for the sum of the numbers from 1 to 39. Perhaps Gauss would have noted that 1+39 s 0, as s +38. Ths s true for all the pars, of whch there are 19, and the number 0 s left on ts own. Nneteen 40 s s 760 and the remanng 0 makes 780. Example: Let s consder the seres If we add the frst term to the last we get 0. If we add the second term to the secondtolast we get 0 agan. Now we see that the seres adds up to four 0s, or 80. Now the queston s  wll ths trck work for all seres? If so, why? If not, whch seres wll t work for? Answer: It wll work for all arthmetc seres. The reason that the second par added up the same as the frst par was that we went up by two on the left, and down by two on the rght. As long as you go up by the same as you go down, the sum wll stay the same and ths s just what happens for arthmetc seres. Arthmetc sequence: s a sequence a 1, a,... a n such that a n a n 1 = d for all n. So the dstance between the two followng elements of the sequence s constant. For example: 1,,3,... (d = 1);,4,6, (d = ); 0,3,6, (d = 3) Arthmetc seres: s a sum of elements of arthmetc sequence. The sum s gven by: Σ n =1a = S n = (a 1 + a n ) n Example: Fnd the sum of the followng arthmetc seres: a 1 =1, a 5 =17, d=4, n=5. Σ n =1a = Σ 5 =1a = (a 1 + a n ) n = (1 + 17) 5 = 45 45
2 Example: Now consder the followng sum: Clearly the arthmetc seres trck wll not work here: s not We need a whole new trck. Here t comes. S = where S s the sum we are lookng for. Now, we multply the whole equaton by 3: 3S = Now let s subtract the frst equaton from the second one: S = 4374 whch means that S = 186. Ths trck wll work for all geometrc seres. Geometrc sequence: s a sequence a 1, a,... a n such that between the two followng elements s constant. For example:,4,8,... (r = ); 1,3,9,7,51 (r = 3) a n a n 1 = r for all n. So the rato Geometrc seres: s a sum of elements of geometrc sequence. The sum s gven by: S n = a 1 1 r n 1 r Example: Fnd the followng sum: In ths example, a 1 = 1, a 5 = 16, n = 5, r =. S n = a 1 1 r n 1 r = = 31 Problem: Fnd the sum of the numbers: 3, 7, 11, 15,..., 99. Soluton: In ths example, a 1 = 3, a n = 99, and d = 4. To fnd n we solve the followng equaton: 3 + (n 1)4 = 99 to get n = 5. Then the sum of the numbers s: Σ n =1 = (a 1 + a n ) n = (3 + 99) 5 = 175 Example: Infnte geometrc seres. Fnd the sum of the numbers: 1, 1, 1, In ths case r = 1 < 1. If r < 1 then the sum of nfnte geometrc seres exsts and t can be found as: Σ =1 = a r = 1 1 = 1 (If r > 1 then the sum s equal to nfnty.) 46
3 Example: Fnd a decmal form of the number Notce that: = So r n ths case s 1 10 and hence: Σ =1 = a r = = 7 9 Ths means that 7 9 = Smlarly, = 09 = ; = Problem: Fnd the sum of the frst 30 terms of Soluton: We know that n = 30 and a 1 = 5, and we need the 30th term. Use the defnton of an arthmetc sequence. a 30 = = 11. Therefore, S 30 = 30(5 + 11)/ = Fnancal Mathematcs, Smple and Compound Interest The central theme of these notes s emboded n the queston, What s the value today of a sum of money whch wll be pad at a certan tme n the future? Snce the value of a sum of money depends on the pont n tme at whch the funds are avalable, a method of comparng the value of sums of money whch become avalable at dfferent ponts of tme s needed. Ths methodology s provded by the theory of nterest. A typcal part of most nsurance contracts s that the nsured pays the nsurer a fxed premum on a perodc (usually annual or semannual) bass. Money has tme value, that s, $1 n hand today s more valuable than $1 to be receved one year hence. A careful analyss of nsurance problems must take ths effect nto account. The purpose of ths secton s to examne the basc aspects of the theory of nterest. A thorough understandng of the concepts dscussed here s essental. In ths last context the nterest rate s called the nomnal annual rate of nterest. The effectve annual rate of nterest s the amount of money that one unt nvested at the begnnng of the year wll earn durng the year, when the amount earned s pad at the end of the year Smple and Compound Interest Interest Interest s a fee pad on borrowed captal. The fee s compensaton to the lender for foregong other useful nvestments that could have been made wth the loaned money. Instead of the lender usng 47
4 the assets drectly, they are advanced to the borrower. The borrower then enjoys the beneft of the use of the assets ahead of the effort requred to obtan them, whle the lender enjoys the beneft of the fee pad by the borrower for the prvlege. The amount lent, or the value of the assets lent, s called the prncpal. Ths prncpal value s held by the borrower on credt. Interest s therefore the prce of credt, not the prce of money as t s commonly  and mstakenly  beleved to be. The percentage of the prncpal that s pad as a fee (the nterest), over a certan perod of tme, s called the nterest rate. (wkpeda.org) Smple nterest Smple Interest s calculated only on the prncpal, or on that porton of the prncpal whch remans unpad. The amount of smple nterest s calculated accordng to the followng formula: where A = P (1 + n) A s the amount of money to be pad back P s the prncpal s the nterest rate (expressed as decmal number) n the number of tme perods elapsed snce the loan was taken Smple nterest s often used over short tme ntervals, snce the computatons are easer than wth compound nterest. For example, magne Jm borrows $3,000 to buy a car and that the smple nterest s charged at a rate of 5.5% per annum. After fve years, and assumng none of the loan has been pad off, Jm owes: A = 3000( ) = 935 At ths pont, Jm owes a total of $9,35 (prncpal plus nterest). Compound nterest In the short run, compound Interest s very smlar to Smple Interest, however, as tme goes on dfference becomes consderably larger. The conceptual dfference s that the prncpal changes wth every tme perod, as any nterest ncurred over the perod s added to the prncpal. Put another way, the lender s chargng nterest on the nterest. A = P (1 + ) n In ths case Jm would owe prncpal of $30,060. Generally: If a prncpal P s nvested at an annual rate r (expressed n decmal form) compounded m tmes a year, then the amount A n the account at the end of n years s gven by: ( A = P 1 + ) nm m 48
5 10. Savngs, Loans, Project Evaluatons Tme value of money The tme value of money represents the fact that, loosely speakng, t s better to have money today than tomorrow. Investor prefers to receve a payment today rather than an equal amount n the future, all else beng equal. Ths s because the money receved today can be deposted n a bank account and an nterest s receved. Present value of a future sum where: P V = F V (1 + ) n P V s the value at tme 0 F V s the value at tme n s the rate at whch the amount wll be compounded each perod n s the number of perods Present value of an annuty The term annuty s used n fnance theory to refer to any termnatng stream of fxed payments over a specfed perod of tme. Payments are made at the end of each perod. P V (A) = A 1 (1 + ) + A 1 (1 + ) A 1 (1 + ) = A (1+) n n (1 + ) = A 1 1 (1+) n where: P V (A) s the value of the annuty at tme 0 A s the value of the ndvdual payments n each compoundng perod s the nterest rate that would be compounded for each perod of tme n s the number of payment perods Present value of a perpetuty A perpetuty s an annuty n whch the perodc payments begn on a fxed date and contnue ndefntely. It s sometmes referred to as a perpetual annuty (UK government bonds). The value of the perpetuty s fnte because recepts that are antcpated far n the future have extremely low present value (present value of the future cash flows). Unlke a typcal bond, because 49
6 the prncpal s never repad, there s no present value for the prncpal. The prce of a perpetuty s smply the coupon amount over the approprate dscount rate or yeld, that s P V (P ) = A (1 + ) + A (1 + ) + A (1 + ) = A = A Future value of a present sum F V = P V (1 + ) n Future value of an annuty F V (A) = A(1 + ) (n 1) + A(1 + ) (n ) 1 (1 + )n A = A 1 (1 + ) = A(1 + )n 1 Example: One hundred euros to be pad 1 year from now, where the expected rate of return s 5% per year, s worth n today s money: P V = F V (1 + ) n = = So the present value of 100 euro one year from now at 5% s Example: Consder a 10 year mortgage where the prncpal amount P s $00,000 and the annual nterest rate s 6%. What wll be a monthly payment? The number of monthly payments s n = 10 years 1 months = 10 months The monthly nterest rate s = 6% per year 1 monhs per year = 0.5% per month P V (A) = A 1 1 (1+) n A = P V (A) 1 1 (1+) n (1 + ) n = P V (A) (1 + ) n ( )10 A = ( ) 10 1 = $0.41 per month. 50
7 Example: Consder a depost of $ 100 placed at 10% annually. How many years are needed for the value of the depost to double? F V = P V (1 + ) n 00 = 100( ) n 1.1 n = = ln 1.1 n = ln n ln 1.1 = ln n = ln = 7.7 years ln 1.1 Example: Smlarly, the present value formula can be rearranged to determne what rate of return s needed to accumulate a gven amount from an nvestment. For example, $100 s nvested today and $00 return s expected n fve years; what rate of return (nterest rate) does ths represent? F V = P V (1 + ) n 00 = 100(1 + ) 5 (1 + ) 5 = = (1 + ) = 1/5 = 1/5 1 = 0.15 = 15% Example: A manager of a company has to choose one of two possble projects. Project A requres mmedate nvestment $500 and yelds returns $00, $300, and $400 n the followng three years. For project B t s necessary to nvest $400 now and the expected returns n the next three years are $400, $100 and $50. Supposed that an nterest rate s 10%. Whch project should the manager choose? Havng tme value of money n mnd, manager should choose project wth a hgher present value. P V A = (1 + ) = (1 + ) = 3 = = 30 P V B = (1 + ) = (1 + ) = 3 = = 85 Project A has a hgher present value and hence should be chosen. 51
8 Cash flow: Some of the problems consst of analyzng specal cases of the followng stuaton. Cash payments of amounts C 0, C 1,..., C n are to be receved at tmes 0, 1,..., n. The payment amounts may be ether postve or negatve. A postve amount denotes a cash nflow; a negatve amount denotes a cash outflow. There are 3 types of questons about ths general settng. (1) If the cash amounts and nterest rate are gven, what s the value of the cash flow at a gven tme pont? () If the nterest rate and all but one of the cash amounts are gven, what should the remanng amount be n order to make the value of the cash flow equal to a gven value? (3) What nterest rate makes the value of the cash flow equal to a gven value? Problem: Instead of makng payments of 300, 400, and 700 at the end of years 1,, and 3, the borrower prefers to make a sngle payment of At what tme should ths payment be made f the nterest rate s 6% compounded annually? Soluton: Computng all of the present values at tme 0 shows that the requred tme t satsfes the equaton of value: = t 1.06 t 1400 = = = log 1.06 t = log t = log log Problem: An nvestor purchases an nvestment whch wll pay 000 at the end of one year and 5000 at the end of four years. The nvestor pays 1000 now and agrees to pay X at the end of the thrd year. If the nvestor uses an nterest rate of 7% compounded annually, what s X? Soluton: The equaton of value today s: = X = X X = = = 5737 Thus X = Problem: A three year certfcate of depost carres an nterest rate 7% compounded annually. The certfcate has an early wthdrawal penalty whch, at the nvestors dscreton, s ether a 5
9 reducton n the nterest rate to 5% or the loss of 3 months nterest. Whch opton should the nvestor choose f the depost s wthdrawn after 9 months? After 7 months? Soluton: Frst of all notce that t does not matter at all what s the amount nvested n ths case. The answer wll be the same whether we depost $1 or 350$ or any other amount of money. If the depost s not wthdrawn for the whole 3 years than t would brng: P P ( ) 3 If the depost s wthdrawn after 9 months n whch opton do we lose less? reducton n nterest: P P ( ) 9/1 = 1.037P loss of 3 months nterest: P P ( ) 6/1 = P In ths case reducton n the nterest s better because we get hgher amount of money wth ths opton. If the depost s wthdrawn after 7 months n whch opton do we lose less? reducton n nterest: P P ( ) 7/1 = P loss of 3 months nterest: P P ( ) 4/1 = P In ths case we should choose the second opton  lose 3 months nterest because ths way we get hgher amount of money Bond Prcng In fnance, a bond s a debt securty, n whch the ssuer (borrower) owes the holders (lenders) a debt and s oblged to pay nterest (the coupon) and to repay the prncpal at a later date  maturty. Par Value (as stated on the face of the bond, F ) s the amount that the ssung frm s to pay to the bond holder at the maturty date. Coupon Yeld s smply the coupon payment (C) as a percentage of the face value (F ). Coupon yeld = C/F 53
10 Current Yeld s smply the coupon payment (C) as a percentage of the (current) bond prce (P ). Current yeld = C/P. Yeld to Maturty (YTM) s the dscount rate r whch returns the market prce of the bond. YTM s thus the nternal rate of return of an nvestment n the bond made at the observed prce. Snce YTM can be used to prce a bond, bond prces are often quoted n terms of YTM. Whatever r s, f you use t to calculate the present values of all payouts and then add up these present values, the sum wll equal your ntal nvestment. In an equaton, P = C(1 + r) 1 + C(1 + r) C(1 + r) n + F (1 + r) n = C[1 (1 + ) n ] where C = annual coupon payment (n dollars, not a percent) n = number of years to maturty F = par value P = purchase prce + F (1 + ) n Problem: Suppose your bond s sellng for $950, and has a coupon rate of 7%; t matures n 4 years, and the par value s $1000. What s the YTM? Soluton: The coupon payment s $70 (that s 7% of $1000), so the equaton to satsfy s 70(1 + r) (1 + r) + 70(1 + r) (1 + r) (1 + r) 4 = 950 We are not really gong to solve ths, but the result s that r equals 8.53% (If you want, you can plug ths number back nto equaton to make sure t s correct). Problem: A $5,000 bond pays the holder an nterest rate of 10% payable semannually. The bond wll be redeemed at par n 10 years. An nvestor wants to purchase the bond on the bond market to yeld a return of 1% payable semannually. What would be the purchase prce of the bond? Soluton: Snce the bond pays 10% on $5,000 semannually, the regular nterest payment wll be: C = = 50 From the nformaton gven, the remanng number of nterest perods s 0. The redempton value of the bond n ten years s the par value or the face value of the bond, $5000. Now to compute the purchase prce, we must calculate the present values of the payments and the redempton value. Snce the yeld rate s the rate the nvestor wants to receve, t s the rate we 54
11 must use to fnd the present values n determnng the purchase prce. Substtutng the values nto our formula, we have: P = C[1 (1 + ) n ] = $, $1, = $4, F (1 + ) n = 50[1 ( ) 0 ] ( ) 0 = Problem: What s the prce of the followng quarterly bond? Face value: $1,000 Maturty: 10 years Coupon rate: 10% Dscount rate: 8% Soluton: /4 [ 1 ] 1 + ( /4) = $ ( /4)
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