172A Ch 2 Q

172A Ch 2 Q - 102 Chapter 2 Equations of value and yield...

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Unformatted text preview: 102 Chapter 2 Equations of value and yield rates exam-LE are Inveshnent period not equal to one year _' Problem: Mohammed had 520.000 io ltis investment account on August 15, 1999. On August 15, 2000 his balance was $11,200 and he deposited an additional $5,000, giving him a new balance of $25,201 Clo August ‘.l5T “2.001l Mohammed's account had a balance of $21,300. Assuming that there are no other contributions to the accident, find the annual time-weightad'yield and note that it is very close to the annual dollar-weighted yield. ' Solution The balance grows as follows: $20,000 $21,200 $27,300 l / $20,200 Simona mom {in the other hand. the dollar-weighted yield i satisfies the equation of value ' 1 Therefore a... = [(flumxmmfii — 1 :1 .asaasstss. sedation + a}? + $5.000{1 + q = season. 'This equation may be solved using the quadratic equation, andi = .05 is the only positive yield rate. I I EHMPLE 2.15 Time-weighted yield less than dollar-weighted yield Problem: Astute Mr. Haywood notices that although the “Tomorrow F'und*I has an excellent performance history, it performed less well when tlte price of gasoline experienced a sharp rise- l-te decides to invest in the fund but, to the extent possible, move his money away from the fund during periods when he anticipates a sharp increase in gasoline prices. Go January 1, Mr. Haywood ' deposits $103,010 in the fund. Cln March 1 his balance is $102,111), and he withdraws $50,000. [in May 1 ltis balance is $52,501 and he deposits $50,000. At the end of the year Mr. Hoyt-Hood’s fund balance is $111,000. Find the time—weighted yield for the “Tomorrow Fun d" for the year, and show that the is lower than Mr. l-laywood‘s dollar-weighted yield. dew-rev ;:_,. =,.a:'.-'_I:-i.sc.u—r-s-' '..-|-:..-vy:::..-:.'.a..-.-....1 “jinn-missu- Sectian 23 Problems, Chapter 1 103 '. solution. .The balance grows as follows: firs-at $100,000 ——r $102,000 I $52,500 $111,000 $51010 - $102,500 I tii1_0.l_»_' 3 nightlife i... = rs. = {Mnfl-‘flnimmi —' 1 e 115205379. The 5 [03.110 £52,000 £102,503 " _ dollar-weighted yield i satisfies ' _.‘.":I':.' '- It itt11.Dflt}= Emmott + i] — $50,000Ij1 + ql—i -_:- sso,ona{1_+ rye. tifiqgoe $100,00tlfl + i} - ssanoou + £30 + 550,000{1 + i]%.Then I? Inc ' I set $110,534.53 -r-'.' 111,000. Therefore, Mt. Haywood‘s dollar-weighted yield is higher than Em. In fact, using the “guess and check" method or Newton’s method, one may determine that Mr. Haywood has a yield rate of approxi- matety .1103. More easily, the Eli-t ll Plus calculator Cash Flow worksheet mayr he used to find a monthly yield rate of 351201439 is' equivalent to an annual yield rate of 1103092019110. The reason that Mr. Hayumd‘s dollar—weighted yield exceeds the Tomorrow Fund’s Lime-weighted yield is that he timed his deposits and withdrawals well. Had they been poorly timed, his dollaraweighted yield would have been worse than the Fund's time-weighted yield. I 2.3 PROBLEMS, CHAPTER 2 11-91 Chapter 2 writing problems :- 11] [following Section (2.3)] Consider the equation Elem“ + o“ — smart + if + sauna + i} = asses. _ Describe a financial situation for which this is the “associated time 4 Equation of value. Give detaiJs explaining the sources of any deposits and the reasons for any withdrawals. [following Section [2.3)] Consider the equation .15 30 . sauna“ + if“ = E $10,000u". it I m=l 104 Chapter 2 Equations of value and yield rates Describe a financial situation for which this is tlte associated time 35 equation of value. (3) [following Section (23)] Write an adverdmm ent for an investment fund, giving annual yields earned by the fund over each of five years and an annual time—weighted yield for the period. {4) Learn about I.l'tree mutual funds and the portfolio focus of each fund. For each fund, note the annual yield rates over various time periods. include the period from the date of date of inception to a recent date and a recent five—year period. Comment on similarities and differencrs in the performance of U113 funds. Speculate on die census of any performance discrepancies. {2.2} Equations of value for intrestments involving a single deposit made under compound interest [1] Mr. Lopes. deposits SE in an accent paying 4% annual effective discount The balance at the end of three years is $002. Find It". (2} Marianne deposiH $1,000 in a live—year certificate ofdeposit. At maturity the balance is $2,530.64. Find the annual effective rate of interest governing the account. {3} Susanne remembers that her only deposit into her savings account was a $1,000 deposit- She knows that the account has had a constant nominal interest rate of 3.2 0's convertible monthly and that the balance is now 51,965.35- How long ago did Suzanne matte her deposit? [4) Use the rule of seventy-two to approximate the length of time it tallies money to double at an annual effective interest rate of 5% and then at an annual effective rate of 10%. Then find d'te exact time it takes for ntoney to double at each of these interest rates. [5] Derive a "rule of rt" to approximate the length of tinte it takes for money to triple. As in the derivation of the “rule of seventy—two," your rule should be derived to give an especially good estimate wlten the annual effective interest rate is 0%. After you have stated your role, compare the approximations it gives for annual effective interest rates of 40.“: and iii—ta with the true values at these rates. {2.3} Equations ofvalue for investments with multiple contributions fl} Sidney borrows $ 12000. The loan is governed by compound interest and the annual effective rate of discount is 6%. Sidney repays $4,000 at thfi end of one year, It at the end of two years, and $3,000 at the end of three years in order to exactly pay oil' the loan. Find Jt'. Section 2.0 Problems, Cl'tapterl 105 _ (2} Rafael opens a savings account with a deposit of $1,501 He deposits . 'r_ . -. $500 one year later and $1,000 a year after that. Just after Rafael's ._ deposit of $1,000, the balance in his account is $3,0i‘0. Find the annual effective interest rate governing the account " {a} Esteban borrows $10,000, and the loan is governed by compound interest at an annual effective interest rate of 0%. Esteban agrees to repay d'tt: loan by making a payment of $10,000 at the end of 1" years and a payment of$12,000 at the end of 2'." years. Find '1". (4} Shaltm'i opens a savings account with a deposit of $3,500. She deposits $500 six months later and $311} nine months after opening the aceounL The balance in Shakari’s account one year after she opened it. is $5,012. Assuming that the account grows by compound ittterest at a constant annual effective interesr rate it, find i. {5] A loan is negotiated with the lender agreeing to accept $0,000 after __ one yEar, $0,0tlfl‘aftet two years. and $20,000 after four years lt1 full repayment of the loan. The loan is renegotiated so that the borrower 'maltes a single payment of $37,111) at time T and this results ill the same total present value of payments when calculated using an annual effective rate of 5%. Estimate '1" using the method of equated time. Also find i" exactly. {t5} Anne and Frank Smith each borrow $12,000 from their father. Anne and Mr. Smith have agreed that she will repay her loan in full by paying $60111 in two years and $0000 in four years. Franlt prefers to make one lump payment of $15000 to fully repay his loan. 1|When should he make that payment so that he and his sister will each have the same effective interest rate?I {1"} Let 111,51. ...,lr,, be positive real numbers. Set a = (Eiflarlffl. 013 l 1.. arithmetic mean of the numbers, and G = (Hg; 1 ed}, the geometric mean of the numbers. The objective of this problem is to establish that d .3 G and that A 3:- 6 whenever i). , fag, , is, are not all equal. [a] Write the poiat—slope equation for the tangent line to y = tax at (A, In A ]. (in) Use {a}. and concavity to show that. 1111' =5 A—1{.r — A] + Inst for all positive 1. Moreover, shoalr that equality holds if and only. if x = A. in] Use (to to provetltatlnG 2: igy=1r|rsfl — a} + tan and that this is a strict inequality unless all the his are equal. (d) straw that 23:, a-lra — A} + In n =1“. {e} Concludc that G c A and G if. A unless all the his are equal. 106 Chapter 2 Equations of value and yield rates [0) Let Cg denote the contribution in cents made at distinct times rt, k = 1.2,... ,n. Suppose that these are all positive so that we have deposits, but no withdrawals. Then the (3,, ’s are positive integers. As in {2.3.9},1et r = lntjiflf—‘lfllfln u. As in {2.3.10}, lat? = Eggtgtja. The objective of dais problem is to use the result of Problem (2.3.?) to salablish tltat T 3 T and that this is a strict inequality unless a = . (a) Consider Cg quantities each equal to a“, = 1, 2. ...n. Then in all we are considering C = C}, + Cu + "If," quantities. Use the result of Problem [2.3.'F(e]] to show that Cris” + can“! + + Cynu‘l" 31;].— -C‘ and that this is a strict inequality unless I: = l. [in] Use {a} to show that the present value of the deposits is at least as large as the present value given by the method of equated time and that it is strictly larger Unless n = l . {1:} Show thatT .3:- T with strict inequality unless u = 1. [9} Suppose that you pay $1,000 at time 0, get $4,000 at time t, and pay $2.000 at time 1. Let Co = $l.000, C1 = —$4,000, and C1 = $2,000. Set C = C3 + Cl +171 = $1,001 — $4,030 + $2,000 = —$l,000. Find T such that getting an inflow of —C at time T has the same present value as the above sequence of financial transactions, assuming that the growth of money is governed by compound interest ati = 1%. Show that T is greater than the weighted average T = L—g'fl + Eel-1 + 5%- ['Ihis shows that Inequality {2.3.1 1} need not hold if you have a negative contn' bntiou.] {10) [recommended For those with a BA Il Plus calculator] Dax borrows $300,030 and the loan is governed by contpound interest at an annual effective interest rate of 4.15%. Dax agrees to repay the loan by ten equally spaced paymenL-a, the first 1' our of which are for $23,000 and the next six of which are for $40,000. When should he mate the first payment"? {11] Find the amount to be paid at the end of eight years that is equivalent to a paytnent of $400 now and a payment of $300 at the end of [our years {a} if 6% simple interest is earned from the date each payment is made and use a comparison date of right now. (b) iffi‘l’u simple interest. is earned from the date each payment is made and use :1 comparison date of eigltt years from now. Section 2.0 Problems, Chapter 2 10? ' [a] EIpifllt’l why the fact you get different answers in parts (a) and [b] does not contradict the fact that equations of value at dit‘l'ernnt times are equivalent equations. {:1} Repeat parts {a} and (b) except replace “simple interest” with f‘cornpound interest." '[calculus needed] Lise Newto n‘s method to solve the problem of Example (233}, More specifically, set fl!) = 525(1.1}—2' + 525(1.1}“’ — _:::'_":1,000, matte an initial guess T1 for a root T, and find a sequence of _1'_' __ approximations {T1} to T that allow you to obtain T to the nearest ' hundredth of apercent. Why might T. = .3 has reasonable initial guess":I {figs} investment return '-(1}'=Payrnents of $3,000 now and $3,000 eight years from now are equivalent ' 's to a payment of $10,000 four years from now at either rate r' or rate j. silt-:1 Find |.i — jl. Explain why the yield rate is not unique in this case. [2] Sumess, Inc- enters into a. financial arrangement in wltich it agrees to - pay $100,000 today and $101,000 two years from new in exchange for ' ' $200,000 one year from now. Show that there is on yield rate that can be assigned to this two—year transaction. {3} Sigmund, Inc. agrees to pay $150,000 today and $40,000 [our years from today in remrn for $110,000 two years from today. What is the yield rate for this fonriyear financial arrangement"? [4] Ftrrns A, B, C, and D enter into a financial arrangement. Money flush firm A will pay expanding firms B and C each $1,000,030 today. B will pay D $2,200,000 three years from today. C will pay 13 SSW,“ two years from today and D $350,000 two years from today. Finally, D will pay A $3,200,001 six years from today. Calculate the yield rate or interest rate, to fl'te nearest hundredth of a percenL. that each firm experiences over llle period of their involvement {6 years [or A, 3 years for B, 2 years for C. and 4 years [or D]. [5) 3015a instcsts $0,572.00 at r = n and season at r = 1. in return, she receives $21,004 at r = 1 and $10,000 at r = 3. Write down a time 0 equation of value and verify that it is satisfied for u = .94, n = .95, and . " U = .05. Find the corresponding threc yield rates. {5] Pedro inves1s $100,000 at t' = 0 and $60,000 at t = 2. In return he gets $60,000att = 1 and $116,500 at! = 3. Write down a time 3 equation of value describing Pedro‘s investment. Explain why there is a unique yield rate and find it. 1ufl Chapter: Equations ot value and yield rates in Show thatil' —1 a y a egg: Cult + ro-rs = Egg Catt + pie and Bull]. Bali} , .,.. B,n_][r'] are all positive [with Huit'} as given in [24.8)], then j = r'. (0] [remmmended For those with a BA ll Plus calculator] On January I, Ezequiel opens an account at Friendly Bank. His opening deposit is for $50 and he makes deposits at the end of each quarter for [our years, then maltes no tnore deposits. He closes the account exactly seven years after he opens it and receives $3423.23. Find his annual yield rate for this seven—year period if his quarterly deposits were $00 in the first year. $25 in the second year. $50 in the third year, and were successively $30!], $450. $000. and $240 in the fourth year. {‘3} [recommended for those with a BA ll Plus calculator] A loan of 520.0001: to be repaid by thirty-three ehd-of—rnonth payntents. The first payment is $400 and then eaclt payment is $25 more than the previous paymenL Find the annual yield rate correct to the nearest hundredth of a percent. (HINT: The Cash Flow worksheet only accepts twenty-four payments. or thirty—two if you have a BA l| Plus Professional calculator. If you are nothing with the EA ll Plus calculator. suppose that the payments beyond the twenty—fourth+ which you do not have registers to accommodate, are made along with the twenty—fourth. Now use the "gums and check" method, obtaining your first estimate by using - This is a challenging problem. especially for those of you with only 24 registers. Hoe-revermhen performing the Euccive calculations required by the "guess and check" method. you may make judicious u5e of the “PU subworI-tsheet to decrease your warlL] [2.5) Reinvestment considerations {1) Angela loans Kathy $0.000. Kathy repays the loan by paying $0,000 at the end of one and a half years and $4,000 at the end of three years. The money received at r = 1% is immediately reinvested at an annual eflective interest rate of “in. Find Kathyis annual clieclive rate of interest and Angela’s annual yield. [2] Kurt loans Randy 514.000.]1andy repays the loan by paying Kurt $4.005 at the end of one year and 56.000 at the end of two years and as well as at the end of three years. The money received at r = 1 and at t = 1 is immediately reinvested at an annual effective interest rate of 3%. Correct to the nearest tenth of a percent. tind Randy’s annual reflective inlerest rate and Kurt‘s annual yield- (3} On January IS, 2000. Enterprise A loans $15,000 to Enterprise l3 and 511000 to Enterprise (.7. Enterprise 13 repays Enterprise A $2,000 on January 15. 2002 trnd this money is reinvested at a 5‘55. annual et'feclllt‘i Section 2.0 Problemsr Chapter 2 109 rate. Enterprise [3 repays Enterprise A $22,500 on January 15, 2W4. What is the annual yield received by Enterprise A over the four-year interval. Compare it to the annual eflective interest rates paid by Enterprises H and C. {2,6} Approximate dollar-weighted yield rates {1} Sandra invests 310.332 in the Wise Investment Fund. Three moutlts later her balance has green to $11,902 and she deposits $2,000. Two months later her fund holdings are 514.300 and she wideraws $2,000. - Two years after her initial investment she learns that her holdings are worth $12,500. {a} Write an equation of value involving die exact dollar—weighted annual yield r' over the twosyeat period. [b] Compute the.appro:ltimate dollar—weighted annual yield over the investment period using {2.6.5} and again using {2.6.3}. {2) On February 1, Arsltalc’s investment account has a balance of $19,300. He deposited $1,200 on April 1 and $2,000 on May 1. He withdrew $0,400 on J uly I. On November 1, Arshalt's balance was $14,320. Find Anhalt’s approximate dollar—Weighted annual yield for this nine—month period using [2.0.5]. ' [3) Franklin's investment fund had a balance of $290000 on January 1. 1995 and a balance of $440,000 two years later. The amount of interest earned during the two years was $34.00}. and the annual yield rate on the fund was 5.4%. Estimate the [dollar-weighted} average date of contributions to die account. [4] The investment balance of a litnt is 3155,00101100 at the beginning of a two—year period and $2,000,000 at the end. The firm makes a. single nontribution during the two-year interval of $200,000. What is the difference between the approximate annual dollar—weighted yield earned by the firm if the contribution is made after t‘i months as opposed to it being made after one year? {2.2} Food perfonnante ill CHI January '| :1930, Antonio invests 59.400 in an investment land. [in January 1, ltlflEl his balance is $0,600 and he deposit: $2,400. On July 1. 1930 his balance is M49100 and he withdraws $1,000. [in January I, 1092 llis balance is 5P. Express: his annual time-weighted yield as a function Of P. {2} Arthur buys Slllllllwortlr ol' stool-t. Six months later, the value of the stock has risen to $2,2tltl and Arthur buys another 3 | .000 worth of stock. 11u ChaptErZ Equations civalue and yield rates After another eight months, Arthur’s holdings are worth $2.900 and lie _ sells off $000 of them. Ten months later, Arthur finds that his stock has ' a value of $2,100. - [a] Colupule the annual time-weighted yield rate of the stock over the two-year period. [it] Compute the annual dollar-weighted yield for Arthur over the _' 2 two-year period. {c} Should the answer in part {a} or part {b} be larger? IWhy? {3} Bright Future [nvesnnent Fund has a balance of “305,000 on January I l. {in May 1, the balance is $1,230,001 Immediately alter this balance is noted. $300,000 is added to the fund. If there are no further contribuLlons to the fund for the year and the lime-weighted annual yield for the fund is 16%, what is the fund balance at the end of die year? Chapter 2 review problems [I] Sohail makes an initial investment of $20,013. In return, he receives $4,003 at the end of one year and another $10,003 at the end of three years. [a] fissilming that the investment is made at simple inlerest at rate 1", write down an equation of value for the investment and find r. {b} Assuming that the investment is made at compound interest at effective interest rate t', Inil-rile down an equation of value for the invesnnent and justify the smlernent that there is a unique yield rate. Use the “guess and checlt method" to estimate f to the nearest hundredth of a percent. [c] Starting with Ilie same initial guess I'ori that you Md in {b}, check you answer using Newlon‘s method. {d} [recommended for those with a 0A ll Plus caICulator] Use the Cash Flow mrltshe-et In find i to the nearest millionth. [1) {:1} Suppose now that invesunents are governed by compound interest at an effective inlerest rate i 3 0. 0y how much does Lhe sum of the time a value oil 55K paid at Lime 0 and the time a value of“: paid all tlute 2n exceed EEK? Express your answer as a function gfn} and show thalgtnj :1 0it'n :1 0. {b} Suppose that investments are governed by the simple interest accumulation function air} = l + rr. r :e 0. Does the sum of the time a value oIEK paid at lime0 and the time a value oIEK paid at time In exceed $2}: for all r and n? Justit'y your answer. Settion 2.8 Problems, Chapter 2 111 :. (3} Elyse invests $10,312 at I = 0. In return, she gets $3,000 a: I = 1 :._.' - '- $10,031:! at r = 2. Half ol' the time 1 payment, she reinvests at an annual Effective interest rate of 5%. What is her annual yield rate for the fir‘ELyear period? _ _ _ . Sports Manufacturing needs capital for expansion. It borrows 31.000000 *3 '[mm Venture Bani-t for three years at 6% nominal interest convertible _ quarterly, and $500.01} for live years from a private investor at a 5% ' :*'¢f|‘_eaclive discount rate. At 01c end of two years, Sports Manufacmring ' makes a $200,000 three-year loan to its supplier of titanium (for basehall hats] at Titi- annuaI effective interest. What annual internal rate of return should 5110115 Mantlfflcltu'ing report for the combined cashflows over the ' fiveryear period? :15] Abiyote invested $31,500 on January 1|, 1994 in the Utopia Fund. On - May 1. 1995, his balance was $28,212 and he withdrew $10,000. {in _ December 1, 1995-; his balance was $15,392, and he deposiled $3,000. On _ ' January 1, 199ir his balance was $30,309. [it] Find the annual time-weighted yield for the Utopia Fund for the three-year period Iron: January 1,1994 until January 1, 199?. [b] Find an approximate annual dollar-weighted yield received by Abiyote [or the three-year period from January 1, 1994 until Junuary 1. 1991' using (“2.155) [c] [recommended for those with a BA ll Plus calculator} Fmd the dollar- weighled yield received by Ahiyote for the threeryear period from January 1. 199-1 un tilJanuary 1, 199?, correct to the nearest millionth ofa percent. {d} Compare the nose-weighted yield experienced by the Utopia Fund and the dollar-weighted yield received by Ahiyote from his investr ment in the Utopia Fund Discuss why the inequality between them is in the direction it is. {El King and Dmitry are friends. They agree that Xiang will pay Dniitry ' ' Slim immediately and another $200 at the end of three years. In return. Dmitry will pay Xiang $15: in exactly one year and again at the end of ' exactly two years. {it} Find E ii 01c Lrensaction is based on compound interest at a nominal _ _ discount tale of 0% convertible monthly. -' {'1} If H = 000. is there a unique positive yield rate for the transaction? Justify your answer. ...
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172A Ch 2 Q - 102 Chapter 2 Equations of value and yield...

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