Finance 402: Problem Set 1 Solutions
|
|
- Christopher Small
- 5 years ago
- Views:
Transcription
1 Fnance 402: Problem Set 1 Solutons Note: Where approprate, the fnal answer for each problem s gven n bold talcs for those not nterested n the dscusson of the soluton. 1. The annual coupon rate s 6%. A sem-annual coupon payment mples we get 6%/2 = 3% of the face value every sx months, or $3. Gven the APR s 12% per annum compounded sem-annually, the perodc nterest rate s 12%/2 = 6%. The cash flow stream conssts of 12 years of semannual coupons n the amount of $3 and 1 balloon payment at maturty for the face value of $100. The cash flows are presented graphcally n Fgure 1. Fgure 1: Cash Flow 0 $3 $3 $3 $3 $3 $3 + $100 Tme Perod We can recognze these cash flows as correspondng to two other types of bonds. The stream of coupons can be thought of as an annuty and the prncpal payment can be thought of as a zero coupon bond. Thus, we can value the coupon payments usng our annuty formula, the prncpal payment usng our zero coupon formula and add the results to get the value of the coupon bond. 1
2 Valung the annuty porton yelds The value of the zero porton s 1 (1 + ) 24 c 1 ( ) 24 = $ = $37.65 $ = $24.70 (1.06) Addng these two fgures gves the current value of the bond: $ The queston s askng us to fnd the payment that would make us ndfferent between recevng that lump-sum amount at the end of the year and monthly payments of $10,000 throughout the year. The most drect way to do ths s to smply compute the value of the monthly payments at the end of one year, whch amounts to computng the future value of an annuty. ( ) $10, 000 = $126, Thus, we are ndfferent between recevng a lump-sum payment of $126, at the end of the year or $10,000 each month a Usng the same logc as n problem 1., we can value ths bond as the sum of an annuty and a zero. For the annuty porton, the perodc payment, a, s $25, whch follows from a 5% coupon rate, $1,000 face value and sem-annual coupons. The perodc nterest rate = 3% follows from the market rate (R) beng 6% and a sem-annual compoundng perod (m = 2). Fnally, the total number of payments (or perods n the annuty) s 7 years (T ) tmes 2 2
3 payments per year so that N = T m = 14 perods. The present value of the annuty, usng our annuty formula s c 1 (1 + ) N $25 1 (1.03) 14 = $ The value of the zero porton of the bond s smply the present value of the par amount, or $1, 000 = $ (1.03) 14 The current prce s the sum of the annuty and zero values, or $ Thus, the bond s currently sellng at a dscount relatve to par (.e. face value). 3.b The effectve annual rate may be found from the followng relatonshp: (1 + r) = (1 + ) 2 Substtutng for, whch s just the perodc nterest rate of 3%, computed above, yelds an effectve annual rate r equal to 6.09%. Note that ths s not equal to the annual market rate of 6% and, n fact, s greater than 6%. Ths a consequence of nterest earnng nterest. That s, the nterest you earn on your money after the frst compoundng perod earns nterest durng the second compoundng perod. By ths logc, we would lke our nvestments to be compounded as frequently as possble. 3.c Ths prce was calculated n part 3.a: B 0 = $1, 000 = $ (1.03) 14 3
4 3.d The bond from part 3.a s now worth $25 1 (1.035) (the value of the annuty porton) plus = $ $1, 000 = $ (1.035) 14 (the value of the zero porton). The current prce s thus $ The bond from part 3.c was calculated n valung the coupon bond and s worth $ a The queston s askng to fnd the perodc payment of a 180-perod annuty wth a present value of $400,000. A summary of the nformaton usng our notaton s: T = 15 m = 12 N = 180 A 0 = $400, 000 R = = = The present value formula for an N-perod annuty s c A 0 = (1 + ) + c (1 + ) c 2 (1 + ) + c N 1 (1 + ) N = c ( ) 1 1 (1 + ) N 1 = c (1 + ) N Multplyng both sdes of above equaton by the perodc payment, c: c = A 0 1 (1 + ) N 1 (1+) N yelds an equaton for 4
5 Substtutng produces the answer: c = $400, ( ) 180 = $4, b Ths s most easly done n Excel but here s an explanaton of what s happenng. Consder the end of the frst month when the frst payment s due. The prncpal s $400,000 and one month s worth of nterest has accrued 400, = 3, You pay 4, whch means the prncpal s reduced by 4, , = 1, Contnue ths process 179 more tmes or let Excel do the rest of the work. 4.c After you make your 12th payment you are left wth an annuty that has 168 payments of $4, each. Thus, the prncpal left on your loan can be found by usng the annuty formula: $4, ( ) = $387, You apply your $200,000 bonus to ths amount so that the new prncpal remanng s $187, You can contnue wth the same amortzaton dea presented n 4.b n your Excel spreadsheet. You should fnd that you wll need another 56 months for a total payback perod of 68 months. The last month wll not requre the full $4,176.90, only $2, Let s just compare the monthly payments snce the terms (lengths) of the two contracts are the same. Ths requres fndng the fxed payment of an annuty. Usng formula A 0 = = c c (1 + ) + c (1 + ) c 2 (1 + ) + c N 1 (1 + ) ( ) N 1 = c 1 (1 + ) N 5 1 (1 + ) N
6 .and rearrangng above equaton yelds: c = A 0 1 (1 + ) N The present value of the annuty s smply the sze of the loan, $58,000, snce that s what you receve today. The perodc nterest rate,, s R/m or 5.5%/12. The maturty s 5 years wth monthly payments mplyng 60 perods. Thus, the monthly payments for the frst deal are 0.055/12 c = $58, = $1, ( /12) The monthly payments for the second deal, wth the $4,000 dscount subtracted from the loan prncpal and the hgher nterest rate, s 0.06/12 c = $54, = $1, ( /12) Therefore, snce the monthly payments are lower, you should take the $4,000 dscount and hgher rate. 6. Begn by fndng the monthly payment requred on an annuty wth present value $155, /12 c = $155, = $1, ( /12) (Ths s just algebrac manpulaton of the annuty formula) Next we need to fnd the amount of remanng prncpal after 10 years of repayment, whch s the value of an annuty wth 240 perods remanng and a perodc payment of $1, or $1, (1 +.11/12) /12 = $143, The prepayment penaltes and closng costs equal 5% $143, = $7, Ths mples that ther new prncpal amount, nclusve of penaltes and closng costs, s $143, $7, = $150,
7 To see f ths opton s better than the orgnal mortgage, we just need to see f the monthly payment s bgger or smaller than the orgnal. 0.09/12 c = $150, = $1, ( /12) The cost s less so they should refnance. 7. Let s value ths by creatng a replcatng portfolo. That s, let s buy some amount (possbly 0) of each bond so that the cash flows of our portfolo exactly match the cash flows of the bond we are tryng to value. By the prncple of no-arbtrage, the value of our replcatng portfolo should exactly equal the value of the bond. See Table 1. Table 1: Perodc Cash Flows Securty Cost Coupon Bond 5,000 5, ,000? Replcatng Portfolo 50 unts of bond A 5, $95.24 = $4, unts of bond B 0 5, $89.85 = $4, ,050 unts of bond C , $83.96 = $88, Total Cash Flows 5,000 5, ,000 $97, Snce the portfolo perfectly replcates the cash flows of the coupon bond, the values of the two must be equal by arbtrage arguments. Therefore, the bond prce s $97, Fgure 2 presents a tme lne of the cash flows: One way to value ths bond s to fnd the present value of the cash flows as of 3 months ago and then fnd the 3-month future value of ths number (ths mplctly assumes that the nterest rate dd not change between 3 months ago and today). Three months ago, our bond was maturng n sx and a half years, mplyng a total of 13 semannual payments of $0.12 mllon. 7
8 Fgure 2: Cash Flow 0 $0.12 $0.12 $0.12 $ $2 (Mllons) Tme Perod (Months) Recognzng that the value of the coupon bond s the sum of the annuty porton and zero porton we get: $ ( ) = $2.64 mllon (1.03) 13 Ths was the value of our bond, 3 months ago. To get the value today, we must fnd the future value of ths number, 3 months hence. $2.64 (1.03) 0.5 = $2.68 mllon An alternatve approach s to value the bond as of 3 months from today and then dscount ths value back to today. In 3 months we wll receve $0.12 mllon. Includng that ntal payment, whch need not be dscounted, the value of our bond n 3 months s: $ $ ( ) $2 = $2.72 mllon Now we need to dscount ths value back half a perod. $2.72 = $2.68 mllon 1.031/2 The value of the bond s $2.68 mllon 9. The tme lne of our cash outlays for the college expense are presented n Fgure 3. 8
9 Fgure 3: Cash Flow 0 $20,000 $20,000 Tme Perod a The present value of these cash flows, and thus the amount we must nvest today, s: $20, 000 $20, = $35, The dscount rate follows from the 8% yeld-to-maturty (R = 0.08) and sem-annual compoundng (m = 2), mplyng = 4%. 9.b Scenaro 1: Unchanged Interest Rates. If we spend $35, on 4-year zeros today, ths means we own bonds wth a total face value (or par amount) of: $35, (1.04) 8 = $48, At the end of year 1, we need $20,000 for the frst payment. Thus we need to sell some fracton of our bond holdngs; that fracton s determned by the market n the followng manner: par = $20, 000 (1) (1.04) 6 We are just solvng the zero formula n reverse to determne the par value we must sell n the market to receve $20,000 n year 1. The soluton s par = $25, After the sale, we are left wth bonds wth a total face value of $48, $25, = $23, At the end of year 2, we need to make another payment of $20,000. Solvng a smlar equaton to (1): par = $20, 000 (2) (1.04) 4 9
10 yelds a par value of $23,397.17, exactly equal to the par amount of our remanng bonds. So we smply sell all of the remanng bonds, collect the $20,000 and make the payment. Scenaro 2: Instantaneous Decrease n Interest Rates to 7%. Before solvng ths part, here s some ntuton. We have locked n (.e. guaranteed) a return of 8% whle the nterest rate has decreased to 7%. Ths s good for us because we are gettng a hgher return on our nvestment relatve to the new rate. Therefore, we should expect to have excess cash after we pay off our expenses, gven that the above dervaton shows we exactly meet our expenses when the nterest rate does not change. From above, we own bonds wth a par amount of $48, At the end of year one, we must go to the market and redeem some fracton of our holdngs to make the $20,000 payment. The par amount of bonds we need to sell s gven by: par = $20, 000 (1.035) 6 whch yelds $24, (note the use of the new dscount rate). Ths leave us wth bonds wth a total face value of $48, $24, = $24, In year two, we must go to the market agan and sell some fracton of our bonds to make the second payment of $20,000. Solvng par = $20, 000 (1.035) 4 yelds a par amount of $22, whch mples we wll have bonds wth a par value of $24, $22, = $1, left over. Note that ths s not the same as havng $1, n cash. To get cash we have to go to the market and sell these bonds. If we dd, we would receve: $1, , 44 (1.035) 4 = $1, Thus, we have an excess of $1, Scenaro 3: Instantaneous Increase n Interest Rates to 9% Before solvng ths part, here s some ntuton. We have locked n at 8% whle the nterest rate has ncreased to 9%. Ths s not good for us because we are gettng a lower return on our nvestment. Thus, we should expect to have a shortage of cash for our expenses. 10
11 Approachng the problem as before, we must sell: par = $20, 000 (1.045) 6 $26,045.20, par value, of our bonds. Ths transacton leaves us wth bonds totalng $48, $26, = $22, n par value. In the second year, f we sell all of our remanng bonds, we wll receve: mplyng a shortfall of $ $22, (1.045) 4 = $19, c Scenaro 1: Unchanged Interest Rates. From part 9.b, the present value of our expense s $35, Spendng ths amount on 1-year zeros mples a face value of $35, (1.04) 2 = $38, At the end of year one, when the bonds mature and pay ther par value, we must sell $20,000 and renvest the remanng $18, n one-year zeros agan. At then end of the second year, our bonds wll be worth $18, (1.04) 2 = $20, 000, exactly equal to the second payment. Scenaro 2: Instantaneous Decrease n Interest Rates to 7%. Before solvng ths part, here s some ntuton. We have locked n (.e. guaranteed) a return of 8% only for the frst year, whle the nterest rate has decreased to 7%. Thus, our nvestment n the second year wll suffer from ths lower rate of return and we should have a shortage of funds. We know from above that we are left wth $18, after the frst payment. Investng ths n one-year zeros at the new nterest rate of 7% ensures a payoff at the end of year two of equal to $18, (1.035) 2 = $19, Thus, we wll have nsuffcent funds n the amount of $20, 000 $19, = $ Scenaro 3: Instantaneous Increase n Interest Rates to 9%. Before solvng ths part, here s some ntuton. We have locked n (.e. guaranteed) a return of 8% only for the frst year, whle the nterest rate has ncreased to 11
12 9%. Thus, our nvestment n the second year wll beneft from ths hgher rate of return and we should have a surplus of funds. We know from above that we are left wth $18, after the frst payment. Investng ths n one-year zeros at the new nterest rate of 9% ensures a payoff at the end of year two equal to $18, (1.045) 2 = $20, Thus, we wll have excess funds n the amount of $20, $20, 000 = $ a Recall that the yeld (.e. yeld-to-maturty, YTM) s the one rate that dscounts all of the bonds cash flows to gve the present value of the bond. The yeld may be found by solvng the followng equaton. 100 = y + 8 (1 + y) (1 + y) 3 You may solve ths equaton by tral and error or usng Excel s solver. 1 The yeld s 8%. In fact, we ddn t really need to solve ths equaton to fnd the yeld. Snce the bond s tradng at par and we are valung t on a payment date, the yeld has to equal the coupon rate. 10.b The Duraton 2 s computed accordng to the followng duraton formula 1 [T 1 P V (c 1 ) + T 2 P V (c 2 ) T N P C(c N )] B 0 ( ) = The modfed duraton measure adjusts ths fgure by dvdng by (1 + ) = 1.08, whch equals DV01, or dollar value of one bass pont, s computed usng equaton DV 01 = B(R 0.01%) B(R) 1 A techncal note: In fact, there s an analytc soluton to the cubc equaton. You wll get three answers correspondng to the order of the polynomal, but only one of the solutons wll be real. The other two wll be a complex conjugate par. 2 Duraton s also referred to as Macaulay Duraton. 12
13 The bond prce at R = 8% s gven as $100. The prce at R = 7.99% s: B(7.99%) = 8 (1.0799) + 8 (1.0799) (1.0799) 3 = Therefore, DV01 s = c We know from above that a one bass pont declne results n a prce ncrease of $ Therefore, a 10 bass pont declne results n an approxmate prce ncrease of 10 $ = $
Final Examination MATH NOTE TO PRINTER
Fnal Examnaton MATH 329 2005 01 1 NOTE TO PRINTER (These nstructons are for the prnter. They should not be duplcated.) Ths examnaton should be prnted on 8 1 2 14 paper, and stapled wth 3 sde staples, so
More informationFinite Math - Fall Section Future Value of an Annuity; Sinking Funds
Fnte Math - Fall 2016 Lecture Notes - 9/19/2016 Secton 3.3 - Future Value of an Annuty; Snkng Funds Snkng Funds. We can turn the annutes pcture around and ask how much we would need to depost nto an account
More informationSOCIETY OF ACTUARIES FINANCIAL MATHEMATICS. EXAM FM SAMPLE SOLUTIONS Interest Theory
SOCIETY OF ACTUARIES EXAM FM FINANCIAL MATHEMATICS EXAM FM SAMPLE SOLUTIONS Interest Theory Ths page ndcates changes made to Study Note FM-09-05. January 14, 014: Questons and solutons 58 60 were added.
More informationLecture Note 2 Time Value of Money
Seg250 Management Prncples for Engneerng Managers Lecture ote 2 Tme Value of Money Department of Systems Engneerng and Engneerng Management The Chnese Unversty of Hong Kong Interest: The Cost of Money
More informationA Php 5,000 loan is being repaid in 10 yearly payments. If interest is 8% effective, find the annual payment. 1 ( ) 10) 0.
Amortzaton If a loan s repad on nstalment (whch s usually n equal amounts); then the loan s sad to be repad by the amortzaton method. Under ths method, each nstalment ncludes the repayment of prncpal and
More informationSurvey of Math: Chapter 22: Consumer Finance Borrowing Page 1
Survey of Math: Chapter 22: Consumer Fnance Borrowng Page 1 APR and EAR Borrowng s savng looked at from a dfferent perspectve. The dea of smple nterest and compound nterest stll apply. A new term s the
More informationFORD MOTOR CREDIT COMPANY SUGGESTED ANSWERS. Richard M. Levich. New York University Stern School of Business. Revised, February 1999
FORD MOTOR CREDIT COMPANY SUGGESTED ANSWERS by Rchard M. Levch New York Unversty Stern School of Busness Revsed, February 1999 1 SETTING UP THE PROBLEM The bond s beng sold to Swss nvestors for a prce
More informationActuarial Science: Financial Mathematics
STAT 485 Actuaral Scence: Fnancal Mathematcs 1.1.1 Effectve Rates of Interest Defnton Defnton lender. An nterest s money earned by deposted funds. An nterest rate s the rate at whch nterest s pad to the
More informationUnderstanding Annuities. Some Algebraic Terminology.
Understandng Annutes Ma 162 Sprng 2010 Ma 162 Sprng 2010 March 22, 2010 Some Algebrac Termnology We recall some terms and calculatons from elementary algebra A fnte sequence of numbers s a functon of natural
More informationYORK UNIVERSITY Faculty of Science Department of Mathematics and Statistics MATH A Test #2 November 03, 2014
Famly Name prnt): YORK UNIVERSITY Faculty of Scence Department of Mathematcs and Statstcs MATH 2280.00 A Test #2 November 0, 2014 Solutons Gven Name: Student No: Sgnature: INSTRUCTIONS: 1. Please wrte
More informationFinal Exam. 7. (10 points) Please state whether each of the following statements is true or false. No explanation needed.
Fnal Exam Fall 4 Econ 8-67 Closed Book. Formula Sheet Provded. Calculators OK. Tme Allowed: hours Please wrte your answers on the page below each queston. (5 ponts) Assume that the rsk-free nterest rate
More informationSurvey of Math Test #3 Practice Questions Page 1 of 5
Test #3 Practce Questons Page 1 of 5 You wll be able to use a calculator, and wll have to use one to answer some questons. Informaton Provded on Test: Smple Interest: Compound Interest: Deprecaton: A =
More informationFinancial mathematics
Fnancal mathematcs Jean-Luc Bouchot jean-luc.bouchot@drexel.edu February 19, 2013 Warnng Ths s a work n progress. I can not ensure t to be mstake free at the moment. It s also lackng some nformaton. But
More informationMoney, Banking, and Financial Markets (Econ 353) Midterm Examination I June 27, Name Univ. Id #
Money, Bankng, and Fnancal Markets (Econ 353) Mdterm Examnaton I June 27, 2005 Name Unv. Id # Note: Each multple-choce queston s worth 4 ponts. Problems 20, 21, and 22 carry 10, 8, and 10 ponts, respectvely.
More informationFinite Mathematics for Business Economics Life Sciences Social Sciences Barnett Ziegler Byleen Twelfth Edition
Fnte Mathematcs for Busness Economcs Lfe Scences Socal Scences Barnett Zegler Byleen Twelfth Edton Pearson Educaton Lmted Ednburgh Gate Harlow Essex CM20 2JE England and Assocated Companes throughout the
More informationAn annuity is a series of payments made at equal intervals. There are many practical examples of financial transactions involving annuities, such as
2 Annutes An annuty s a seres of payments made at equal ntervals. There are many practcal examples of fnancal transactons nvolvng annutes, such as a car loan beng repad wth equal monthly nstallments a
More informationMATH 373 Quiz 5 Fall 2018 November 20, 2018
MATH 373 Quz 5 Fall 218 November 2, 218 1. A callable bond matures at the end of 2 years for 1,. The bond pays coupons at a rate of 7% convertble sem-annually. The bond can be called at the end of 14 year
More informationDept of Mathematics and Statistics King Fahd University of Petroleum & Minerals
Dept of Mathematcs and Statstcs Kng Fahd Unversty of Petroleum & Mnerals AS201: Fnancal Mathematcs Dr. Mohammad H. Omar Major Exam 2 FORM B Soluton Aprl 16 2012 6.30pm-8.00pm Name ID#: Seral #: Instructons.
More informationTHIRD MIDTERM EXAM EC26102: MONEY, BANKING AND FINANCIAL MARKETS MARCH 24, 2004
THIRD MIDTERM EXAM EC26102: MONEY, BANKING AND FINANCIAL MARKETS MARCH 24, 2004 Ths exam has questons on eght pages. Before you begn, please check to make sure that your copy has all questons and all eght
More informationMathematical Thinking Exam 1 09 October 2017
Mathematcal Thnkng Exam 1 09 October 2017 Name: Instructons: Be sure to read each problem s drectons. Wrte clearly durng the exam and fully erase or mark out anythng you do not want graded. You may use
More informationTime Value of Money, Part 2 Future Value aueof a $1 (Single Sum) Learning Outcomes. Future Value
Tme Value of Money, Part 2 Future Value aueof a $1 (Sngle Sum) Intermedate Accountng I Dr. Chula Kng 1 Learnng Outcomes The concept of future value Future value of a sngle sum How to set up the problem
More informationChapter 5 Bonds, Bond Prices and the Determination of Interest Rates
Chapter 5 Bonds, Bond Prces and the Determnaton of Interest Rates Problems and Solutons 1. Consder a U.S. Treasury Bll wth 270 days to maturty. If the annual yeld s 3.8 percent, what s the prce? $100 P
More informationDept of Mathematics and Statistics King Fahd University of Petroleum & Minerals
Dept of Mathematcs and Statstcs Kng Fahd Unversty of Petroleum & Mnerals AS201: Fnancal Mathematcs Dr. Mohammad H. Omar Major Exam 2 FORM B Soluton November 27 2012 6.30pm-8.00pm Name ID#: Seral #: Instructons.
More information2) In the medium-run/long-run, a decrease in the budget deficit will produce:
4.02 Quz 2 Solutons Fall 2004 Multple-Choce Questons ) Consder the wage-settng and prce-settng equatons we studed n class. Suppose the markup, µ, equals 0.25, and F(u,z) = -u. What s the natural rate of
More informationAnswers to exercises in Macroeconomics by Nils Gottfries 2013
. a) C C b C C s the ntercept o the consumpton uncton, how much consumpton wll be at zero ncome. We can thnk that, at zero ncome, the typcal consumer would consume out o hs assets. The slope b s the margnal
More informationFuture value of an annuity
Announcements The secon hour-exam wll be hel on Fray, July 12. The use of cell phones an other wreless evces s not permtte on the exam. You wll nee to brng a separate calculator for the exam. Sharng of
More information7.4. Annuities. Investigate
7.4 Annutes How would you lke to be a mllonare wthout workng all your lfe to earn t? Perhaps f you were lucky enough to wn a lottery or have an amazng run on a televson game show, t would happen. For most
More informationiii) pay F P 0,T = S 0 e δt when stock has dividend yield δ.
Fnal s Wed May 7, 12:50-2:50 You are allowed 15 sheets of notes and a calculator The fnal s cumulatve, so you should know everythng on the frst 4 revews Ths materal not on those revews 184) Suppose S t
More informationISE High Income Index Methodology
ISE Hgh Income Index Methodology Index Descrpton The ISE Hgh Income Index s desgned to track the returns and ncome of the top 30 U.S lsted Closed-End Funds. Index Calculaton The ISE Hgh Income Index s
More informationCHAPTER 1: MATHEMATICS OF INVESTMENT
Why do you need to know nvestments, bonds, stocks, nterests? Why s there a need to nvest your hard earned money? Whether you just want to save for that phone or tablet that you wanted to buy; or you re
More informationProblem Set 6 Finance 1,
Carnege Mellon Unversty Graduate School of Industral Admnstraton Chrs Telmer Wnter 2006 Problem Set 6 Fnance, 47-720. (representatve agent constructon) Consder the followng two-perod, two-agent economy.
More informationPrice and Quantity Competition Revisited. Abstract
rce and uantty Competton Revsted X. Henry Wang Unversty of Mssour - Columba Abstract By enlargng the parameter space orgnally consdered by Sngh and Vves (984 to allow for a wder range of cost asymmetry,
More informationElton, Gruber, Brown, and Goetzmann. Modern Portfolio Theory and Investment Analysis, 7th Edition. Solutions to Text Problems: Chapter 16
lton, Gruer, rown, and Goetzmann Modern Portfolo Theory and Investment nalyss, 7th dton Solutons to Text Prolems: hapter 6 hapter 6: Prolem From the text we know that three ponts determne a plane. The
More information15-451/651: Design & Analysis of Algorithms January 22, 2019 Lecture #3: Amortized Analysis last changed: January 18, 2019
5-45/65: Desgn & Analyss of Algorthms January, 09 Lecture #3: Amortzed Analyss last changed: January 8, 09 Introducton In ths lecture we dscuss a useful form of analyss, called amortzed analyss, for problems
More informationPrinciples of Finance
Prncples of Fnance Grzegorz Trojanowsk Lecture 6: Captal Asset Prcng Model Prncples of Fnance - Lecture 6 1 Lecture 6 materal Requred readng: Elton et al., Chapters 13, 14, and 15 Supplementary readng:
More informationCreating a zero coupon curve by bootstrapping with cubic splines.
MMA 708 Analytcal Fnance II Creatng a zero coupon curve by bootstrappng wth cubc splnes. erg Gryshkevych Professor: Jan R. M. Röman 0.2.200 Dvson of Appled Mathematcs chool of Educaton, Culture and Communcaton
More informationMULTIPLE CURVE CONSTRUCTION
MULTIPLE CURVE CONSTRUCTION RICHARD WHITE 1. Introducton In the post-credt-crunch world, swaps are generally collateralzed under a ISDA Master Agreement Andersen and Pterbarg p266, wth collateral rates
More informationEvaluating Performance
5 Chapter Evaluatng Performance In Ths Chapter Dollar-Weghted Rate of Return Tme-Weghted Rate of Return Income Rate of Return Prncpal Rate of Return Daly Returns MPT Statstcs 5- Measurng Rates of Return
More informationSTUDY GUIDE FOR TOPIC 1: FUNDAMENTAL CONCEPTS OF FINANCIAL MATHEMATICS. Learning objectives
Study Gude for Topc 1 1 STUDY GUIDE FOR TOPIC 1: FUNDAMENTAL CONCEPTS OF FINANCIAL MATHEMATICS Learnng objectves After studyng ths topc you should be able to: apprecate the ever-changng envronment n whch
More informationChapter 10 Making Choices: The Method, MARR, and Multiple Attributes
Chapter 0 Makng Choces: The Method, MARR, and Multple Attrbutes INEN 303 Sergy Butenko Industral & Systems Engneerng Texas A&M Unversty Comparng Mutually Exclusve Alternatves by Dfferent Evaluaton Methods
More informationConsumption Based Asset Pricing
Consumpton Based Asset Prcng Mchael Bar Aprl 25, 208 Contents Introducton 2 Model 2. Prcng rsk-free asset............................... 3 2.2 Prcng rsky assets................................ 4 2.3 Bubbles......................................
More informationFall 2017 Social Sciences 7418 University of Wisconsin-Madison Problem Set 3 Answers
ublc Affars 854 enze D. Chnn Fall 07 Socal Scences 748 Unversty of Wsconsn-adson roblem Set 3 Answers Due n Lecture on Wednesday, November st. " Box n" your answers to the algebrac questons.. Fscal polcy
More information4. Greek Letters, Value-at-Risk
4 Greek Letters, Value-at-Rsk 4 Value-at-Rsk (Hull s, Chapter 8) Math443 W08, HM Zhu Outlne (Hull, Chap 8) What s Value at Rsk (VaR)? Hstorcal smulatons Monte Carlo smulatons Model based approach Varance-covarance
More informationValue of L = V L = VL = VU =$48,000,000 (ii) Owning 1% of firm U provides a dollar return of.01 [EBIT(1-T C )] =.01 x 6,000,000 = $60,000.
OLUTION 1. A company wll call a bond when the market prce of the bond s at or above the call prce. For a zero-coupon bond, ths wll never happen because the market prce wll always be below the face value.
More informationElton, Gruber, Brown and Goetzmann. Modern Portfolio Theory and Investment Analysis, 7th Edition. Solutions to Text Problems: Chapter 4
Elton, Gruber, Brown and Goetzmann Modern ortfolo Theory and Investment Analyss, 7th Edton Solutons to Text roblems: Chapter 4 Chapter 4: roblem 1 A. Expected return s the sum of each outcome tmes ts assocated
More informationCS 286r: Matching and Market Design Lecture 2 Combinatorial Markets, Walrasian Equilibrium, Tâtonnement
CS 286r: Matchng and Market Desgn Lecture 2 Combnatoral Markets, Walrasan Equlbrum, Tâtonnement Matchng and Money Recall: Last tme we descrbed the Hungaran Method for computng a maxmumweght bpartte matchng.
More informationElton, Gruber, Brown, and Goetzmann. Modern Portfolio Theory and Investment Analysis, 7th Edition. Solutions to Text Problems: Chapter 9
Elton, Gruber, Brown, and Goetzmann Modern Portfolo Theory and Investment Analyss, 7th Edton Solutons to Text Problems: Chapter 9 Chapter 9: Problem In the table below, gven that the rskless rate equals
More informationIND E 250 Final Exam Solutions June 8, Section A. Multiple choice and simple computation. [5 points each] (Version A)
IND E 20 Fnal Exam Solutons June 8, 2006 Secton A. Multple choce and smple computaton. [ ponts each] (Verson A) (-) Four ndependent projects, each wth rsk free cash flows, have the followng B/C ratos:
More information2. Compute Compound Interest
The Mathematcs of Fnance Careers and Mathematcs 9 In ths chapter, we wll dscuss the mathematcs of finance the rules that govern nvestng and borrowng money. 9.1 Interest Actuary Actuares use ther broad knowledge
More informationMATH 373 Quiz 3 Fall 2017 October 12, 2017
MATH 373 Quz 3 Fall 2017 October, 2017 1. Alex wants to nvest for hs retrement. Today s hs 22 nd brthday. He wll make a payment of 10,000 on each brthday begnnng wth hs 30 th brthday. Hs last payment wll
More informationHomework 9: due Monday, 27 October, 2008
PROBLEM ONE Homework 9: due Monday, 7 October, 008. (Exercses from the book, 6 th edton, 6.6, -3.) Determne the number of dstnct orderngs of the letters gven: (a) GUIDE (b) SCHOOL (c) SALESPERSONS. (Exercses
More informationFM303. CHAPTERS COVERED : CHAPTERS 5, 8 and 9. LEARNER GUIDE : UNITS 1, 2 and 3.1 to 3.3. DUE DATE : 3:00 p.m. 19 MARCH 2013
Page 1 of 11 ASSIGNMENT 1 ST SEMESTER : FINANCIAL MANAGEMENT 3 () CHAPTERS COVERED : CHAPTERS 5, 8 and 9 LEARNER GUIDE : UNITS 1, 2 and 3.1 to 3.3 DUE DATE : 3:00 p.m. 19 MARCH 2013 TOTAL MARKS : 100 INSTRUCTIONS
More informationElements of Economic Analysis II Lecture VI: Industry Supply
Elements of Economc Analyss II Lecture VI: Industry Supply Ka Hao Yang 10/12/2017 In the prevous lecture, we analyzed the frm s supply decson usng a set of smple graphcal analyses. In fact, the dscusson
More informationMgtOp 215 Chapter 13 Dr. Ahn
MgtOp 5 Chapter 3 Dr Ahn Consder two random varables X and Y wth,,, In order to study the relatonshp between the two random varables, we need a numercal measure that descrbes the relatonshp The covarance
More informationOCR Statistics 1 Working with data. Section 2: Measures of location
OCR Statstcs 1 Workng wth data Secton 2: Measures of locaton Notes and Examples These notes have sub-sectons on: The medan Estmatng the medan from grouped data The mean Estmatng the mean from grouped data
More informationPivot Points for CQG - Overview
Pvot Ponts for CQG - Overvew By Bran Bell Introducton Pvot ponts are a well-known technque used by floor traders to calculate ntraday support and resstance levels. Ths technque has been around for decades,
More informationSIMPLE FIXED-POINT ITERATION
SIMPLE FIXED-POINT ITERATION The fed-pont teraton method s an open root fndng method. The method starts wth the equaton f ( The equaton s then rearranged so that one s one the left hand sde of the equaton
More informationProblem Set #4 Solutions
4.0 Sprng 00 Page Problem Set #4 Solutons Problem : a) The extensve form of the game s as follows: (,) Inc. (-,-) Entrant (0,0) Inc (5,0) Usng backwards nducton, the ncumbent wll always set hgh prces,
More informationLecture 10: Valuation Models (with an Introduction to Capital Budgeting).
Foundatons of Fnance Lecture 10: Valuaton Models (wth an Introducton to Captal Budgetng). I. Readng. II. Introducton. III. Dscounted Cash Flow Models. IV. Relatve Valuaton Approaches. V. Contngent Clam
More informationMultifactor Term Structure Models
1 Multfactor Term Structure Models A. Lmtatons of One-Factor Models 1. Returns on bonds of all maturtes are perfectly correlated. 2. Term structure (and prces of every other dervatves) are unquely determned
More informationECE 586GT: Problem Set 2: Problems and Solutions Uniqueness of Nash equilibria, zero sum games, evolutionary dynamics
Unversty of Illnos Fall 08 ECE 586GT: Problem Set : Problems and Solutons Unqueness of Nash equlbra, zero sum games, evolutonary dynamcs Due: Tuesday, Sept. 5, at begnnng of class Readng: Course notes,
More informationMacroeconomic Theory and Policy
ECO 209 Macroeconomc Theory and Polcy Lecture 7: The Open Economy wth Fxed Exchange Rates Gustavo Indart Slde 1 Open Economy under Fxed Exchange Rates Let s consder an open economy wth no captal moblty
More informationFacility Location Problem. Learning objectives. Antti Salonen Farzaneh Ahmadzadeh
Antt Salonen Farzaneh Ahmadzadeh 1 Faclty Locaton Problem The study of faclty locaton problems, also known as locaton analyss, s a branch of operatons research concerned wth the optmal placement of facltes
More informationFinance 402: Problem Set 1
Finance 402: Problem Set 1 1. A 6% corporate bond is due in 12 years. What is the price of the bond if the annual percentage rate (APR) is 12% per annum compounded semiannually? (note that the bond pays
More informationUniversity of Toronto November 9, 2006 ECO 209Y MACROECONOMIC THEORY. Term Test #1 L0101 L0201 L0401 L5101 MW MW 1-2 MW 2-3 W 6-8
Department of Economcs Prof. Gustavo Indart Unversty of Toronto November 9, 2006 SOLUTION ECO 209Y MACROECONOMIC THEORY Term Test #1 A LAST NAME FIRST NAME STUDENT NUMBER Crcle your secton of the course:
More informationUniversity of Toronto November 9, 2006 ECO 209Y MACROECONOMIC THEORY. Term Test #1 L0101 L0201 L0401 L5101 MW MW 1-2 MW 2-3 W 6-8
Department of Economcs Prof. Gustavo Indart Unversty of Toronto November 9, 2006 SOLUTION ECO 209Y MACROECONOMIC THEORY Term Test #1 C LAST NAME FIRST NAME STUDENT NUMBER Crcle your secton of the course:
More informationSpring 2018 Social Sciences 7418 University of Wisconsin-Madison. Transactions and Portfolio Crowding Out
Economcs 44 Menze D. Cnn Sprng 8 Socal Scences 748 Unversty of Wsconsn-Madson. Standard IS-LM Transactons and Portfolo Crowdng Out Transactons crowdng out of nvestment s te reducton n nvestment attrbutable
More informationMacroeconomic Theory and Policy
ECO 209 Macroeconomc Theory and Polcy Lecture 7: The Open Economy wth Fxed Exchange Rates Gustavo Indart Slde 1 Open Economy under Fxed Exchange Rates Let s consder an open economy wth no captal moblty
More informationQuiz on Deterministic part of course October 22, 2002
Engneerng ystems Analyss for Desgn Quz on Determnstc part of course October 22, 2002 Ths s a closed book exercse. You may use calculators Grade Tables There are 90 ponts possble for the regular test, or
More informationCh Rival Pure private goods (most retail goods) Non-Rival Impure public goods (internet service)
h 7 1 Publc Goods o Rval goods: a good s rval f ts consumpton by one person precludes ts consumpton by another o Excludable goods: a good s excludable f you can reasonably prevent a person from consumng
More informationProblems to be discussed at the 5 th seminar Suggested solutions
ECON4260 Behavoral Economcs Problems to be dscussed at the 5 th semnar Suggested solutons Problem 1 a) Consder an ultmatum game n whch the proposer gets, ntally, 100 NOK. Assume that both the proposer
More informationImpact of CDO Tranches on Economic Capital of Credit Portfolios
Impact of CDO Tranches on Economc Captal of Credt Portfolos Ym T. Lee Market & Investment Bankng UnCredt Group Moor House, 120 London Wall London, EC2Y 5ET KEYWORDS: Credt rsk, Collateralzaton Debt Oblgaton,
More informationS yi a bx i cx yi a bx i cx 2 i =0. yi a bx i cx 2 i xi =0. yi a bx i cx 2 i x
LEAST-SQUARES FIT (Chapter 8) Ft the best straght lne (parabola, etc.) to a gven set of ponts. Ths wll be done by mnmzng the sum of squares of the vertcal dstances (called resduals) from the ponts to the
More informationFinance 100 Problem Set Bonds
Finance 100 Problem Set Bonds 1. You have a liability for paying college fees for your children of $20,000 at the end of each of the next 2 years (1998-1999). You can invest your money now (January 1 1998)
More informationNote on Cubic Spline Valuation Methodology
Note on Cubc Splne Valuaton Methodology Regd. Offce: The Internatonal, 2 nd Floor THE CUBIC SPLINE METHODOLOGY A model for yeld curve takes traded yelds for avalable tenors as nput and generates the curve
More informationINTERNATIONAL CAPITAL BUDGETING
INTERNATIONAL CAPITAL BUDGETING Sources: Internatonal Fnancal Management; Eun and Resnck Multnatonal Fnancal Management; Shapro Modern Corporate Fnance; Shapro Internatonal Fnancal Management; P G Apte
More informationFROM THE ANSWER SERIES
FINNCIL MTHS QUESTIONS & NSWERS 1 FROM THE NSWER SERIES publshed by : The nswer 210 Man Road CLREMONT 7708 Cape Town tel: (021) 671 0837 fa: (021) 671 256 e-mal: nfo@theanswerseres.co.za www.theanswerseres.co.za
More informationMutual Funds and Management Styles. Active Portfolio Management
utual Funds and anagement Styles ctve Portfolo anagement ctve Portfolo anagement What s actve portfolo management? How can we measure the contrbuton of actve portfolo management? We start out wth the CP
More informationTaxation and Externalities. - Much recent discussion of policy towards externalities, e.g., global warming debate/kyoto
Taxaton and Externaltes - Much recent dscusson of polcy towards externaltes, e.g., global warmng debate/kyoto - Increasng share of tax revenue from envronmental taxaton 6 percent n OECD - Envronmental
More informationRisk and Return: The Security Markets Line
FIN 614 Rsk and Return 3: Markets Professor Robert B.H. Hauswald Kogod School of Busness, AU 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 1 Rsk and Return: The Securty Markets Lne From securtes
More informationTests for Two Correlations
PASS Sample Sze Software Chapter 805 Tests for Two Correlatons Introducton The correlaton coeffcent (or correlaton), ρ, s a popular parameter for descrbng the strength of the assocaton between two varables.
More informationTHE VOLATILITY OF EQUITY MUTUAL FUND RETURNS
North Amercan Journal of Fnance and Bankng Research Vol. 4. No. 4. 010. THE VOLATILITY OF EQUITY MUTUAL FUND RETURNS Central Connectcut State Unversty, USA. E-mal: BelloZ@mal.ccsu.edu ABSTRACT I nvestgated
More informationEDC Introduction
.0 Introducton EDC3 In the last set of notes (EDC), we saw how to use penalty factors n solvng the EDC problem wth losses. In ths set of notes, we want to address two closely related ssues. What are, exactly,
More informationFall 2016 Social Sciences 7418 University of Wisconsin-Madison. Transactions and Portfolio Crowding Out
Economcs 435 Menze D. Cnn Fall 6 Socal Scences 748 Unversty of Wsconsn-Madson. Standard IS-LM Transactons and ortfolo Crowdng Out Transactons crowdng out of nvestment s te reducton n nvestment attrbutable
More informationTCOM501 Networking: Theory & Fundamentals Final Examination Professor Yannis A. Korilis April 26, 2002
TO5 Networng: Theory & undamentals nal xamnaton Professor Yanns. orls prl, Problem [ ponts]: onsder a rng networ wth nodes,,,. In ths networ, a customer that completes servce at node exts the networ wth
More informationInternational Financial Management
Multnatonal Corporatons (MNC Internatonal nancal Management nance ummer 006 xed versus loatng Exchange Rates loatng xed Managed floatng rate Currences float freely n ths, and s (prces are set by supply
More informationEconomics 330 Money and Banking Problem Set No. 3 Due Tuesday April 3, 2018 at the beginning of class
Economcs 0 Money and Bankng Problem Set No. Due Tuesday Aprl, 08 at the begnnng of class Fall 08 Dr. Ner I. A. The followng table shows the prce of $000 face value -year, -year, -year, 9-year and 0- year
More informationHewlett Packard 10BII Calculator
Hewlett Packard 0BII Calculator Keystrokes for the HP 0BII are shown n the tet. However, takng a mnute to revew the Quk Start secton, below, wll be very helpful n gettng started wth your calculator. Note:
More informationWelfare Aspects in the Realignment of Commercial Framework. between Japan and China
Prepared for the 13 th INFORUM World Conference n Huangshan, Chna, July 3 9, 2005 Welfare Aspects n the Realgnment of Commercal Framework between Japan and Chna Toshak Hasegawa Chuo Unversty, Japan Introducton
More informationChapter 11: Optimal Portfolio Choice and the Capital Asset Pricing Model
Chapter 11: Optmal Portolo Choce and the CAPM-1 Chapter 11: Optmal Portolo Choce and the Captal Asset Prcng Model Goal: determne the relatonshp between rsk and return key to ths process: examne how nvestors
More informationMorningstar After-Tax Return Methodology
Mornngstar After-Tax Return Methodology Mornngstar Research Report 24 October 2003 2003 Mornngstar, Inc. All rghts reserved. The nformaton n ths document s the property of Mornngstar, Inc. Reproducton
More informationProject Management Project Phases the S curve
Project lfe cycle and resource usage Phases Project Management Project Phases the S curve Eng. Gorgo Locatell RATE OF RESOURCE ES Conceptual Defnton Realzaton Release TIME Cumulated resource usage and
More informationStochastic ALM models - General Methodology
Stochastc ALM models - General Methodology Stochastc ALM models are generally mplemented wthn separate modules: A stochastc scenaros generator (ESG) A cash-flow projecton tool (or ALM projecton) For projectng
More informationCHAPTER 9 FUNCTIONAL FORMS OF REGRESSION MODELS
CHAPTER 9 FUNCTIONAL FORMS OF REGRESSION MODELS QUESTIONS 9.1. (a) In a log-log model the dependent and all explanatory varables are n the logarthmc form. (b) In the log-ln model the dependent varable
More informationReview. Time Series Models
Revew Topcs & Authors Fnal exam s not cumulatve, but stll need to know IS-LM (no multplers) Shfts n IS - IS (.e., curve shfts to the rght) f T (C ), or π e (I ), or G ; results n larger output () for gven
More informationISE Cloud Computing Index Methodology
ISE Cloud Computng Index Methodology Index Descrpton The ISE Cloud Computng Index s desgned to track the performance of companes nvolved n the cloud computng ndustry. Index Calculaton The ISE Cloud Computng
More informationREFINITIV INDICES PRIVATE EQUITY BUYOUT INDEX METHODOLOGY
REFINITIV INDICES PRIVATE EQUITY BUYOUT INDEX METHODOLOGY 1 Table of Contents INTRODUCTION 3 TR Prvate Equty Buyout Index 3 INDEX COMPOSITION 3 Sector Portfolos 4 Sector Weghtng 5 Index Rebalance 5 Index
More informationApplications of Myerson s Lemma
Applcatons of Myerson s Lemma Professor Greenwald 28-2-7 We apply Myerson s lemma to solve the sngle-good aucton, and the generalzaton n whch there are k dentcal copes of the good. Our objectve s welfare
More informationCOS 511: Theoretical Machine Learning. Lecturer: Rob Schapire Lecture #21 Scribe: Lawrence Diao April 23, 2013
COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture #21 Scrbe: Lawrence Dao Aprl 23, 2013 1 On-Lne Log Loss To recap the end of the last lecture, we have the followng on-lne problem wth N
More informationChapter 15: Debt and Taxes
Chapter 15: Debt and Taxes-1 Chapter 15: Debt and Taxes I. Basc Ideas 1. Corporate Taxes => nterest expense s tax deductble => as debt ncreases, corporate taxes fall => ncentve to fund the frm wth debt
More information