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# Financial Models FIN 4453

University of Central Florida

GPA 3.79

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This 19 page Class Notes was uploaded by Haylee Spencer on Thursday October 22, 2015. The Class Notes belongs to FIN 4453 at University of Central Florida taught by Vladimir Gatchev in Fall. Since its upload, it has received 91 views. For similar materials see /class/227496/fin-4453-university-of-central-florida in Finance at University of Central Florida.

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Date Created: 10/22/15

FIN 4453 Financial Models Spring 2009 Vladimir Gatchev PORTFOLIO THEORY OPTIMIZATION AND ASSET ALLOCATION NOT IN TEXT Up until now we ve looked at assets in isolation That is we have measured the risk and return associated with a particular stock or an asset but we have not examined how these assets work together to form a portfolio First we will see how to calculate measures Of historic and expected risk and return Of a portfolio Second we will examine how investors should nd Optimal portfolios Of risky assets And nally we will see what choices investors have to make when they can also invest in a riskfree asset I THE BASIC IDEA BEHIND PORTFOLIO THEORY Suppose that you want to invest in three stocks How much should you invest in each In Other words what is the Optimal way to combine assets in a portfolio TO answer this we want to derive a measure for the expected rate Of return and the expected risk associated with any portfolio Of assets we want to analyze TO do this we will need to analyze and quantify how assets work together as they form a portfolio We will see that there are two effects Of adding a stock to a portfolio O Own effect 7 the stocks own inherent level Of risk and potential return O Interaction effect 7 the stocks interaction with every Other stock within the portfolio 0 Harry Markowitz is credited for much Of the work in the area Of Portfolio Theory April 2009 1 II PORTFOLIO RETURN A Historic Returns of a Portfolio De nition The average historic rate of return for a portfolio is calculated as the valueweighted average of the historic returns of the individual investments in the portfolio K RPORT 2W1 R 7 11 Wherej1 to K indicates the individual investments in the portfolio and w is the market value weight of investment j in the portfolio at the beginning of the investment period Money invested in security j Total money invested in the portfolio w Examgle For the past year IBM had a return of 10 and Microsoft had a return of 15 Find the historic return of a portfolio that one year ago invested 30 in IBM and 70 in MSFT B Expected Returns of a Portfolio De nition Portfolio eXpected return is the valueweighted average of the eXpected returns of the individual assets in that portfolio K ERPORT Z W ERj 11 Wherej1 to K indicates the individual investments in the portfolio and w is the market value weight of investment j in the portfolio at the beginning of the investment period Money invested in security j w Total money invested in the portfolio Examgle For the neXt year IBM is eXpected to return 8 and Microsoft is eXpected to return 6 Find the eXpected return of a 200 portfolio that today invests 60 in IBM and 140 in MSFT April 2009 2 III PORTFOLIO VARIANCE AND STANDARD DEVIATION It would be nice if we could calculate the standard deviation of a portfolio by taking the weighted average of each stock s standard deviation But that would ignore the interaction between each pair of stocks So we must incorporate the concept of Covariance A Historic Variance and Standard Deviation 0 Two Assets Case 2 2 2 2 2 UPORT WAUA WBUB 2wAwBC0vAYB Identically 2 2 2 2 2 UPORT WAG1 WBUB ZWAWBUAUBPAB l 2 UPORT UPORT 0 Multiple Assets Case In general when we have N assets in the portfolio the variance of the portfolio is given by N N N 2 2 2 UPON 2 wt 0391 E E wleCovw 21 11 11 12 Examgle Consider the following returns of IBM Sun and Oracle What is the variance and standard deviation of a portfolio that has 50000 invested in IBM 75000 invested in Sun and 35000 invested in Oracle YEAR IBM SUN ORCL 2000 25 5 35 2001 5 28 15 2002 28 51 30 2003 30 80 20 Standard Dev 2352 4950 2610 April 2009 3 0 Number of Assets and Portfolio Standard Deviation rr nne nnrtfnlin 39 drop in a way similar to the one shown in the graph below NUMBER 0 r STO c KS AND P0 RTFO Llo sTANDARD DEleno N STANDARD DElenoN 0 20 w on so 100 120 NUMBER or STO CKS B Expected Variance and Standard Deviation The easiest w quot L r 39 quot quot quot quot 39 39 39 39 blend the assets into one portfolio Then we can use our method from the previous lecture to nd the variance and standard deviation of one asset our one portfolio Emmglz Let s calculate the standard deviation of a portfolio of two stocks 7 25 invested in a beer maker and 75 invested in an automobile maker Annmahile Reruns n unwmsnn stmananemam 436 First calculate the Blended Portfolio Returns per economic state April 2009 IV PORTFOLIO THEORY ASSUMPTIONS A Non Satiation B Risk Aversion C Rationality D Mean Variance April 2009 V De nition EFFICIENT PORTFOLIOS A single asset or a portfolio of assets is considered to be EFFICIENT if it has the highest return for a given level of risk or the lowest risk for a given level of return Examgle In each of the following scenarios which portfolio is the ef cient one April 2009 Scenario 1 PortfolioA Portfolio B 10 15 5 Scenario 2 PortfolioA Portfolio B 13 E 39 10 0 Scenario 3 PortfolioA Portfolio B VI POSSIBLE PORTFOLIOS WITH MULTIPLE ASSETS A Possible Portfolios with Three Assets Here is an example with three assets The three assets are indicated with stars The rest of the points are the possible portfolios that combine the three assets in different ways 30 25 20 15 10 Expected Return 5 0 0 5 10 15 20 25 30 35 Standard Dev iation Which portfolios would investors that satisfy the assumptions discussed before want to invest in B Investor Preferences 0 Indifference Curve A curve that contains all combinations of risk and return that yield the same utility level 0 The steepness of an indifference curve indicates the level of risk aversion 25 20 15 10 Expected Return 5 0 0 2 4 6 8 10 12 Standard Deviation April 2009 7 VII INTRODUCING A RISK FREE ASSET A Risk Free Asset Characteristics 0 What is the standard deviation of a riskfree asset s return 0 What is the covariance of the returns of a riskfree asset with the returns of a risky asset B Combining a Risky Portfolio with a Risk Free Asset the Capital Market Line 0 What is the expected return of a portfolio that consists of the riskfree asset and a risky portfolio 0 What is the expected standard deviation of a portfolio that consists of the riskfree asset and a risky portfolio Assuming that we can lend m borrow at the riskfree rate our Efficient Frontier now becomes what is termed the Capital Market Line CML 25 20 15 10 Expected Return 5 0 0 2 4 6 8 10 12 Standard Deviation April 2009 8 VIII PORTFOLIO STYLE ANALYSIS A Index Portfolios For index portfolios the most common approach to examining the performance of the portfolio is to examine hoW close it tmcks the index Let s examine hoW the Vanguard 500 portfolio compares to the SampP 500 index Vanguard 500 Correlation With SampP 500 9999 B Style Portfolios For style portfolios the idea is to check hoW closely the portfolio correlates With the style Let s look at the Vanguard Value portfolio Vanguard Value Correlation With SampP 500 9254 Correlation With SMB small vs big 485 Correlation With HML value vs growth 1 15 Correlation With momentum 1 143 Vanguard Value Style 100 4 3 V 033 E o a E 033 5 pa 100 100 033 033 100 VALUE 7 NEUTRAL 7 GROWTH OVanguard Value Does the Vanguard Value portfolio act like a value portfolio April 2009 9 Some Practice Problems For maximum bene t you should try to solve these problems with Excel rather than your calculator Problem 1 Your portfolio consists of three stocks IBM Sun Microsystems and Oracle Here are your positions in each stock along with the purchase price and the expected return of each stock Stock shares in portfolio Purchase Price Expected Return IBM 100 9971 10 SUN 700 563 16 ORCL 400 1379 13 Calculate the expected rate of return on your portfolio Answer 127 o Problem 2 Consider the following returns for the three stocks in your portfolio IBM Sun and Oracle over the last 4 years Calculate the covariance and the correlation between each pair of stocks YEAR IBM SUN ORCL 2000 25 5 3 5 2001 5 28 15 2002 28 51 30 2003 30 80 20 Answer Covariance Correlation IBMvs SUN 01015 08718 IBM vs ORCL 00584 09509 SUN vs ORCL 00928 07179 Problem 3 Suppose you purchase 100 shares of Intel C0 Nasdaq INTC at 16share and 200 shares of Amazoncom Nasdaq AMZN at 15share One year from now you sell your holdings of Intel for 24share and also sell your holdings of Amazoncom for 13share What is the holding period return of your portfolio Answer Initial Investment 16100 15200 4600 Final Value 24 100 13200 5000 Return 5000 7 4600 4600 87 April 2009 Problem 4 Consider the following returns for the three stocks in your portfolio IBM Sun and Oracle over the last 4 years same as in the previous example YEAR IBM SUN ORCL 2000 25 5 3 5 2001 5 28 15 2002 28 51 30 2003 30 80 20 Standard Dev 23 52 49 50 2610 Calculate the standard deviation and variance of a portfolio that has 100000 invested in IBM and 50000 invested in Sun Hint Calculate the dollar amount of each stock every year and sum them up to get the value of the portfolio every year Then you can calculate the return of the portfolio every year Once you have the returns of the portfolio by year nding the variance and the standard deviation is straight forward Answer Year IBM SUN TOTAL Portfolio Return 100000 50000 150000 2000 125000 52500 177500 1833 2001 118750 37800 156550 1180 2002 85500 18522 104022 3355 2003 111150 33340 144490 3890 I Variance I 01026 I I Standard Deviation I 3204 I April 2009 Problem 5 You have the following two assets to invest in I Asset A I Asset B EReturn 020 EReturn 020 Egstd Dev 010 Egstd Dev 010 Let s put them in a portfolio equally weights both equal 50 a If the correlation coefficient between A and B is 100 calculate the portfolio standard deviatron b If the correlation coefficient between A and B is 050 calculate the portfolio standard deviation c If the correlation coefficient between A and B is 000 calculate the portfolio standard deviation d If the correlation coefficient between A and B is 4150 calculate the portfolio standard deviation e If the correlation coefficient between A and B is 100 calculate the portfolio standard deviatron Answer a b c d e Correlation 1 05 0 05 1 Variance 00100 00075 00050 00025 0000 Standard Deviation 1000 866 707 500 000 April 2009 12 FIN 4453 Financial Models Spring 2010 Vladimir Gatchev CHAPTER 7 THE TIME VALUE OF MONEY This chapter contains one of the main nancial concepts No pressure there The central idea is that 1 today is not worth the same as 1 tomorrow It is more valuable to us today This is because we can invest it today allowing it to grow to a value greater than 1 tomorrow ie money has time value Q Which would you rather have given to you 10000 today or 15000 in 5 years A It depends What does it depend on Suppose the interest rate is 10 Suppose the interest rate is 5 M M 1 10000 110 11000 1 10000 105 10500 2 11000 110 12100 2 10050 105 11025 3 12100 110 13310 3 11025 105 11576 4 13310 110 14641 4 11576 105 12155 5 14641110 16105 5 12155 105 12763 Here we d rather have 10000 today Here we d rather have 15000 in 5 years Growth of 10000 US 17000 16000 15000 14000 13000 12000 11000 10000 Dollar Valu 0 l 2 Year 3 4 5 El Interest 5 El Interest 10 FebruaryMarch 2010 l Time Lines These are very useful 0015 e eelally for complex ume value ofmoney Th SP problems Get useol to uslng ulem ey Wlll help keep your gures strmght A umellne for the above problems would have looked llke ths n l 2 z A 5 IEIEIEIEI 777 10000 ls our PRESENT VALUE he unknown value ls the FUTU39RE LUE The NUMBER OF PERIODS we39re deallng wth ls 5 n or 107 Kyou have any 3 ofthese varlables you can solve forthe form one I FUTURE VALUE A a l Yulur 39nhleLnkuhuinns resenrxslue slcce u H PRESENT VALUE 7 Preseurl39uluecpleulunuus a 9 senr39l39slur slam a Yzars 5 la lnrrresraere we ll rure39lslue slam FebruarerarchZEll 2 m ANNUITIES What if we invest money every year and we want to nd the future value of our Annuity problems will allow us to answer such questions 2 i i i i i i HUD HUD mm mm mm A Present Value at an Annuity 13 Prrsen 39amrnhnAnnuii 15 ntzrzstRazz 17 9xz52nt aiue B Future Value at an Annuity ta Fuiure39aluennlA unit sin 12 m c Payment at an Annuity 25 Paymmlnhnzhmnil 25 xzszm39mluz so 39 mm 7287 39rars s 29 Lmzxzsl au me 30 FebrunyMnenznin D Number anerindsin an Annuity 32 XI hrrnIYrrindsinnnA nil V 3 31252m1u2 so nrrevaure 51m 5 mg 53 a 53 Yzars a nreresr 3 am in 37 Annual vamrenr 5 Wu E Interest Rate in an Annuity 39 lluereisale m mAmudl a P1 serrr39raur 4 s 44 ruremue 5mm em was4243444 54 43 IntuzstRau p 44 AnnualPawnznt 5150 F Perpetuity e formulas to solve the presentvame ofaperpetmty The formulaxs G Deferred Annuities Deferred annurtaes are annurtaes that don39t start rrnrnedrately but rather start at some future porntrn tame See Example on next page FebrunyMnenzmn Emplz You wrsn to endow a enarr m nance at a busmess school Ithe rnterest rate rs 10 per year and your ar s to proyrde 200000 a year In erpeturty how much must be set asrde Loday7 The rst payment W111 be made 1 year from today 5 a 5102 m 1 ampem 1 2 a1n2nzpzszar 3 4 szsmtmluzTnda Emplz Now uppose you don t want the endowment to start for 3 years Ithe rnterest rate 15 still 10 per year and yo ar 15 to proyrde 200000 a year mpa pemltyw 1d needto set asrde rnoreor esst ay7 what 15 ount Th1 15 a deferred annurty orrnorepreersely perpeturty problem I 1 2 3 4 s n x rump m a an amp yam Febmreramh 2mm H Simple versus Cnmpnund Interest Starting Pomt 12 eompounded quarterly 15 notthe same as 12 eompounded annua11y 12 eompounded quanerprovxdes you Wth 12 4 or 3 permonth annua1 compoundmg Annua1 Percentage Rate APR are rates reponed on annua1 basxs but eou1dbe an r wvannua ye1e monthly So APRs are not neeessaruy sxmple mterest rates FPF h nun rea11y mterested m r eredu eard requmng monthly payments eames an APR of18 n What Example You 15 the Effective Annua1 Rate EAR7 not pay 7 m 1e 13 Outstandmg alantz o a 19 monermn my W 5150 zu Annu x1121stmmAnnu Cmnpnundmg 21 Annualinzzxzsunmllndm anpaundmg 5195 7 BIEKIYIBIQ IZE MZI39 1 Remember The more frequentthe compoundmg me hxgherthe EAR FebruaryMnrenzmn

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