Consider the following Collateralized Mortgage Obligations (CMOs) with three classes of securities, Tranche A, Tranche B and Tranche Z.

Please help to finish the 3 question in the file

Take Home Assignment Seminar course in Real Estate and Banking Attempt All three Questions. Please follow the instruction below when you submit your work. Work submitted to the wrong address will not be marked. Please send in a sof copy of your work in WORD (NOT PDF) format to my email address, \"KTSE@HKU.HK\" before 23:59 of July 9 (Sunday), 2017. Question 1: Collateralized Mortgage Obligations (40 points) Consider the following Collateralized Mortgage Obligations (CMOs) with three classes of securities, Tranche A, Tranche B and Tranche Z. Tranche A B Z Assets 10-year, 11% fixed rate Mortgages = $75,000,000 Overcollateralization = $3,000,000 Stated maturity Coupon Rate 4 years 9% 5 years 10% 10 years 11% All cash flows are annual. Amount Issued $27,000,000 $15,000,000 $30,000,000 It is expected that the mortgage pool will experience constant prepayment rate of either 0% or 10% per year depending on the mortgage interest rate level. a. Determine the market value of Tranche A, B, and Z securities if the mortgage interest rate tomorrow afer the issue of the securities increases to 15%. b. Determine the market value of Tranche A, B, and Z securities if the mortgage interest rate tomorrow afer the issue of the securities decreases to 6%. k s maurice tse 2 Question 2: Collateralized Debt Obligations (30 points) Nopay Bank is going to issue collateralized debt obligations (CDOs) against the entire issue of the 10-year risky debt of Fortune Limited as presented here: Fortune Limited Market Value of Assets Market Value of Debt and Equity Assets $100 10-yr Debt (Face = $66.8) ?? Equity ?? The risky debt is a zero-coupon bond that pays no interest. At maturity, the bond will pay the investors just the face value. The underlying assets of Fortune Limited are valued at $100 today. The volatility of asset value in terms of annualized standard deviation is estimated to be 25%. The risk free rate of interest is 5% per year compounded continuously. The entire issue of the 10-year risky debt with face value equal to $66.8 is to be placed into a debt pool. Against the debt pool, Nopay Bank will issue 10-year CDO with face value equal to $50 that pays no interest. The rest of the 10-year risky debt will be held in the debt pool as over-collateralization (equity). (a) Determine the market value of the 10-year risky debt. (b) Determine the market value of Fortune's equity. (c) Determine the market value of the 10-year CDO that pays no interest. (d) Determine the market value of the over-collateralization. (e) Determine the cost of insurance to protect the CDO investors against Fortune's default. (f) Determine the market value of Fortune's equity and the market value of the overcollateralization (equity of the debt pool) when the volatility of the asset value increases from 25% to 50%. Explain your findings with the results of part (b) and part (d) above. The Black-Scholes equation is as follows: Call Value S N (d1 ) Xe rT N (d 2 ) ln( S / X ) (r 0.5 2 )T d1 ; d 2 d1 T T k s maurice tse 3 Question 3: Real Estate Bubble (30 points) Following our discussion on stochastic rational bubble in the Hong Kong property market, repeat the same analysis for year 2015 and 2016 by looking up the relevant data from the Census and Statistics Department web site or other sources. (a) Evaluate the size of \"bubble\" for 2015 and 2016. (b) How likely the real estate bubble will burst in 2017? (c) How likely the real estate bubble will burst in the next two years, 2018 and 2019? k s maurice tse 4 Seminar\tCourse\tin\tReal\tEstate\tand\tBanking Bubbles and\tCrashes Kwok-Sang\t(Maurice)\tTse\t Faculty\tof\tBusiness\tand\tEconomics The\tUniversity\tof\tHong\tKong ktse@hku.hk k s\ttse, hku 1 Bubbles Bubbles\thave\tviolent\tcollapses. They\tcan\tlead\tto\tdistortions\tin\tresource\tallocation. They\tmay\tbe\tassociated\twith\tfinancial\tcollapses. k s\ttse 2 What\tis\tbubble A\tbubble\tis\ta\tsituation\twhere\tasset\tprices\tmove because\tthey\tare\texpected\tto\tmove. In\ta\tbubble\tthe\tprice\tmoves\taway\tfrom\tfundamentals based\tsolely\ton\texpectations\tof\tfurther\tmovements. Expectations\tbecome\tself-confirming. Examples\tabound.... - housing\tbubble,\ttelecoms\tbubble,\ttulipmania, South\tSea\tBubble,\tGreat\tCrash\tof\t1929, commodities\tetc.... k s\ttse 3 Example:\tJapan\tStock\tPrice\tBubble 1985-1991 Nikkei\t225 45000 40000 35000 30000 25000 20000 15000 10000 5000 0 1970 1975 1980 1985 1990 1995 2000 2005 The\tJapanese\tasset\tprice\tbubble was\tan\teconomic\tbubble\tfrom\t1986\tto\t1991\twith real\testate\tand\tstock\tprices\tgreatly\tinflated.\tThe\tbubble's\tsubsequent\tcollapse\tlasted for\tmore\tthan\ta\tdecade\twith\tstock\tprices\tinitially\tbottoming\tin\t2003,\talthough\tthey would\tdescend\teven\tfurther\tamidst\tthe\tglobal\tfinancial\tcrisis\tin\t2008. The\tJapanese asset\tprice\tbubble\tcontributed\tto\twhat\tsome\trefer\tto\tas\tthe\tLost\tDecade.\tEconomist Paul\tKrugman has\targued\tthat\tJapan\tfell\tinto\ta\tliquidity\ttrap during\tthese\tyears. k s\ttse 4 Example:\tNasdaq\tBubble\t1999-2000 Between\t1999\tand\tearly\t2000,\tthe\tU.S.\tFederal\tReserve\tincreased\tinterest\trates\tsix times: The\teconomy\tbegan\tto\tlose\tspeed. The\tdot-com\tbubble\tburst\ton\tFriday,\tMarch\t10,\t2000,\twhen\tthe\ttechnology\theavy NASDAQ\tComposite\tindex,\tpeaked\tat\t5,048.62\t(intra-day\tpeak\t5,132.52),\tmore than\tdouble\tits\tvalue\tjust\ta\tyear\tbefore. k s\ttse 5 Hong\tKong\tPrice\tand\tRental\tIndex\tTrend Private\tDomestic\tPrice\tIndex (1999\tPrice) 250.0 200.0 Post-Crisis\tLevels 1997\tLevels 150.0 100.0 50.0 REAL ESTATE BUBBLE FORMED ?? Class\tA,\tB,\tC Class\tD,\tE 93Q1 93Q3 94Q1 94Q3 95Q1 95Q3 96Q1 96Q3 97Q1 97Q3 98Q1 98Q3 99Q1 99Q3 00Q1 00Q3 01Q1 01Q3 02Q1 02Q3 03Q1 03Q3 04Q1 04Q3 05Q1 05Q3 06Q1 06Q3 07Q1 07Q3 08Q1 08Q3 09Q1 09Q3 10Q1 10Q3 11Q1 11Q3 12Q1 12Q3 0.0 k s\ttse 6 What\tis\tbubble Bubbles - Where\tdo\tthey\tcome\tfrom?\tWhat\tto\tdo\tabout them? k s\ttse 7 Bubbles Two\tbroad\t(and\tpolar)\tviews: - There\tis\ta\tshortage\tof\tstore\tof\tvalue\t bubbles help\tfixing\tthis\tproblem. - Agents\tmisbehave\t(either\tan\tagency\tproblem\tor\ta behavioral\tproblem). Other:\t\"irrational\texuberance\"\tand\tmore\tformal behavioral\tstories - More\tlikely\tto\tarise\twhen\tthe\tabove\tconditions\tare present. k s\ttse 8 Bubbles Allen-Gale\t(2000)\t Bubbles\tand\tcrises There\tis\ta\tpattern: Phase\t1:\tfinancial liberalization\tor some\texpansionary policy\tfuels\ta bubble Phase\t2:\tthe\tbubble bursts\tand\tasset prices\tcollapse Phase\t3:\twidespread defaults\tby\tleveraged asset\tbuyers,\tleading\tto a\tbanking\tand/or exchange\trate\tcrisis,\tand a\tpersistent\trecession Main\tingredient\t(we'll\tdiscuss\there): - Uncertainty\tabout\tpayoffs\t(real\tor\tfinancial\tsector)\tcan lead\tto\tbubbles\tin\tan\tintermediated\tfinancial\tsystem\t(risk shifting/asset\tsubstitution) k s\ttse 9 A\tSimple\tModel Two\tdates,\tt\t=\t0,\t1\tand\ta\tsingle\tconsumption\tgood Two\tinvestment\tassets: - Safe\tasset\tand\tin\tvariable\tsupply\tat\ta\trate\t5% - Risky\tasset\tand\tin\tfixed\tsupply\twith\tuncertain\tvalue\t(R)\tat\tt\t=\t1\tgiven\tby\t: State of\tEconomy Boom Recession Value\t(R) 1.20 0 Probability 0.4 0.6 The\treturn\ton\tthe\tsafe\tasset\tis\t5%. Non-pecuniary\tconvex\tcost\tof\tinvesting\tin\t\"x\"\tamount\tof\tthe risky\tassets\tis\t0.05x2 Example:\tCost\tdue\tto\twaiting\ttime\t(delay\tin\tconsumption) To\trestrict\tportfolio\tsizes\tand\tto\tensure\tequilibrium\tprofit for\tborrowers k s\ttse 10 A\tSimple\tModel There\tare\tmany\tsmall,\trisk\tneutral investors;\tsame\tfor\tbanks. Investors\thave\tno\twealth\twhile\tbanks\thave\ta\tfixed\tamount\tB to\tlend\t(which\tthey\tsupply\tinelastically). Only\tinvestors\tknow\thow\tto\tinvest,\tso\tbanks'\tonly\tchoice\tis to\tlend\tto\tinvestors. Banks\tand\tinvestors\tuse\tsimple\tdebt\tcontracts\t(in\tparticular, they\tdon't\tdepend\ton\tsize). Since\tinvestors\tcan\tborrow\tas\tmuch\tas\tthey\twant\tat\tthe going\trate\t5%,\tin\tequilibrium\tthe\tcontracted rate\ton\tloans must\tbe\tequal\tto\tthe\triskless\tinterest\trate\t5%. Symmetric\tequilibrium: All\tinvestors\tare\tidentical\tex-post. k s\ttse 11 A\tSimple\tModel How\twould\tthe\trepresentative\tinvestor\tinvest? The\tinvestor\twill\tinvest\tin\ta\tportfolio of\tsafe\tand\trisky assets. Xs and\tXR are\tthe\trepresentative\tinvestor's\tholdings\tof\tthe safe\tand\trisky\tassets,\tboth\tin\tunits\tof\tconsumption\tgood. Banks\tuse\tdebt\tcontracts\tand\tcannot\tobserve\tinvestment decisions\tby\tborrowers. Borrower\tgets\tthe\tbenefit\tif\tthe\toutcome\tis\tgood. However,\tthe\tborrower does\tnot\tbear\tthe\tfull\tcost\tof investment\tif\tthe\toutcome\tis\tbad;\the\tcan\tjust\tshift\tthe\tloss to\tthe\tbank\tby\tgoing\tbankrupt. k s\ttse 12 A\tSimple\tModel Borrow\ta\tbank\tloan Buy\tsafe\tand\trisky\tasset Price\tof\trisky\tasset\t=\tP t=0,\ttoday t=1 Suppose\trepresentative\tinvestor\tbuys\tXs units\tof\tthe safe\tasset\tand\tXR units\tof\tthe\trisky\tasset\ttoday\t(t=0)\tby borrowing\ta\tbank\tloan. Let\tP\tbe\tthe\tprice\tof\tthe\trisky\tasset\trelative\tto\tthe\tsafe asset. And\tsuppose\tthe\tunit\tprice\tof\tsafe\tasset\tis\t$1. He\tborrows\ta\tbank\tloan: - Bank\tLoan\t=\t1\t(Xs)\t+\tP\t(XR) k s\ttse 13 A\tSimple\tModel Borrow\ta\tbank\tloan Buy\tsafe\tasset\trisky\tasset Price\tof\trisky\tasset\t=\tP Sell\tthe\tsafe\tand\trisky\tasset Repay\tbank\tloan\twith\tinterest Realize\tprofit\tor\tloss t=0,\ttoday t=1 The\tbank\tloan\twith\tinterest\tdue\tat\tt\t=\t1\tis: 1.05\t(Xs+\tPXR) The\tliquidation\tvalue\tof\tthe\tportfolio\tat\tdate\tt\t=\t1\tis: 1.05X S + RX R Value\tof Safe\tAsset Value\tof Risky\tAsset So,\tthe\tpayoff\tfor\tthe\tinvestor\tat\tdate\tt\t=\t1\tis [1.05X S + RX R ]1.05(X S + PX R ) = (R 1.05P)X R What\tis\tthe\tvalue\tof\tthe\trisky\tasset\tR\tat\tt\t=1\t? k s\ttse 14 A\tSimple\tModel What\tis\tthe\tvalue\tof\tthe risky\tasset\tR\tat\tt\t=\t1? 1.2\t;\tProb =\t0.4 Risky Asset\tR t=0,\ttoday 0\t; Prob =\t0.6 t=1 Recall\tthe\tpayoff\tto\tinvestor\tat\tt\t=\t1\tis: (R 1.05P)X R The\trisky\tasset\tR\tis\teither\t1.2\twith\tprob 0.4 or\t0\twith\tprob 0.6. - If\tR\tis\tzero,\tthen\tthe\tpayoff\tis\ta\tloss. The\tinvestor\twill\tjust\tgo bankrupt\t(limited\tliability) and\tthe\tbank\twill\tbear\tthe\tloss. - That\tmeans\tthe\tpayoff\tto\tthe\tinvestor\tis: Either (1.2\t\t1.05P)\tXR >\t0 OR ZERO ! The\texpected\tpayoff\tto\tinvestor\tat\tt\t=\t1\tis\ttherefore: (1.21.05P)X R 0.4 + 0 0.6 k s\ttse 15 A\tSimple\tModel Recall\tthe\texpected\tpayoff\tto\tinvestor\tat\tt\t=\t1\tis: (1.2 1.05P)X R 0.4 What\tis\tthe\tOPTIMAL\tinvestment\tdecision\tfor\tthe\tinvestor? The\tdecision\tfor\tthe\tinvestor\tis\tto\tdetermine\tthe\toptimal holding\tof\tthe\trisky\tasset\tXR\tthat\twill\tmaximize\this\texpected wealth\tat\tt\t=\t1. (1.2 1.05P)X R 0.4 0.05(X R )2 Expected\tPayoff Non-pecuniary\tCost The\tinvestment\tdecision\tcan\talso\tbe\tdescribed\tby: Max $%(1.2 1.05P)X R 0.4 0.05(X R )2 &' X R 0 k s\ttse 16 A\tSimple\tModel And\tthe\tdecision\tproblem\tis\tto\tdetermine\tthe\toptimal holding\tof\tthe\trisky\tasset\tXR: Max $%(1.2 1.05P)X R 0.4 0.05(X R )2 &' X R 0 Differentiation:\t1st order\tcondition\tfor\tmaximization: (1.2 1.05P) 0.4 0.1X R = 0 Solving\tfor\tthe\tequilibrium\tprice\tof\trisky\tasset\tP,\twe have: 1 " 0.1X R % P= $#1.2 '& 1.05 0.4 k s\ttse 17 A\tSimple\tModel To\tsee\tthe\tbubble,\twe\tneed\tto\tcompare\tthe\tequilibrium price\tof\trisky\tasset\tP\twith\tthe\tfundamental\tvalue\tP*. What\tis\tthe\tfundamental\tvalue\tP*? - Define\tthe\tfundamental\tas\tthe\tprice\t(P*)\tan\tinvestor would\tbe\twilling\tto\tpay\tin\tthe\tabsence\tof risk\tshifting. - In\tother\twords,\tthe\tinvestor\tcannot\tpass\tthe investment\tloss\tto\tthe\tbank\tby\tgoing\tbankrupt. k s\ttse 18 A\tSimple\tModel What\tis\tthe\tvalue\tof\tthe risky\tasset\tR\tat\tt\t=\t1? 1.2\t;\tProb =\t0.4 Risky Asset\tR t=0,\ttoday 0 ; Prob =\t0.6 t=1 Recall\tthe\tpayoff\tto\tinvestor\tat\tt\t=\t1\tis: (R 1.05P)X R The\trisky\tasset\tR\tis\teither\t1.2\twith\tprob 0.4 or\t0\twith\tprob 0.6. - If\tthe\tinvestor\tis\tnot\tallowed to\tgo\tbankrupt,\tthen\tthe payoff\tto\tthe\tinvestor\tis: Either (1.2\t\t1.05P)\tXR >\t0 OR (0\t\t1.05P)\tXR \t0.4 >\t0. Hence,\tdue\tto\trisk\tshifting,\tP\tis\thigher\tthan fundamental\tP* (bubble!); P\t>\tP* The\tcounterpart\tof\tthe\tbubble\tis\tthe\tbank\tlosses,\tand hence\tthe\trest\tof\tthe\tstory... Any\tirrationality\tinvolved?? In\ta\tsense\tit\tis\tnot\ta\tGE\tbubble,\tas\tthe\tprice\tof\tbanks\tshould\tgo\tdown...\tbut\tit\tmay\twell be\tthat\thouseholds\tare\tstuck...\tthis\ttakes\tus\tto\tthe\tstandard\tmodel\tof\tRE\tbubbles\tin macro,\twhich\thighlights\tthe\tshortage\tof\tassets.. k s\ttse 22 A\tSimple\tModel What\tare\tthe\timplications? - Abundant\tliquidity\tin\tthe\tfinancial\tsector\t(banks) - Investment\trisk\tinvolveduncertainty\ton\tprice - Investors\tcan\tshift\ttheir\tlosses\tto\tbanks - Interest\trate\tdoes\tnot\treflect\tunderlying\trisk - Investors\tare\twilling\tto\tpay\ta\thigher\tprice\tfor\trisky investment\tthan\tthe\tfundamental\tvalue,\tleading to\tthe\tformation\tof\ta\tbubble. - No\tirrationality\tinvolved. k s\ttse 23 Rational\tBubble Investors\trealize\tprice\tis\tdivorced\tfrom\tfundamentals, but\tbelieve\tthat\tprice\trises\twill\tpersist\tfor\tsome\ttime, and\tthat\tprice\tgrowth\twill\tcompensate\tfor\trisk\tof collapse. - Investors\tknow\tthat\tthe\tbubble\twill\tcollapse\tbut believe\tthey\tcan\tget\tout\tbefore\tit\tdoes. k s\ttse 24 Rational\tBubble Irrational\tbubbles\tthen\tinvolve\tunrealistic\texpectations about\tasset's\tfuture\tprospects - fad - mania - Irrational\texuberance\t(Unsustainable\tinvestor enthusiasm\tthat\tdrives\tasset\tprices\tup\tto\tlevels\tthat aren't\tsupported\tby\tfundamentals.) Let\tus\tconsider\ta\trational\tbubble\t.... k s\ttse 25 Rational\tBubbleSimple Characterization Let\tvt be\tthe\tproperty\tprice;\tsuppose fundamentals imply\tthat vt =\tv* for\tall\tt. At\tsome\ttime\tt0 the\tprice\tjumps\tby\ta\tbubble\tb0 to\tv0 - suppose\tagents\texpect\tvt =\t1\t+\tr\tin\teach\tfuture period\tt. i.e.,\tbt =\tb0(1\t+\tr)t for\tarbitrary\tb0. See\tFigure. Why\tare\tagents\twilling\tto\tpay\tincreasing\tprices\tfor\tthe property? k s\ttse 26 Rational\tBubbleSimple Characterization Why\tare\tagents\twilling\tto\tpay\tincreasing\tprices\tfor\tthe property? - expected\tcapital\tgains\tare\tself-fulfilling\t-- this\tis\ta rational\tbubble,\tthe\tcapital\tgain\ton\tproperty compensates\tfor\tthe\talternative\treturns. Notice\tthis\trequires\tthe\tprice\tto\tgrow\tforever. - If\teveryone\tknew\tthat\tat\tperiod\t(T\t+\tj)\tthat\tthe\tbubble would\tburst\t(that\tis,\tv\t v* or\tbt 0),\tthen\tno\tone would\tpay\tthe\tbubble\tprice\tat\tperiod\t(T\t+\tj\t- 1). The\tbubble\tunravels. k s\ttse 27 Rational\tBubbleSimple Characterization A\tbubble\tin\thousing\tmarket If\teveryone\tknew\tthat\tat\tperiod\tT\t+\tj\tthat\tthe\tbubble\twould\tburst\t(v\t v* or\tbt 0),\tthen\tno\tone\twould\tpay\tthe\tbubble\tprice\tat\tT\t+\tj - 1. The\tbubble unravels. v bt =\tb0(1+r)t V0 =\tv* +\tb0 v* t0 T+j k s\ttse Time 28 Rational\tBubbleSimple Characterization Since\tprices\tcannot\trise\tforever,\tare\tbubbles\truled\tout? - No,\tas\tlong\tas\tthe\tdate\tof\tprice\tcollapse\tis\tuncertain! v bt =\tb0(1+r)t V0 =\tv* +\tb0 v* t0 T+j k s\ttse Time 29 Stochastic\tRational\tBubble* Since\tprices\tcannot\trise\tforever, are\tbubbles\truled\tout? - no,\tas\tlong\tas\tthe\tdate\tof\tprice\tcollapse\tis\tuncertain Suppose\tthat\tin\teach\tperiod\tagents\tbelieve\tthat\tthe\tprobability\tthe\tbubble will\tnot\tburst is\tq. bt (1+r)bt q No\tBurst:\tExpected\tBubble\t=\tbt+1 1-q Burst:\tExpected\tBubble\t=\t0 Since\tthe\tbubble\tis\texpected\tto\tgrow\tat\tgrowth\trate\tr,\tthen\twe\thave: (1+ r)bt q bt+1 = (1+ r)bt bt+1 = q The\tACTUAL\tbubble\tat\tt+1\tis\twritten\tas\t: - t+1 is\ta\twhite\tnoise\twith\tmean\t0. k s\ttse b!t+1 = (1+ r)bt q + !t+1 30 Stochastic\tRational\tBubble Since\tprices\tcannot\trise\tforever, are\tbubbles\truled\tout? - no,\tas\tlong\tas\tthe\tdate\tof\tprice\tcollapse\tis\tuncertain Suppose\tthat\tin\teach\tperiod\tagents\tbelieve\tthat\tthe\tprobability\tthe\tbubble will\tnot\tburst is\tq. Then\twe\thave ! (1+ r)b t # !b = " q + !t+1 $$$$prob$of$NO$burst$=$q t+1 #! $t+1 $$$$$$$$$$$$$$$$$$$$$prob$of$burst$=$(11q)$ - t+1 is\ta\twhite\tnoise\twith\tmean\t0. If\tthe\tbubble\tfollows\tthis\tpath\tit\tis\trational. - notice\tthat\tthe\texpected\tvalue\tof\tthe\tbubble\tin\tperiod\tT\t+1\tis\texactly\tbt+1. - To\tsee\tthis,\tnote\tthat (1+ r)bt ! Et [bt+1 ] = q + q(0)+(1 q)(0) = (1+ r)bt q --our\tinitial\texpression\tfor\tthe\tbubble\tpath. See\tExcel. k s\ttse 31 Stochastic\tRational\tBubble Notice\tthat\tthe\tstochastic\tbubble\tgrows\tfaster\tthan\tunder certainty,\tbecause\tinvestors\tmust\tbe\tcompensated\tfor\tthe\trisk\tof the\tbubble\tbursting. (1+ r)bt > (1+ r)bt q Analysis:\tFrom\trational\tbubble\twe\tcan\tback\tout\tthe\tmarket's expectation\tof\tit\tbursting. - At\tany\ttime\tt,\tyou\tknow\tthe\tactual\tprice\tand\tthe\tinterest\trate, and\tif\tyou\tknow\tv*,\tthen\tyou\tcan\tcalculate\tbt =\tvt - v*. - If\tthe\tbubble\thas\tnot\tburst\tyet\tthen (1+ r)bt bt+1 = q (1+ r)bt q= bt+1 k s\ttse 32 Stochastic\tRational\tBubble If\tthe\tbubble\thas\tnot\tburst\tyet,\tthen (1+ r)bt bt+1 = q (1+ r)bt q= bt+1 This\tprobability\tcan\tbe\tthen\tcompared\tto\tthe\tamount\tof\ttime\tthe bubble\thas\tbeen\tgrowing. - This\tnotion\tof\ta\trational\tbubble\tis\tused\tfrequently\tin\tanalysis\tof asset\tmarkets. Ad\thoc\tExample:\tHong\tKong\tProperty\tMarket k s\ttse 33 Example:\tHong\tKong\tProperty\tMarket Median\tHH\tIncome/Month 25000 19872 20667 20000 15000 10000 5000 0 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 Suppose\tan\taverage\thousehold\tcan\tonly\tcommit\tup\tto\t1/3\tof\tits\tincome\tto mortgage\tpayment. Suppose\tmortgage\tterms\tare\t20-year,\tmonthly\tpayments,\twith\tmortgage\trate equal\tto\tPrime\tRate\t(5%)\tminus\t2.5%,\tand\t30%\tdown\tpayment. k s\ttse 34 Example:\tHong\tKong\tProperty\tMarket Suppose\tan\taverage\thousehold\tcan\tonly\tcommit\tup\tto\t1/3\tof\tits income\tto\tmortgage\tpayment. Suppose\tmortgage\tterms\tare\t20-year,\tmonthly\tpayments,\tmortgage rate\tequal\tto\tPrime\tRate\t(5%)\tminus\t2.5%,\tand\t30%\tdown\tpayment. What\tis\tthe\tmaximum\thome\tprice\tthe\thousehold\tcan\tafford\tin\t2012? Based\ton\tthe\tmedian\thousehold\tincome\tand\tthe\tmortgage\tterms,\tthe maximum\tmortgage\tan\taverage household\tcan\tborrow\tis\tdetermined as\tfollows: Mortgage\tLoan\t= 12n ( " ( " 5% 2.5% %240 + r % + *1 $1+ ' *1 $1+ ' 20, 667 # & # & 12 12 -= * PMT * = 1, 300, 042 r 5% 2.5% * * 3 * * 12 12 ) , ) , Affordable\thome\tprice\t=\tHK$\t1,300,042/0.7\t= HK$\t1,857,203 k s\ttse 35 Stochastic\tRational\tBubble Property\tPrice: Class\tA\tResidential\t(40\tsquare\tmeters) $3,500,000 3224262.8 Afforable Price $3,000,000 2931148 Actual Price $2,500,000 Size\tof\tBubble $1,857,203 $2,000,000 $1,857,203 $1,500,000 $1,000,000 $500,000 $0 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 Affordable\tProperty\tPrice\tis\tcalculated\tbased\ton\t1/3\tof\tmedian\thousehold income,\t20\tyear\tmortgage,\t5%\tminus\t2.5%\tmortgage\trate,\tand\t30%\tdown payment. k s\ttse 36 Stochastic\tRational\tBubble Property\tPrice: Class\tA\tResidential\t(40\tsquare\tmeters) $3,500,000 What\tis\tthe $3,000,000 market expectation $2,500,000 that\tproperty price\twill\tfall $2,000,000 to\tthe $1,500,000 affordable price\tlevel? $1,000,000 Afforable Price 2931148 Actual Price $1,857,203 $1,857,203 b2012 =\t2931148\t- 1857203\t=\t1073945 b2013 =\t3224262\t- 1857203\t=\t1367060 Mortgage\trate\t=\t2.5% $500,000 $0 2002 (1+ r)b2012 b2013 = q 3224262.8 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 (1+ r)b2012 1.025 1073945 q= = = 0.805 b2013 1367060 k s\ttse 37 Stochastic\tRational\tBubble Property\tPrice: Class\tA\tResidential\t(40\tsquare\tmeters) What\tis\tthe $3,500,000 market $3,000,000 expectation $2,500,000 that\tthe property\tprice $2,000,000 bubble\twill burst\twithin $1,500,000 the\tnext\t5 $1,000,000 years? Afforable Price 3224262.8 2931148 Actual Price $1,857,203 $1,857,203 (1+ r)b2012 1.025 1073945 q= = = 0.805 b2013 1367060 $500,000 $0 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 Probability\tthat\tthe\tprice\tbubble\twill\tNOT\tburst\tat\tthe\tend\tof\t5\tyears\tis (0.805)5 =\t0.339 Probability\tit\twill\tburst\twithin\tthe\tnext\t5\tyears\tis\t66.1% k s\ttse 38 Stochastic\tRational\tBubble:\tImplications Fact\tthat\tbubble\thas\tto\tgrow\tat\tan\texpected\trate\tof\tr\tallows one\tto\teliminate\tmany\tpotential\trational\tbubbles. - commodities\twith\tclose\tsubstitutes\tputs\tlimits - a\tbubble\ton\ta\tnon-zero\tsupply\tasset\tcannot\tarise\tif\tr exceeds\tthe\tgrowth\trate\tof\tthe\teconomy,\tsince\tthe\tbubble would\toutgrow\tthe\taggregate\twealth\tin\tthe\teconomy (Tokyo\tImperial\tPalace\tworth\tmore\tthan\tthe\tsum\tof\tall\tthe real\testates\tin\tCalifornia\tin\tthe\t1980s\t??). Hence,\tbubbles\tcan\tonly\texist\tin\ta\tworld\tin\twhich\tthe\tr \t0 OR ZERO\t! The\tdecision\tproblem\tfor\tthe\tinvestor\tis\tto\tdetermine\tthe optimal\tholding\tof\tthe\trisky\tasset\tXR: Max [ X R (RH rP) c(X R )] X R 0 k s\ttse 47 A\tSimple\tModel And\tthe\tdecision\tproblem\tis\tto\tdetermine\tthe optimal\tholding\tof\tthe\trisky\tasset\tXR: Max [ X R (RH rP) c(X R )] X R 0 First\torder\tcondition\tfor\tmaximization: (RH rP) c'(X R ) = 0 Solving\tfor\tthe\tequilibrium\tprice\tof\trisky\tasset\tP,\twe have: 1" c'(X R ) % P = $ RH '& # r k s\ttse 48 A\tSimple\tModel To\tsee\tthe\tbubble,\twe\tneed\tto\tcompare\tthe equilibrium\tprice\tof\trisky\tasset\tP\twith\tthe fundamental\tvalue\tP*. What\tis\tthe\tfundamental\tvalue\tP*? - Define\tthe\tfundamental\tas\tthe\tprice\t(P*)\tan\tagent would\tbe\twilling\tto\tpay\tin\tthe\tabsence\tof risk shifting. k s\ttse 49 A\tSimple\tModel Define\tthe\tfundamental\tas\tthe\tprice\t(P*)\tan\tagent\twould\tbe willing\tto\tpay\tin\tthe\tabsence\tof risk\tshifting. Then\tthe\tdecision\tproblem\tis: Max { X R [(RH rP * ) + (0 rP * )(1 )] c(X R )} X R 0 Note\tthat\tR\tis\teither\t\"RH with\tprob \" or\t\"0\twith\tprob (1-)\". First\torder\tcondition\tfor\tmaximization: (RH rP * ) rP * (1 ) c'(X R ) = 0 Solving\tfor\tfundamental\tvalue\tof\tthe\trisky\tasset\tP*,\twe\thave 1 P = [ RH c'(X R )] r * k s\ttse 50 A\tSimple\tModel Comparing\tP\twith\tP*,\tit\tis\teasy\tto\tsee\tthat P =\tP* Note\tthat\t is\tprobability\tof\teconomic\tboom\tand 1\t>\t >\t0. Hence,\tdue\tto\trisk\tshifting,\tP\tis\thigher\tthan fundamental\tP* (bubble!); P\t>\tP* The\tcounterpart\tof\tthe\tbubble\tis\tthe\tbank\tlosses,\tand hence\tthe\trest\tof\tthe\tstory... Any\tirrationality\tinvolved?? In\ta\tsense\tit\tis\tnot\ta\tGE\tbubble,\tas\tthe\tprice\tof\tbanks\tshould\tgo\tdown...\tbut\tit\tmay\twell be\tthat\thouseholds\tare\tstuck...\tthis\ttakes\tus\tto\tthe\tstandard\tmodel\tof\tRE\tbubbles\tin macro,\twhich\thighlights\tthe\tshortage\tof\tassets.. k s\ttse 51 a. Determine the market value of Tranche A, B, and Z securities if the mortgage interest rate tomorrow after the is Value = Amount Issued*(1+i)+(Interest rate*Increase rate) Tranche A TrancheB Tranche C $ $ $ 26,640,355.20 15,225,000.00 30,495,000.00 a. Determine the market value of Tranche A, B, and Z securities if the mortgage interest rate tomorrow after the is Value = Amount Issued*(1+i)-(Interest rate*Decrease rate) Tranche A $ 27,000,001.50 TrancheB $ 15,000,000.01 Tranche C $ 30,000,000.01 t rate tomorrow after the issue of the securities increases to 15%. t rate tomorrow after the issue of the securities decreases to 6%. a. Determine the market value of the 10-year risky debt. Market Value = cash flow/(1+rdebt)^t Market Value 38.4615385 b. Determine the market value of Fortune's equity. Market Value 76.9230769 c. Determine the market value of the 10-year CDO that pays no interest. Market Value 51.3846154 d. Determine the market value of the over-collateralization. Market Value 89.8461538 e. Determine the cost of insurance to protect the CDO investors against Fortune's default. Cost of insurance 70 f. Determine the market value of Fortune's equity and the market value of the over-collateralization (equity of th asset value increases from 25% to 50%. Explain your findings with the results of part (b) and part (d) above. Call Value S N (d1 ) Xe rT N (d 2 ) S N d1 d2 Call value 100 1 0.005 -0.045 $48.88 alization (equity of the debt pool) when the volatility of the nd part (d) above. Question 3 Hong Kong Property Market for 2015 and 2016 2015 2016 Residential property (HK$ billion) 416.5 395.1 Non-residential property (HK$ billion) 132.1 90.1 Yr 2015 Yr2016 548.7 211.25 485.3 193.30 DATE Actual Price Affordable prices Affordable Property Price = (1/3*medain household income)+20 year mortgage+(2.5%*Mortgage rate)+(30%*down payme 2015 2016 Median household income 43.55 44.525 20 year mortagage 3.015006 3.1045274027 2.5% of mortgage interest exp 0.075375 0.0776131851 30% of downpayment 164.61 145.59 Total 211.2504 193.2971405878 Wages and earnings 2011 2015 Nominal 177.2 (+9.9) 211.3 (+4.4) Real(2) 127.1 (+4.4) 120.7 (+2.3) Nominal 138.0 (+5.9) 163.3 (+4.8) Real(4) 117.0 (+0.5) 122.2 (+2.8) Nominal 178.1 (+7.5) 222.5 (+5.7) Real(4) 151.1 (+2.1) 166.4 (+3.8) Nominal 111.9 (+8.3) 136.1 (+4.5) Real(6) 108.7 (+1.6) 113.3 (+2.0) Wage Index (Sep. 1992=100) (1) Salary index(3) (Jun. 1995=100) Salary Index (A) Salary Index (B) Index of Payroll per Person Engaged(5) (1st quarter 1999=100) Average median household income Total loans and advances to customers(1) (HK$ billion) Number of households ('000) 130.7 5,080.7 2 386 7,534.5 2 499 Mortgage per household 2.129379715 3.015006 Hong Kong Property Market for 2015 and 2016 800 700 600 500 400 300 200 100 0 Yr 2015 Actual Price (a) Evaluate the size of \"bubble\" for 2015 and 2016. Yr2016 Affordable prices $ 200,000,000.00 (b) How likely the real estate bubble will burst in 2017? The bubble is not likely burst in 2017 becase the size od the bubble is 2016 is still very high. (c) How likely the real estate bubble will burst in the next two years, 2018 and 2019? The bubble is likely to burst in the next two years since the affordable prices and actual price of the properti ate)+(30%*down payment) 2016 219.6 (+3.6) 120.7 (+2.5) 169.8 (+4.0) 124.2 (+1.7) 234.1 (+5.2) 171.3 (+2.9) 147.2 (+3.8) 118.1 (+2.5) 133.6 7,817.2 2 518 3.104527 Yr2016 al price of the properties are closing in. Hong Kong Property Market for 2015 and 2016 800 700 600 500 400 300 200 100 0 Yr 2015 Actual Price Yr2016 Affordable prices Seminar\tCourse\tin\tReal\tEstate\tand\tBanking Bubbles and\tCrashes Kwok-Sang\t(Maurice)\tTse\t Faculty\tof\tBusiness\tand\tEconomics The\tUniversity\tof\tHong\tKong ktse@hku.hk k s\ttse, hku 1 Bubbles Bubbles\thave\tviolent\tcollapses. They\tcan\tlead\tto\tdistortions\tin\tresource\tallocation. They\tmay\tbe\tassociated\twith\tfinancial\tcollapses. k s\ttse 2 What\tis\tbubble A\tbubble\tis\ta\tsituation\twhere\tasset\tprices\tmove because\tthey\tare\texpected\tto\tmove. In\ta\tbubble\tthe\tprice\tmoves\taway\tfrom\tfundamentals based\tsolely\ton\texpectations\tof\tfurther\tmovements. Expectations\tbecome\tself-confirming. Examples\tabound.... - housing\tbubble,\ttelecoms\tbubble,\ttulipmania, South\tSea\tBubble,\tGreat\tCrash\tof\t1929, commodities\tetc.... k s\ttse 3 Example:\tJapan\tStock\tPrice\tBubble 1985-1991 Nikkei\t225 45000 40000 35000 30000 25000 20000 15000 10000 5000 0 1970 1975 1980 1985 1990 1995 2000 2005 The\tJapanese\tasset\tprice\tbubble was\tan\teconomic\tbubble\tfrom\t1986\tto\t1991\twith real\testate\tand\tstock\tprices\tgreatly\tinflated.\tThe\tbubble's\tsubsequent\tcollapse\tlasted for\tmore\tthan\ta\tdecade\twith\tstock\tprices\tinitially\tbottoming\tin\t2003,\talthough\tthey would\tdescend\teven\tfurther\tamidst\tthe\tglobal\tfinancial\tcrisis\tin\t2008. The\tJapanese asset\tprice\tbubble\tcontributed\tto\twhat\tsome\trefer\tto\tas\tthe\tLost\tDecade.\tEconomist Paul\tKrugman has\targued\tthat\tJapan\tfell\tinto\ta\tliquidity\ttrap during\tthese\tyears. k s\ttse 4 Example:\tNasdaq\tBubble\t1999-2000 Between\t1999\tand\tearly\t2000,\tthe\tU.S.\tFederal\tReserve\tincreased\tinterest\trates\tsix times: The\teconomy\tbegan\tto\tlose\tspeed. The\tdot-com\tbubble\tburst\ton\tFriday,\tMarch\t10,\t2000,\twhen\tthe\ttechnology\theavy NASDAQ\tComposite\tindex,\tpeaked\tat\t5,048.62\t(intra-day\tpeak\t5,132.52),\tmore than\tdouble\tits\tvalue\tjust\ta\tyear\tbefore. k s\ttse 5 Hong\tKong\tPrice\tand\tRental\tIndex\tTrend Private\tDomestic\tPrice\tIndex (1999\tPrice) 250.0 200.0 Post-Crisis\tLevels 1997\tLevels 150.0 100.0 50.0 REAL ESTATE BUBBLE FORMED ?? Class\tA,\tB,\tC Class\tD,\tE 93Q1 93Q3 94Q1 94Q3 95Q1 95Q3 96Q1 96Q3 97Q1 97Q3 98Q1 98Q3 99Q1 99Q3 00Q1 00Q3 01Q1 01Q3 02Q1 02Q3 03Q1 03Q3 04Q1 04Q3 05Q1 05Q3 06Q1 06Q3 07Q1 07Q3 08Q1 08Q3 09Q1 09Q3 10Q1 10Q3 11Q1 11Q3 12Q1 12Q3 0.0 k s\ttse 6 What\tis\tbubble Bubbles - Where\tdo\tthey\tcome\tfrom?\tWhat\tto\tdo\tabout them? k s\ttse 7 Bubbles Two\tbroad\t(and\tpolar)\tviews: - There\tis\ta\tshortage\tof\tstore\tof\tvalue\t bubbles help\tfixing\tthis\tproblem. - Agents\tmisbehave\t(either\tan\tagency\tproblem\tor\ta behavioral\tproblem). Other:\t\"irrational\texuberance\"\tand\tmore\tformal behavioral\tstories - More\tlikely\tto\tarise\twhen\tthe\tabove\tconditions\tare present. k s\ttse 8 Bubbles Allen-Gale\t(2000)\t Bubbles\tand\tcrises There\tis\ta\tpattern: Phase\t1:\tfinancial liberalization\tor some\texpansionary policy\tfuels\ta bubble Phase\t2:\tthe\tbubble bursts\tand\tasset prices\tcollapse Phase\t3:\twidespread defaults\tby\tleveraged asset\tbuyers,\tleading\tto a\tbanking\tand/or exchange\trate\tcrisis,\tand a\tpersistent\trecession Main\tingredient\t(we'll\tdiscuss\there): - Uncertainty\tabout\tpayoffs\t(real\tor\tfinancial\tsector)\tcan lead\tto\tbubbles\tin\tan\tintermediated\tfinancial\tsystem\t(risk shifting/asset\tsubstitution) k s\ttse 9 A\tSimple\tModel Two\tdates,\tt\t=\t0,\t1\tand\ta\tsingle\tconsumption\tgood Two\tinvestment\tassets: - Safe\tasset\tand\tin\tvariable\tsupply\tat\ta\trate\t5% - Risky\tasset\tand\tin\tfixed\tsupply\twith\tuncertain\tvalue\t(R)\tat\tt\t=\t1\tgiven\tby\t: State of\tEconomy Boom Recession Value\t(R) 1.20 0 Probability 0.4 0.6 The\treturn\ton\tthe\tsafe\tasset\tis\t5%. Non-pecuniary\tconvex\tcost\tof\tinvesting\tin\t\"x\"\tamount\tof\tthe risky\tassets\tis\t0.05x2 Example:\tCost\tdue\tto\twaiting\ttime\t(delay\tin\tconsumption) To\trestrict\tportfolio\tsizes\tand\tto\tensure\tequilibrium\tprofit for\tborrowers k s\ttse 10 A\tSimple\tModel There\tare\tmany\tsmall,\trisk\tneutral investors;\tsame\tfor\tbanks. Investors\thave\tno\twealth\twhile\tbanks\thave\ta\tfixed\tamount\tB to\tlend\t(which\tthey\tsupply\tinelastically). Only\tinvestors\tknow\thow\tto\tinvest,\tso\tbanks'\tonly\tchoice\tis to\tlend\tto\tinvestors. Banks\tand\tinvestors\tuse\tsimple\tdebt\tcontracts\t(in\tparticular, they\tdon't\tdepend\ton\tsize). Since\tinvestors\tcan\tborrow\tas\tmuch\tas\tthey\twant\tat\tthe going\trate\t5%,\tin\tequilibrium\tthe\tcontracted rate\ton\tloans must\tbe\tequal\tto\tthe\triskless\tinterest\trate\t5%. Symmetric\tequilibrium: All\tinvestors\tare\tidentical\tex-post. k s\ttse 11 A\tSimple\tModel How\twould\tthe\trepresentative\tinvestor\tinvest? The\tinvestor\twill\tinvest\tin\ta\tportfolio of\tsafe\tand\trisky assets. Xs and\tXR are\tthe\trepresentative\tinvestor's\tholdings\tof\tthe safe\tand\trisky\tassets,\tboth\tin\tunits\tof\tconsumption\tgood. Banks\tuse\tdebt\tcontracts\tand\tcannot\tobserve\tinvestment decisions\tby\tborrowers. Borrower\tgets\tthe\tbenefit\tif\tthe\toutcome\tis\tgood. However,\tthe\tborrower does\tnot\tbear\tthe\tfull\tcost\tof investment\tif\tthe\toutcome\tis\tbad;\the\tcan\tjust\tshift\tthe\tloss to\tthe\tbank\tby\tgoing\tbankrupt. k s\ttse 12 A\tSimple\tModel Borrow\ta\tbank\tloan Buy\tsafe\tand\trisky\tasset Price\tof\trisky\tasset\t=\tP t=0,\ttoday t=1 Suppose\trepresentative\tinvestor\tbuys\tXs units\tof\tthe safe\tasset\tand\tXR units\tof\tthe\trisky\tasset\ttoday\t(t=0)\tby borrowing\ta\tbank\tloan. Let\tP\tbe\tthe\tprice\tof\tthe\trisky\tasset\trelative\tto\tthe\tsafe asset. And\tsuppose\tthe\tunit\tprice\tof\tsafe\tasset\tis\t$1. He\tborrows\ta\tbank\tloan: - Bank\tLoan\t=\t1\t(Xs)\t+\tP\t(XR) k s\ttse 13 A\tSimple\tModel Borrow\ta\tbank\tloan Buy\tsafe\tasset\trisky\tasset Price\tof\trisky\tasset\t=\tP Sell\tthe\tsafe\tand\trisky\tasset Repay\tbank\tloan\twith\tinterest Realize\tprofit\tor\tloss t=0,\ttoday t=1 The\tbank\tloan\twith\tinterest\tdue\tat\tt\t=\t1\tis: 1.05\t(Xs+\tPXR) The\tliquidation\tvalue\tof\tthe\tportfolio\tat\tdate\tt\t=\t1\tis: 1.05X S + RX R Value\tof Safe\tAsset Value\tof Risky\tAsset So,\tthe\tpayoff\tfor\tthe\tinvestor\tat\tdate\tt\t=\t1\tis [1.05X S + RX R ]1.05(X S + PX R ) = (R 1.05P)X R What\tis\tthe\tvalue\tof\tthe\trisky\tasset\tR\tat\tt\t=1\t? k s\ttse 14 A\tSimple\tModel What\tis\tthe\tvalue\tof\tthe risky\tasset\tR\tat\tt\t=\t1? 1.2\t;\tProb =\t0.4 Risky Asset\tR t=0,\ttoday 0\t; Prob =\t0.6 t=1 Recall\tthe\tpayoff\tto\tinvestor\tat\tt\t=\t1\tis: (R 1.05P)X R The\trisky\tasset\tR\tis\teither\t1.2\twith\tprob 0.4 or\t0\twith\tprob 0.6. - If\tR\tis\tzero,\tthen\tthe\tpayoff\tis\ta\tloss. The\tinvestor\twill\tjust\tgo bankrupt\t(limited\tliability) and\tthe\tbank\twill\tbear\tthe\tloss. - That\tmeans\tthe\tpayoff\tto\tthe\tinvestor\tis: Either (1.2\t\t1.05P)\tXR >\t0 OR ZERO ! The\texpected\tpayoff\tto\tinvestor\tat\tt\t=\t1\tis\ttherefore: (1.21.05P)X R 0.4 + 0 0.6 k s\ttse 15 A\tSimple\tModel Recall\tthe\texpected\tpayoff\tto\tinvestor\tat\tt\t=\t1\tis: (1.2 1.05P)X R 0.4 What\tis\tthe\tOPTIMAL\tinvestment\tdecision\tfor\tthe\tinvestor? The\tdecision\tfor\tthe\tinvestor\tis\tto\tdetermine\tthe\toptimal holding\tof\tthe\trisky\tasset\tXR\tthat\twill\tmaximize\this\texpected wealth\tat\tt\t=\t1. (1.2 1.05P)X R 0.4 0.05(X R )2 Expected\tPayoff Non-pecuniary\tCost The\tinvestment\tdecision\tcan\talso\tbe\tdescribed\tby: Max $%(1.2 1.05P)X R 0.4 0.05(X R )2 &' X R 0 k s\ttse 16 A\tSimple\tModel And\tthe\tdecision\tproblem\tis\tto\tdetermine\tthe\toptimal holding\tof\tthe\trisky\tasset\tXR: Max $%(1.2 1.05P)X R 0.4 0.05(X R )2 &' X R 0 Differentiation:\t1st order\tcondition\tfor\tmaximization: (1.2 1.05P) 0.4 0.1X R = 0 Solving\tfor\tthe\tequilibrium\tprice\tof\trisky\tasset\tP,\twe have: 1 " 0.1X R % P= $#1.2 '& 1.05 0.4 k s\ttse 17 A\tSimple\tModel To\tsee\tthe\tbubble,\twe\tneed\tto\tcompare\tthe\tequilibrium price\tof\trisky\tasset\tP\twith\tthe\tfundamental\tvalue\tP*. What\tis\tthe\tfundamental\tvalue\tP*? - Define\tthe\tfundamental\tas\tthe\tprice\t(P*)\tan\tinvestor would\tbe\twilling\tto\tpay\tin\tthe\tabsence\tof risk\tshifting. - In\tother\twords,\tthe\tinvestor\tcannot\tpass\tthe investment\tloss\tto\tthe\tbank\tby\tgoing\tbankrupt. k s\ttse 18 A\tSimple\tModel What\tis\tthe\tvalue\tof\tthe risky\tasset\tR\tat\tt\t=\t1? 1.2\t;\tProb =\t0.4 Risky Asset\tR t=0,\ttoday 0 ; Prob =\t0.6 t=1 Recall\tthe\tpayoff\tto\tinvestor\tat\tt\t=\t1\tis: (R 1.05P)X R The\trisky\tasset\tR\tis\teither\t1.2\twith\tprob 0.4 or\t0\twith\tprob 0.6. - If\tthe\tinvestor\tis\tnot\tallowed to\tgo\tbankrupt,\tthen\tthe payoff\tto\tthe\tinvestor\tis: Either (1.2\t\t1.05P)\tXR >\t0 OR (0\t\t1.05P)\tXR \t0.4 >\t0. Hence,\tdue\tto\trisk\tshifting,\tP\tis\thigher\tthan fundamental\tP* (bubble!); P\t>\tP* The\tcounterpart\tof\tthe\tbubble\tis\tthe\tbank\tlosses,\tand hence\tthe\trest\tof\tthe\tstory... Any\tirrationality\tinvolved?? In\ta\tsense\tit\tis\tnot\ta\tGE\tbubble,\tas\tthe\tprice\tof\tbanks\tshould\tgo\tdown...\tbut\tit\tmay\twell be\tthat\thouseholds\tare\tstuck...\tthis\ttakes\tus\tto\tthe\tstandard\tmodel\tof\tRE\tbubbles\tin macro,\twhich\thighlights\tthe\tshortage\tof\tassets.. k s\ttse 22 A\tSimple\tModel What\tare\tthe\timplications? - Abundant\tliquidity\tin\tthe\tfinancial\tsector\t(banks) - Investment\trisk\tinvolveduncertainty\ton\tprice - Investors\tcan\tshift\ttheir\tlosses\tto\tbanks - Interest\trate\tdoes\tnot\treflect\tunderlying\trisk - Investors\tare\twilling\tto\tpay\ta\thigher\tprice\tfor\trisky investment\tthan\tthe\tfundamental\tvalue,\tleading to\tthe\tformation\tof\ta\tbubble. - No\tirrationality\tinvolved. k s\ttse 23 Rational\tBubble Investors\trealize\tprice\tis\tdivorced\tfrom\tfundamentals, but\tbelieve\tthat\tprice\trises\twill\tpersist\tfor\tsome\ttime, and\tthat\tprice\tgrowth\twill\tcompensate\tfor\trisk\tof collapse. - Investors\tknow\tthat\tthe\tbubble\twill\tcollapse\tbut believe\tthey\tcan\tget\tout\tbefore\tit\tdoes. k s\ttse 24 Rational\tBubble Irrational\tbubbles\tthen\tinvolve\tunrealistic\texpectations about\tasset's\tfuture\tprospects - fad - mania - Irrational\texuberance\t(Unsustainable\tinvestor enthusiasm\tthat\tdrives\tasset\tprices\tup\tto\tlevels\tthat aren't\tsupported\tby\tfundamentals.) Let\tus\tconsider\ta\trational\tbubble\t.... k s\ttse 25 Rational\tBubbleSimple Characterization Let\tvt be\tthe\tproperty\tprice;\tsuppose fundamentals imply\tthat vt =\tv* for\tall\tt. At\tsome\ttime\tt0 the\tprice\tjumps\tby\ta\tbubble\tb0 to\tv0 - suppose\tagents\texpect\tvt =\t1\t+\tr\tin\teach\tfuture period\tt. i.e.,\tbt =\tb0(1\t+\tr)t for\tarbitrary\tb0. See\tFigure. Why\tare\tagents\twilling\tto\tpay\tincreasing\tprices\tfor\tthe property? k s\ttse 26 Rational\tBubbleSimple Characterization Why\tare\tagents\twilling\tto\tpay\tincreasing\tprices\tfor\tthe property? - expected\tcapital\tgains\tare\tself-fulfilling\t-- this\tis\ta rational\tbubble,\tthe\tcapital\tgain\ton\tproperty compensates\tfor\tthe\talternative\treturns. Notice\tthis\trequires\tthe\tprice\tto\tgrow\tforever. - If\teveryone\tknew\tthat\tat\tperiod\t(T\t+\tj)\tthat\tthe\tbubble would\tburst\t(that\tis,\tv\t v* or\tbt 0),\tthen\tno\tone would\tpay\tthe\tbubble\tprice\tat\tperiod\t(T\t+\tj\t- 1). The\tbubble\tunravels. k s\ttse 27 Rational\tBubbleSimple Characterization A\tbubble\tin\thousing\tmarket If\teveryone\tknew\tthat\tat\tperiod\tT\t+\tj\tthat\tthe\tbubble\twould\tburst\t(v\t v* or\tbt 0),\tthen\tno\tone\twould\tpay\tthe\tbubble\tprice\tat\tT\t+\tj - 1. The\tbubble unravels. v bt =\tb0(1+r)t V0 =\tv* +\tb0 v* t0 T+j k s\ttse Time 28 Rational\tBubbleSimple Characterization Since\tprices\tcannot\trise\tforever,\tare\tbubbles\truled\tout? - No,\tas\tlong\tas\tthe\tdate\tof\tprice\tcollapse\tis\tuncertain! v bt =\tb0(1+r)t V0 =\tv* +\tb0 v* t0 T+j k s\ttse Time 29 Stochastic\tRational\tBubble* Since\tprices\tcannot\trise\tforever, are\tbubbles\truled\tout? - no,\tas\tlong\tas\tthe\tdate\tof\tprice\tcollapse\tis\tuncertain Suppose\tthat\tin\teach\tperiod\tagents\tbelieve\tthat\tthe\tprobability\tthe\tbubble will\tnot\tburst is\tq. bt (1+r)bt q No\tBurst:\tExpected\tBubble\t=\tbt+1 1-q Burst:\tExpected\tBubble\t=\t0 Since\tthe\tbubble\tis\texpected\tto\tgrow\tat\tgrowth\trate\tr,\tthen\twe\thave: (1+ r)bt q bt+1 = (1+ r)bt bt+1 = q The\tACTUAL\tbubble\tat\tt+1\tis\twritten\tas\t: - t+1 is\ta\twhite\tnoise\twith\tmean\t0. k s\ttse b!t+1 = (1+ r)bt q + !t+1 30 Stochastic\tRational\tBubble Since\tprices\tcannot\trise\tforever, are\tbubbles\truled\tout? - no,\tas\tlong\tas\tthe\tdate\tof\tprice\tcollapse\tis\tuncertain Suppose\tthat\tin\teach\tperiod\tagents\tbelieve\tthat\tthe\tprobability\tthe\tbubble will\tnot\tburst is\tq. Then\twe\thave ! (1+ r)b t # !b = " q + !t+1 $$$$prob$of$NO$burst$=$q t+1 #! $t+1 $$$$$$$$$$$$$$$$$$$$$prob$of$burst$=$(11q)$ - t+1 is\ta\twhite\tnoise\twith\tmean\t0. If\tthe\tbubble\tfollows\tthis\tpath\tit\tis\trational. - notice\tthat\tthe\texpected\tvalue\tof\tthe\tbubble\tin\tperiod\tT\t+1\tis\texactly\tbt+1. - To\tsee\tthis,\tnote\tthat (1+ r)bt ! Et [bt+1 ] = q + q(0)+(1 q)(0) = (1+ r)bt q --our\tinitial\texpression\tfor\tthe\tbubble\tpath. See\tExcel. k s\ttse 31 Stochastic\tRational\tBubble Notice\tthat\tthe\tstochastic\tbubble\tgrows\tfaster\tthan\tunder certainty,\tbecause\tinvestors\tmust\tbe\tcompensated\tfor\tthe\trisk\tof the\tbubble\tbursting. (1+ r)bt > (1+ r)bt q Analysis:\tFrom\trational\tbubble\twe\tcan\tback\tout\tthe\tmarket's expectation\tof\tit\tbursting. - At\tany\ttime\tt,\tyou\tknow\tthe\tactual\tprice\tand\tthe\tinterest\trate, and\tif\tyou\tknow\tv*,\tthen\tyou\tcan\tcalculate\tbt =\tvt - v*. - If\tthe\tbubble\thas\tnot\tburst\tyet\tthen (1+ r)bt bt+1 = q (1+ r)bt q= bt+1 k s\ttse 32 Stochastic\tRational\tBubble If\tthe\tbubble\thas\tnot\tburst\tyet,\tthen (1+ r)bt bt+1 = q (1+ r)bt q= bt+1 This\tprobability\tcan\tbe\tthen\tcompared\tto\tthe\tamount\tof\ttime\tthe bubble\thas\tbeen\tgrowing. - This\tnotion\tof\ta\trational\tbubble\tis\tused\tfrequently\tin\tanalysis\tof asset\tmarkets. Ad\thoc\tExample:\tHong\tKong\tProperty\tMarket k s\ttse 33 Example:\tHong\tKong\tProperty\tMarket Median\tHH\tIncome/Month 25000 19872 20667 20000 15000 10000 5000 0 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 Suppose\tan\taverage\thousehold\tcan\tonly\tcommit\tup\tto\t1/3\tof\tits\tincome\tto mortgage\tpayment. Suppose\tmortgage\tterms\tare\t20-year,\tmonthly\tpayments,\twith\tmortgage\trate equal\tto\tPrime\tRate\t(5%)\tminus\t2.5%,\tand\t30%\tdown\tpayment. k s\ttse 34 Example:\tHong\tKong\tProperty\tMarket Suppose\tan\taverage\thousehold\tcan\tonly\tcommit\tup\tto\t1/3\tof\tits income\tto\tmortgage\tpayment. Suppose\tmortgage\tterms\tare\t20-year,\tmonthly\tpayments,\tmortgage rate\tequal\tto\tPrime\tRate\t(5%)\tminus\t2.5%,\tand\t30%\tdown\tpayment. What\tis\tthe\tmaximum\thome\tprice\tthe\thousehold\tcan\tafford\tin\t2012? Based\ton\tthe\tmedian\thousehold\tincome\tand\tthe\tmortgage\tterms,\tthe maximum\tmortgage\tan\taverage household\tcan\tborrow\tis\tdetermined as\tfollows: Mortgage\tLoan\t= 12n ( " ( " 5% 2.5% %240 + r % + *1 $1+ ' *1 $1+ ' 20, 667 # & # & 12 12 -= * PMT * = 1, 300, 042 r 5% 2.5% * * 3 * * 12 12 ) , ) , Affordable\thome\tprice\t=\tHK$\t1,300,042/0.7\t= HK$\t1,857,203 k s\ttse 35 Stochastic\tRational\tBubble Property\tPrice: Class\tA\tResidential\t(40\tsquare\tmeters) $3,500,000 3224262.8 Afforable Price $3,000,000 2931148 Actual Price $2,500,000 Size\tof\tBubble $1,857,203 $2,000,000 $1,857,203 $1,500,000 $1,000,000 $500,000 $0 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 Affordable\tProperty\tPrice\tis\tcalculated\tbased\ton\t1/3\tof\tmedian\thousehold income,\t20\tyear\tmortgage,\t5%\tminus\t2.5%\tmortgage\trate,\tand\t30%\tdown payment. k s\ttse 36 Stochastic\tRational\tBubble Property\tPrice: Class\tA\tResidential\t(40\tsquare\tmeters) $3,500,000 What\tis\tthe $3,000,000 market expectation $2,500,000 that\tproperty price\twill\tfall $2,000,000 to\tthe $1,500,000 affordable price\tlevel? $1,000,000 Afforable Price 2931148 Actual Price $1,857,203 $1,857,203 b2012 =\t2931148\t- 1857203\t=\t1073945 b2013 =\t3224262\t- 1857203\t=\t1367060 Mortgage\trate\t=\t2.5% $500,000 $0 2002 (1+ r)b2012 b2013 = q 3224262.8 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 (1+ r)b2012 1.025 1073945 q= = = 0.805 b2013 1367060 k s\ttse 37 Stochastic\tRational\tBubble Property\tPrice: Class\tA\tResidential\t(40\tsquare\tmeters) What\tis\tthe $3,500,000 market $3,000,000 expectation $2,500,000 that\tthe property\tprice $2,000,000 bubble\twill burst\twithin $1,500,000 the\tnext\t5 $1,000,000 years? Afforable Price 3224262.8 2931148 Actual Price $1,857,203 $1,857,203 (1+ r)b2012 1.025 1073945 q= = = 0.805 b2013 1367060 $500,000 $0 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 Probability\tthat\tthe\tprice\tbubble\twill\tNOT\tburst\tat\tthe\tend\tof\t5\tyears\tis (0.805)5 =\t0.339 Probability\tit\twill\tburst\twithin\tthe\tnext\t5\tyears\tis\t66.1% k s\ttse 38 Stochastic\tRational\tBubble:\tImplications Fact\tthat\tbubble\thas\tto\tgrow\tat\tan\texpected\trate\tof\tr\tallows one\tto\teliminate\tmany\tpotential\trational\tbubbles. - commodities\twith\tclose\tsubstitutes\tputs\tlimits - a\tbubble\ton\ta\tnon-zero\tsupply\tasset\tcannot\tarise\tif\tr exceeds\tthe\tgrowth\trate\tof\tthe\teconomy,\tsince\tthe\tbubble would\toutgrow\tthe\taggregate\twealth\tin\tthe\teconomy (Tokyo\tImperial\tPalace\tworth\tmore\tthan\tthe\tsum\tof\tall\tthe real\testates\tin\tCalifornia\tin\tthe\t1980s\t??). Hence,\tbubbles\tcan\tonly\texist\tin\ta\tworld\tin\twhich\tthe\tr \t0 OR ZERO\t! The\tdecision\tproblem\tfor\tthe\tinvestor\tis\tto\tdetermine\tthe optimal\tholding\tof\tthe\trisky\tasset\tXR: Max [ X R (RH rP) c(X R )] X R 0 k s\ttse 47 A\tSimple\tModel And\tthe\tdecision\tproblem\tis\tto\tdetermine\tthe optimal\tholding\tof\tthe\trisky\tasset\tXR: Max [ X R (RH rP) c(X R )] X R 0 First\torder\tcondition\tfor\tmaximization: (RH rP) c'(X R ) = 0 Solving\tfor\tthe\tequilibrium\tprice\tof\trisky\tasset\tP,\twe have: 1" c'(X R ) % P = $ RH '& # r k s\ttse 48 A\tSimple\tModel To\tsee\tthe\tbubble,\twe\tneed\tto\tcompare\tthe equilibrium\tprice\tof\trisky\tasset\tP\twith\tthe fundamental\tvalue\tP*. What\tis\tthe\tfundamental\tvalue\tP*? - Define\tthe\tfundamental\tas\tthe\tprice\t(P*)\tan\tagent would\tbe\twilling\tto\tpay\tin\tthe\tabsence\tof risk shifting. k s\ttse 49 A\tSimple\tModel Define\tthe\tfundamental\tas\tthe\tprice\t(P*)\tan\tagent\twould\tbe willing\tto\tpay\tin\tthe\tabsence\tof risk\tshifting. Then\tthe\tdecision\tproblem\tis: Max { X R [(RH rP * ) + (0 rP * )(1 )] c(X R )} X R 0 Note\tthat\tR\tis\teither\t\"RH with\tprob \" or\t\"0\twith\tprob (1-)\". First\torder\tcondition\tfor\tmaximization: (RH rP * ) rP * (1 ) c'(X R ) = 0 Solving\tfor\tfundamental\tvalue\tof\tthe\trisky\tasset\tP*,\twe\thave 1 P = [ RH c'(X R )] r * k s\ttse 50 A\tSimple\tModel Comparing\tP\twith\tP*,\tit\tis\teasy\tto\tsee\tthat P =\tP* Note\tthat\t is\tprobability\tof\teconomic\tboom\tand 1\t>\t >\t0. Hence,\tdue\tto\trisk\tshifting,\tP\tis\thigher\tthan fundamental\tP* (bubble!); P\t>\tP* The\tcounterpart\tof\tthe\tbubble\tis\tthe\tbank\tlosses,\tand hence\tthe\trest\tof\tthe\tstory... Any\tirrationality\tinvolved?? In\ta\tsense\tit\tis\tnot\ta\tGE\tbubble,\tas\tthe\tprice\tof\tbanks\tshould\tgo\tdown...\tbut\tit\tmay\twell be\tthat\thouseholds\tare\tstuck...\tthis\ttakes\tus\tto\tthe\tstandard\tmodel\tof\tRE\tbubbles\tin macro,\twhich\thighlights\tthe\tshortage\tof\tassets.. k s\ttse 51 a. Determine the market value of Tranche A, B, and Z securities if the mortgage interest rate tomorrow after the is Value = Amount Issued*(1+i)+(Interest rate*Increase rate) Tranche A TrancheB Tranche C $ $ $ 26,640,355.20 15,225,000.00 30,495,000.00 a. Determine the market value of Tranche A, B, and Z securities if the mortgage interest rate tomorrow after the is Value = Amount Issued*(1+i)-(Interest rate*Decrease rate) Tranche A $ 27,000,001.50 TrancheB $ 15,000,000.01 Tranche C $ 30,000,000.01 t rate tomorrow after the issue of the securities increases to 15%. t rate tomorrow after the issue of the securities decreases to 6%. a. Determine the market value of the 10-year risky debt. Market Value = cash flow/(1+rdebt)^t Market Value 38.4615385 b. Determine the market value of Fortune's equity. Market Value 76.9230769 c. Determine the market value of the 10-year CDO that pays no interest. Market Value 51.3846154 d. Determine the market value of the over-collateralization. Market Value 89.8461538 e. Determine the cost of insurance to protect the CDO investors against Fortune's default. Cost of insurance 70 f. Determine the market value of Fortune's equity and the market value of the over-collateralization (equity of th asset value increases from 25% to 50%. Explain your findings with the results of part (b) and part (d) above. Call Value S N (d1 ) Xe rT N (d 2 ) S N d1 d2 Call value 100 1 0.005 -0.045 $48.88 alization (equity of the debt pool) when the volatility of the nd part (d) above. Question 3 Hong Kong Property Market for 2015 and 2016 2015 2016 Residential property (HK$ billion) 416.5 395.1 Non-residential property (HK$ billion) 132.1 90.1 Yr 2015 Yr2016 548.7 211.25 485.3 193.30 DATE Actual Price Affordable prices Affordable Property Price = (1/3*medain household income)+20 year mortgage+(2.5%*Mortgage rate)+(30%*down payme 2015 2016 Median household income 43.55 44.525 20 year mortagage 3.015006 3.1045274027 2.5% of mortgage interest exp 0.075375 0.0776131851 30% of downpayment 164.61 145.59 Total 211.2504 193.2971405878 Wages and earnings 2011 2015 Nominal 177.2 (+9.9) 211.3 (+4.4) Real(2) 127.1 (+4.4) 120.7 (+2.3) Nominal 138.0 (+5.9) 163.3 (+4.8) Real(4) 117.0 (+0.5) 122.2 (+2.8) Nominal 178.1 (+7.5) 222.5 (+5.7) Real(4) 151.1 (+2.1) 166.4 (+3.8) Nominal 111.9 (+8.3) 136.1 (+4.5) Real(6) 108.7 (+1.6) 113.3 (+2.0) Wage Index (Sep. 1992=100) (1) Salary index(3) (Jun. 1995=100) Salary Index (A) Salary Index (B) Index of Payroll per Person Engaged(5) (1st quarter 1999=100) Average median household income Total loans and advances to customers(1) (HK$ billion) Number of households ('000) 130.7 5,080.7 2 386 7,534.5 2 499 Mortgage per household 2.129379715 3.015006 Hong Kong Property Market for 2015 and 2016 800 700 600 500 400 300 200 100 0 Yr 2015 Actual Price (a) Evaluate the size of \"bubble\" for 2015 and 2016. Yr2016 Affordable prices $ 200,000,000.00 (b) How likely the real estate bubble will burst in 2017? The bubble is not likely burst in 2017 becase the size od the bubble is 2016 is still very high. (c) How likely the real estate bubble will burst in the next two years, 2018 and 2019? The bubble is likely to burst in the next two years since the affordable prices and actual price of the properti ate)+(30%*down payment) 2016 219.6 (+3.6) 120.7 (+2.5) 169.8 (+4.0) 124.2 (+1.7) 234.1 (+5.2) 171.3 (+2.9) 147.2 (+3.8) 118.1 (+2.5) 133.6 7,817.2 2 518 3.104527 Yr2016 al price of the properties are closing in. Hong Kong Property Market for 2015 and 2016 800 700 600 500 400 300 200 100 0 Yr 2015 Actual Price Yr2016 Affordable prices