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[1]
[null,"Tokyo earthquake Cat Bond:"]
[{"$nulls":3},"Coupon:",{"v":0.07,"d":"7%"}," annual payments",null,"Treasury ytm:",null,0.03," (cc)"]
[{"$nulls":3},"Hazard rate:",{"v":0.015,"d":"1.50%"},"(continuous rate)"]
[{"$nulls":3},"Promised cash flow timeline:"]
[{"$nulls":4},"Time (Yrs:)",0,1,2,3,4,5]
[{"$nulls":4},"CF",null,{"$f":"100*$E$3","v":7.000000000000001},{"$f":"100*$E$3","v":7.000000000000001},7.000000000000001,7.000000000000001,{"$f":"100*(1+$E$3)","v":107}]
[{"$nulls":4},"Surv Prob:",{"$f":"EXP(-$E$4*F6)","v":1},{"$f":"EXP(-$E$4*G6)","v":0.9851119396030626},0.9704455335485082,0.9559974818331,0.9417645335842487,0.9277434863285529,null,"Common mistake:",null,"Putting 100 as the Year 0 cash flow."]
[{"$nulls":4},"Exp CF:",null,{"$f":"G7*G8","v":6.895783577221439},{"$f":"H7*H8","v":6.793118734839558},6.691982372831701,6.592351735089742,99.26855303715516,{"$nulls":3},"We are not told the price, so we don't know what the Time 0 cash flow is to get this stream of future cash flows."]
[{"$nulls":4},"df:",null,{"$f":"EXP(-$J$3*G6)","v":0.9704455335485082},{"$f":"EXP(-$J$3*H6)","v":0.9417645335842487},0.9139311852712282,0.8869204367171575,0.8607079764250578,{"$nulls":3},"  We find that the bond's price is 110.49,  so",{"$nulls":3},{"$f":"-F12","v":-110.4936389386788}," Is the correct Time 0 outflow."]
[{"$nulls":4},"PV[E(CF)]",null,{"$f":"G9*G10","v":6.6919823728317},{"$f":"H9*H10","v":6.397518296898598},6.116011381816242,5.846891479878905,85.44123540725334]
[{"$nulls":5},{"$f":"SUM(G11:K11)","v":110.4936389386788},{"$nulls":8},"Also read the question -- it only asks for the future cash flows."]
[]
[{"$nulls":3},"The bond will be selling at a premium because its coupon rate of 7% is above its ytm.  We know its ytm is 4.5%.",{"$nulls":8},"Common mistake:",null,"Yes, we know that the ytm is less than the coupon rate because the bond is selling at a premium."]
[{"$nulls":4},"Verify:",{"$f":"J3+E4","v":{"v":0.045,"d":"4.50%"}},{"$nulls":8},"We know more: the cc ytm on a cat bond equals the sum of the cc ytm on the term-matched Treasury plus the hazard rate."]
[]
[{"$nulls":4},"df(@ytm):",null,{"$f":"EXP(-$F$15*G6)","v":0.9559974818331},{"$f":"EXP(-$F$15*H6)","v":0.9139311852712282},0.8737159116880344,0.835270211411272,0.7985162187593771]
[{"$nulls":4},"PV:",null,{"$f":"G7*G17","v":6.691982372831701},{"$f":"H7*H17","v":6.397518296898598},6.116011381816242,5.846891479878905,85.44123540725334]
[{"$nulls":5},{"$f":"SUM(G18:K18)","v":110.4936389386788}]
[{"$nulls":5},{"$f":"F19*200","v":22098.72778773576}]
[]
[null,"Discussion:","Before we look at the bond in any detail, we know that its cc ytm is equal to the sum of the Treasury and the relevant hazard rate (both on a cc basis). "]
[null,null,"So the bond's cc ytm is:",null,{"$f":"E4+J3","v":{"v":0.045,"d":"4.50%"}},"  We know that it will sell at a premium (its price will be above 100) since this rate in equivalent to:",{"$nulls":7},{"$f":"EXP(E23)-1","v":{"v":0.04602785990871694,"d":"4.60%"}}," on an annually-compounded basis, which is below the coupon rate (paid annually) of 7%."]
[]
[null,null,"We can value the bond in several ways:"]
[{"$nulls":5},"1. As the problem suggests: first obtain the expected cash flows -- which equal the promised cash flows times the probability that we will get the cash.  Then we discount these expected cash flows at the US Treasury ytm.  This is done above."]
[{"$nulls":5},"2. Discount the promised cash flows at the bond's ytm.  This combines the survival probability and the discounting in a single step."]
[]
[null,null,"This highlights the importance of reading the question carefully and thinking about what you are told, and what you are asked before you start writing."]
[null,null,"In this case we do not know the bond's price, but we are told the hazard rate that the market is using and the ytm on comparable-term Treasuries."]
[]
[null,null,"Remember the ytm is the single discount rate that reconciles the promised cash flows with the market price.  For cat bonds this includes the hazard rate--to adjust the promised cash flows to expected cash flows.  "]
[null,null,"   The rate of return that you expect to earn-- the expected return-- is the same as the yield to maturity on a comparable-term Treasury security since there is no risk premium on cat bonds."]
[2]
[null,"FEMA Flood-linked cat bond:"]
[null,null,"Coupon:",{"v":0.1125,"d":"11.25%"}," annual payments",null,"Treasury ytm:",null,0.02," (cc)"]
[null,null,"Price:","Par"]
[{"$nulls":3},"We know that the bond's ytm on an annually-compounded basis is the same as its coupon rate, since it is selling for par.",{"$nulls":8},"  This is not something that we can assume differently.  We know the bond pays annual coupons and is selling for par."]
[{"$nulls":3},"(We know it's on an annually compounded basis because the coupons are paid annually.)",{"$nulls":8},"     ==> This means that its ytm on an annually-compounded  basis is the coupon rate."]
[{"$nulls":3},"We need the ytm on a continuously-compounded basis:",{"$nulls":4},"e^(rcc) = 1+rann"]
[{"$nulls":8},"So the cc ytm is:",null,{"$f":"LN(1+D36)","v":0.10660973505825827}]
[{"$nulls":8},"and therefore the continuous hazard rate is:",{"$nulls":3},{"$f":"K41-I36","v":0.08660973505825827}]
[{"$nulls":3},"The 4-year survival probability is:",null,null,{"$f":"EXP(-M42*4)","v":0.7072019995582608}]
[{"$nulls":3},"So the probability that there is at least one flood over the next 4 years is:",{"$nulls":5},{"$f":"1-G43","v":0.2927980004417392}]
[]
[{"$nulls":3},"The expected return on a natural disaster cat bond is the ytm on the term-matched Treasury.",{"$nulls":6},"  We see this by asking what yield equates the future expected cash flows with the cat bond's price."]
[{"$nulls":10},"--This is an extremely important point!"]
[{"$nulls":10},"There is no risk premium on natural disaster cat bonds as the risk that they bear is diversifiable -- it is not related to the economy.]"]
[{"$nulls":10},"For this reason the expected return--which is the yield you expect to earn--on a cat bond is the same as the expected return on a comparable-term US Treasury."]
[]
[3,"Saga of Jim and Jane:"]
[{"$nulls":3},"House value",700000]
[{"$nulls":3},"hazard rate",0.005]
[{"$nulls":3},"1-year survival probability:",null,{"$f":"EXP(-E53)","v":0.9950124791926823}]
[{"$nulls":3},"Fire risk:",null,{"$f":"1-F54","v":0.00498752080731768}]
[{"$nulls":3},"Expected loss:",null,{"$f":"E52*F55","v":3491.264565122376}]
[]
[null,"Discussion:"]
[null,null,"Jane's wealth is",null,10000000000,"So the ",700000,"house represents ",null,{"$f":"G59/E59","v":{"v":7e-05,"d":"0.0070%"}}," of her wealth"]
[null,null,"Jim's is",null,1000000,{"$nulls":4},{"$f":"G59/E60","v":{"v":0.7,"d":"70.0000%"}}," of his wealth"]
[]
[]
[null,null,"If Jane's house burns down there will be no change in her lifestyle -- by virtue of the comparative size of this loss to her total wealth, she has diversified away the fire risk."]
[null,null,"By contrast if Jim's house burns down, and he has no insurance he loses 70% of his wealth -- which will negatively affect his life style.  Given his resources, there is nothing he can do to diversify the risk of his home.  "]
[{"$nulls":3},"--If it burns down his lifestyle will be adversely affected--no matter what he does with thhe rest of his assets."]
[null,null,"Jane would be willing to pay an actuarially fair price for insurance on her Tucson house, but not a penny more."]
[null,null,"Jim is willing to pay a risk premium because he has not diversified away the fire risk.",{"$nulls":6},"Therefore insurance companies can earn a risk premium by selling insurance to people like Jim."]