Moetary Ecoomics oblem Set #5 Moetary Ecoomics: oblem Set #5 Solutios This problem set is marked out of 1 poits. The weight give to each part is idicated below. Please cotact me asap if you have ay questios. 1. Structured fiace. Suppose there are two bods ad that each pays $1 cash or ot. The probability of gettig $1 is.95 ad is idepedet across bods. (a) Explai how a fiacial itermediary ca sell prioritised juior j ad seior s claims to $1 agaist the possible cash flows from a portfolio of these two bods. I your aswer, give the possible realizatios of the cash flows, the probabilities of these evets, ad the paymets made to juior ad seior claims i each evet. How much would a risk eutral ivestor be prepared to pay for the j ad s claims. Is this more or less tha they would pay for the uderlyig bods? Explai. (1 poits) (b) Now suppose there are three bods, each as above. Explai how a itermediary ca sell three prioritised claims (juior j, mezzaie m ad seior s) agaist the possible cash flows from the three bods. Give the possible realizatios of cash flows, the probabilities of these evets, ad the paymets made to juior ad seior claims i each evet. (1 poits) (c) Now suppose there are two pools each of two bods each as i part (a) above. Each pool has juior ad seior claims. Explai how a fiacial itermediary ca sell prioritised juior j j ad seior s j claims to $1 agaist the possible cash flows from a portfolio formed from the juior traches j 1 ad j 2 from each pool. What patter of cash flows leads to seior claim i the secod roud of securitizatio beig paid or ot paid? Give the possible realizatios of the cash flows, the probabilities of these evets, ad the paymets made to the j j ad s j claims from the secod roud of securitizatio. Would a risk eutral ivestor pay more for a seior claim i the first roud of securitizatio (s 1 or s 2 ) or for a seior claim i the secod roud (s j )? Explai. (15 poits) (d) Now suppose there are two bods as i part (a) except that the uderlyig bods paymets are perfectly positively correlated. Give the possible realizatios of the cash flows, the probabilities of these evets, ad the paymets made to juior ad seior claims i each evet. Would a risk eutral ivestor be prepared to pay a premium for seior claims? Explai. What if the uderlyig bod paymets are istead perfectly egatively correlated, would your aswers chage? Would a risk averse ivestor view thigs differetly? (5 poits)
Moetary Ecoomics: oblem Set #5 2 Solutios: (a) The possible realizatios ad their probabilities are give i the table below. The calculatios of the probabilities of each state use the fact that the probability of gettig 1 is idepedet across bods. I ay state of the world where either bod pays out, we pay 1 to the seior claim. Oly if both bods pay out do we pay 1 to the juior claim. I this sese, the juior claim is the residual claimat to the cash flow from the package of bods (like equity). realizatio {, } {, 1} {1, } {1, 1} probability.25.475.475.925 paymet {j, s} {, } {, 1} {, 1} {1, 1} The probability of the juior claim beig paid is therefore (j = 1) =.925 while the probability of the seior claim beig paid is (s = 1) =.925 +.475 +.475 =.9975. A risk eutral ivestor would be willig to pay at most.925 for the juior claim ad at most.9975 for the seior claim. Therefore they would be willig to pay more for the seior claim tha for oe of the uderlyig bods (due to the protectio offered by the juior claim) but the juior claim is worth less tha oe of the uderlyig bods. (b) There are ow 2 3 = 8 possible realizatios. These realizatios are of 4 mutually exclusive types, illustrated i the table below: (i) o bods pay, (ii) oe out of three bods pay, (iii) two out of three bods pay, (iv) all bods pay. realizatio {,, } {,, 1} {1, 1, } {1, 1, 1} probability.125.2375.45125.857375 paymet {j, m, s} {,, } {,, 1} {, 1, 1} {1, 1, 1} The probability of the juior claim beig paid is therefore just (j = 1) =.857375, the probability of the mezzaie claim beig paid is (m = 1) =.857375 +.45125 3 =.99275, while the probability of the seior claim beig paid is (s = 1) =.99275 +.2375 3 =.999875. A risk eutral ivestor would be willig to pay at most.857375 for the juior claim,.99275 for the mezzaie ad.999875 for the seior. Both the mezz ad the seior are more valuable tha the uderlyig bods. (c) Now we take the paymets agaist the juior claims j i for i = 1, 2 pools each of 2 bods as i part (a). Below are the realizatios, probabilities ad cash flows i the secod roud of securitizatio. realizatio {j 1, j 2 } {, } {, 1} {1, } {1, 1} probability.95.88.88.8145 paymet {j j, s j } {, } {, 1} {, 1} {1, 1} The probability of the juior claim i the secod roud beig paid is therefore (j j = 1) =.8145 while the probability of the seior claim i the secod roud beig paid is (s j = 1) =.8145 +.88 2 =.991. I order for the juior claim i the first roud to be paid out, there has to be o default i the pool o which that claim is writte. So i
Moetary Ecoomics: oblem Set #5 3 order for the seior claim i the secod roud to be paid out, there has to be o default i at least oe of the two pools of bods. A risk eutral ivestor would pay at most.991 for a seior claim i the secod roud, i.e., less tha the.9975 they d be willig to pay for a seior claim i the first roud. Although safer tha the uderlyig bods ad the juior claims from the first roud, the seior claim i the secod roud is still riskier tha the seior claims i the first roud. (d) If the uderlyig bods are perfectly positively correlated, the either both bods pay out (with probability.95) or either does (with probability.5). I this case there is o possibility of usig prioritizatio (i.e., a capital structure) to protect a seior claim. Sice there is o possibility of usig prioritizatio, a risk eutral ivestor would pay at most.95 for a claim, the same as for the uderlyig bods. If the bods are istead perfectly egatively correlated, the a pool of two such bods pays out 1 with probability 1. (sice if oe does t pay, the other does). Thus a claim to a pool of these two bods ca deliver 1 for sure ad a risk eutral ivestor would be willig to pay 1 for such a claim (more tha.95). Notice therefore that it is ot correlatio per se across the uderlyig bods that destroys the ability to protect a seior claim, it is more specifically positive correlatio that is the problem. Negative correlatio across the uderlyig bods makes it easier ot harder to protect the seior claim (as always, at the cost of makig the juior claim worth less). I geeral, a risk averse ivestor will always eed to be compesated for risk by beig able to buy a security at a price lower tha the risk eutral ivestor would be prepared to pay. How much of a discout depeds o the curvature i their utility fuctio. For CRRA utility with coefficiet σ, the required discout is proportioal to σ/2 times the variace of the cash flow [at least for small risks]. 2. Default risk i a portfolio of mortgages. Cosider a mortgage pool that cosists of i = 1,..., mortgages X i. The X i are IID Beroulli trials which default X i = 1 with probability p ad do ot default X i = with probability 1 p. The average default from a mortgage pool is p with variace p(1 p). Now suppose that mortgage pools come i a variety of types each characterized by a particular value of the parameter p. These types of pools are distributed accordig to a probability desity f(p) > for p [, 1]. Suppose also that we have a represetative portfolio of these mortgage pools. Let p deote the portfolio average p, that is p E[p] = 1 pf(p) dp Notice that i this portfolio the variatio i mortgage paymets comes i two ways: withi pool variatio due to idiosycratic realizatios of X i, ad betwee pool variatio due to differeces i p. Coditioal o p, the X i withi a pool are idepedet. (a) Derive formulas for the portfolio average X i, the portfolio variace of X i ad the correlatio of two radomly chose mortgages X i ad X j from the portfolio. What is the correlatio if all mortgage pools have p = p? Explai. (15 poits) Hit: recall that for two radom variables Y ad Z, the law of iterated expectatios says that E[Y ] = E[E(Y Z)] ad the aalysis of variace decompositio gives Var[Y ] = Var[E(Y Z)] + E[Var(Y Z)].
Moetary Ecoomics: oblem Set #5 4 Now let D deote the umber of defaults withi a give mortgage pool D ad let D / deote the correspodig default rate. X i (b) Derive the portfolio average umber of defaults ad the portfolio variace of the umber of defaults. What values do these statistics take whe all mortgage pools have p = p? What values do they take if p = 1 with probability p ad p = with probability 1 p? Explai. (15 poits) (c) Derive the portfolio average default rate ad the portfolio variace of the default rate. Cosider the case where there are may mortgages withi a give pool, i.e., where. I this case, how much of the variatio i default rates comes from withi a pool ad how much from variatio betwee pools? Explai. (15 poits) (d) Explai ituitively why the portfolio s frequecy distributio of defaults approaches ( ) < θ F (θ) as where F ( ) is the cumulative probability distributio associated with f( ), that is F (θ) θ f(p) dp What role does the distributio of mortgage pool types f(p) play i makig it possible for a fiacial itermediary to carve out traches of differetly-rated juior ad seior claims to the mortgage paymets? (15 poits) Solutios: (a) Applyig the law of iterated expectatios we have E[X i ] = E[E(X i p)] = E[p] = p ad similarly usig the ANOVA decompositio Var[X i ] = Var[E[X i p]] + E[Var(X i p)] = Var[p] + E[p(1 p)] Now recall that for ay radom variable Y the variace is Var[Y ] = E[Y 2 ] E[Y ] 2 so Var[X i ] = Var[p] + E[p(1 p)] = E[p 2 ] E[p] 2 + E[p] E[p 2 ] = p(1 p)
Moetary Ecoomics: oblem Set #5 5 The correlatio betwee X i ad X j is defied by Corr[X i, X j ] Cov[X i, X j ] Var[Xi ] Var[X j ] Sice Var[X i ] = Var[X j ] = p(1 p) this is just Corr[X i, X j ] = Cov[X i, X j ] p(1 p) Now recall that the covariace of ay two radom variables Y, Z is Cov[Y, Z] = E[Y Z] E[Y ]E[Z] so that Cov[X i, X j ] = E[X i X j ] p 2 = E[E(X i X j p)] p 2 = E[E(X i p)e(x j p)] p 2 = E[p 2 ] p 2 = Var[p] where the secod equality uses the law of iterated expectatios ad the third equality uses the fact that, coditioal o p, the X i ad X j are idepedet so that the (coditioal) expectatio of the product is the product of the (coditioal) expectatios. This tells us that the overall correlatio betwee outcomes is ultimately drive by variatio i p across pools. Also, perhaps surprisigly, this model ca oly give us positively correlated default outcomes (sice Var[p] ). Sice Var[X i ] = Var[X j ] = Var[X] we ca also write the correlatio as Corr[X i, X j ] = Var[p] Var[X] I short, the correlatio coefficiet is the proportio of the ucoditioal variatio i default outcomes that is accouted for by variatio across pools. I the special case of all p = p (so that the distributio f(p) is degeerate), there is o variatio across mortgage pools ad the correlatio coefficiet is zero. I this case, the overall correlatio is the same as the withi pool variatio (sice all pools are idetical), amely zero. (b) For the average umber of defaults we have [ ] E[D ] = E X i = E[X i ] = p = p Now for the variace, a directio calculatio gives [ ] Var[D ] = Var X i = Var[X i ] + Cov[X i, X j ] j=1,j i
Moetary Ecoomics: oblem Set #5 6 (this is the variable geeralizatio of the familiar formula that for ay two radom variables Y, Z we have Var[Y + Z] = Var[Y ] + Var[Z] + 2Cov[Y, Z]). So usig the results from part (a) we have Var[D ] = p(1 p) + Var[p] j=1,j i = p(1 p) + ( 1)Var[p] This ca be writte equivaletly i the form of a ANOVA decompositio Var[D ] = Var[E[D p]] + E[Var(D p)] = Var[p] + E[p(1 p)] = 2 Var[p] }{{} + E[p(1 p)] }{{} betwee pool variatio withi pool variatio If there is o variatio across mortgage pools p = p, the clearly Var[p] = ad E[p(1 p) = p(1 p) ad the variace of D is the same as the variace of a idividual pool, amely p(1 p) (all the covariace terms are zero). I this case, all the variatio is withi pool variatio. At the other extreme, if p = 1 with probability p or p = with probability 1 p, the defaults are perfectly correlated withi a give pool ad so there is o variatio withi a pool. We ca calculate directly that (equivaletly, E[p 2 ] = p) so that E[p(1 p)] = (1) () ( p) + () (1) (1 p) = Var[D ] = 2 Var[p] ad ideed all the variatio is betwee pool variatio. (c) For the average default rate we have [ ] E ad for the variace Var [ ] = 1 Var [D ] = 2 = 1 E[D ] = 1 p = p p(1 p) + 1 Var[p] As, the mea default rate remais p while the variace reduces to [ ] lim Var = Var[p] I the limit of a large umber of mortgages, all the variatio i default rates comes from variatio i p betwee pools (ad oe from withi a pool). If p = p for all pools, the the variace falls to zero sice there is o loger ay variatio across pools.
Moetary Ecoomics: oblem Set #5 7 (d) The basic idea is to use ( ) < θ = 1 ( < θ ) p f(p) dp ad the to use a law of large umbers, D / p for, so that ( ) { < θ if p > θ p 1 if p < θ, as so that the frequecy distributio of defaults is just give by F ( ), amely ( ) 1 ( ) < θ = < θ p f(p) dp θ { if p > θ 1 if p < θ } f(p) dp, as = θ f(p) dp = F (θ) The ability to create seior claims depeds crucially o there ot beig too much correlatio. As the correlatio icreases, the value of seior claims falls ad the value of juior claims rises util, i the limit of perfect correlatio, the two are perfect substitutes. There is less correlatio ad hece more ability to create seior claims the more variatio there is i p across pools, ad this is a property of the f(p) distributio.