(Practice Problems) Mortgage Options

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Apr 3, 2024

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1) True or False: Recourse for a lender reduces the value of the put option for the mortgage borrower. True. In mortgages, the default option is considered a “put” option and the prepayment option is considered a “call” option. In the above question, because the lender can go after the borrower’s other assets in the case of default, this makes the default option less valuable to the borrower. 2) A borrower takes a $500,000, 4 year IO loan (with annual payments) and an interest rate of 7%. If the borrower defaults in the fourth year, the lender will recover $321,000. What is the lender’s recovery rate if the borrower defaults? What is the lender’s loss rate? Recovery rate = $321,000 / [1.07 * $500,000] = 60% Loss rate = 1 – recovery rate = 40% 3) You work for a commercial mortgage lender. Your company is planning to lend $1.2m on an IO 4 year balloon at 4.5% with annual payments. There are no points on the loan. You expect that there is a 70% chance the loan will pay as promised, and a 30% chance the loan will default in year 3. You anticipate a loss rate of 40% if the loan defaults. a. What is the contractual yield (YTM) on the loan? 4.5% b. What is the yield if the loan defaults? Year 0 Year 1 Year 2 Year 3 Year 4 CFs -1.2 .054 .054 .7524 0 -11.10% c. What is the yield degradation if the loan defaults in year 3? 4.5 – (-11.1 ) = 15.6% d. What is the expected yield / return on the loan? .7 * 4.5 + .3 * -11.1 = -0.18% e. What is the S.D. of returns? Var = .7 (4.5 – (-.18))^2 + .3(-11.1 – (-.18))^2 S.D. = sqrt(Var) = 7.15% f. What is the IRR(expected cash flows)? Year 0 Year 1 Year 2 Year 3 Year 4 No def (.7) -1.2 .054 .054 .054 1.254 DEF (.3) -1.2 .054 .054 .7524 0 E(CF) -1.2 .054 .054 .26352 .8778
IRR[E(CF)] = 1.13% g. Should you (the lender) make this loan if the OCC on investments with similar risk is 3%? No, both the lender’s expected return and the IRR[E(CFs)] are less than the lender’s OCC. 4) A borrower owes $150,000 on a property that is currently worth $125,000. He lost his job and currently does not have the income to service the debt, but he does have money in a retirement account he could use to make up any shortfall on mortgage. His costs of default are $15,000. Should the borrower default or repay the loan? Compare benefit of walking away (defaulting) with cost of default ($150,000 - $125,000) vs. $15,000 $25,000 > $15,000 should default and let the lender foreclose 5) True or false: All else equal, if the lender expects potential default to occur sooner, the lender needs to charge a higher rate to get the same expected return. True. Earlier default hurts the lender (because the PV of the loss is higher). To counter this, the lender needs to charge a higher rate to get the same expected return. 6) Explain any potential issues with the following statement: If a lender increases the contract rate on the loans it offers, the expected return will be higher as well. Higher interest rates might increase the probability of default, which, in turn could actually lower the expected return. Also, if the lender increases rates, it’s possible that the best borrowers will select out of the pool (e.g., not take a loan) because they can get a better loan elsewhere. This remaining borrowers that want a loan will be riskier on average, and the increase in interest rates may be offset by additional losses on default. 7) Explain the relationship between the value of a mortgage and the value of the default and prepayment options. The value of a mortgage (to the lender/investor) can be thought of as: Value of mortgage = PV of pmts at market rate – Value of prepayment option – Value of default option = Value of mtg w/out options – Value of prepayment option – Value of default option The value of the prepayment (default) option has positive value to the mortgage borrower, and thus reduces the value of the mortgage to the lender. Since options always have positive value, the value of the mortgage (if it has the default and prepayment options) will always be worth less than the present value of the remaining payments on a mortgage that doesn’t have the default and prepayment options. The value of the options also changes over time. Generally speaking, decreasing interest rates increases the value of the prepayment option while decreasing property values increase the value of the default option.
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