1、A New View of Mortgages (and life),Scene 1,A farmer owns a horse farm outside Lexington on Richmond Road. Demographic trends indicate that this part of Lexington is booming and is projected to continue to grow. Problem: Current local government is hostile to development.,Scene 2,Local developer noti
2、ces the horse farm and thinks that the site is an excellent candidate for a new shopping mall. Developer knows that the local mayor is up for re-election next year. Outcome of election is uncertain, but has potential to install new mayor with pro-growth views.,Scene 3,What can the developer do to ta
3、ke advantage of this opportunity? Approach farmer with an offer to buy an option to purchase the horse farm.,The Option,Developer pays the farmer $X for the right to purchase the horse farm after the election for $Y. If pro-growth mayor wins, then horse farm will be worth $Z1 (where EZ1 Y). develope
4、r exercises the right to purchase the land for $Y and either develops the shopping mall or sells to another for $Z1 (profit = Z1-Y).,The Option,If current mayor wins, horse farm will be worth $Z2 where $Z2 $Y. Can assume that Z2 is probably the value of the land as a farm. Developer lets option expi
5、re without purchasing land Farmer keeps the payment $X.,Next Example,An insurance company has a large real estate portfolio. The insurance company projects that it will need $1 million next year to fund possible claims. What can it do to protect itself from changes in value to its real estate portfo
6、lio between now and when the claims will have to be paid.,Answer,Purchase an option to sell one of its properties for $1 million. If prices go down then protected If prices go up, will lose the appreciation but still locked in with enough funds to pay the claims.,Options,In the first example, the de
7、veloper purchased a CALL option. the right to buy an asset In the second example, the insurance company purchased a PUT option. the right to sell an asset.,Call Option,Contract giving its owner the right to purchase a fixed number of shares of a specified common stock at a fixed price by a certain d
8、ate Stock = underlying security (ST = market price) price = strike price (K) date = expiration date writer = person who issues the call (the seller) buyer = person who purchases the call call price = market price of the call, (CT),Types of Call Options,European Call = exercise only at maturity Ameri
9、can Call = exercise at any time up to maturity,Call Option Payoff at Maturity,Put Option,Contract giving its owner the right to sell a fixed number of shares of a specified stock at a fixed price at any time by a certain date.,Put Option Payoff at Maturity,Mortgages as Options,A mortgage is a promis
10、e to repay a debt secured by property. property = collateral = underlying security = stock However, a mortgage is much more complex than a simple stock option. mortgage is a contract with several options,Mortgages as Options,Default Option right of borrower to stop making payments in exchange for th
11、e property default = exercise of a PUT option,Mortgages as Options,Prepayment Option right of borrower to prepay the mortgage at any time prepayment = exercise of a CALL option,Default,Mortgage Default is defined as a failure to fulfill a contract Technical default = breech of any provision of the m
12、ortgage contract 1 day late on payment failure to pay property taxes failure to pay insurance premiums,Default,Industry Standards: Delinquency: missed payment Default = 90 days delinquent (3 missed payments) Foreclosure: process of selling the property to pay off the debt takes many months to forecl
13、osure,Default,Default is considered a European put option. Borrowers will only default when a payment is due Thus, the mortgage can be thought of as a string of default options. Every time you make a payment, you are purchasing a put option giving you the right to sell the house to the lender for th
14、e mortgage balance next month.,Mortgage Default,Simplistic Default Example,Assume the following: a house has a current value of $100. The standard deviation of the return to housing is 0.22314355 The risk-free interest rate is 4% per annum. In order to purchase the house, we promise to repay a lende
15、r $95 in 2 years. Note: This is a zero-coupon bond no monthly or yearly payments are made.,Given the previous assumptions, we assume that the house value will either rise to $125 or fall to $80 by the end of the first year (with equal probability). By the end of the second year, the value of the hou
16、se will be $156.25, $100, or $64.,House Price Paths,Binomial Model,Cox, Ross, and Rubinstein (CRR) (discrete time version),Default Values,At end of year 2, we owe $95 to lender. If house value = $156.25, then our equity is $61.25 and we should repay the loan (not default).($156.25 - $95 = $61.25) If
17、 house value = $64.00, then our equity is $-31.00 and we should default (lender gets to keep house). ($64.00 - $95 = $-31) D = minK,H,Mortgage Value,Starting with the terminal payoffs, we need to calculate the present value of the mortgage. Thus, we need to calculate the pseudo-probability of a chan
18、ge in house prices.,Mortgage Value,At the end of year 1, the present values of the terminal pay-offs are calculated as:,Mortgage Value,Finally, at mortgage origination, the present value of the loan is calculated as:,Mortgage Value,Mortgage Value,Note: Based on our assumptions of changes in house pr
19、ices, the lender will originate a mortgage of $81.59 at Year 0. We borrower $81.59 and promise to repay $95 at the end of Year 2. What is our effective interest rate on this mortgage?,Mortgage Value,Interest Rate Answer:r = 7.9054%,Note: since the risk-free rate is 4% this implies that the default r
20、isk premium for this mortgage is 3.9054%,Prepayment,Paying off mortgage early (prior to maturity date) Financial = when interest rates fall below contract rate Non-financial = borrower moves, divorce, (not optimal with respect to interest rates) prepayment is considered to be an American option borr
21、ower may prepay at any time prior to maturity,Default and Prepayment,Default and Prepayment are substitutes. If borrower prepays the mortgage, then he cant default implies that default has no value If borrower defaults on the mortgage, then she cant prepay implies that prepayment has no value,Mortga
22、ge Pricing,Ten years ago Enterprise S&L made a 30 year mortgage for $100,000 at an annual interest rate of 8%. The current market rate for an equivalent loan is 12%. What is the market value of this loan?,Mortgage Pricing,Simplistic Answer: Price = $66,640 More Complex (realistic) Price = PV of Payments - Value of Default Option - Value of Prepayment Option,
