A New View of Mortgages and life

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 notices 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 take 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 $Z 1 (where E[Z 1] > Y). – developer exercises the right to purchase the land for $Y and either develops the shopping mall or sells to another for $Z 1 (profit = Z 1 -Y).

The Option • If current mayor wins, horse farm will be worth $Z 2 where $Z 2 < $Y. – Can assume that Z 2 is probably the value of the land as a farm. – Developer lets option expire 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 portfolio 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 developer 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 date • • • 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 price = market price of the call, (CT)

Types of Call Options • European Call = exercise only at maturity • American 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 promise 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 the 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 mortgage 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 foreclosure

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 the 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 lender $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 house will be $156. 25, $100, or $64.

House Price Paths Year 0 Year 1 Year 2 $156. 25 $125. 00 $100. 00 $80. 00 $64. 00

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 house value = $64. 00, then our equity is $-31. 00 and we should default (lender gets to keep house). • ($64. 00 - $95 = $-31) – D = min[K, H]

Mortgage Value • Starting with the terminal payoffs, we need to calculate the present value of the mortgage. – Thus, we need to calculate the pseudoprobability of a change 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 Year 0 Year 1 Year 2 $95. 00 $91. 35 $81. 59 $95. 00 $77. 44 $64. 00

Mortgage Value • Note: Based on our assumptions of changes in house prices, 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 risk 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 • borrower may prepay at any time prior to maturity

Default and Prepayment • Default and Prepayment are substitutes. – If borrower prepays the mortgage, then he can’t default • implies that default has no value – If borrower defaults on the mortgage, then she can’t prepay • implies that prepayment has no value

Mortgage 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