More Advanced Mortgage Math Real Estate Finance, Spring, 2017

Overview Risks in residential and commercial mortgages Yield degradation (commercial) Hazard rate Prepayments? Lender Restrictions Pricing pools of mortgages Sensitivity of pools to interest rates Tranching: IO/PO Strips and Creating AAA

Major Risks for Pricing Pools of Mortgages Commercial: Default Default is bad because typically a borrower only defaults if some bad event has happened and the building is worth less than the mortgage Residential: Prepay Prepayment is bad because a borrower (should) only prepay if the stream of expected future payments under the current mortgage terms is worth more than the par value of the mortgage Note: borrowers often move, so prepayment also occurs randomly

Language of Commercial Mortgage Default Equation Contractual Yield – Yield Degradation (credit losses) = Realized Yield Credit losses: Shortfalls to lender as a result of default and foreclosure Yield Degradation: Lender’s loss measured as a multi-period lifetime return on the original investment (IRR impact)

Numerical Example $100 loan 3 years, annual payments in arrears 10% interest rate Interest only loan Contract Yield to Maturity = 10.00%

Numerical Example Continued Now suppose loan defaults in year 3 Bank takes property and sells it in foreclosure Bank only gets 70% of outstanding balance: $77 = 0.70*$110 credit losses: $33 recovery rate: 70% loss severity: 30% Realized Cash Flows imply an IRR of -1.12% (Use cash flow buttons) Yield degradation = 11.12% = 10.00 – (-1.12%)

Numerical Example Continued Suppose instead the loan defaults in year 2 bank gets 70% of outstanding balance: $77 = 0.70*$110 (why is outstanding balance $110 in year 2 ???) Realized Cash Flows imply an IRR of -7.11% Yield degradation = 17.11% = 10.00 – (-7.11%) All else equal, the earlier defaults occur the more costly they are

On Yield Degradation In general, Expected Return = Contract Yield – (Pr. Default)*Yield degradation Suppose 10% chance of default in year 3 With a 70% recovery rate in such a default No other chance of any other default Expected return:

More on Yield Degradation Assume 10% chance of default in year 2 with a 70% conditional recovery 10% chance of default in year 3 with a 70% conditional recovery 80% chance of no default Note these are “unconditional probabilities” Do not depend on any pre-conditioning event Describe mutually exclusive and exhaustive set of possibilities for the mortgage Probabilities sum to 100%

In the real world … More realistic analysis uses hazard functions Hazard functions specify the probability of default at each point given default has not already occurred Note probability of default exactly in year 2 = 99% * 2% = 1.98% Year Hazard Prob Loan is Still Active at Year End 1 1% 99% = (100% - 1%) 2 2% 97% = (100% - 2%) * 99% 3 3% 94.1% = (100% - 3%) * 97%

Cumulative Default Probability What is the probability a loan defaults by end of year 3? = Probability of default in year 1 = 1% + Probability of default in year 2 = 99% * 2% = 1.98% + Probability of default in year 3 = 97% * 3% = 2.91% = 5.91% The probability a loan does not default = 100% - 5.91% = 94.1%

Example Question You issue a 3-year 10% mortgage assuming this about default: What is the expected return? Year Hazard Recovery Rate 1 1% 80% 2 2% 70% 3 3%

Lender Restrictions Lenders put into place restrictions or boundaries to limit the possibility of default. These limits keep interest rates low. The three most common are Loan-to-value ratio (LTV) = loan amount divided by property value Debt-service coverage ratio (DCR) = NOI divided by Debt Service (DS) DS includes interest and principal. Typically DCR must be at least 1.2 Related: Break Even Ratio (BER) BER = (DS + Operating Expenses) / Potential Gross Income A typical requirement is that BER < 85% or Mkt Avg Occupancy less 5% buffer In a multi-year proforma, lenders want to see positive net cash in every year

Lender Underwriting Problem 1 Suppose 10-year yields in the bond market are 7.00% effective annual rate, and the market for commercial mortgage loans requires a contract yield risk premium of 175 basis points at an annual rate. If a property has an annual net operating income (NOI) of $400,000 and the underwriting criteria require a debt coverage ratio (DCR) of at least 130%, then what is the maximum loan that can be offered assuming a 15-year amortization rate and annual payments on the mortgage? Notice everything here is annual to make the problem a bit easier; remember to set P/Y = 1 on your calculator.

Lender Underwriting Problem 1 Suppose 10-year yields in the bond market are 7.00% effective annual rate, and the market for commercial mortgage loans requires a contract yield risk premium of 175 basis points at an annual rate. If a property has an annual net operating income (NOI) of $400,000 and the underwriting criteria require a debt coverage ratio (DCR) of at least 130%, then what is the maximum loan that can be offered assuming a 15-year amortization rate and annual payments on the mortgage? The contract interest rate on the mortgage must be 8.75% = 7.0% + 1.75% Max pmt determined by DCR 1.3 = $400,000/x, x = $307,692.30 Max loan determined N=15, I=8.75, PMT=307,692.30, FV=0, PV = -$2,517,243.94

Lender Underwriting Problem 2 On your proforma, you purchase a property on 1/1/2017 for $1,000,000 with NOI accruing on 12/31/2017 of $40,000 and then immediately sell the building for $1,050,000. A lender has offered you a thirty-year fixed rate mortgage with annual payments at 4.5%. The maximum debt service coverage ratio the lender will allow is 1.3. What is the maximum LTV of the property?

Lender Underwriting Problem 2 On your proforma, you purchase a property on 1/1/2017 for $1,000,000 with NOI accruing on 12/31/2017 of $40,000 and then immediately sell the building for $1,050,000. A lender has offered you a thirty-year fixed rate mortgage with annual payments at 4.5%. The maximum debt service coverage ratio the lender will allow is 1.3. What is the maximum LTV of the property? DCR = NOI/debt service. 1.3 = $40,000/x, x = $30,769.23 The maximum size mortgage is N = 30, I/YR = 4.5, PMT = $30,769.23, PV = $501,196.56 The LTV = $501,196.56 / $1,000,000 = 0.5012

Why is prepayment bad? Suppose a lender issues an interest only mortgage for $100 at 10% for 3 years. The sequence of expected payments is: Now suppose that interest rates drop to 8%. This sequence of cash flows from the mortgage is worth

Why is prepayment bad? (continued) Now consider cash a mortgage with a prepayment in year 2. The cash flows are only worth The mortgage is worth 105.1542 if no prepayment occurs 103.5665 if a prepayment occurs Suppose the probability is 50%. The price will be 104.3604 If a prepayment occurs, the purchaser will take a loss.

Pools Rather than talk about an expected return on an individual mortgage, it makes sense to talk about expected return on a pool. A coin flip is either heads or tails. It is not 50% heads. Analogously, a mortgage will either default or prepay or it will not. However, 50% of a large pool of coins all flipped at once will be heads. And x% of a pool or mortgages can be counted on to default or prepay.

Pools Assume you buy a pool of 100 residential mortgages. Each mortgage is interest only for $100 at 10% for 3 years. The prepayment hazard rate is 10% in year 1 and 20% in year 2. Secondary market participants discount these cash flows at 8% What is the pool worth?

Pools Here are the cash flows from the pool The pool is worth $104.54 per mortgage Year No Prepay Prepay Payment # Cash Total Cash 1 $10 90 $900 $110 10 $1,100 $2,000 2 72 $720 18 $1,980 $2,700 3 $7,920

Sensitivity of Pools to Interest Rates: How pools change in value when interest rates change depends on the relationship of prepayments with interest rates Illustrative Example: Assume you buy a pool of 100 residential mortgages. Each mortgage is interest only for $100 at 10% for 3 years. Secondary market participants discount these cash flows at 6% The prepayment hazard rate is 10% in year 1 and 20% in year 2. What is the pool worth? Now assume: The prepayment hazard rate jumps to 15% in year 1 and stays at 20% in year 2.

Example Relationship of Interest Rates and Prepay No change in prepay behavior: Change in prepay behavior Year No Prepay Prepay Payment # Cash Total Cash 1 $10 $110 2 3 Year No Prepay Prepay Payment # Cash Total Cash 1 $10 $110 2 3

Tranching Tranche is French for “slice.” Tranching is slicing up cash flows from a pool. The example you know is debt vs equity. In a pool of debt instruments, cash flows can still be sliced up 2 Examples Principal vs. Interest Strips (PO vs IO) Prioritization of cash flows (AAA vs residual)

Example 1: PO vs IO strips Consider a pool of 100 30-year fixed rate mortgages with par value of $100,000, a coupon of 5%, and annual year-end payments Assume that 10% of the pool pre-pays in year 1 and the other 90% pre-pays in year 2 How much are investors willing to pay for the PO and IO strips?

Example 1 – PO and IO strips continued Step 1: Compute annual payments P/YR = 1 N = 30, I/YR = 5, PV = -$100,000, FV = 0 PMT =?= $6,505.1435 Step 2: Compute int. and prin. paid and payoff amount per loan Year 1, no prepay: PMT = $6,505.1435 INT = $5,000, PRINC = $1,505.1435, OLB = $98,494.8565 Year 1, prepay: INT = $5,000, PRINC = $100,000 Year 2 (everyone remaining prepays): INT = $4,924.7428 = 0.05 * OLB from year 1 PRINC = $98,494.8565

Example 1 – PO and IO strips continued Step 3: compute table of payoffs of the pool The Value of the Pools is (set P/YR = 1 and I/YR = 5) IO Strip: CF0 = 0, CF1 = 500,000, CF2 = 443,226.85 NPV = $878,210.2948 PO Strip: CF0 = 0, CF1 = 1,135,462.92, CF2 = 8,864,537.09 NPV = $9,121,789.7107 IO Strip + PO Strip = $10m

Example 1 – PO and IO strips continued Suppose the pre-pay rate jumps to 20% in year 1! The Value of the Pools is (set P/YR = 1 and I/YR = 5) IO Strip: CF0 = 0, CF1 = 500,000, CF2 =393,979.42 NPV = $833,541.4240 (this falls) PO Strip: CF0 = 0, CF1 = 2,120,411.49, CF2 =7,879,588.52 NPV = 9,166,458.5801$ (this rises) IO Strip + PO Strip = $10m

Example 2: Prioritization of Cash Flows Consider a pool of 100 commercial mortgages, each with one payment remaining to be made at the end of the year. The contractual payment is $105,000. If the mortgage defaults, the amount collected is only $73,500. There are 2 possible states of the world, each with a 50% chance Good Economy: 10% of the mortgages default Bad Economy: 30% of the mortgages default What is the maximum amount of bonds that can be sold as AAA?

Example 2: Prioritization of Cash Flows Contd. In the worst case scenario, $9,555,000 of cash is paid into the pool. This is the amount that can be sold as AAA Contractually, the residual (equity) gets $10,500,000 - $9,555,000 = $945,000 In expectation, the equity gets: 0.50*(10,185,000 – 9,555,000) + 0.50*0 = $315,000 Upside $630K, Downside $0K