1. More Advanced Mortgage Math
Real Estate Finance, Spring, 2017

2. 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

3. 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

4. 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)

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

6. 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% = – (-1.12%)

7. 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% = – (-7.11%)
All else equal, the earlier defaults occur the more costly they are

8. 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:

9. 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%

10. 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%

11. 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% % = 94.1%

12. 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%

13. 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

14. 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.

15. 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% % Max pmt determined by DCR 1.3 = $400,000/x, x = $307, Max loan determined N=15, I=8.75, PMT=307,692.30, FV=0, PV = -$2,517,243.94

16. 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 What is the maximum LTV of the property?

17. 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 What is the maximum LTV of the property? DCR = NOI/debt service = $40,000/x, x = $30, The maximum size mortgage is N = 30, I/YR = 4.5, PMT = $30,769.23, PV = $501, The LTV = $501, / $1,000,000 =

18. 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

19. 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
if no prepayment occurs
if a prepayment occurs
Suppose the probability is 50%. The price will be
If a prepayment occurs, the purchaser will take a loss.

20. 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.

21. 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?

22. Pools Here are the cash flows from the pool
The pool is worth $ 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

23. 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.

24. 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

25. 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)

26. Example 1: PO vs IO strips
Consider a pool of 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?

27. 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,
Step 2: Compute int. and prin. paid and payoff amount per loan
Year 1, no prepay:
PMT = $6,
INT = $5,000, PRINC = $1, , OLB = $98,
Year 1, prepay: INT = $5,000, PRINC = $100,000
Year 2 (everyone remaining prepays):
INT = $4, = 0.05 * OLB from year 1
PRINC = $98,

28. 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, NPV = $878,
PO Strip: CF0 = 0, CF1 = 1,135,462.92, CF2 = 8,864, NPV = $9,121,
IO Strip + PO Strip = $10m

29. 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, NPV = $833, (this falls)
PO Strip: CF0 = 0, CF1 = 2,120,411.49, CF2 =7,879, NPV = 9,166, $ (this rises)
IO Strip + PO Strip = $10m

30. 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?

31. 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 = $315,000
Upside $630K, Downside $0K