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
2
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
3
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)
4
Numerical Example




$100 loan
3 years, annual payments in arrears
10% interest rate
Interest only loan
$10 $10 $110
0  100 


2
1.10 1.10 1.103
 Contract Yield to Maturity = 10.00%
5
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:
• recovery rate:
• loss severity:
$33
70%
30%
 Realized Cash Flows imply an IRR of -1.12% (Use cash flow buttons)
$10
$10
$77
0  100 


2
1  x (1  x ) (1  x )3
 Yield degradation = 11.12% = 10.00 – (-1.12%)
6
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%
0  100 
$10
$77

1  x (1  x ) 2
 Yield degradation = 17.11% = 10.00 – (-7.11%)
 All else equal, the earlier defaults occur the more costly they are
7
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:
8.89%  (0.9) * 10%  (0.1) * 1.12%
 (0.9) * 10%  (0.1) * (10%  11.12%)
 10%  (0.1) * 11.12%
8
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%
E r   10%  (0.10) * 11.12%  (0.10) * 17.11%
 10%  2.82%
 7.18%
9
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
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%
 Note probability of default exactly in year 2 = 99% * 2% = 1.98%
10
Cumulative Default Probability
 What is the probability a loan defaults by end of year 3?




= Probability of default in year 1
+ Probability of default in year 2
+ Probability of default in year 3
= 5.91%
= 1%
= 99% * 2% = 1.98%
= 97% * 3% = 2.91%
 The probability a loan does not default = 100% - 5.91% = 94.1%
11
Example Question
 You issue a 3-year 10% mortgage assuming this about default:
Year
Hazard
Recovery Rate
1
1%
80%
2
2%
70%
3
3%
70%
 What is the expected return?
E r   10%  (0.01) * 22.00%  (0.0198) * 17.11%  (0.0291) * 11.12%
 10%  0.88%
 9.12%
12
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
1. Loan-to-value ratio (LTV) = loan amount divided by property value
2. 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
3. In a multi-year proforma, lenders want to see positive net cash in every year
13
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.
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?
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
15
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?
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 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
17
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:
$10 $10 $110
100 


2
1.10 1.10 1.103
 Now suppose that interest rates drop to 8%. This sequence of cash
flows from the mortgage is worth
$10 $10 $110
105.1542 


2
1.08 1.08 1.083
18
Why is prepayment bad? (continued)
 Now consider cash a mortgage with a prepayment in year 2.
The cash flows are only worth
103.5665 
$10 $110

1.08 1.082
 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.
19
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.
20
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?
21
Pools
 Here are the cash flows from the pool
Year
No Prepay
Prepay
Payment
#
Cash
Payment
#
Cash
Total Cash
1
$10
90
$900
$110
10
$1,100
$2,000
2
$10
72
$720
$110
18
$1,980
$2,700
3
$110
72
$7,920
$7,920
 The pool is worth $104.54 per mortgage
$10,453.82 
$2000 $2700 $7920


2
1.08
1.08
1.083
22
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.
• What is the pool worth?
23
Example Relationship of Interest Rates and Prepay
 No change in prepay behavior:
Year
No Prepay
Payment
#
Prepay
Cash
Payment
1
$10
$110
2
$10
$110
3
$110
#
Cash
Total Cash
Cash
Total Cash
 Change in prepay behavior
Year
No Prepay
Payment
#
Prepay
Cash
Payment
1
$10
$110
2
$10
$110
3
$110
#
24
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)
25
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?
26
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
27
Example 1 – PO and IO strips continued
 Step 3: compute table of payoffs of the pool
Year
1
2
# Loans
10
90
Loans that Prepay
Interest
Principal
$50,000.00
$1,000,000.00
$ 443,226.85 $ 8,864,537.09
Loans that do not Prepay
Totals
# Loans
Interest
Principal
Interest
Principal
90
$450,000.00 $135,462.92 $500,000.00 $1,135,462.92
$ 443,226.85 $ 8,864,537.09
 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
28
Example 1 – PO and IO strips continued
 Suppose the pre-pay rate jumps to 20% in year 1!
Year
1
2
# Loans
20
80
Loans that Prepay
Interest
Principal
$100,000.00 $2,000,000.00
$ 393,979.42 $ 7,879,588.52
Loans that do not Prepay
Totals
# Loans
Interest
Principal
Interest
Principal
80
$400,000.00 $120,411.48 $500,000.00 $2,120,411.48
$ 393,979.42 $ 7,879,588.52
 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
29
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?
30
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
Contractual
Default
No Default
Total Cash
# Loans
0
100
Payment
$ 73,500.00
$ 105,000.00
Total
# Loans
10
90
Payment
Total
$ 73,500.00 $ 735,000.00
$ 105,000.00 $ 9,450,000.00
$ 10,185,000.00
# Loans
30
70
Payment
Total
$ 73,500.00 $ 2,205,000.00
$ 105,000.00 $ 7,350,000.00
$ 9,555,000.00
$
$ 10,500,000.00
$ 10,500,000.00
Good Economy
 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
Default
No Default
Total Cash
Bad Economy
Default
No Default
Total Cash
 Upside $630K, Downside $0K
31