Exam 2 Review
Baruch College: RES3200
1
Baruch College: RES3200
RES-3200
2
 Intro
 ARM
 Commercial RE
 Legal
 Refi
 Cap rates
 Finance
 Underwriting
 Leverage
 FRM
 Buy vs Lease
 Risk
 Exam 1
 Exam 2
 Development
 Ownership Entity
 MBS (MPT, CMO)
 REIT
 Portfolio
 Exam 3
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Exam 2
3
 How to study for Exam 2
 Slides & re-do examples in slides
 re-do HWs
 practice Exam 2
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Review
Roadmap
4
 Finance review (NPV, IRR)
 Adjustable Rate Mortgages
 REFI decision
 Residential Mortgage Underwriting (3 C’s)
 Residential Valuation Buy vs Lease (Rent + TS - PITIM)
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Finance Review
Investment Definition
5
 Investment: claim to a future (possibly risky) stream of cash flows
𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡𝑡 ≡ 𝐶𝐹𝑡 , 𝐶𝐹𝑡+1 , 𝐶𝐹𝑡+2 , … , 𝐶𝐹𝑡+𝑘 , …
DCF: 𝑁𝑃𝑉 𝑖 = σ𝑇𝑡=0
𝐶𝐹𝑡
𝐶𝐹𝑡+1
𝐶𝐹𝑡+2
𝑡
𝑡+1
𝑡+2
𝐶𝐹𝑡
1+𝑖 𝑡
⋯
𝐶𝐹𝑡+𝑘
⋯
𝑡+k
 30yr Bond 𝐶𝐹0 = −300, 𝐶𝐹1 = ⋯ = 𝐶𝐹29 = 0, 𝐶𝐹30 = 1000
 House: 𝐶𝐹0 = −100𝑘 (price), 𝐶𝐹𝑡 = 𝑅𝑒𝑛𝑡𝑡 + 𝑇𝑆𝑡 − 𝑃𝐼𝑇𝐼𝑀𝑡
 Mortgage: 𝐶𝐹0 = −100𝑘 (loan), 𝐶𝐹1 = 𝐶𝐹360 = 𝑃𝑀𝑇 (frm)
 Stock: 𝐶𝐹0 = −90 (price), 𝐶𝐹𝑡 = 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑𝑡 (net of taxes)
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Finance Review
Internal Rate of Return aka Yield
6
 Buy asset for $100 now (t=0), 𝐶𝐹0 = −$100
 Asset pays $104 in one year (t=1), 𝐶𝐹1 = +$104
 The Net Present Value (NPV) of the investment is a function of the discount rate i
 𝑁𝑃𝑉 𝑖 = −$100 +

$104
1+𝑖 1
For every discount rate i, there is a different NPV(i)
 IRR: the discount rate that makes NPV equal zero 𝑁𝑃𝑉 𝑖 = 𝐼𝑅𝑅 = 0
 CF0=-100, CF1=+104, F1=1, IRR, CPT, 𝐼𝑅𝑅 = 4%
 Two equivalent ways of thinking about this investment:
 A: pay $100 now, get $104 in 1 year, the Yield on investment=IRR=4%
 B: save $100 now at 4%, in 1 year get back principal $100 plus interest 4% × $100 = $4
 If the required return is 12% do you invest? What is the NPV?
 NPV, I=12, down, CPT, 𝑁𝑃𝑉 = −7.14 < 0
 Invest when NPV ≥ 0 ↔ required return ≤ IRR
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Finance Review
Internal Rate of Return
7
Buy office building for 100k now (t=0)
 Spend 50k developing in the first year (t=1), 30k second year (t=2)
 Assume you can rent it out while developing
 𝑅𝑒𝑛𝑡1 = 25𝑘, 𝑅𝑒𝑛𝑡2 = 30𝑘, 𝑅𝑒𝑛𝑡3 = 35𝑘 and sell for 200k in (t=3)


Net cash-flows





𝐶𝐹0 = −100𝑘,
𝐶𝐹1 = −50𝑘 + 25𝑘 = −25𝑘,
𝐶𝐹2 = −30𝑘 + 30𝑘 = 0
𝐶𝐹3 = 35𝑘 + 200𝑘 = 235𝑘
𝑁𝑃𝑉 𝑖 = −100𝑘 −
25𝑘
1+𝑖 1
+
0
1+𝑖 2
+
235𝑘
1+𝑖 3
(ON EXAM ONLY WRITE FINAL CASH FLOW!!!)
Year
0
1
2
3
Rent
$ 25,000.00 $ 30,000.00 $ 35,000.00
Sale Price
$ 200,000.00
Purchase Price
$ (100,000.00)
Development Cost
$ (50,000.00) $ (30,000.00)
Net Cash flow
$ (100,000.00) $ (25,000.00) $
-
$ 235,000.00
IRR
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25.12%
Finance Review
Internal Rate of Return
8
 𝑁𝑃𝑉 𝑖 = −100𝑘 −
25𝑘
1+𝑖 1
+
0
1+𝑖 2
+
235𝑘
1+𝑖 3
 IRR: the discount rate that makes NPV equal zero 𝑁𝑃𝑉 𝑖 = 𝐼𝑅𝑅 = 0
 CF0=-100k, CF1=-25k,F1=1,CF2=0,F2=1,CF3=235k, IRR, CPT

𝐼𝑅𝑅 ≈ 25.12%
 If the required return is 12% do you invest? What is the NPV?

NPV, I=12, down, CPT, 𝑁𝑃𝑉 = 44,946.93 > 0
 Invest when NPV ≥ 0 ↔ required return ≤ IRR
 What if the required return is 27%?
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Review
Roadmap
9
 Finance review (NPV, IRR)
 Adjustable Rate Mortgages
 REFI decision
 Residential Mortgage Underwriting (3 C’s)
 Residential Valuation Buy vs Lease (Rent + TS - PITIM)
Baruch College: RES3200
Mortgage
10
 Loan=$𝐿, Down payment=$𝐷𝑃, House price=𝑉0
𝐿 + 𝐷𝑃 = 𝑉0
 Loan to Value ratio (LTV) at origination,
𝐿
𝐿𝑇𝑉0 =
𝑉0
 Q: Ann gets a $200𝑘 loan, her 𝐿𝑇𝑉 = 75%, how much did her house cost?
 A: .75 =
200𝑘
,
𝑉
hence 𝑉 =
Baruch College: RES3200
200𝑘
.75
= 266,666.67
𝐿
𝑉0 =
𝐿𝑇𝑉0
Mortgage
11
• The balance after k-monthly payments is 𝐵𝑘 .
• The LTV after the 𝑘 𝑡ℎ payment: 𝐿𝑇𝑉𝑘 =
𝐵𝑘
𝑉𝑘
Initial balance 𝐵0 = 𝐿.
equity: 𝐻𝐸𝑄𝑘 = 𝑉𝑘 − 𝐵𝑘
• The law of motion for ALL debt:
𝑖𝑡+1
𝐵𝑡+1 = 𝐵𝑡 +
𝐵 − 𝑃𝑀𝑇𝑡+1
𝑚 𝑡
• 30 year FA-FRM: 𝑖𝑡 = 𝑖 for all t, 𝑃𝑀𝑇𝑡 = 𝑃𝑀𝑇, FA: 𝐵360 = 0, 𝐻𝐸𝑄360 = 𝑉360
• 30 year IO-FRM, Balloon: 𝑃𝑀𝑇 =
𝑖
𝐵 ,
𝑚 0
𝐵1 = 𝐵2 = ⋯ = 𝐵360 = 𝐵0
• 𝐵0 = 100𝑘, 𝑖/𝑚 = 6%/12, 𝑃𝑀𝑇 = .5 ∗ 100𝑘 = 500, 𝐵360 = 100𝑘
• If FRM, 𝑃𝑀𝑇 = 599.55, 𝐵360 = 0
• 30 year FA-ARM: 𝑖𝑡 resets periodically subject to caps and floors
•
𝑃𝑀𝑇𝑡 diddo, same LOM
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Adjustable Rate Mortgage
12
 1/1: initial rate fixed for 1 year, resets every year
 Hybrid Arms: 3/1, 5/1, A/B: initial rate fixed for A years, resets every B years
 Index – The rate series that is used to determine the base interest rate on the
loan. (LIBOR, treasury CMT)
 Margin – The spread that is added to the index. If 𝑀𝑎𝑟𝑔𝑖𝑛 = 250𝑏𝑝, the
mortgage resets 250 basis points or 2.50% above the index.
 Fully Indexed Rate (FIR)= Index + Margin

Initial Rate – The rate on the mortgage before the 1st Reset date.

Initial Rate Cap – The highest rate rise above the initial rate on the 1st rate reset date.
Periodic Rate Cap – The highest rate rise between resets after the 1st rate reset.
Life Rate Cap – The highest rate rise above the initial rate during the life of the loan.


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Adjustable Rate Mortgages
1/1 ARM: Year 1 and Year 2
13

Compute PMT for 1st year

𝟐𝟓𝟎𝒌 =
𝑷𝑴𝑻
𝟏+.𝟎𝟑/𝟏𝟐
𝟏
+ ⋯+
𝑷𝑴𝑻
𝟏+.𝟎𝟑/𝟏𝟐 𝟑𝟔𝟎
Balance =
Term
=
Rate
=
PMT
=

ARM Loan
PV
Term
Initial Rate
Index
Reset Frequency
Current Index
Margin
$250,000
360
3.00%
$1,054.01
=
=
=
=
=
=
=
$
250,000
30 years
3.000%
1 year CMT
Annual
1.00%
2.50%
To find PMT for 2nd year – RECAST the loan
 STEP 1: compute balance in 12 months
𝟏𝟎𝟓𝟒
𝟏𝟎𝟓𝟒

𝐵12 =

STEP 2: Mortgage Rate for 2nd year, suppose index still = 1.00%,

STEP 3: compute the PMT at the new rate 3.5% as if PMTs are made until maturity!
𝟏+.𝟎𝟑/𝟏𝟐
 𝟐𝟒𝟒, 𝟕𝟖𝟎

𝟏
=
+ ⋯+
𝟏+.𝟎𝟑/𝟏𝟐 𝟑𝟔𝟎−𝟏𝟐=𝟑𝟒𝟖
𝑷𝑴𝑻
𝟏+.𝟎𝟑𝟓/𝟏𝟐
Year 2 PMT = $1,120.68
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= 𝟐𝟒𝟒, 𝟕𝟖𝟎. 𝟓𝟎,
𝟏 +⋯+
[N, 348], [I, 3/12], [PMT, -1054], [CPT, PV]
𝑷𝑴𝑻
𝟏+.𝟎𝟑𝟓/𝟏𝟐
𝟑𝟒𝟖
𝑟 = 1.00 + 2.50 = 3.50%
Balance =
Term
=
Rate
=
PMT
=
$ 244,780
348
3.50%
$1,120.68
Adjustable Rate Mortgages
The Definition of APR For an ARM
14
The TRUE APR a lender must report to a borrower is based on the fully indexed rate of the ARM
at origination and assumes no prepayment.
 True APR= just IRR assuming index stays the same. (reflects all points etc)

ARM Loan
PV
Term
Initial Rate
Index
Reset Frequency
Current Index
Margin
Caps
PMT

$
250,000
30 years
3.000%
1 year CMT
Annual
1.00%
2.50%
5/2/5
$
1,054.01
1st year 𝑃𝑚𝑡 = 1054, years 2-30 based on 3.5%, 1,120.68
𝑁𝑃𝑉 = +250𝑘 −
•
•
=
=
=
=
=
=
=
=
=
1054
𝐼𝑅𝑅
1+
12
1 +⋯+
1054
𝐼𝑅𝑅
1+
12
12 +
1,120.68
𝐼𝑅𝑅
1+
12
13 + ⋯ +
CF0=250k, CF1=-1054, F1=12, CF2=-1120.68, F2=348, CPT IRR*12
Annual IRR = 3.46% higher than the teaser rate (3%).
•
If there are fees subtract from the balance.
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1,120.68
𝐼𝑅𝑅
1+
12
360
=0
Review
Roadmap
15
 Finance review (NPV, IRR)
 Adjustable Rate Mortgages
 REFI decision
 Residential Mortgage Underwriting (3 C’s)
 Residential Valuation Buy vs Lease (Rent + TS - PITIM)
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The Refinancing Decision Process
The Economics: costs vs benefits
16
 Suppose rates fall and you are thinking of refinancing.
 Costs: fees of getting a new mortgage (pain in the neck)
 Benefits: lower monthly payment at new lower rate
 (how long will you enjoy savings)
 What other investment opportunities should be considered.
 What is a variable cost and what is a sunk cost?
 Past points and fees paid are sunk costs. At the time a borrower is
considering whether to refinance past costs are irrelevant, only current and
future costs are relevant.
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The Refinancing Decision Process
Current Loan vs Refinance Option
17
Original Loan: 30yr 7.0%
New Loan: 25yr 6.5%
 PV
$250,000
360
7.00%
$1663.26
 PV
 PMT
=
=
=
=
=
=
=
=
$235,328
300
6.50%
$1588.95
 UPB 𝑩𝟔𝟎
=
$235,328
 Origination Fee=
2.0 points
$1,500
 N (30 yr)
 I
 N (25 yr)
 I
 PMT
 Closing Costs =

Time remaining =
 5 years left
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300 months
 Total Costs
=
$6,206
 Save: 1663.26 -1588.96=74.30/mo
The Refinancing Decision Process
Current Loan vs Refinance Option
18
 Determine the IRR that equates the 2 options (either keeping the current loan OR
refinance into new one).
 If I paid $6,206 today for a 25 year annuity that paid me $74.30/month would I
want to make that investment? What would I be earning?
𝑁𝑃𝑉 𝑖 = −6,206 +
 Solve for I which is the IRR.
 IRR =13.91%
74.30
𝑖
1+
12
1
+ ⋯+
N
PV
PMT
I
74.30
𝑖
1+
12
=
=
=
=
300
300
(6,206.00)
74.30
??
 Is that a good return? Would you buy this asset?
 If a borrower has a required return 𝑟 < 𝐼𝑅𝑅 = 13.91% he should refinance!
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=
13.91%
The Refinancing Decision Process
Current Loan vs Refinance Option
19
 What if the borrower was only going to be in the house for another 10 years?
 Loan balance of the old loan at t=180

= 185,047
[N, 180], [I, 7/12], [PMT, -1663.26], [CPT, PV] 185,047.58
 Loan balance of new loan at t=120

= 182,405
[N, 300-120=180], [I, 6.5/12], [PMT, -1588.95], [CPT, PV] 182,405.76
 Difference (advantage of a lower loan balance)
 𝑁𝑃𝑉 = −6,206 +
74.30
𝑖
1+12
1 + ⋯+
=
74.30
𝑖
1+12
120 +
2,643
2643
𝑖 120
1+12
 N=120, PV=-6206, PMT=74.30, FV=2643, CPT I*12 = 11.23%
 Since the mortgage is not kept for the full 25 years the savings of
$74.30/month is not as valuable.
 The IRR falls from 13.91% to 11.23%
 If your required return was 𝑟 = 12%, would you still refi?
 Cashout refi/ HEL/ HELOC
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Review
Roadmap
20
 Finance review (NPV, IRR)
 Adjustable Rate Mortgages
 REFI decision
 Residential Mortgage Underwriting (3 C’s)
 Residential Valuation Buy vs Lease (Rent + TS - PITIM)
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Residential Loan Underwriting
3 Cs of Underwriting
21
 CREDIT: credit report (Transunion, Experian, Equifax)


FICO score 300-850
Current and past debts, tax liens, bankruptcies, deficiencies
 Capacity: borrower’s capacity to make loan payments



Employment history, income statements, bank asset statements
DTI front end and back-end, (PITI: Principal+Interest+Taxes+Insurance)
Reserve Requirement: e.g. enough assets in bank to pay mortgage for 5 months
 Collateral: the value of the collateral in case of default



Appraisal report
LTV requirement
Conforming loans: if LTV>80% need Private Mortgage Insurance (PMI)

Loans with higher LTVs are riskier, often have higher interest rates
 Lenders use the 3 Cs to determine if you will be approved for a loan and
under what terms (rate, points etc)
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Residential Loan Underwriting
Credit: FICO Score
22
 What factors influence the FICO score?

35% - Payment history


30% - Credit Utilization – credit usage/credit availability ratio


The longer you have been a borrower the better.
10% - Types of credit used


Closing an unused credit card actually hurts.
15% - Length of credit history


On time payments improve score.
The more types of borrowing the better/ unless high cost borrowing.
10% - Recent Credit searches

The more often you apply the lower the score temporarily.
 Income/employment doesn’t affect FICO score.
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Residential Loan Underwriting
Capacity: Income Analysis
23
monthly PITI
Front End DTI =
monthly pretax Income
monthly PITI + 𝑜𝑡ℎ𝑒𝑟 𝑑𝑒𝑏𝑡
Back End DTI =
monthly pretax Income
• E.g. 28%/36% (conventional)
• Front End DTI =
monthly PITI
monthly pretax income
𝑃𝑀𝑇+𝑇+𝐼
= monthly pretax income ≤ .28
• monthly PITI = 𝑃𝑀𝑇 + 𝑇 + 𝐼 ≤ .28 × monthly pretax income
• 𝑃𝑀𝑇 ≤ .28 × monthly pretax income − T − I
• Back End DTI =
monthly PITI+𝑜𝑡ℎ𝑒𝑟 𝑑𝑒𝑏𝑡
monthly pretax income
≤ .36
• monthly PITI + 𝑜𝑡ℎ𝑒𝑟 𝑑𝑒𝑏𝑡 ≤ .36 × monthly pretax income
• 𝑃𝑀𝑇 ≤ .36 × monthly pretax income − T − I − other debt
• Both the front end and back end must be satisfied!
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Residential Loan Underwriting
Capacity
24
 Alice earns $120k/year or 10k/month
 Property taxes $6k/year, $500/month. Insurance $2400/yr, $200/mo.
 Monthly T+I=$700
 Q1: if front end DTI=28%, what is the most she can spend on monthly PITI?
 A1: .28 × $10𝑘 = $2,800 so 𝑃𝐼𝑇𝐼 ≤ $2,800
 Q2: if back end DTI=28%, what is the biggest monthly mortgage payment
Alice can make?
 A2: 𝑃 + 𝐼 + 𝑇 + 𝐼 = 𝑃𝑀𝑇 + $700 ≤ $2,800 so 𝑃𝑀𝑇 ≤ $2,100
Baruch College: RES3200
Residential Loan Underwriting
Capacity
25
 Alice earns $120k/year or 10k/month
 Property taxes $6k/year, $500/month. Insurance $2400/yr, $200/mo.
 Front End 28%, 𝑃𝐼𝑇𝐼 ≤ $2,800, 𝑃𝑀𝑇 ≤ $2,100
 Q1: if back end DTI=36%, what is the most she can spend on monthly PITI
and other debt?
 A1: .36 × $10𝑘 = $3,600 so 𝑃𝐼𝑇𝐼 + 𝑜𝑡ℎ𝑒𝑟 𝑑𝑒𝑏𝑡 ≤ $3,600
𝑃𝑀𝑇 + 𝑇 + 𝐼 + 𝑜𝑡ℎ𝑒𝑟 𝑑𝑒𝑏𝑡 ≤ $3,600
𝑃𝑀𝑇 ≤ $2,900 − 𝑜𝑡ℎ𝑒𝑟 𝑑𝑒𝑏𝑡
 Q2: if front end DTI=36%, what is the biggest monthly mortgage payment
Alice can make?



If other debt =$200, 𝑃𝑀𝑇 ≤ $2,900 − $200 = $2,700
If other debt =$800, 𝑃𝑀𝑇 ≤ $2,900 − $800 = $2,100
If other debt =$1k, 𝑃𝑀𝑇 ≤ $2,900 − $1𝑘 = $1,900
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Residential Loan Underwriting
Capacity
26
 Alice earns $10k/month
 Property taxes + Insurance = $700/mo.
 Front End 28%, 𝑃𝐼𝑇𝐼 ≤ $2,800, 𝑃𝑀𝑇 ≤ $2,100
 Back End 36%,𝑃𝐼𝑇𝐼 + 𝑜𝑡ℎ𝑒𝑟 𝑑𝑒𝑏𝑡 ≤ $3,600, 𝑃𝑀𝑇 ≤ $2,900 − 𝑜𝑡ℎ𝑒𝑟 𝑑𝑒𝑏𝑡



If other debt =$200, 𝑃𝑀𝑇 ≤ $2,900 − $200 = $2,700
If other debt =$800, 𝑃𝑀𝑇 ≤ $2,900 − $800 = $2,100
If other debt =$1k, 𝑃𝑀𝑇 ≤ $2,900 − $1𝑘 = $1,900
 BOTH the Front End and Back End must be satisfied!
 If other debt =$200, Front End 𝑃𝑀𝑇 ≤ $2,100, Back End 𝑃𝑀𝑇 ≤ $2,700

Maximum monthly mortgage 𝑃𝑀𝑇 = 𝑀𝑖𝑛 $2100, $2700 = $2100 (FE binds)
 If other debt =$800, Front End 𝑃𝑀𝑇 ≤ $2,100, Back End 𝑃𝑀𝑇 ≤ $2,100

Maximum monthly mortgage 𝑃𝑀𝑇 = 𝑀𝑖𝑛 $2100, $2100 = $2100 (both bind)
 If other debt =$1k, Front End 𝑃𝑀𝑇 ≤ $2,100, Back End 𝑃𝑀𝑇 ≤ $1,900

Maximum monthly mortgage 𝑃𝑀𝑇 = 𝑀𝑖𝑛 $2100, $1900 = $1900 (BE binds)
Baruch College: RES3200
Residential Loan Underwriting
Capacity
27
 Alice earns $10k/month. Property taxes + Insurance = $700/mo.
 Front End 28%, 𝑃𝐼𝑇𝐼 ≤ $2,800, 𝑃𝑀𝑇 ≤ $2,100
 Back End 36%,𝑃𝐼𝑇𝐼 + 𝑜𝑡ℎ𝑒𝑟 𝑑𝑒𝑏𝑡 ≤ $3,600, 𝑃𝑀𝑇 ≤ $2,900 − 𝑜𝑡ℎ𝑒𝑟 𝑑𝑒𝑏𝑡


If other debt =$200, 𝑃𝑀𝑇 ≤ $2,900 − $200 = $2,700
Maximum monthly mortgage 𝑃𝑀𝑇 = 𝑀𝑖𝑛 $2100, $2700 = $2100 (FE binds)
 Q: w/ 30yr FA-FRM at 5%, what is the biggest loan Alice can get?
 𝑃𝑉 =
2100
.05
1+ 12
1 + ⋯+
2100
.05 360
1+ 12
,
 N,360, I,5/12, PMT,-2100, CPT, PV=391,191.40
 Note: this is the biggest loan Alice can get exclusively based on income
requirements (DTI). Ignoring FICO and LTV.
Baruch College: RES3200
Residential Loan Underwriting
Collateral
28
 Alice wants to buy a property that costs $450k and she can afford up to a
100k down payment.
 If the bank’s maximum LTV requirement is 80%, will Alice be approved?
 . 8 ∗ 450𝑘 = 360𝑘, she needs a 90k down payment, she has 100k so yes!
 Suppose the appraiser says the property is only worth 300k, what is the
biggest loan she will get if the maximum LTV is 80%?
 300𝑘 ∗ .8 = 240𝑘, combined with Alice’s down payment she can afford to
spend 240𝑘 + 100𝑘 = 340𝑘 ≪ 450𝑘 asking price. NO!
 The appraiser’s low appraisal reduces Alice’s loan size by so much that she
can’t afford the property.
Baruch College: RES3200
Residential Loan Underwriting
Mortgage Insurance
29
 Example: 𝐻𝑃0 = $100𝑘, 𝐵0 = $95𝑘, 𝐿𝑇𝑉0 = 95%
 Suppose the bank requires Private Mortgage Insurance (PMI) for the portion of
the loan above 80% LTV (portion above $80k)
 The borrower must insure: $95𝑘 − $80𝑘 = $15𝑘
 The borrower pays the insurer a premium, the insurer will pay the lender up to
$15k if there is a deficiency.
 Suppose it’s an IO loan, after 5 years 𝐻𝑃60 = $83𝑘, 𝐵60 = $95𝑘



𝐷𝑒𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦60 = 𝐵60 − 𝐻𝑃60 = $95𝑘 − $83𝑘 = $12𝑘
The insurer pays the bank $12k, since 𝐷𝑒𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦60 < $15𝑘
The bank doesn’t suffer a loss if HP falls up to 20%. (Insurer up to 5%)
 If 𝐻𝑃60 = $78𝑘, 𝐵60 = $95𝑘


𝐷𝑒𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦60 = 𝐵60 − 𝐻𝑃60 = $95𝑘 − $78𝑘 = $17𝑘
The insurer pays the bank $15k, the bank’s loss =$2k
 With an IO and PMI, HP must fall at least 20% before the bank suffers a loss
Baruch College: RES3200
Review
Roadmap
30
 Finance review (NPV, IRR)
 Adjustable Rate Mortgages
 REFI decision
 Residential Mortgage Underwriting (3 C’s)
 Residential Valuation Buy vs Lease (Rent + TS - PITIM)
Baruch College: RES3200
Valuation
Real Estate Appraisal & Tax Shields
31
 These apply to residential & commercial valuation.
1) Sales Comparison Approach (Comparables)
•
What does a similar property sell for? uses recent sales data
•
𝑃෠𝑠𝑢𝑏𝑗𝑒𝑐𝑡 = 𝑃𝑐𝑜𝑚𝑝𝑎𝑟𝑎𝑏𝑙𝑒 + 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑏𝑒𝑡𝑡𝑒𝑟 − 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑤𝑜𝑟𝑠𝑒
2) Replacement Cost (what does it cost to replace?)
•
Land Value + Hard Costs + Soft Costs = Replacement Cost
3) Discounted Cash Flow Approach (Income)
𝑅1 +𝑇𝑆1−𝑃𝐼𝑇𝐼𝑀1
1+𝑖 1
+
𝑃5 −𝐵5 −𝐸𝑥𝑝5
1+𝑖 5
𝑁𝑃𝑉 = − 𝐷𝑃0 + 𝐸𝑥𝑝0 +

TS=Tax Shield
PITIM= Principal + Interest + Taxes + Insurance + Maintenance

+ ⋯+
𝑅5 +𝑇𝑆5 −𝑃𝐼𝑇𝐼𝑀5
1+𝑖 5

 Tax shield depends on your marginal income tax rate.

Property taxes and mortgage interest are income tax deductible.
 𝑇𝑎𝑥 𝑆ℎ𝑖𝑒𝑙𝑑 = 𝑚𝑎𝑟𝑔𝑖𝑛𝑎𝑙 𝑡𝑎𝑥 𝑟𝑎𝑡𝑒 × 𝐷𝑒𝑑𝑢𝑐𝑡𝑖𝑜𝑛
Baruch College: RES3200
BUY VS LEASE
Tenure Choice
32
• 𝑁𝑃𝑉 𝑜𝑤𝑛 = − 𝐷𝑃0 + 𝐸𝑥𝑝0 +
• 𝑁𝑃𝑉 𝑟𝑒𝑛𝑡 = 0 +
•
−𝑅1
1+𝑖 1
+ ⋯+
𝑁𝑃𝑉 𝑜𝑤𝑛−𝑟𝑒𝑛𝑡 = − 𝐷𝑃0 + 𝐸𝑥𝑝0 +
𝑇𝑆1 −𝑃𝐼𝑇𝐼𝑀1
1+𝑖 1
+ ⋯+
𝑇𝑆𝑇 −𝑃𝐼𝑇𝐼𝑀𝑇
1+𝑖 𝑇
+
𝑃𝑇 −𝐵𝑇 −𝐸𝑥𝑝𝑇
1+𝑖 𝑇
−𝑅𝑇
1+𝑖 𝑇
𝑅1 +𝑇𝑆1 −𝑃𝐼𝑇𝐼𝑀1
1+𝑖 1
+ ⋯+
𝑅𝑇 +𝑇𝑆𝑇 −𝑃𝐼𝑇𝐼𝑀𝑇
1+𝑖 𝑇
+
𝑃𝑇 −𝐵𝑇 −𝐸𝑥𝑝𝑇
1+𝑖 𝑇
• Just like stocks there are two sources of income:
• Dividend Income= periodic income (unlike stocks can be negative)
• Capital Gains Income: owner has a residual claim on CF, 𝑃𝑇 − 𝐵𝑇 − 𝐸𝑥𝑝𝑇 , the
lender has a senior claim. The owner is an equity investor.
Baruch College: RES3200
BUY VS LEASE
Tenure Choice: Wealth Accumulation
33
 Lucy is moving to Tulsa. She can own or rent.
 She can buy for: 𝑃0 = $126,247.44
 100% LTV, FA-FRM, 30 years, annual payments, r=5%,
 [30,N], [5,I], [126247.44, PV], CPT, PMT=$8,212.58
 𝑇 + 𝐼 + 𝑀 = 3% ∗ 𝑃0 = 3787.42 (assume no tax shield)
 𝐶𝑜𝑠𝑡 𝑜𝑓 𝑂𝑤𝑛𝑒𝑟𝑠ℎ𝑖𝑝 = 𝑃𝐼𝑇𝐼𝑀 = 8,212.58+3,787.42=$12,000
 She can rent the same house for: 𝑅𝑒𝑛𝑡 = $12,000
𝑃𝑟𝑖𝑐𝑒

𝑅𝑒𝑛𝑡
 𝑈=
$126,247.44
=
= 10.52,
$12,000
𝑅𝑒𝑛𝑡
$12,000
𝑃𝑟𝑖𝑐𝑒
=
$126,247.44
= 9.5%
 By assumption, Lucy makes no down payment to buy.
 It costs her $12k/year to live in the same house whether she owns
or rents.
 Q: is Lucy better off owning or renting?
Baruch College: RES3200
BUY VS LEASE
Tenure Choice
34
 Lucy is moving to Tulsa. She can own or rent.
 own: 𝑃0 = $126,247.44, PITIM = 𝑃𝑀𝑇 + 𝑇𝐼𝑀 = $12𝑘
 Rent: 𝑅𝑒𝑛𝑡 = $12𝑘 (assume rent is constant)
 𝑁𝑃𝑉 𝑜𝑤𝑛 = 0 +
−𝑃𝐼𝑇𝐼𝑀
1+𝑖 1
𝑟𝑒𝑛𝑡
−𝑅𝑒𝑛𝑡1
1+𝑖 1
 𝑁𝑃𝑉
=0+
 𝑁𝑃𝑉 𝑜𝑤𝑛−𝑟𝑒𝑛𝑡
=0+
+⋯+
−𝑃𝐼𝑇𝐼𝑀
1+𝑖 𝑇
+⋯+
−𝑅𝑒𝑛𝑡𝑇
1+𝑖 𝑇
0
1+𝑖 1
+⋯+
+
𝑃𝑇 −𝐵𝑇 −𝐸𝑥𝑝𝑇
1+𝑖 𝑇
0
1+𝑖 𝑇
+
𝑃𝑇 −𝐵𝑇 −𝐸𝑥𝑝𝑇
1+𝑖 𝑇
 The only difference between owning and renting is that Lucy will walk
away with home equity after expenses
 𝑃𝑇 − 𝐵𝑇 − 𝐸𝑥𝑝𝑇
Baruch College: RES3200
BUY VS LEASE
Tenure Choice
35




Lucy is moving to Tulsa. She can own or rent.
own: 𝑃0 = $126,247.44, PITIM = 𝑃𝑀𝑇 + 𝑇𝐼𝑀 = $12𝑘
Rent: 𝑅𝑒𝑛𝑡 = $12𝑘 (assume rent is constant)
home equity after expenses: 𝑃𝑇 − 𝐵𝑇 − 𝐸𝑥𝑝𝑇
 Suppose house prices grow 𝑔 = 4.5% (Demers-Eisfeldt)
 Suppose selling expenses =8% of 𝑃𝑇 : 𝐸𝑥𝑝𝑇 = .08 × 𝑃𝑇 (Trulia)
 𝑇 = 30, 𝐵30 = 0, 𝑃30 = 472,836.83, 𝑃30 − 𝐵30 − 𝐸𝑥𝑝30 = 435,009.88
 Lucy lived in the same house for 30 years but left w/ an extra $435k from owning
rather than renting.
 𝑇 = 1, 𝐵1 = 124,347.28, 𝑃1 = 131,928.57,

[N, 29], [I, 5], [PMT, -8212.58], [CPT, PV]
 𝐸𝑥𝑝1 = .08 ∗ 131,928.57 = 10,554.29, 𝑃1 − 𝐵1 − 𝐸𝑥𝑝1 = −2972.98
 Q: if rent grew over this time period, would Lucy be better off owning?
 Piketty-Rognlie: housing explains a lot of wealth inequality
Baruch College: RES3200
Sample Exam – multiple choice
36
 1. What is a typical LTV constraint on mortgages?
(A) 10%
(B) 20%
(C) 40%
(D) 80%
 2. Which of the following factors will affect your FICO credit score?
(A) gender
(B) salary
(C) recent credit searches
(D) race
 3. All of the following are reasons a borrower may want to refinance EXCEPT?
(A) lenders demand lower debt-to-income (payment-to-income) ratios
(B) the borrowing rate declines
(C) the borrower’s credit score rises
(D) the property value rises
Baruch College: RES3200
Sample Exam – multiple choice
37
 1. What is a typical LTV constraint on mortgages?
(A) 10%
(B) 20%
(C) 40%
(D) 80%
 2. Which of the following factors will affect your FICO credit score?
(A) gender
(B) salary
(C) recent credit searches
(D) race
 3. All of the following are reasons a borrower may want to refinance EXCEPT?
(A) lenders demand lower debt-to-income (payment-to-income) ratios
(B) the borrowing rate declines
(C) the borrower’s credit score rises
(D) the property value rises
Baruch College: RES3200
Sample Exam
38

4. A bank makes a 30 year Fully Amortizing FRM for $1,000,000 at an annual interest
rate of 4.25% compounded monthly, with monthly payments. What is the market value of
this loan after 7 years of payments if the annual interest rate for this loan is 7%
compounded monthly?
Scientific calculator
 Payment = 1,000,000*.0425/12/(1-1/(1+.0425/12)^360) = 4,919.40
 Value = 4,919.40/.07*12*(1-1/(1+.07/12)^(23*12)) = 673,964.99

Financial calculator
 Payment: N = 360, I = 4.25/12, PV = - 1,000,000, FV = 0, CPT PMT
 Value: N = 23*12, I = 7/12, PMT = 4919.40, FV = 0, CPT PV (change sign)

Baruch College: RES3200
Sample Exam
39
 5. A bank makes a 30 year Fully Amortizing FRM for $1,000,000
at an annual interest rate of 4.25% compounded monthly, with
monthly payments. Suppose inflation is 1% per year,
compounded monthly. What is the real value of the 60th
payment?
 4,919.40/(1+.01/12)^60 = 4,679.57
Baruch College: RES3200
Sample Exam
40
 6. A 30 year 1/1 ARM has an initial rate of 1.5%. In the future the
rate will reset to 225 basis points above the LIBOR index with no
rate caps or floors. In 1 year, at the time of the first reset, the
LIBOR is 6.75%. What interest rate will be used to compute
payments at the first reset?
 2.25%+6.75% = 9.00%
Baruch College: RES3200
Sample Exam
41

7. Ann gets a 30 year 1/1 Fully Amortizing ARM for $1,000,000, with monthly payments and
monthly compounding. The initial rate is 4%. In the future, the rate will reset to 250 basis
points above the LIBOR. There are no rate caps or floors. Suppose at the first reset, the
LIBOR was 1%. What is the monthly mortgage payment for the second year?
Scientific calculator
 1,000,000*(1-1/(1+.04/12)^348)/(1-1/(1+.04/12)^360)*.035/12/(1-1/(1+.035/12)^348)) =
4,497.68

Financial calculator
 Payment for 1st year: N = 360, I = 4/12, PV = - 1,000,000, FV = 0, CPT PMT
 Balance after 1st year: N = 348, I = 4/12, PMT = 4774.15, FV = 0, CPT PV
 Payment for 2nd year: N = 348, I = 3.5/12, PV = - 982,389.64, FV = 0, CPT PMT

Baruch College: RES3200
Sample Exam
42
 8. Ann got a 30 year Fully Amortizing FRM for $1,000,000 at an annual
interest rate of 7% compounded monthly, with monthly payments. After 5
years of payments, Ann can refinance the balance into a 25 year Fully
Amortizing FRM at an annual interest rate of 5.25% compounded monthly,
with monthly payments. If Ann refinances, what will be her monthly savings on
her mortgage payment?




Payment = 1,000,000*.07/12/(1-1/(1+.07/12)^360) = 6,653.02
Balance = 6,653.02/.07*12*(1-1/(1+.07/12)^300) = 941,315.20
New payment = 941,315.20*.0525/12/(1-1/(1+.0525/12)^300) = 5,640.81
6,653.02 – 5,640.81 = 1,012.21





Financial calculator
Payment: N = 360, I = 7/12, PV = - 1,000,000, FV = 0, CPT PMT
Balance 5 years: N = 300, I = 7/12, PMT = 6653.02, FV = 0, CPT PV
New payment: N = 300, I = 5.25/12, PV = - 941,315.20, FV = 0, CPT PMT
6653.02 - 5640.81 = 1,012.21
Baruch College: RES3200
Sample Exam
43
 9. Ann got a 30 year Fully Amortizing FRM for $1,000,000 at an annual
interest rate of 7% compounded monthly, with monthly payments. After 5
years of payments, Ann can refinance the balance into a 25 year Fully
Amortizing FRM at an annual interest rate of 6% compounded monthly,
with monthly payments.
 Refinancing will cost Ann 1 point and $1,500 in closing costs. If Ann
refinances into this loan after 5 years, what will be her total cost of
refinancing (i.e. the cost paid immediately at the moment of refinancing)?
 941,315.20*.01+1,500 = 10,913.16
Baruch College: RES3200
Sample Exam
44
 10. Jim has an annual income of $240,000. Jim is looking to buy a
house with monthly property taxes of $300 and monthly homeowner’s
insurance of $200. Apple bank has a maximum front end PTI limit of
28%. Considering only the front end PTI limit, what is the most they will
allow Jim to spend on a monthly mortgage payment?
 (A) 5,100
(B) 5,558.33
(C) 5,600
 240,000*.28/12-300-200 = 5,100
Baruch College: RES3200
(D) 66,700
Sample Exam
45
 11. Jim has an annual income of $240,000. Jim is looking to buy a house
with monthly property taxes of $300 and monthly homeowner’s insurance
of $200. Jim has $1,500 in monthly student loan payments and an average
monthly credit card bill of $1,000. Apple bank has a maximum front end
PTI limit of 28% and a maximum back end PTI limit of 36%. Considering
both the front end PTI limit and the back end PTI limit, what is the most
they will allow Jim to spend on a monthly mortgage payment?
 min(5,100.00, 240,000*.36/12-300-200-1,500-1,000) = 4,200
 240,000*.36/12-300-200-1,500-1,000=4,200
 min(5100; 4200) = 4,200
 Another example: in case your answer to Q10 was 3100
 min(3,100.00, 240,000*.36/12-300-200-1,500-1,000) = 4,200
 min(3100, 4200)=3,100
Baruch College: RES3200
Sample Exam
46
 12. Jim has an annual income of $240,000. Jim is looking to buy a house with
monthly property taxes of $300 and monthly homeowner’s insurance of $200. Jim
has $1,500 in monthly student loan payments and an average monthly credit card
bill of $1,000. Apple bank has a maximum front end PTI limit of 25% and a
maximum back end PTI limit of 35%. Jim will make the biggest mortgage payment
he can that satisfies both the front end PTI limit and the back end PTI limit. Jim will
get a fully amortizing 30 year FRM at an annual rate of 3.40%, with monthly
payments, compounded monthly. What is the biggest mortgage loan Jim can get?
 Payment =
= min(240,000*.25/12-300-200, 240,000*.35/12-300-200-1,000-1,500) = 4,000
 4,000/.034*12*(1-1/(1+.034/12)^360) = 901,954.74
 Financial Calculator solution for loan size
 Payment: N = 360, I = 3.4/12, PMT = 4000, FV = 0, CPT PV (change sign)
Baruch College: RES3200
Sample Exam
47
 13. Sam bought a house that costs $500,000. Sam got a 92% LTV loan. The
lender demanded that Sam buy private mortgage insurance to insure the
portion of the loan over 80% LTV. Suppose 5 years later, Sam’s mortgage
balance is $450,000. However, Sam defaults and his house sells for $400,000
in a foreclosure auction. How much will the mortgage insurance company pay
Sam’s lender?
 deficiency = 450,000 – 400,000 = 50,000
 private insurance coverage = 0.92 * 500,000 - 0.8 * 500,000 = 60,000
 Lender needs 50,000 and has coverage up to 60,000 → lender gets 50,000
Baruch College: RES3200
Sample Exam
48
 14. Ann’s average income tax rate is 30%, and her marginal income tax
rate is 35%. Her property tax bill for 2015 was $4,000 and her total
interest payment on her mortgage was $4,500. Based on this
information, what is Ann’s total income tax shield (i.e. how much is Ann
saving on her income taxes, in dollars)?
 .35*(4,000+4,500) = 2,975
Baruch College: RES3200
15-16. Refinance
49
15. Ann got a 30 year FRM with annual payments equal to $24,000.
 After 2 years of payments Ann will refinance the balance into a 28-year FRM with annual
payments equal to $20,000. Refinancing will cost Ann $5,500.
 Ann will prepay the new loan 3 years after refinancing. She will save $2,500 on her loan
balance when she prepays.
 Using all the information given, find the NPV for Ann's refinancing decision if her annual
discount rate is 20%.


Solution
Cost of REFI = 5,500.00
PMT savings = 24,000-20,000=4,000 (for 3 years)
Balance saving: 2,500
Final year total CF= 4,000 + 2,500 = 6,500

𝑵𝑷𝑽 𝒊 = −𝟓, 𝟓𝟎𝟎 +

Plug in i = 20% to get NPV = 4,372.69




Baruch College: RES3200
𝟒,𝟎𝟎𝟎
𝟏+𝒊 𝟏
+
𝟒,𝟎𝟎𝟎
𝟏+𝒊 𝟐
+
𝟔,𝟓𝟎𝟎
𝟏+𝒊 𝟑
50
 16. Find the IRR of the refinancing decision in Q15. Hint: this question is
designed to be done with financial calculator, not with a regular calculator.
 Solution: CF0 = -5500, CF1 = 4000, CF2 = 4000, CF3 = 6500, CPT IRR
 Answer: 62.36%
Baruch College: RES3200
17-19. Buy vs Lease
51
17. Lucy bought a house that costs $100,000. She financed it with an 80% LTV mortgage. The
mortgage is a 30 year fully amortizing FRM with annual compounding and annual payments.
Lucy’s annual cost of ownership net of tax savings is exactly equal to the annual rent she
would have paid to live in the same house. Lucy will sell the house 30 years after purchase.
Suppose the house price grows 4.5% annually (compounded annually). Buying costs are 5%
of the purchase price of the house. Selling costs are 8% of the selling price of the house.
 Using all the information given, find the NPV for Lucy's buy vs lease decision if her annual
discount rate is 10%.


Solution
Sale price: 100,000 * 1.045^30= 374,531.81
Lucy’s sale CF= 374,531.81*.92= 344,569.27
CF0= -.20*100,000 – .05*100,000= -25,000
CF1 = 0, (t=1-29)
CF30=344,569.27

𝑵𝑷𝑽 𝒊 = −𝟐𝟓, 𝟎𝟎𝟎 +

Plug in i = 10% to get NPV = -5,253.23





Baruch College: RES3200
𝟑𝟒𝟒,𝟓𝟔𝟗.𝟐𝟕
𝟏+𝒊 𝟑𝟎
17-19. Buy vs Lease
52

18. Find the IRR of the buying decision in Q17.

Solve: −𝟐𝟓, 𝟎𝟎𝟎 +

19. Ann’s forecast for the growth rate of her house price was incorrect and she actually got
IRR = 2% in Q18. What was the actual growth rate of Ann’s house price? Write your answer in
percent but without the “%” sign. Hint: this question is designed to be done with a regular
calculator, not with a financial calculator.

Solve: −𝟐𝟓, 𝟎𝟎𝟎 +
𝟑𝟒𝟒,𝟓𝟔𝟗.𝟐𝟕
𝟏+𝑰𝑹𝑹 𝟑𝟎
= 𝟎 ⇒ 𝑰𝑹𝑹 =
𝟏𝟎𝟎,𝟎𝟎𝟎∗ 𝟏+𝐱 𝟑𝟎 ∗𝟎.𝟗𝟐
𝟏+𝟎.𝟎𝟐 𝟑𝟎
𝟑𝟒𝟒,𝟓𝟔𝟗.𝟐𝟕 𝟏/𝟑𝟎
𝟐𝟓,𝟎𝟎𝟎
− 𝟏 = 𝟎. 𝟎𝟗𝟏𝟒 = 𝟗. 𝟏𝟒%
=𝟎
𝟏 + 𝐱 𝟑𝟎 = 𝟎. 𝟒𝟗𝟐𝟐𝟏𝟖
 𝟏 + 𝐱 = 𝟎. 𝟒𝟗𝟐𝟐𝟏𝟖𝟏/𝟑𝟎 = 𝟎. 𝟗𝟕𝟔𝟔𝟒𝟗
 𝐱 = −𝟎. 𝟎𝟐𝟑𝟒 = −𝟐. 𝟑𝟒%

The actual exam will not be the same as above, but roughly similar.
 Hint: check out HWs 4-5 and lecture slides 7-10!

Baruch College: RES3200
The Refinancing Decision Process
Current Loan vs Refinance Option: Prepay+ Longer term
53
 You have lived in your home for 5 years and still have the
original mortgage.
 A lender has approached you with an offer to refinance.
 You plan to live in the house another 10 years.
 What should you do?
Current Loan
$250,000
30 years
7.00%
$1,663.26
0
0
Baruch College: RES3200
Refinance Option
Loan Amount
$ ??
Term
30 years
Rate
6.00%
Payment
$ ??
Bank Fees
2 points
Other Closing Costs
$1,500
The Refinancing Decision Process
Current Loan vs Refinance Option: Prepay+ Longer term
54




You have lived in your home for 5 years and still have the original mortgage.
A lender has approached you with an offer to refinance.
You plan to live in the house another 10 years.
What should you do?
Current Loan
$250,000
30 years
7.00%
$1,663.26
Not relevant
Not relevant





Loan Amount
Term
Rate
Payment
Bank Fees
Other Closing Costs
Refinance Option
$ ??
30 years
6.00%
$ ??
2 points
$1,500
Balance after 5 years = 1,663.26/.07*12*(1-1/(1+.07/12)^300)=235,329.50
Cost of refinancing = 1,500+.02* 235,329.50= 6,206.59
New payment = 235,329.50*.06/12/(1-1/(1+.06/12)^360)=1,410.92
Balance in 10 years = 1,410.92/.06*12*(1-1/(1+.06/12)^240)=196,937.30
if not refinance, balance in 10 years = 1,663.26/.07*12*(1-1/(1+.07/12)^(15*12))=
185,047.58
Time, months 0
1-59
A CF if not refi
250000 -1663.26
B CF if refi
250000 -1663.26
Baruch College: RES3200
B-A
60
-1663.26
-1663.26-6206.59
-6206.59
61-179
180
-1663.26 -1663.26-185047.58
-1410.92 -1410.92-196937.30
252.34
-11637.38
The Refinancing Decision Process
Current Loan vs Refinance Option: Prepay+ Longer term
55




You have lived in your home for 5 years and still have the original mortgage.
A lender has approached you with an offer to refinance.
You plan to live in the house another 10 years.
What should you do?
Current Loan
$250,000
30 years
7.00%
$1,663.26
Not relevant
Not relevant





Loan Amount
Term
Rate
Payment
Bank Fees
Other Closing Costs
Refinance Option
$ ??
30 years
6.00%
$ ??
2 points
$1,500
Balance after 5 years = 1,663.26/.07*12*(1-1/(1+.07/12)^300)=235,329.50
Cost of refinancing = 1,500+.02* 235,329.50= 6,206.59
New payment = 235,329.50*.06/12/(1-1/(1+.06/12)^360)=1,410.92
Balance in 10 years = 1,410.92/.06*12*(1-1/(1+.06/12)^240)=196,937.30
if not refinance, balance in 10 years = 1,663.26/.07*12*(1-1/(1+.07/12)^(15*12))=
185,047.58
Time, months 0
1-59
A CF if not refi
250000 -1663.26
BBaruch
CF if College:
refi
250000 -1663.26
RES3200
B-A
60
-1663.26
-7869.85
-6206.59
61-179
180
-1663.26 -186710.84
-1410.92 -198348.22
252.34
-11637.38
IRR mnth
0.583%
0.463%
-1.76; 3.95
IRR year
7.00%
5.56%
-21.1% or 47.4%?
56
Baruch College: RES3200
NPV(r) plot
57
Baruch College: RES3200