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Canvas Example MCQ — Q33

• A) 1.0 percentage point
• B) 1.4 percentage points
• C) 2.0 percentage points
• D) 40%
• E) None of A-D is close; write the correct answer.

Initial rr = .05(.70)-.02 = .015 = 1.5%. New rr = .07(.70)-.02 = .029 = 2.9%. Change = 2.9%-1.5% = 1.4 percentage points.

Rule: A rate moving from 1.5% to 2.9% rises by 1.4 percentage points.

Canvas Example MCQ — Q34

• A) It increases by 1.4%.
• B) It increases by 1.4 percentage points, corresponding to about 93.3%.
• C) It increases by 93.3 percentage points.
• D) It increases by 2.0 percentage points, corresponding to 40%.
• E) None of A-D is close; write the correct answer.

Percentage-point change = 2.9-1.5 = 1.4 pp. Percentage change = 1.4/1.5 = 0.9333 = 93.3%.

Rule: Never confuse percentage points with relative percent change.

Canvas Example MCQ — Q35

• A) The real after-tax mortgage rate increases by 2 percentage points.
• B) The real after-tax mortgage rate decreases by 2 percentage points.
• C) The real after-tax mortgage rate decreases by 30%.
• D) Inflation has no effect because mortgage interest is tax deductible.
• E) None of A-D is close; write the correct answer.

Before: .06(.70)-.02 = 2.2%. After: .06(.70)-.04 = 0.2%. Change = -2.0 pp.

Rule: Higher inflation lowers the approximate real borrowing cost when nominal rate and tax treatment are fixed.

Canvas Example MCQ — Q36

• A) 1.5%
• B) 1.9%
• C) 2.9%
• D) 4.9%
• E) None of A-D is close; write the correct answer.

.07(.70)-.03 = .049-.03 = .019 = 1.9%.

Rule: Apply the formula to the new values, not the changes.

Canvas Example MCQ — Q93

Given: Logit model for choosing an adjustable-rate mortgage. At FIXRATE=12, MARGIN=2, NETWORTH=3 the index is z = −0.80. MARGIN coefficient = −0.35. Logit: P = 1/(1+e^(−z)); marginal effect = b·P·(1−P).

• A) approximately -35.0 percentage points
• B) approximately -12.8 percentage points
• C) approximately -7.5 percentage points
• D) approximately +7.5 percentage points
• E) None of A-D is close; write the correct answer.

z=-0.80, P≈0.3100. ME = (-.35)(.31)(.69) = -.0749 ≈ -7.49 pp.

Rule: Logit ME = β x P(1-P).

Old Exam — Oct 2023, Q16 · adapted

Given: Probit Model 1: z = −7.0166 + 0.5301·FIXRATE − 0.2675·MARGIN + 0.0864·NETWORTH; P(ARM) = Φ(z); marginal effect of X = b_X·φ(z).

• A) ‒0.0030
• B) ‒0.0092
• C) ‒0.0343
• D) ‒0.2675
• E) None of the above (A, B, C, D) is close to be correct.

Model 1: z=-7.0166+0.5301(10)-0.2675(3)+0.0864(3.5). Then ME_MARGIN=-0.2675 phi(z)≈-0.0092 -> B.

Rule: Phi(.) gives probability; phi(.) enters Probit marginal effects. Raw Probit coefficients are not percentage-point effects.

Old Exam — Oct 2022, Q22

Given: Probit Model 1: z=-7.0166+0.5301 FIXRATE-0.2675 MARGIN+0.0864 NETWORTH.

• A) 0.1762
• B) 0.3707
• C) 0.6837 This is when FIXRATE = 15, which is not what the question is about. Correct answer is 0.478 with FIXRATE = 14, therefore D is the correct answer.
• D) 0.4780
• E) None of the above (A, B, C, D) is close to be correct.

z=-7.0166+0.5301(14)-0.2675(3)+0.0864(4)≈-0.055. Phi(z)≈0.478 -> D.

Rule: Phi(.) gives probability; phi(.) enters Probit marginal effects. Raw Probit coefficients are not percentage-point effects.

Old Exam — Oct 2022, Q23

Given: Probit Model 2: z=-9.8200+0.7535 FIXRATE-0.1945 MARGIN; ME_MARGIN=b_MARGIN x phi(z).

• A) ‒0.0202
• B) ‒1.641
• C) 0.1038
• D) 0.039
• E) None of the above (A, B, C, D) is close to be correct.

Model 2: z=-9.82+0.7535(11.5)-0.1945(2.5)≈-1.64. phi(z)≈0.1038. ME_MARGIN=-0.1945x0.1038≈-0.0202 -> A.

Rule: Phi(.) gives probability; phi(.) enters Probit marginal effects. Raw Probit coefficients are not percentage-point effects.

Old Exam — Oct 2022, Q24 · adapted

Given: Probit Model 1: z = −7.0166 + 0.5301·FIXRATE − 0.2675·MARGIN + 0.0864·NETWORTH; P(ARM) = Φ(z); marginal effect of X = b_X·φ(z).

• A) ‒0.092
• B) ‒0.0441
• C) 0.0142
• D) 0.039
• E) None of the above (A, B, C, D) is close to be correct.

Model 1: compute z at FIXRATE=11.5, MARGIN=2.5, NETWORTH=3; then ME_NETWORTH=0.0864 phi(z)≈0.0142 -> C.

Rule: Phi(.) gives probability; phi(.) enters Probit marginal effects. Raw Probit coefficients are not percentage-point effects.

Computer Lab 1 — Exercise 2B, Step 1

• A) $241,162.50
• B) $221,250.00
• C) $663,750.00
• D) $259,912.50

Loan = 885,000×75% = 663,750. Financing cost = 663,750×3% = 19,912.50. Net loan proceeds = 663,750−19,912.50 = 643,837.50. Initial Equity = 885,000 − 643,837.50 = $241,162.50.

Rule: Initial Equity = Purchase Price − Net Loan Proceeds, where Net Loan Proceeds = Loan − up-front financing cost.

Computer Lab 1 — Exercise 2B, Step 2

• A) $4,870.36
• B) $5,530.00
• C) $4,425.00
• D) $58,444.35

Monthly i=8%/12=0.66667%, n=360. PMT = Loan×i/[1−(1+i)^-n] = 663,750×0.0066667/[1−(1.0066667)^-360] = $4,870.36.

Rule: CPM monthly payment: PMT = Loan × i / [1 − (1+i)^(−n)], with i and n expressed monthly.

Computer Lab 1 — Exercise 2B, Step 3

• A) $631,026.17, the PV of the remaining 300 payments
• B) $371,715.00, original loan minus 60 payments made
• C) $663,750.00, unchanged since CPM keeps balance constant
• D) $58,444.35, one year of debt service

RMB after 60 payments = PV of remaining 300 payments = PMT×[1−(1+i)^-300]/i = 4,870.36×[1−(1.0066667)^-300]/0.0066667 = $631,026.17.

Rule: Remaining mortgage balance = present value, at that point in time, of all remaining scheduled payments — not loan minus cash paid.

Computer Lab 1 — Exercise 2B, Question 8

• A) $385,425.51
• B) $343,587.03
• C) $1,074,896.04
• D) $41,838.49

BTCF5 = NOI5 − debt service = 100,282.83−58,444.35 = 41,838.49. BTER = NSP−RMB = 974,613.20−631,026.17 = 343,587.03. Year5 equity CF = BTCF5+BTER = 41,838.49+343,587.03 = $385,425.51.

Rule: Year-of-sale equity cash flow = BTCF + BTER, where BTER = Net Sale Proceeds − Remaining Mortgage Balance.

Computer Lab 1 — Exercise 2B, Question 9

• A) NPV ≈ +$38,149.47 → invest
• B) NPV ≈ -$38,149.47 → reject
• C) NPV ≈ +$76,789.79 → invest
• D) NPV ≈ $0 → indifferent

Discounting the equity CFs at 16%: PV1=26,427.28, PV2=24,768.62, PV3=23,116.11, PV4=21,493.86, PV5=183,506.10, sum=279,311.97. NPV=-241,162.50+279,311.97=+$38,149.47 → invest.

Rule: Levered NPV decision rule: discount equity cash flows at the required EQUITY return (not the unlevered return).

Computer Lab 1 — Exercise 2B, Question 10

• A) Invest — 20.25% exceeds the required equity return of 16%
• B) Reject — 20.25% is below the required equity return
• C) Indifferent — IRR equals the required return
• D) Cannot decide without the unlevered IRR

Levered IRR (~20.25%) > required equity return (16%), so by the IRR decision rule the investor should invest.

Rule: IRR decision rule: invest if IRR > required (hurdle) return for that cash-flow stream.

Computer Lab 1 — Exercise 3 Problem 4

• A) ≈$1,219,749.29
• B) ≈$1,000,000.00
• C) ≈$967,145.20
• D) ≈$1,500,000.00 (unchanged)

RMB5 = PMT × [1−(1+i)^-120]/i with i=10%/12=0.833333%. RMB5 = 16,119.08×[1−(1.00833333)^-120]/0.00833333 ≈ $1,219,749.29.

Rule: Remaining balance after k payments = PV, at that time, of the remaining (n−k) scheduled payments, discounted at the monthly rate.

Computer Lab 2 — Exercise 1, Step 1

• A) $125,000
• B) $180,000
• C) $150,000
• D) $108,000

DCR = NOI/Debt Service → Debt Service = NOI/DCR = 150,000/1.2 = $125,000.

Rule: Debt Service = NOI / DCR when DCR-based underwriting determines the maximum loan debt service.

Computer Lab 2 — Exercise 1(a)

• A) ≈$666,035.37
• B) ≈$1,134,145.51 (undiscounted sum)
• C) ≈$1,745,458.48
• D) ≈$571,172.70

PV = 25,000/1.12 + 29,500/1.12² + 34,135/1.12³ + 38,909.05/1.12⁴ + 1,006,601.46/1.12⁵ = 22,321.43+23,517.22+24,296.62+24,727.40+571,172.70 = $666,035.37.

Rule: PV of equity interest = sum of each year's equity cash flow discounted at the required equity return.

Computer Lab 2 — Exercise 1(b)(c)

• A) Property value ≈ $1,745,458.48; cap rate ≈ 8.59%
• B) Property value ≈ $1,745,458.48; cap rate ≈ 12.00%
• C) Property value ≈ $1,079,423.11; cap rate ≈ 13.90%
• D) Property value ≈ $413,387.74; cap rate ≈ 36.29%

Property Value = PV(equity)+PV(mortgage) = 666,035.37+1,079,423.11 = $1,745,458.48. Going-in cap rate = NOI1/Property Value = 150,000/1,745,458.48 ≈ 8.59%.

Rule: Total property value = PV of debt claim + PV of equity claim. Going-in cap rate = Year-1 NOI ÷ total property value.

Computer Lab 2 — Exercise 1 trap

• A) Convert to a monthly payment (annual debt service/12), use the monthly interest rate, and discount the remaining number of MONTHLY payments
• B) Use the annual debt service and annual interest rate directly in the annuity formula with the number of years remaining
• C) Simply subtract total payments made from the original loan amount
• D) The remaining balance equals the original loan amount since DCR fixes it

Since payments occur monthly, the correct method divides annual debt service by 12 to get the monthly PMT, uses the monthly rate (annual rate/12), and finds RMB as the PV of the remaining monthly payments.

Rule: Always match the payment frequency, compounding period, and remaining number of periods — convert annual rates/figures to monthly when payments are monthly.

Computer Lab 3 — Problem 4

• A) $16,119.08
• B) $12,500.00
• C) $10,000.00
• D) $193,428.92

i=10%/12=0.833333%, n=180. PMT=1,500,000×0.00833333/[1−(1.00833333)^-180]=$16,119.08.

Rule: CPM monthly payment = Loan×i/[1−(1+i)^(−n)], with i and n in months.

Computer Lab 5 — Exercise 1, Task 2

• A) 2.00%
• B) 6.00%
• C) 2.80%
• D) 0.80%

Real rate before tax = nominal rate − inflation = 4% − 2% = 2.00%.

Rule: Real rate (before tax) = nominal rate − inflation.

Computer Lab 5 — Exercise 1, Task 2

• A) 0.80%
• B) -26.00% (4% − 30% − 2%)
• C) 2.00%
• D) 2.80%

Nominal after-tax rate = 4%×(1−0.30) = 2.80%. Real after-tax rate = 2.80% − 2.00% = 0.80%.

Rule: The tax deduction multiplies the interest cost by (1−t); it does not subtract t directly from the rate in percentage points.

Computer Lab 5 — Exercise 1, Task 3

• A) ≈3.30%
• B) 3.60% (simple average of the three rates)
• C) 3.27% (simple average of 3.2% and 3.6% only)
• D) 4.00%, the highest rate category

r_bar = [3300(0.032)+900(0.036)+100(0.040)]/4300 = (105.6+32.4+4.0)/4300 = 142/4300 ≈ 0.03302 = 3.30%.

Rule: Use amount-weighted averages, not simple averages, when categories differ greatly in size.

Computer Lab 5 — Exercise 2, Step 2

• A) $3,850,000 annual interest; full $70,000,000 principal repaid as a balloon payment at maturity
• B) $3,850,000 annual interest; principal amortizes gradually like a standard CPM loan
• C) $7,000,000 annual interest; no principal repayment needed
• D) $385,000 annual interest; balloon payment at maturity

Annual interest = 70,000,000 × 5.5% = 3,850,000. Since the loan is interest-only, no amortization occurs; the entire $70,000,000 principal is repaid as a balloon payment at maturity.

Rule: Interest-only loans: fixed interest each year during the hold, full principal repaid at maturity (no CPM amortization).

Computer Lab 5 — Exercise 2, Task 1

• A) ≈$4,333,058
• B) ≈$21,665,290
• C) ≈$24,333,058
• D) $0, because appreciation participation only applies to NOI

Appreciation = 121,665,290.24 − 100,000,000 = 21,665,290.24. Lender participation = 20% × 21,665,290.24 ≈ 4,333,058.05.

Rule: Participation in appreciation = share × max(sale price − purchase price, 0); apply the share only to the appreciation amount, not the full sale price.

Computer Lab 5 — Exercise 2, Task 2

• A) $1,260,000
• B) $1,400,000
• C) $6,300,000
• D) $0, since there's no appreciation in Year 1

Participation_1 = 20% × (7,000,000 − 700,000) = 20% × 6,300,000 = 1,260,000.

Rule: NOI participation = share × max(NOI − threshold, 0); subtract the threshold before applying the share.

Computer Lab 5 — Exercise 2, Task 2

• A) It creates upside-only participation: if appreciation or NOI above the threshold is negative, the lender's participation payment is zero, never negative
• B) It caps the lender's participation at a fixed dollar maximum
• C) It ensures the lender always receives at least the base interest payment
• D) It guarantees the borrower a minimum cash flow

MAX(x,0) ensures the lender only participates in positive upside; if the appreciation or NOI-above-threshold value is negative, the lender receives zero participation rather than a negative payment.

Rule: Participation formulas use MAX(...,0) to make participation upside-only, never a cost to the lender.

Computer Lab 6 — Exercise 3, Model 1 interpretation · adapted

• A) A higher ARM margin makes choosing an adjustable-rate mortgage less attractive (negative sign); the raw coefficient is NOT a direct percentage-point probability effect
• B) A higher ARM margin makes an ARM more attractive, and -0.2675 means exactly a 26.75 percentage-point decrease in probability
• C) The coefficient has no interpretable meaning in a Probit model
• D) -0.2675 is the exact percentage-point change in probability for a one-unit increase in MARGIN

The negative sign indicates direction (higher MARGIN lowers the attractiveness/probability of choosing ARM), but Probit coefficients are index-scale, not probability-scale; the actual probability effect requires multiplying by φ(Xβ), the marginal effect formula.

Rule: Probit coefficient signs indicate direction only; the probability-scale marginal effect requires β_j×φ(Xβ).

Formula Battle — method recognition (built from the Formula Book)

• A) L × i
• B) L / n
• C) L × i / [1 − (1+i)^(−n)]
• D) L(1+i)^n

Annuity formula. Correct method: L × i / [1 − (1+i)^(−n)].

Rule: PMT = L·i / [1 − (1+i)^(−n)]