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15 questions · 2 formulas

Terminal value (sale price)

TV_T = NOI(T+1) / exit cap

Trap: Use NOI in year T+1, not year T. Add TV to year-T cash flow.

Solve for sale price

PV(sale) = Price + target NPV − PV(NOIs); Sale = PV(sale) × (1+r)^T

DCF / NPV / IRRE-level★ High priority — official Canvas example

Price today = $250m. Year-1 NOI = $12.8m, NOI grows 3% annually through year 5, discount rate = 9%. What year-5 sale price gives NPV = $5m?

→ C) $311,421,134

Canvas Example MCQ — Q4

  • A) $255,873,456
  • B) $357,905,558
  • C) $311,421,134
  • D) $329,447,946
  • E) None of A-D is close; write the correct answer.

NOI1-5 = 12.800, 13.184, 13.580, 13.987, 14.407m. PV of those NOIs at 9% is about 52.598m. Required PV of sale = 250 + 5 - 52.598 = 202.402m. Sale price = 202.402 x 1.09^5 = 311.421m.

Rule: When solving for a terminal value, first isolate its required present value, then compound it forward to the sale date.

Regression BasicsE-level★ High priority — official Canvas example

Actual sale price = $2,800,000; predicted price = $1,207,607. Residual?

→ C) $1,592,393

Canvas Example MCQ — Q52

  • A) $4,392,393
  • B) $1,207,607
  • C) $1,592,393
  • D) $3,602,607
  • E) None of A-D is close; write the correct answer.

Residual = actual - predicted = 2,800,000 - 1,207,607 = 1,592,393.

Rule: Positive residual means the actual outcome was above the fitted value.

Normal DistributionD–A level★ High priority — official Canvas example

In ln(price) on ln(sqft), coefficient on ln(sqft) = 0.7244. Interpret.

→ A) A 1% increase in house size is associated with about a 0.7244% increase in sales price.

Canvas Example MCQ — Q54

  • A) A 1% increase in house size is associated with about a 0.7244% increase in sales price.
  • B) A 1,000 sqft increase in size increases sales price by about $7,244.
  • C) A 1% increase in size increases sales price by about $72.44.
  • D) A 10% increase in size increases sales price by about 0.8925%.
  • E) None of A-D is close; write the correct answer.

This is a log-log specification. The slope is elasticity directly.

Rule: ln(Y) on ln(X): coefficient = elasticity.

DCF / NPV / IRRE-levelVery high priority

Suppose that you want to buy a property today with expected future net cash flows (Net Operating Income) of $250 000 per year for the next five years (starting one year from now). After that, you expect the net cash flow to step up 20 percent, to $300 000, for the following five years. If you expect to sell the property 10 years from now, for a price 10 times the net cash flow at that time, what is the value of your property if the required rate of return is 12%?

→ E) None of the above (A, B, C, D) is close to be correct.

Old Exam — Dec 2022, Q2

  • A) $2 750 000
  • B) $1 214 100
  • C) $ 1 514 828
  • D) $1 412 556
  • E) None of the above (A, B, C, D) is close to be correct.

PV = sum[250,000/(1.12)^t, t=1..5] + sum[300,000/(1.12)^t, t=6..10] + 3,000,000/(1.12)^10 = about $2,480,748. The listed A-D values miss it, so E.

Rule: Recognize cash-flow timing first. Discount each cash flow; terminal/sale proceeds are added in the sale year. IRR is the rate that sets NPV=0.

DCF / NPV / IRRE-levelVery high priority

A property is expected to give NOI of $250 000 per year in years 1–5 and $300 000 per year in years 6–10, and is sold at the end of year 10 for 10× that year's NOI. At a 12% required return its value is about $2.481m. The seller asks $1 500 000 today. Should you do the deal according to the IRR rule?

→ A) Yes, because the IRR is higher than the required rate of return.

Old Exam — Dec 2022, Q3 · adapted

Given: Value at 12% ≈ $2 480 748; asking price $1 500 000.

  • A) Yes, because the IRR is higher than the required rate of return.
  • B) Yes, because the IRR is lower than the required rate of return.
  • C) No, because the IRR is higher than the required rate of return.
  • D) No, because the IRR is lower than the required rate of return.
  • E) None of the above (A, B, C, D) is close to be correct.

The investment value at the 12% required return is about $2.481m, above the $1.5m asking price, so NPV>0. Equivalent IRR rule: IRR>required return -> accept.

Rule: Recognize cash-flow timing first. Discount each cash flow; terminal/sale proceeds are added in the sale year. IRR is the rate that sets NPV=0.

DCF / NPV / IRRE-levelVery high priority

Your investment pays you annual cash flows of $200 000 for the next 10 years. Year 10, your investment also pays you $850 000 (in addition to the $200 000). As usual all payments occur at the end of each year. How much must you pay for the investment today, if you want the IRR to be 15.0 %?

→ B) $1 214 100

Old Exam — Oct 2022, Q2

  • A) $2 011 200
  • B) $1 214 100
  • C) $2 358 300
  • D) $1 252 500
  • E) None of the above (A, B, C, D) is close to be correct.

Price that gives IRR=15% equals PV at 15%: 200,000 annuity for 10 years + 850,000 discounted from Year 10 ≈ $1.214m -> B.

Rule: Recognize cash-flow timing first. Discount each cash flow; terminal/sale proceeds are added in the sale year. IRR is the rate that sets NPV=0.

DCF / NPV / IRRD–A levelVery high priority

Centre Point: Year 6 NOI (projected at 3% annual growth from Year-1 NOI of $89,100) is $103,291.32. Terminal cap rate = 10%, selling expenses = $58,300. What is the Net Sale Proceeds (NSP) in Year 5?

→ A) $974,613.20

Computer Lab 1 — Exercise 2A, Question 2

  • A) $974,613.20
  • B) $1,032,913.20
  • C) $891,000.00
  • D) $1,091,213.20

Sale Price = NOI6/0.10 = 103,291.32/0.10 = 1,032,913.20. NSP = Sale Price − Selling Expenses = 1,032,913.20 − 58,300 = $974,613.20.

Rule: NSP = Sale Price − Selling Expenses, and Sale Price = NOI_(T+1) / terminal cap rate.

Scenario RiskD–A levelVery high priority

A $2,000,000 property, pessimistic scenario: Year-1 NOI=$200,000 declining 2%/year, sale price in Year5=$1.8m (cash flows: -2,000,000; 200,000; 196,000; 192,080; 188,238.40; 1,984,473.63). NPV at 7% = +$73,412.82 and NPV at 8% = -$5,336.86. What is the approximate IRR?

→ A) ≈7.93%

Computer Lab 1 — Exercise 3 Problem 3

  • A) ≈7.93%
  • B) ≈7.07%
  • C) ≈8.00%
  • D) ≈6.84%

IRR ≈ 7% + [73,412.82/(73,412.82+5,336.86)]×1% = 7% + 0.93% = 7.93%.

Rule: Linear interpolation between a low rate with positive NPV and a high rate with negative NPV approximates IRR.

Scenario RiskE-levelVery high priority

A $2,000,000 property has Most Likely scenario: NOI = $200,000 flat every year, sale price after 5 years = $2,000,000 (equal to purchase price). Why is the IRR exactly 10.00% in this scenario without needing trial-and-error?

→ A) Because annual NOI (10% of price) plus an unchanged resale price means a 10% discount rate makes NPV exactly zero

Computer Lab 3 — Problem 3 (Most likely)

  • A) Because annual NOI (10% of price) plus an unchanged resale price means a 10% discount rate makes NPV exactly zero
  • B) Because IRR always equals NOI/Purchase price regardless of other cash flows
  • C) Because the sale price must always equal the purchase price for any IRR to exist
  • D) Because flat NOI implies zero IRR

NOI/Purchase price = 200,000/2,000,000 = 10%, exactly matching a constant 'coupon-like' yield, and with the exit value unchanged, the investment behaves like a perpetual-income bond returning exactly 10% — so NPV(10%)=0 trivially.

Rule: When periodic income is a constant percentage of invested capital and the capital value is returned unchanged at the end, the IRR equals that constant income yield.

Scenario RiskD–A levelVery high priority

Optimistic scenario for a $2,000,000 property: Year-1 NOI=$200,000 growing 3%/yr, sale price Year5=$2,200,000. Total Year-5 cash flow (NOI5+sale) = $2,425,101.76. NPV at 12.1% ≈ +$1,299 and at 12.2% ≈ -$6,142. What is the approximate IRR?

→ A) ≈12.12%

Computer Lab 3 — Problem 3 (Optimistic)

  • A) ≈12.12%
  • B) ≈12.17%
  • C) ≈12.20%
  • D) ≈12.10%

IRR ≈ 12.1% + [1,299/(1,299+6,142)]×0.1% = 12.1% + 0.017% ≈ 12.12%.

Rule: Linear interpolation between adjacent trial rates with opposite-signed NPV gives a close IRR approximation.

Scenario RiskD–A levelVery high priority

Comparing unlevered (E[IRR]=10.01%, σ=1.62%) to levered (E[IRR]=10.14%, σ=5.52%) for the same property, has the $1.5m loan increased risk, and why?

→ A) Yes — σ rises from 1.62% to 5.52% because fixed debt service makes equity a residual claim, amplifying the effect of NOI/sale-price changes on equity returns

Computer Lab 3 — Problem 4B

  • A) Yes — σ rises from 1.62% to 5.52% because fixed debt service makes equity a residual claim, amplifying the effect of NOI/sale-price changes on equity returns
  • B) No — expected return only rose slightly (10.01%→10.14%), so risk is essentially unchanged
  • C) No — leverage reduces risk by diversifying the capital structure
  • D) Risk is unchanged because E[IRR]/σ actually increased with leverage

σ more than triples (1.62%→5.52%). Because debt service is fixed, all variability in property performance (NOI, sale price) flows through to the smaller equity base, magnifying the percentage swings in equity returns.

Rule: Leverage magnifies equity-return volatility because debt holders are paid first (fixed claim) while equity absorbs all residual variability.

DCF / NPV / IRRD–A levelVery high priority

A property is purchased for $100,000,000. NOI in Year 1 is $7,000,000, growing at 3% per year, and the property value grows at 4% per year. If held for 5 years, what is the approximate sale price at the end of Year 5?

→ A) ≈$121,665,290

Computer Lab 5 — Exercise 2, Setup

  • A) ≈$121,665,290
  • B) ≈$120,000,000
  • C) ≈$108,243,216
  • D) ≈$135,000,000

Sale price = 100,000,000 × (1.04)^5 ≈ 121,665,290.24, applying 4% compound growth over 5 years to the purchase price.

Rule: Compound the purchase price forward at the property value growth rate to estimate a future sale price.

DCF / NPV / IRRD–A levelVery high priority

A property's value grows from $100,000,000 at purchase to $121,665,290.24 at sale in Year 5 (a participating mortgage with 20% lender participation in appreciation). What is the lender's participation payment at sale, using Part = share × max(Sale price − Purchase price, 0)?

→ A) ≈$4,333,058

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.

DCF / NPV / IRRD–A levelVery high priority

With a $100,000,000 purchase financed by a $70,000,000 interest-only loan at 5.5%, lender participation of $4,333,058 in appreciation at sale, and a Year 5 sale price of $121,665,290.24 plus NOI of $7,878,561.67, the levered IRR is 19.52% versus an unlevered IRR of 10.88%. What does this indicate?

→ A) Positive financial leverage, since the levered IRR exceeds the unlevered IRR

Computer Lab 5 — Exercise 2, Task 1

  • A) Positive financial leverage, since the levered IRR exceeds the unlevered IRR
  • B) Negative financial leverage, since borrowing reduced the return
  • C) The loan had no effect on investor returns
  • D) The participation feature eliminated all leverage benefit

Because 19.52% (levered) > 10.88% (unlevered), using debt increased the equity investor's return — this is positive financial leverage, meaning the asset's unlevered return exceeds the cost of debt.

Rule: Positive leverage occurs when the levered IRR to equity exceeds the unlevered IRR; this happens when borrowing cost is below the asset's return.

DCF / NPV / IRRE-levelVery high priority

A property is sold at the end of Year 5 and the exit cap rate is 6%. Which formula gives the sale price?

→ B) NOI6 / cap rate

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

  • A) NOI5 / cap rate
  • B) NOI6 / cap rate
  • C) Price × cap rate
  • D) NOI1 / cap rate

Sale at end of year T uses next year's NOI. Correct method: NOI6 / cap rate.

Rule: TV5 = NOI6 / exit cap