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16 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

Property price today = 20,000,000. NOI in years 1-6 = 2.00, 2.05, 2.10, 2.15, 2.20, 2.25 million; terminal value in year 6 = 25.0 million; required return = 12%. Find NPV.

→ B) 1,335,101

Canvas Example MCQ — Q2

  • A) -12,470,597
  • B) 1,335,101
  • C) 17,750,000
  • D) 5,526,164
  • E) None of A-D is close; write the correct answer.

Discount each NOI. In year 6 discount NOI6 + terminal value = 27.25m. Present value of all future cash flows = 21,335,101. NPV = 21,335,101 - 20,000,000 = 1,335,101.

Rule: Final-year cash flow = operating cash flow + terminal value.

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.

DCF / NPV / IRRE-levelVery high priority

Suppose that the net present value (NPV) of an investment is zero. Then the internal rate of return (IRR) of this investment must be_____

→ B) equal to the discount rate (required rate of return).

Old Exam — Oct 2023, Q1

  • A) zero %.
  • B) equal to the discount rate (required rate of return).
  • C) higher than the discount rate (required rate of return).
  • D) lower than the discount rate (required rate of return).
  • E) None of the above (A, B, C, D) is close to be correct.

IRR is defined as the discount rate that makes NPV=0. If NPV is zero at the stated required return, IRR equals that return -> 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 / IRRE-levelVery high priority

You are a rich student as usual, and therefore you will invest a small portion of your huge wealth in a Drisky office building in downtown San Fransico. You assume that the office building will generate annual cash flows forever (in perpetuity). One year from now, your investment will pay you $2 million. You also assume that the yearly payments will grow with a nice constant annual growth rate in perpetuity of 3% (g = 3%). The present value of the growing cash flow is $25 000 000. You assume that the risk- free rate of return (the risk-free discount rate) is 6%. Then the risk premia must be_____.

→ B) 5.00 %

Old Exam — Oct 2023, Q2

  • A) 5.24 %
  • B) 5.00 %
  • C) 11.24 %.
  • D) 11.00 %.
  • E) None of the above (A, B, C, D) is close to be correct.

Growing perpetuity: PV=CF1/(r-g). 25=2/(r-g) -> r-g=8%. Total required return r=11%. Risk premium =11%-6%=5% -> 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 / IRRE-levelVery high priority

Suppose that the net present value (NPV) of an investment is positive. Then the internal rate of return (IRR) of this investment must be_____

→ C) higher than the discount rate (required rate of return).

Old Exam — Dec 2022, Q1

  • A) zero %.
  • B) equal to the discount rate (required rate of return).
  • C) higher than the discount rate (required rate of return).
  • D) lower than the discount rate (required rate of return).
  • E) None of the above (A, B, C, D) is close to be correct.

NPV is evaluated at the required return r. If NPV(r)>0 for a conventional investment, the zero-NPV rate lies above r. Therefore IRR>r.

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

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

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 $2 480 748. New information: the nominal risk-free rate is 8%, inflation is 2% and the risk premium is 3%. By what percentage does the property value change when you switch to the new required return? (You may not need all the information.)

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

Old Exam — Dec 2022, Q4 · adapted

Given: Original required return 12%. New: nominal risk-free 8%, inflation 2%, risk premium 3%.

  • A) The percentage change is 4.43%.
  • B) The percentage change is ‒4.43%.
  • C) The percentage change is 4.33%.
  • D) The percentage change is ‒4.15%.
  • E) None of the above (A, B, C, D) is close to be correct.

Nominal risk-free rate is already nominal, so new required return = 8%+3%=11% (do not subtract inflation). Revalue the same cash flows at 11%: about $2.639m. Change=(2.639/2.481)-1≈+6.36% -> 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

Suppose that the net present value (NPV) of an investment is negative. Then the internal rate of return (IRR) of this investment must be_____

→ D) lower than the discount rate.

Old Exam — Oct 2022, Q1

  • A) zero %.
  • B) equal to the discount rate.
  • C) higher than the discount rate.
  • D) lower than the discount rate.
  • E) None of the above (A, B, C, D) is close to be correct.

For a conventional investment, NPV(r)<0 means the discount rate is above the zero-NPV rate, so IRR<required return.

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 $35 000 for the next 50 year. Luckily, in year 10, year 20, year 30, year 40, and year 50, your investment also pays you $800 000 (in addition to the $35 000). As usual all payments occur at the end of each year. Your required rate of return (the discount rate) is 2.5%. The present value of the cash flow is_______.

→ B) $3 017 948.8

Old Exam — Dec 2021 re-exam, Q1

  • A) $5 750 000.0
  • B) $3 017 948.8
  • C) $2 671 111.2
  • D) $5 356 751.4
  • E) None of the above (A, B, C, D) is close to be correct.

PV=35,000 annuity for 50 years at 2.5% + 800,000 discounted from years 10,20,30,40,50 ≈ $3,017,948.8 -> 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 / IRRE-levelVery high priority

You assume that your new investment will generate annual cash flow forever (in perpetuity). Today (now), your investment will pay you $1 million. Thereafter, you assume that the yearly payments grow with an incredible (according to you) constant annual growth rate in perpetuity. Your required rate of return (the discount rate) is 15%. The present value of the growing cash flow is $45 000 000. Then the constant annual growth rate must be_____.

→ A) 12.500 %

Old Exam — Dec 2021 re-exam, Q2

  • A) 12.500 %
  • B) 12.778 %
  • C) 2.500 %.
  • D) 13.5444 %.
  • E) None of the above (A, B, C, D) is close to be correct.

Payment today is 1m, then growing stream starts next year. 45 = 1 + [1(1+g)/(0.15-g)]. Solving gives g=12.5% -> A.

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

You assume that your new investment will generate annual cash flow forever (in perpetuity). Next year (year one) your investment will pay you $5 million. Thereafter, you assume that the yearly payments grow with an incredible (according to you) constant annual growth rate in perpetuity. Your required rate of return (the discount rate) is 12%. The present value of the growing cash flow is $62 500 000. Then the constant annual growth rate must be_____.

→ C) 4 %.

Old Exam — Oct 2021, Q2

  • A) 6 %
  • B) 5 %
  • C) 4 %.
  • D) 3 %.
  • E) None of the above (A, B, C, D) is close to be correct.

Growing perpetuity PV=CF1/(r-g): 62.5=5/(0.12-g) -> 0.12-g=0.08 -> g=4% -> C.

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 has Initial NOI = $100,000, with 3% constant annual NOI growth. Required return r = 12%, terminal cap rate = 9%, holding period = 5 years. Calculate the property value using a 5-year DCF where the Year 5 reversion value uses Year 6 NOI.

→ A) $1,111,111.11

Computer Lab 1 — Exercise 1, Base Case

  • A) $1,111,111.11
  • B) $1,288,082.30
  • C) $1,000,000.00
  • D) $833,333.33
  • E) $1,400,633.19

NOI grows at 3%/yr: NOI1=100,000...NOI6=115,927.41. Reversion value = NOI6/0.09 = 1,288,082.30. Year5 total CF = NOI5+RV5 = 1,400,633.19. Discounting all CFs at 12% and summing PVs gives Property Value = $1,111,111.11.

Rule: DCF property value = sum of PV(NOI1..5) + PV(reversion), where reversion uses NOI of the year AFTER sale divided by terminal cap rate.

DCF / NPV / IRRE-levelVery high priority

Using the Base Case DCF (Initial NOI $100,000, 3% growth, r=12%, terminal cap rate 9%, 5-year hold), the property value is $1,111,111.11. What is the going-in cap rate?

→ B) 9.00%

Computer Lab 1 — Exercise 1, Base Case

  • A) 12.00%
  • B) 9.00%
  • C) 9.48%
  • D) 8.57%

Going-in cap rate = NOI1 / Property Value = 100,000 / 1,111,111.11 = 9.00%.

Rule: Going-in cap rate = Year 1 NOI ÷ Property Value.

DCF / NPV / IRRD–A levelVery high priority

Case A (excess supply): Initial NOI = $100,000 with 0% growth in Years 1-3, then 3% growth thereafter. Required return = 12%, terminal cap rate = 9%, 5-year hold. NOI6 = $109,272.70. What is the property value?

→ A) $1,054,776.08

Computer Lab 1 — Exercise 1, Case A

  • A) $1,054,776.08
  • B) $1,111,111.11
  • C) $1,214,141.11
  • D) $1,166,992.19

RV5 = NOI6/0.09 = 109,272.70/0.09 = 1,214,141.11. Year5 total CF = 106,090+1,214,141.11=1,320,231.11. Discount all years 1-5 at 12%: PVs are 89,285.71+79,719.39+71,178.02+65,458.36+749,134.59 = $1,054,776.08.

Rule: Flat short-term NOI growth before resuming long-term growth lowers both NOI and reversion value, reducing property value versus the Base Case.

DCF / NPV / IRRE-levelVery high priority

In a 5-year DCF property valuation where the asset is sold at the end of Year 5, which NOI figure should be used to calculate the reversion (sale) value via the terminal cap rate?

→ A) Year 6 NOI (the NOI expected in the year after sale)

Computer Lab 1 — Exercise 1 concept

  • A) Year 6 NOI (the NOI expected in the year after sale)
  • B) Year 5 NOI (the NOI in the year of sale)
  • C) Year 1 NOI (initial NOI)
  • D) The average NOI over Years 1-5

Reversion value = NOI_(T+1)/terminal cap rate, because the buyer at the end of Year 5 is purchasing rights to future (Year 6 onward) income.

Rule: Terminal/reversion value always capitalizes the NEXT year's NOI (forward-looking), not the NOI in the sale year.