Search

25 questions · 5 formulas

NPV

NPV = −I₀ + CF1/(1+r) + CF2/(1+r)² + … + CFT/(1+r)^T

Trap: Forgetting −I₀. Decision: NPV > 0 → accept.

IRR

IRR = r such that NPV(r) = 0

Trap: Compare IRR with the required return, not 0%. For MCQs, test the alternatives.

Solve for sale price

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

Expected return (3 scenarios)

E[R] = p1·R1 + p2·R2 + p3·R3

Trap: Don't take a simple average.

Variance & SD

Var = p1(R1−E)² + p2(R2−E)² + p3(R3−E)² σ = √Var

Trap: Don't divide by n — probabilities do the weighting.

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

If an investment has NPV = 0 at a particular discount rate, what must the IRR be?

→ B) equal to the discount rate.

Canvas Example MCQ — Q1

A) zero %.
B) equal to the discount rate.
C) higher than the discount rate.
D) lower than the discount rate.

By definition, IRR is the rate that makes NPV exactly zero. If NPV is zero at 12%, then IRR = 12%.

Rule: NPV(r)=0 -> r is the IRR.

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

Required return is 10% and project IRR is 5%. Invest or not?

→ C) Because the IRR is lower than the required return, you should not invest.

Canvas Example MCQ — Q3

A) Because the IRR is higher than 0%, you should invest.
B) Because the IRR is lower than the required return, you should invest.
C) Because the IRR is lower than the required return, you should not invest.
D) The NPV of this project is positive.
E) None of A-D is close; write the correct answer.

IRR 5% is below the required 10%. Equivalently, NPV at the 10% required return is negative for a conventional project.

Rule: Accept when IRR > required return; reject when IRR < required return.

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) 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 $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 / 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.

Scenario RiskE-levelVery high priority

The expected IRR is closest to______.

→ B) 6.83 %

Old Exam — Dec 2022, Q6

Given: Scenario setup: purchase price $45m; 15-year holding period; Year-1 NOI $2.5m; growth rates -0.5%, +1.5%, +2.5%; exit value at Year 15 uses Year-16 NOI / scenario exit cap (6%, 5.5%, 5%); probabilities 28%,45%,27%.

A) 6.79 %
B) 6.83 %
C) 7.05 %
D) 7.10 %
E) None of the above (A, B, C, D) is close to be correct.

Method: compute the return/NPV separately in every scenario, then E[X]=sum p_s X_s. Risk: SD=sqrt(sum p_s(X_s-E[X])^2). Source correct answer: B) 6.83 %.

Rule: Probabilities sum to 1. Expected value is probability-weighted; variance uses squared deviations around the expected value.

DCF / NPV / IRRD–A levelVery high priority

Centre Point unlevered cash flows: CF0=-$885,000, CF1=$89,100, CF2=$91,773, CF3=$94,526.19, CF4=$97,361.98, CF5=$1,074,896.04 (NOI5+NSP5). At a required unlevered return of 10%, what is the NPV, and should the investor invest?

→ A) NPV ≈ +$76,789.79 → invest

Computer Lab 1 — Exercise 2A, Question 3

A) NPV ≈ +$76,789.79 → invest
B) NPV ≈ -$76,789.79 → reject
C) NPV ≈ +$174,896.04 → invest
D) NPV ≈ $0 → indifferent

Discounting at 10%: PV1=81,000, PV2=75,845.45, PV3=71,018.93, PV4=66,499.54, PV5=667,425.87. Sum=1,038,789.79−885,000=+$76,789.79. Positive NPV → invest.

Rule: NPV > 0 at the required return → invest; NPV < 0 → reject.

Returns & CompoundingD–A levelVery high priority

For the Centre Point unlevered cash flows (CF0=-$885,000; CF1-5 as given), NPV at 12% = +$6,796.50 and NPV at 13% = -$25,642.90. Using linear interpolation, approximate the unlevered IRR, and compare it against the required unlevered return of 10%.

→ A) ≈12.21%; exceeds required return → invest

Computer Lab 1 — Exercise 2A, Question 5

A) ≈12.21%; exceeds required return → invest
B) ≈12.21%; below required return → reject
C) ≈10.00%; exactly meets the required return
D) ≈13.00%; exceeds required return → invest

IRR ≈ 12% + [6,796.50/(6,796.50+25,642.90)]×1% = 12% + 0.21% = 12.21%. Since 12.21% > 10% required unlevered return, invest.

Rule: Interpolation: IRR ≈ r_L + [NPV_L/(NPV_L − NPV_H)]×(r_H − r_L).

Returns & CompoundingE-levelVery high priority

By definition, what is the NPV of a project when the discount rate used equals the project's IRR?

→ A) Zero

Computer Lab 1 — Exercise 2A, Question 7

A) Zero
B) Equal to the initial investment
C) Always positive
D) Equal to the sum of undiscounted cash flows

IRR is defined as the discount rate that sets NPV = 0, so plugging IRR back into the NPV formula always yields NPV = 0.

Rule: IRR is the rate r such that NPV(r) = 0 — a definitional identity.

Mortgage & LeverageD–A levelHigh priority

Centre Point levered (equity) cash flows: CF0=-$241,162.50; CF1=$30,655.65; CF2=$33,328.65; CF3=$36,081.84; CF4=$38,917.63; CF5=$385,425.51. At a required equity return of 16%, calculate the levered NPV and decide.

→ A) NPV ≈ +$38,149.47 → invest

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).

Mortgage & LeverageD–A levelHigh priority

For Centre Point's levered cash flows, NPV at 20% = +$2,071.48 and NPV at 21% = -$5,942.50. Interpolating, the levered IRR is approximately 20.25%. Given a required equity return of 16%, what is the investment decision?

→ A) Invest — 20.25% exceeds the required equity return of 16%

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.

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

In the Pessimistic scenario with $1,500,000 debt financing (10%, 15yr), the levered equity cash flows are CF0=-500,000, CF1=6,571.08, CF2=2,571.08, CF3=-1,348.92, CF4=-5,190.52, CF5=571,295.42. NPV at 2.8% ≈ +$553 and at 2.9% ≈ -$1,850. What is the approximate levered IRR?

→ A) ≈2.82%

Computer Lab 3 — Problem 4A

A) ≈2.82%
B) ≈2.85%
C) ≈7.93% (same as the unlevered pessimistic IRR)
D) ≈2.90%

IRR ≈ 2.8% + [553/(553+1,850)]×0.1% = 2.8%+0.023% ≈ 2.82%.

Rule: Leverage can sharply reduce equity IRR in weak scenarios because fixed debt service erodes a larger share of a smaller NOI base.