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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.

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.

Canvas Example MCQ — Q8

• A) 0.10%
• B) 2.55%
• C) 10.00%
• D) 15%
• E) None of A-D is close; write the correct answer.

E[R] = 0.20(15%) + 0.60(10%) + 0.20(5%) = 3% + 6% + 1% = 10%.

Rule: Expected return is a probability-weighted average.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Old Exam — Oct 2023, Q7 · adapted

Given: Severe recession: p = 10%, IRR = −4.75% Moderate recession: p = 30%, IRR = 0.55% Baseline: p = 35%, IRR = 6.53% Moderate expansion: p = 20%, IRR = 9.80% Strong boom: p = 5%, IRR = 18.94%

• A) 6.22%
• B) 4.53 %
• C) 4.88%
• D) 3.55 %
• E) None of the above (A, B, C, D) is close to be correct.

E[IRR] = 0.10(−4.75) + 0.30(0.55) + 0.35(6.53) + 0.20(9.80) + 0.05(18.94) = −0.475 + 0.165 + 2.286 + 1.960 + 0.947 ≈ 4.88% → C.

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

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.

Old Exam — Dec 2021 re-exam, Q6

• A) sensitive to changes in the price of that good.
• B) not sensitive to changes in the price of that good.
• C) irrational.
• D) not interested in that good.

A vertical demand curve has zero price elasticity: quantity demanded does not respond to price -> B.

Rule: Midpoint elasticity uses percentage changes relative to midpoint values; elasticity is unit-free, not a percent itself.

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).

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.

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 2C

• A) ≈20.03%
• B) ≈19.86% (simple average)
• C) ≈25.95%
• D) ≈5.83%

E[IRR] = 0.05(5.89%)+0.20(13.74%)+0.50(20.25%)+0.20(25.95%)+0.05(33.50%) = 20.03%, a probability-weighted average, not a simple average.

Rule: Expected value under scenario analysis = Σ (probability × outcome), not a simple unweighted average.

Computer Lab 1 — Exercise 2C

• A) Leverage increases expected return but increases risk proportionally even more, lowering the risk-adjusted return ratio
• B) Leverage increases expected return with no effect on risk
• C) Leverage decreases both expected return and risk
• D) Leverage has no effect on return or risk, only on cash flow timing

Both mean IRR and σ rise with LTV, but σ rises faster (1.98%→10.21%, a ~5.2x increase vs. mean IRR's ~2.4x increase), so the Mean/Std.Dev ratio falls from 6.18x to 2.91x — risk grows faster than return.

Rule: Financial leverage magnifies both expected equity returns and the variability (risk) of those returns; risk generally grows faster than return as leverage increases.

Computer Lab 1 — Exercise 3 Problem 2

• A) Investment I (CV=0.316) is lower risk per unit of return than Investment II (CV=0.350)
• B) Investment II is lower risk per unit of return because it has a lower standard deviation
• C) Both investments have identical CV
• D) Investment II is better because its expected IRR (14%) is higher

E[IRR]_I=10%, σ_I=3.16%, CV_I=0.316. E[IRR]_II=14%, σ_II=4.90%, CV_II=0.350. Lower CV means less risk per unit of expected return, so Investment I has the better risk-adjusted profile by this measure, even though II has a higher expected IRR.

Rule: Coefficient of Variation (σ/E[R]) measures risk per unit of expected return; lower CV means better reward-to-risk (though this does not mean universally 'better').

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.

Computer Lab 1 — Exercise 3 Problem 3

• A) E[IRR]≈10.01%, σ≈1.62%
• B) E[IRR]≈10.02%, σ≈2.19%
• C) E[IRR]≈10.68%, σ≈1.62%
• D) E[IRR]≈10.01%, σ≈4.90%

E[IRR]=0.30(7.93%)+0.40(10.00%)+0.30(12.12%)=10.01%. Variance=Σp(IRR−E[IRR])²≈0.00026296, σ=√0.00026296≈1.62%.

Rule: Standard deviation = √[Σ p_i × (R_i − E[R])²] using scenario probabilities.

Computer Lab 1 — Exercise 3 Problem 3(d)

• A) The alternative has higher expected return, but the lab property has much lower risk per unit of return (lower CV)
• B) The lab property dominates on every metric: higher return and lower risk
• C) The alternative dominates on every metric: higher return and lower risk
• D) They are financially identical because CV differs by less than 0.2

The alternative's E[R]=12% exceeds 10.01%, but its CV (0.333) is roughly double the lab property's CV (0.162), meaning it carries much more risk per unit of expected return.

Rule: When comparing investments with different expected returns AND different risks, use CV = σ/E[R] to assess risk per unit of return.

Computer Lab 1 — Exercise 3 Problem 4

• A) Yes — standard deviation rises from 1.62% to 5.52% while expected return rises only slightly, because fixed debt service makes equity a residual claim
• B) No — leverage has almost no effect since expected return barely changed
• C) No — standard deviation actually falls with leverage because debt diversifies risk
• D) Cannot be determined without knowing the loan's credit rating

Fixed debt service must be paid regardless of NOI outcome, so equity cash flows (the residual) swing proportionally more with property performance. Standard deviation more than triples (1.62%→5.52%) while expected return only rises slightly (10.01%→10.14%).

Rule: Financial leverage magnifies equity risk because debt service is a fixed prior claim; equity absorbs the full variability on a proportionally smaller base.

Computer Lab 2 — Exercise 2A, Scenario 1

• A) 28.47%
• B) 35.00%
• C) 70.00%
• D) 3.50%

-100+350/(1+IRR)^5=0 → (1+IRR)^5=3.5 → IRR=3.5^(1/5)-1=28.47%.

Rule: For a single lump-sum return after n years: IRR = (ending value/beginning value)^(1/n) − 1.

Computer Lab 2 — Exercise 2A

• A) CAGR and Multiple depend only on beginning and ending wealth over the whole period, while IRR also accounts for the timing of each interim cash flow
• B) The calculations contain an error — equal multiples must always produce equal IRRs
• C) IRR is independent of timing; the difference must come from rounding
• D) Multiple and CAGR used a different total cash amount in each scenario

Multiple = total cash returned/initial investment = 350/100 = 3.5x and CAGR = 3.5^(1/5)-1 = 28.47% for all three, since total dollars and duration are identical. IRR, however, is sensitive to WHEN cash is received: receiving money earlier allows it to be 'reinvested' implicitly at the IRR, raising the computed IRR.

Rule: Multiple and CAGR are driven only by beginning/ending wealth; IRR is additionally sensitive to the timing of interim cash flows.

Computer Lab 3 — Problem 2

• A) 10.00%
• B) 10.00% is wrong, it should be a simple average of 10%
• C) 11.67%
• D) 15.00%

E[IRR]=0.20(15%)+0.60(10%)+0.20(5%)=3%+6%+1%=10.00%.

Rule: Expected value = Σ (probability × outcome) across all scenarios.

Computer Lab 3 — Problem 2

• A) ≈3.16%
• B) ≈5.00%
• C) ≈1.00%
• D) ≈10.00%

Variance = 0.20×(0.15−0.10)²+0.60×(0.10−0.10)²+0.20×(0.05−0.10)² = 0.20×0.0025+0+0.20×0.0025 = 0.0010. σ=√0.0010=0.03162=3.16%.

Rule: σ = √[Σ p_i(R_i−E[R])²].

Computer Lab 3 — Problem 2

• A) Investment II first-order stochastically dominates Investment I; any investor who prefers more return to less should choose II despite its higher standard deviation
• B) Investment I dominates because it has lower standard deviation (3.16% vs 4.90%)
• C) Neither dominates since II has higher risk
• D) They are equally good since both have the same Pessimistic-scenario IRR (5%)

Because Investment II's IRR is strictly ≥ Investment I's IRR in every state of the world (20%>15%, 15%>10%, 5%=5%), II dominates I regardless of risk preferences — a rational investor would never choose I over II here.

Rule: If one investment's payoff is at least as good in every scenario (and better in at least one), it first-order stochastically dominates — risk aversion is irrelevant to this comparison.

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.

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.

Computer Lab 3 — Problem 3(b)

• A) ≈10.01%
• B) ≈10.02%
• C) ≈10.68%
• D) ≈11.00%

E[IRR]=0.30(7.93%)+0.40(10.00%)+0.30(12.12%)=2.379%+4.00%+3.636%=10.0144%≈10.01%.

Rule: Probability-weighted expected value: Σ p_i × IRR_i.

Computer Lab 3 — Problem 3(d)

• A) The property is stronger by E[R]/σ (6.18 vs 3.00), but the alternative offers higher absolute expected return with more risk — the best choice depends on the investor's risk aversion
• B) The alternative is stronger on every metric, so it is unambiguously better
• C) The property is stronger on every metric, so it is unambiguously better
• D) They are equally good since both ratios round to about 3-6x

E[R]/σ favors the property (6.18 > 3.00), reflecting better return-per-unit-of-risk. However, the alternative has a higher absolute expected return (12% vs 10.01%), so a return-maximizing investor willing to bear more risk might prefer it. No single metric makes one universally dominant.

Rule: When neither investment dominates on both return and risk simultaneously, the final choice depends on the investor's degree of risk aversion.

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.

Computer Lab 3 — Problem 4A-B

• A) ≈10.14%
• B) ≈10.01%
• C) ≈10.42%
• D) ≈13.44%

E[IRRe]=0.30(2.82%)+0.40(10.42%)+0.30(17.07%)=0.846%+4.168%+5.121%=10.1356%≈10.14%.

Rule: Probability-weighted expected value across all scenarios: Σ p_i × IRR_i.

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.

Computer Lab 4 — Exercise 1, Task 5

• A) The property should offer a higher expected return/cap rate than the Treasury yield to compensate for added risk
• B) They should offer identical returns since both are low-risk
• C) The Treasury should offer a higher return because it is less liquid
• D) Property risk is irrelevant once a cap rate is quoted

Real estate carries vacancy, operating, regulatory, liquidity and asset-specific risks beyond those of government bonds, so investors typically require a risk premium—i.e., a cap rate/expected return above the Treasury yield.

Rule: Riskier assets require a return premium above the risk-free (Treasury) rate.

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.

Computer Lab 6 — Exercise 1, Task 1

• A) R² never decreases when more regressors are added, even if they add no real explanatory power, so it can reward unnecessary complexity
• B) R² always decreases when regressors are added
• C) R² is undefined for models with more than 3 regressors
• D) R² automatically accounts for overfitting

Ordinary R² is non-decreasing in the number of regressors by construction, so adding irrelevant variables can never lower it, making it a poor tool for comparing models of different sizes. Adjusted R² corrects for this with a penalty for extra regressors.

Rule: Use adjusted R², not plain R², to compare models with differing numbers of regressors.

Canvas Example MCQ — Q9

• A) 0.10%
• B) 5.00%
• C) 31.60%
• D) 3.16%
• E) None of the above is correct.

Mean = 10%. Variance = 0.20(15−10)² + 0.60(10−10)² + 0.20(5−10)² = 10. SD = √10 ≈ 3.16%.

Rule: Risk = √[Σ p(R − E[R])²]

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

• A) (IRR1 + IRR2 + IRR3) / 3
• B) p1·IRR1 + p2·IRR2 + p3·IRR3
• C) √[Σ p(IRR − E)²]
• D) The most likely scenario's IRR

Expected value is probability-weighted. Correct method: p1·IRR1 + p2·IRR2 + p3·IRR3.

Rule: E = p1x1 + p2x2 + p3x3