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Old Exam — Dec 2021 re-exam, Q8

Given: Historical HPI dataset question; keep the concept of comparing standard deviations.

• A) Slovenia with standard deviation of 24.97 %
• B) Slovenia with standard deviation of 58.79 %
• C) Croatia with standard deviation of 35.20 %
• D) Croatia with standard deviation of 24.05 %
• E) None of the above (A, B, C, D) is close to be correct.

Historical external-data result: D) Croatia with standard deviation of 24.05 %. Current paper-exam use: if the needed index levels/returns are supplied, apply the relevant return, compounding or correlation formula by hand; do not depend on the old spreadsheet.

Rule: The old exam supplied a spreadsheet/software. For the 2026 paper exam, expect the necessary values/table/output to be printed. Use the method, not the historical data file.

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) 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 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 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 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 3 — Problem 2 recap

• A) Direct scenario-by-scenario (state-by-state) comparison / first-order stochastic dominance — it requires no assumption about risk preference
• B) The coefficient of variation (σ/E[R]), since it always gives the definitive ranking
• C) Standard deviation alone, since lower risk always wins
• D) Expected return alone, since higher return always wins

If an investment's payoff is ≥ in every scenario (state-by-state), it dominates regardless of any risk-preference assumption — this is a stronger and more general conclusion than CV, σ, or E[R] comparisons alone, each of which implicitly assumes something about risk tolerance or ignores risk entirely.

Rule: State-by-state (first-order stochastic) dominance, when it exists, is the strongest possible ranking tool because it holds for all risk preferences.

Computer Lab 6 — Exercise 2, (a)

• A) ≈40.13%
• B) ≈50.00%
• C) ≈25.00%
• D) ≈59.87%

z = (0−0.01)/0.04 = −0.25. P(R<0) = Φ(−0.25) ≈ 0.4013 = 40.13%.

Rule: Standardize with z=(x−μ)/σ, then use Φ(z) for a left-tail probability.

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]