Corporate Finance · Quiz 2 · Jan 2025

**1. Distinguish debt and equity:** Debt normally promises interest/principal and has priority over equity in distributions. Equity is a residual ownership claim: owners receive what remains after contractual obligations. **2. Apply the distinction:** Ordinary creditors do not automatically share in the project’s residual profits as owners do. Their contractual payment can remain the same even if profits rise greatly. **3. Answer the negative wording:** The question asks for the incorrect statement, so select the statement that ordinary debt creditors share project profits. **Answer: C — Debt creditors share the profit earned by the concerned project..** **Convention note:** The standard debt/equity distinction is intended; special profit-participating or convertible contracts can add features beyond ordinary debt. A: **Incorrect:** This is true in the usual priority ordering, so it should not be selected as incorrect. B: **Incorrect:** Debt’s priority and contractual payoff generally make it less risky than the residual equity claim in the same project. C: **Correct:** This is the incorrect statement: ordinary debt promises contractual payments rather than a residual profit share. D: **Incorrect:** This is true for ordinary debt: lending alone does not confer equity ownership.

**1. Explain an interior optimum:** A consumer chooses the highest attainable indifference curve subject to a budget. With smooth preferences and positive consumption of both goods, the curve touches the budget line tangentially. **2. State the condition:** For goods x and y with prices Pₓ and Pᵧ, \[\mathrm{MRS}_{xy}=\frac{MU_x}{MU_y}=\frac{P_x}{P_y}.\] Marginal utility per rupee is equalized, not necessarily marginal utility itself. **3. Explain the budget:** Under locally nonsatiated preferences, leaving income unspent would permit an improvement, so the budget is exhausted. **Answer: C — The budget line is tangent to the indifference curve..** **Convention note:** Tangency assumes a smooth interior solution. Corner solutions or kinked indifference curves need not satisfy that geometric condition. A: **Incorrect:** At a smooth interior optimum, MRS equals the magnitude of the budget slope rather than being smaller. B: **Incorrect:** With the usual nonsatiation assumption, the entire budget is used. C: **Correct:** Tangency is the standard smooth interior consumer-optimum condition. D: **Incorrect:** Marginal utilities need not be equal unless the relevant prices are also equal.

**1. Describe a short sale:** A short seller borrows shares, sells them now, then buys replacement shares later to return to the lender. **2. Express the profit:** Ignoring costs, for n shares \[\Pi=n(S_0-S_T).\] A falling final price produces profit; a rising final price produces a loss. **3. Compare the bounds:** Stock prices cannot fall below zero, so the maximum gross profit is limited by the initial sale proceeds. There is no fixed upper bound on a stock price, so the potential short-sale loss is unbounded. **Answer: C — Unlimited loss potential.** A: **Incorrect:** Loss is not capped by the initial share price because the repurchase price can rise much further. B: **Incorrect:** Gross profit is capped when the repurchase price reaches zero. C: **Correct:** An arbitrarily high repurchase price produces an arbitrarily large loss. D: **Incorrect:** The position has substantial price and other risks.

**1. Define the feasible set:** It contains all portfolios permitted by the available assets and constraints. For each target mean return, the minimum-variance portfolio is the feasible portfolio with the smallest variance. **2. Identify the efficient branch:** On the upper branch of the minimum-variance boundary, no other feasible portfolio offers a higher mean at the same risk or lower risk at the same mean. This efficient frontier is part of the minimum-variance set. **3. Distinguish an investor’s choice:** The global minimum-variance portfolio is only the least-risk point. An investor’s preferred allocation depends on preferences and the investment opportunity set. **Answer: C — The efficient frontier is a subset of the minimum variance set..** **Convention note:** D can describe risky-only allocation choices with different risk aversion. The universally intended geometric statement among these choices is C; the common-risk-free-asset interpretation explains why D is not generally true. A: **Incorrect:** The inclusion is reversed: the minimum-variance set lies within the broader feasible set. B: **Incorrect:** Minimum variance alone does not make a portfolio optimal for every investor. C: **Correct:** The efficient frontier is the efficient portion of the minimum-variance boundary. D: **Incorrect:** With a common risk-free asset and the usual separation assumptions, investors can share the same tangency risky portfolio while varying their risk-free allocation.

**1. Write linear utility:** A linear increasing wealth utility has the form \(U(W)=aW+b\) with a > 0. **2. Take expectations:** \[E[U(W)]=aE[W]+b=U(E[W]).\] Thus the investor values a risky wealth distribution the same as its certain mean, rather than paying to avoid or seeking risk for its own sake. **3. Classify the attitude:** Zero curvature, \(U^{\prime\prime}(W)=0\), corresponds to risk neutrality. **Answer: A — The individual is risk neutral.** A: **Correct:** Linear increasing utility makes choices depend on expected wealth without a separate risk penalty. B: **Incorrect:** Risk aversion requires concave utility in this framework. C: **Incorrect:** Risk seeking corresponds to convex rather than linear utility. D: **Incorrect:** Risk neutrality is explicitly available and follows from the linear utility.

**1. Use the balance sheet identity:** Net worth or capital is assets minus liabilities: \[K=A-L.\] **2. Identify balance-sheet insolvency:** If liabilities exceed assets, net worth is negative. Assets are then insufficient to cover all obligations in aggregate, which is the intended insolvency condition. **Answer: B — His liabilities exceed his assets..** **Convention note:** This question uses balance-sheet insolvency. Inability to pay debts when due is a related cash-flow insolvency concept. A: **Incorrect:** Capital exceeding assets is not the stated balance-sheet insolvency test. B: **Correct:** L > A gives negative net worth and balance-sheet insolvency. C: **Incorrect:** A > L means positive net worth, not balance-sheet insolvency. D: **Incorrect:** Liabilities can exceed equity in a solvent leveraged firm; comparison with assets is needed.

**1. State the preference rule:** Sarah’s utility is \(U=2\mu-10\sigma^2\), where mean returns and standard deviations are expressed as decimals. Compare both means and variances rather than only their highest possible returns. **2. Evaluate A:** \(\mu_A=0.4(0.12)+0.6(0.18)=0.156\). Its variance is \[\sigma_A^2=0.4(0.12-0.156)^2+0.6(0.18-0.156)^2=0.000864.\] Hence \(U_A=2(0.156)-10(0.000864)=0.30336\). **3. Evaluate B:** \(\mu_B=0.5(0.28)+0.5(0.08)=0.18\), and \(\sigma_B^2=0.5(0.10)^2+0.5(-0.10)^2=0.01\). Thus \(U_B=0.36-0.10=0.26\). **4. Compare:** Since 0.30336 > 0.26, A has the higher stated utility despite its lower mean return. **Answer: A — Asset A.** A: **Correct:** A has the higher mean-variance utility after accounting for its much lower variance. B: **Incorrect:** B’s higher mean is outweighed by the variance penalty in this utility function. C: **Incorrect:** The computed utilities differ, so Sarah is not indifferent. D: **Incorrect:** Probabilities, returns and the complete preference function are provided.

**1. Interpret a fair bet:** With no discounting or separate risk premium specified, the actuarially fair price is the probability-weighted expected payout. **2. Calculate each contribution:** \[E[X]=0.20(100)+0.40(80)+0.40(60)=20+32+24=76.\] The probabilities sum to 1, so all outcomes have been included. **Answer: D — 76.** **Convention note:** Actuarial fairness uses expected payout. A particular risk-averse person’s maximum willingness to pay could differ; no such utility or time discount is supplied. A: **Incorrect:** 84 exceeds the expected payout and is not obtained from the stated probabilities. B: **Incorrect:** 48 omits part of the expected payout. C: **Incorrect:** 52 is not the probability-weighted sum. D: **Correct:** The three weighted contributions sum to 76.

**1. Compute the safe alternative:** Investing Rs 20,000 at 8% yields \(20000(1.08)=21600\) after one year. **2. Equalize expected repayment:** Let r be Ramesh’s promised rate. A risk-neutral lender requires \[0.75\,[20000(1+r)]+0.25(0)=21600.\] Hence \(1+r=1.08/0.75=1.44\), so r = 44%. **3. Isolate the default premium:** The premium above the safe rate is \[44\%-8\%=36\text{ percentage points}.\] The promised rate and its default premium are distinct quantities. **Answer: B — 36%.** A: **Incorrect:** 37% is not the difference between the required 44% promised rate and the safe 8% rate. B: **Correct:** The required promised rate exceeds the safe rate by 36 percentage points. C: **Incorrect:** 17% would not compensate for the stated 25% zero-recovery default risk. D: **Incorrect:** 10% is too small to equalize expected repayment with the government bond.

**1. Identify the leverage convention:** The asset-to-equity leverage multiple is total assets divided by capital/equity. **2. Substitute the bank values:** \[\text{leverage}=\frac{1500}{750}=2.\] The identity \(1500=750+750\) confirms that the stated assets equal liabilities plus capital. **Answer: C — 2.** **Convention note:** The question uses assets/equity. A regulatory capital-to-exposure leverage ratio uses the reciprocal-style convention; it should not be confused with this multiple. A: **Incorrect:** A multiple of 4 does not follow from the stated assets and equity. B: **Incorrect:** A multiple of 3 does not follow from the stated assets and equity. C: **Correct:** The bank holds two units of assets for each unit of capital. D: **Incorrect:** 0.5 is capital divided by assets, the reciprocal of the leverage multiple.

**1. Write the capital allocation line:** With risky weight w, \(\mu_p=r_f+w(\mu_R-r_f)\) and \(\sigma_p=w\sigma_R\) on the usual nonnegative-risky-weight branch. **2. Eliminate the weight:** \[\mu_p=r_f+\frac{\mu_R-r_f}{\sigma_R}\sigma_p.\] Thus the mean-versus-standard-deviation slope is excess return divided by risky standard deviation. **3. Substitute:** \[\text{slope}=\frac{24\%-6\%}{9\%}=\frac{18}{9}=2.\] **Answer: C — 2.** A: **Incorrect:** 2.5 is not the given excess-return-to-risk ratio. B: **Incorrect:** The expected-return premium is positive, so the usual allocation-line slope is not negative. C: **Correct:** 18 percentage points divided by 9 percentage points gives 2. D: **Incorrect:** The sign is incorrect for this positive risk premium.

**1. Express beta using correlation:** \[\beta_A=\frac{\operatorname{Cov}(R_A,R_M)}{\sigma_M^2}=\rho_{AM}\frac{\sigma_A}{\sigma_M}.\] **2. Substitute the values:** \[\beta_A=0.4\frac{0.16}{0.04}=0.4(4)=1.6.\] A beta above 1 indicates more than unit market sensitivity in the linear-return relation. **Answer: A — 1.6.** A: **Correct:** The correlation-adjusted volatility ratio equals 1.6. B: **Incorrect:** A positive correlation cannot produce negative beta with positive standard deviations. C: **Incorrect:** 0.75 is not the stated volatility ratio multiplied by correlation. D: **Incorrect:** 2.5 does not follow from the supplied quantities.

**1. Specify the equally weighted portfolio:** Nine identical-risk assets are held with weight 1/9 each. Their common variance is \(0.12^2\), and uncorrelated returns make all off-diagonal covariances zero. **2. Add weighted variances:** \[\sigma_p^2=9\left(\frac19\right)^2(0.12^2)=\frac{0.12^2}{9}=0.0016.\] **3. Take the square root:** \[\sigma_p=\frac{0.12}{\sqrt9}=0.04=4\%.\] The common 22% expected return does not affect this variance calculation. **Answer: D — 4%.** **Convention note:** Equal allocation is the intended portfolio assumption. With arbitrary unequal weights, the number of assets alone would not determine the portfolio standard deviation. A: **Incorrect:** 32% greatly exceeds the individual risk and is not the diversification result. B: **Incorrect:** 12% ignores the risk reduction from uncorrelated equal-weight holdings. C: **Incorrect:** 8% does not equal 12% divided by the square root of 9. D: **Correct:** The equal-weight independent-risk scaling gives 4%.

**1. Find the expected two-year value:** All amounts are in thousands. \[E[V_2]=0.35(100)+0.65(200)=35+130=165.\] **2. Discount for two years:** At the stated annual cost of capital r = 12%, \[PV=\frac{E[V_2]}{(1+r)^2}=\frac{165}{1.12^2}=\frac{165}{1.2544}.\] **3. Round the result:** \(PV=131.53699\) thousand, approximately 131.54 thousand. The two-year timing requires two compounding periods. **Answer: B — 131.54 thousand.** A: **Incorrect:** 147.32 discounts the expected value for only one year. B: **Correct:** The expected payoff discounted over two years is 131.54 thousand. C: **Incorrect:** 119.98 is not the two-year discounted value at 12%. D: **Incorrect:** 110.50 is not the discounted value implied by the given probabilities and rate.

**1. Read the market inputs:** The image gives risk-free return 8%, expected market return 18% and asset beta 1.5. **2. Use CAPM:** \[E[R_i]=r_f+\beta_i(E[R_M]-r_f).\] The market premium is \(18\%-8\%=10\%\). **3. Calculate:** \[E[R_i]=8\%+1.5(10\%)=8\%+15\%=23\%.\] **Answer: D — 23%.** A: **Incorrect:** 7% is below the safe rate despite positive beta and positive market premium. B: **Incorrect:** 19% does not match the required beta-scaled premium. C: **Incorrect:** 15% is the risk premium before adding the safe return. D: **Correct:** Adding the 15% premium to 8% gives 23%.

**1. Calculate the initial short-sale proceeds:** Selling 50 borrowed shares at Rs 200 brings in \(50\times200=10000\) rupees. **2. Calculate the repurchase cost:** Buying 50 replacement shares at Rs 190 costs \(50\times190=9500\) rupees. **3. Find the trading profit:** \[\Pi=50(200-190)=10000-9500=\text{Rs }500.\] **Answer: C — Rs 500.** **Convention note:** The problem ignores borrowing fees, dividends, transaction costs and interest on collateral/proceeds. A: **Incorrect:** 190 is the final price of one share, not total profit. B: **Incorrect:** 200 is the initial price of one share, not total profit. C: **Correct:** A gain of 10 per share on 50 shares gives 500. D: **Incorrect:** 1000 would require a 20-rupee price fall per share.

**1. State debt priority and required return:** The creditor lends 400 thousand and requires an expected 20% return under the stated cost-of-capital assumption. The required expected repayment is \(400(1.20)=480\) thousand. **2. Model repayment by state:** Let F be the promised repayment. In a flood, only 240 thousand is available, all going to debt. With no flood, the creditor receives F, provided F ≤ 600. **3. Solve for F:** \[0.04(240)+0.96F=480\quad\Longrightarrow\quad F=\frac{480-9.6}{0.96}=490.\] Since 490 < 600, the assumed repayment in the good state is feasible. **4. Convert to the promised rate:** \[r_D=\frac{490}{400}-1=0.225=22.50\%.\] **Answer: B — 22.50%.** **Convention note:** The intended calculation applies the supplied 20% required return to debt. A project-wide cost of capital alone does not generally identify a debt-specific required return without further risk-pricing assumptions. A: **Incorrect:** 10.69% would not produce the required expected repayment. B: **Correct:** Promising 490 on a loan of 400 gives 22.50%. C: **Incorrect:** 25.50% exceeds the rate implied by the stated expected-return equality. D: **Incorrect:** 31.43% is not the solution of the debt repayment equation.

**1. Recover the debt repayment:** Under the intended 20% creditor expected-return assumption, \(0.04(240)+0.96F=400(1.20)\), giving promised repayment F = 490 thousand. **2. Apply residual equity priority:** In a flood the 240 thousand payoff is insufficient for debt, so equity gets 0. Without a flood, equity gets \(600-490=110\) thousand. **3. Average the equity cash flows:** \[E[E_1]=0.04(0)+0.96(110)=105.6\text{ thousand}.\] This is the future cash payoff, not profit after deducting the initial 100 thousand equity investment. **Answer: A — 105.6 thousand.** **Convention note:** As in the related creditor question, this uses the exam’s intended assumption that the stated 20% required return prices the debt expected payoff. A: **Correct:** The expected residual cash flow is 105.6 thousand. B: **Incorrect:** 250 does not reflect the debt’s priority and promised repayment. C: **Incorrect:** 100.75 is not the weighted residual payoff. D: **Incorrect:** 40 is not the residual payoff in either stated state or its expectation.

**1. Read the income and utility:** Nikita has utility \(U(x)=\sqrt{2x}\). Income is 800 with probability 0.75 and 200 with probability 0.25. **2. Compute state utilities:** \(U(800)=\sqrt{1600}=40\) and \(U(200)=\sqrt{400}=20\). **3. Average utilities, not incomes:** \[E[U]=0.75(40)+0.25(20)=30+5=35.\] Applying utility only after averaging income would give a different quantity because utility is nonlinear. **Answer: D — 35.0.** A: **Incorrect:** 100 is not the probability-weighted utility. B: **Incorrect:** 37.2 is close to utility of mean income, not the expected utility of the risky income. C: **Incorrect:** 24 is not the weighted average of state utilities 40 and 20. D: **Correct:** The weighted state utilities sum to 35.

**1. Compute uninsured expected utility:** With utility \(U(x)=\sqrt{2x}\), \[EU=0.75\sqrt{2(800)}+0.25\sqrt{2(200)}=0.75(40)+0.25(20)=35.\] **2. Express fully insured wealth:** A 600-unit injury benefit raises injured income from 200 to 800 before premium. Paying premium p therefore gives certain income 800 - p in either state. **3. Set the maximum acceptable premium:** At indifference, \[\sqrt{2(800-p)}=35\quad\Longrightarrow\quad1600-2p=1225\quad\Longrightarrow\quad p=187.5.\] Any higher premium lowers certain utility below the uninsured expected utility. **Answer: B — 187.5 units.** A: **Incorrect:** This is below the indifference premium and is not the maximum willingness to pay. B: **Correct:** 187.5 leaves certain income 612.5 with utility 35. C: **Incorrect:** 120.5 is below the maximum premium implied by indifference. D: **Incorrect:** 244.5 leaves utility below 35, so it is too high.

**1. State the data:** Asset expected returns are \(\mu_1=0.15\) and \(\mu_2=0.20\). Portfolio weights are \(w_1=0.60\), \(w_2=0.40\). **2. Use linearity of expectation:** \[\mu_p=w_1\mu_1+w_2\mu_2=0.60(0.15)+0.40(0.20)=0.09+0.08=0.17.\] Covariance is not needed for the mean return. **Answer: C — 0.17.** A: **Incorrect:** 0.14 is below both asset means and cannot be their positive-weight average. B: **Incorrect:** 0.16 does not follow from the given weights. C: **Correct:** The weighted mean is 0.17, or 17%. D: **Incorrect:** 0.18 would require a larger weight in the higher-return asset.

**1. State the risk inputs:** Weights are 0.6 and 0.4, standard deviations are 0.25 and 0.30, and covariance is 0.04. **2. Calculate variance:** \[\begin{aligned}\sigma_p^2&=w_1^2\sigma_1^2+w_2^2\sigma_2^2+2w_1w_2\sigma_{12}\\ &=0.36(0.0625)+0.16(0.09)+0.48(0.04)\\ &=0.0225+0.0144+0.0192=0.0561.\end{aligned}\] **3. Convert variance to standard deviation:** \(\sigma_p=\sqrt{0.0561}=0.236854\), approximately 0.24, or 24%, at the options’ precision. **Answer: C — 0.24.** A: **Incorrect:** 0.15 is too small for the computed variance. B: **Incorrect:** 0.20 is not the square root of 0.0561. C: **Correct:** The calculated standard deviation 0.236854 rounds to 0.24. D: **Incorrect:** 0.29 overstates the risk produced by the given covariance and weights.

**1. Set up the weight constraint:** Let w be Asset 1’s weight, so Asset 2’s weight is 1 - w. Their expected returns are 15% and 20%. **2. Impose the target mean:** \[0.15w+0.20(1-w)=0.19.\] Expanding gives \(0.20-0.05w=0.19\), hence \(w=0.01/0.05=0.20\). **3. State both allocations:** Invest 20% in Asset 1 and 80% in Asset 2. Checking gives \(0.2(15\%)+0.8(20\%)=19\%\). **Answer: D — 20%, 80%.** A: **Incorrect:** 40%/60% produces 18%, not 19%. B: **Incorrect:** 50%/50% produces 17.5%. C: **Incorrect:** 30%/70% produces 18.5%. D: **Correct:** 20%/80% gives the required 19%.

**1. Write variance as a function of one weight:** With w in Asset 1 and 1 - w in Asset 2, \[V(w)=w^2\sigma_1^2+(1-w)^2\sigma_2^2+2w(1-w)\sigma_{12}.\] **2. Differentiate and minimize:** Setting the derivative to zero yields \[w^*=\frac{\sigma_2^2-\sigma_{12}}{\sigma_1^2+\sigma_2^2-2\sigma_{12}}.\] The denominator is positive here, so the stationary point minimizes variance. **3. Substitute:** \[w_1^*=\frac{0.30^2-0.04}{0.25^2+0.30^2-2(0.04)}=\frac{0.05}{0.0725}=0.689655.\] Hence \(w_2^*=1-w_1^*=0.310345\). These are approximately 68.97% and 31.03%. **Answer: C — Approximately 68.97% in Asset 1 and 31.03% in Asset 2 (option C).** **Convention note:** The option gives 68.96%/31.04%, apparently by truncating the first percentage and using its complement. Exact computation gives 68.9655%/31.0345%; ordinary two-decimal rounding gives 68.97%/31.03%. A: **Incorrect:** The stated weights do not solve the variance first-order condition for this covariance. B: **Incorrect:** These weights substantially overweight the higher-variance asset relative to the minimizing solution. C: **Correct:** This is the listed approximation to the minimum-variance weights. D: **Incorrect:** An almost equal allocation is not the variance-minimizing result for the supplied risk inputs.

**1. Recover the linked two-asset data:** The same examination’s Corporate Finance passage supplies expected returns μ₁=0.15, μ₂=0.20, standard deviations σ₁=0.25, σ₂=0.30 and covariance σ₁₂=0.04. Let w be the fraction in Asset 1 and 1−w in Asset 2. **2. Minimize portfolio variance:** Differentiating \(w^2\sigma_1^2+(1-w)^2\sigma_2^2+2w(1-w)\sigma_{12}\) gives \[w_{min}=\frac{\sigma_2^2-\sigma_{12}}{\sigma_1^2+\sigma_2^2-2\sigma_{12}}=\frac{0.09-0.04}{0.0625+0.09-0.08}=0.689655.\] The Asset-2 weight is 0.310345. **3. Calculate the expected return:** \[\mu_p=0.689655(0.15)+0.310345(0.20)=0.16551724\ldots=16.55\%.\] **Answer: B — 16.55%.** **Convention note:** The minimum-variance expected return is 16.5517%, which rounds to 16.55%. A: **Incorrect:** It is below both asset returns and cannot result from the positive minimum-variance weights. B: **Correct:** It matches the weighted return at the minimum-variance weights. C: **Incorrect:** It does not use the minimum-variance weight pair. D: **Incorrect:** It puts too much effective weight on the higher-return asset for the variance-minimizing solution.