Corporate Finance · Quiz 2 · Jan 2026

**1. Identify weights and expected returns:** \(w_1=0.30\), \(w_2=0.70\), \(\mu_1=0.10\), \(\mu_2=0.25\). **2. Use linearity of expectation:** Portfolio expected return is the weighted average: \[\mu_p=w_1\mu_1+w_2\mu_2=0.30(0.10)+0.70(0.25)=0.03+0.175=0.205.\] **3. Interpret the result:** The expected return is 20.5%. Covariance affects portfolio variance, not this weighted-average expectation. **Data note:** The supplied covariance 0.1 exceeds \(\sigma_1\sigma_2=0.2(0.4)=0.08\), so it is not a valid covariance for those standard deviations. This inconsistency does not enter the expected-return calculation. **Answer: B — Selected: 0.205.** A: **Incorrect:** 0.195 is not the weighted average of the given returns. B: **Correct:** 0.205 equals 0.30(0.10)+0.70(0.25). C: **Incorrect:** 0.220 overstates the calculated expectation. D: **Incorrect:** 0.235 is too close to Asset 2’s return for a 30% allocation to Asset 1.

**1. Use diversification:** For N equally weighted, uncorrelated assets with equal standard deviation \(\sigma\), covariance terms vanish. \[\sigma_p^2=N\left(\frac1N\right)^2\sigma^2=\frac{\sigma^2}{N}.\] **2. Take the square root:** \(\sigma_p=\sigma/\sqrt N\). **3. Substitute:** With \(\sigma=25\%\) and \(N=100\), \(\sigma_p=25\%/10=2.5\%\). Standard deviation falls with the square root of asset count, not asset count itself. **Answer: C — Selected: 2.5%.** A: **Incorrect:** Diversification does not multiply the individual risk by ten. B: **Incorrect:** 25% ignores the diversification across uncorrelated assets. C: **Correct:** 25% divided by the square root of 100 is 2.5%. D: **Incorrect:** 0.25% divides standard deviation by 100 instead of by 10.

**1. State the CAL equation:** A capital allocation line relates expected return and standard deviation: \(E(r_p)=r_f+m\sigma_p\), where m is the slope. **2. Substitute consistent units:** \(r_f=0.04\), \(m=0.5\), \(\sigma_p=0.12\). \[E(r_p)=0.04+0.5(0.12)=0.04+0.06=0.10.\] **3. Conclude:** The portfolio offers an expected return of 10%. Its six-percentage-point risk premium is added to the 4% risk-free rate. **Answer: D — Selected: 10%.** A: **Incorrect:** 2% is below the risk-free rate despite a positive CAL slope and positive risk. B: **Incorrect:** 4% omits the compensation for risk. C: **Incorrect:** 8% does not satisfy the stated CAL equation. D: **Correct:** 4%+0.5(12%)=10%.

**1. Identify final wealth in each state:** With probability 0.9 there is no accident and wealth remains 1,000. With probability 0.1, the loss is 500 and final wealth is 500. **2. Apply utility to each state:** The given utility is \(u(c)=\ln c\), so \(u(1000)\approx6.9078\) and \(u(500)\approx6.2146\). **3. Average utility, not wealth:** \[E[u]=0.9\ln(1000)+0.1\ln(500)=6.8384\approx6.84.\] Because utility is nonlinear, \(E[\ln c]\) is not generally \(\ln E[c]\). **Answer: A — Selected: 6.84.** A: **Correct:** 6.84 is the rounded probability-weighted utility. B: **Incorrect:** 6.91 is approximately utility with no accident and ignores the loss probability. C: **Incorrect:** 6.21 is approximately utility in the accident state alone. D: **Incorrect:** 5.60 is below both possible utility values, so cannot be their weighted average.

**1. Use risk neutrality:** A risk-neutral investor values a bet by its expected monetary payoff. No extra risk premium is required. **2. Weight the outcomes:** \[E[X]=0.20(500)+0.30(200)+0.50(-100)=100+60-50=110.\] **3. Interpret fair price:** Paying 110 makes the expected net payoff zero: \(E[X-110]=0\). This assumes no separate discounting or costs, as none are provided. **Answer: C — Selected: 110.** A: **Incorrect:** 90 is below the calculated fair expected payoff. B: **Incorrect:** 100 counts the first weighted outcome but misses the net contribution of the other two. C: **Correct:** 110 equals the expected payoff. D: **Incorrect:** 120 is above the expected payoff.

**1. Normalize the loan principal:** Suppose the lender advances 1 unit. To earn an expected return of 10%, expected repayment must be 1.10. **2. Account for default:** Repayment is \(1+i\) with probability 0.8 and zero with probability 0.2. Thus \[0.8(1+i)+0.2(0)=1.10.\] **3. Solve the promised rate:** \(1+i=1.10/0.8=1.375\), so \(i=0.375=37.5\%\). The promised rate exceeds 10% because repayment is not certain. **Answer: D — Selected: 37.5%.** A: **Incorrect:** A 10% promise yields only 0.8(1.10)=0.88 expected repayment per unit. B: **Incorrect:** A 20% promise yields 0.96 expected repayment, below 1.10. C: **Incorrect:** A 27.5% promise yields 1.02 expected repayment, below the required 1.10. D: **Correct:** 0.8(1.375)=1.10 meets the expected return requirement.

**1. Identify the risk inputs:** Equal weights give \(w_A=w_B=0.5\). Standard deviations are 0.15 and 0.25, with correlation zero. **2. Compute portfolio variance:** \[\sigma_p^2=w_A^2\sigma_A^2+w_B^2\sigma_B^2+2w_Aw_B\rho\sigma_A\sigma_B.\] The last term is zero, so \(\sigma_p^2=0.25(0.0225)+0.25(0.0625)=0.02125\). **3. Take the square root:** \(\sigma_p=\sqrt{0.02125}=0.14577\approx14.58\%\). The expected asset returns are not needed for this risk calculation. **Answer: A — Selected: 14.58%.** A: **Correct:** The square root of 0.02125 is approximately 14.58%. B: **Incorrect:** 17.56% does not follow from the zero-correlation variance formula. C: **Incorrect:** 20% is the simple average of standard deviations and would require perfect positive correlation. D: **Incorrect:** 22.14% overstates the calculated diversified risk.

**1. Define the Sharpe ratio:** It measures expected excess return per unit of total return risk: \[S=\frac{E(r_p)-r_f}{\sigma_p}.\] **2. Substitute:** Expected excess return is \(15\%-3\%=12\%\). Therefore \(S=12\%/20\%=0.60\). **3. Interpret:** Each unit of standard deviation is associated with 0.60 units of expected excess return. The ratio is dimensionless. **Answer: B — Selected: 0.60.** A: **Incorrect:** 0.75 divides 15% by 20% without subtracting the risk-free rate. B: **Correct:** (15%-3%)/20%=0.60. C: **Incorrect:** 0.40 uses the wrong excess-return numerator. D: **Incorrect:** 0.20 is the standard deviation in decimal form, not the Sharpe ratio.

**1. Use perfect negative correlation:** With \(\rho=-1\), portfolio variance becomes \[\sigma_p^2=[w_1\sigma_1-(1-w_1)\sigma_2]^2.\] **2. Set the risk contributions equal:** Zero variance requires \(0.10w_1=0.20(1-w_1)\). Thus \(0.30w_1=0.20\). **3. Solve:** \(w_1=2/3\approx0.67\) and \(w_2=1/3\). The less volatile asset gets twice the weight so that the two perfectly offsetting risk exposures match. **Answer: B — Selected: 0.67.** A: **Incorrect:** 1.5 does not make the two risk contributions equal. B: **Correct:** Two-thirds in Asset 1 and one-third in Asset 2 cancel the risk. C: **Incorrect:** Equal weights leave unequal volatility exposures: 0.05 versus 0.10. D: **Incorrect:** Perfect negative correlation allows a zero-risk combination with the stated weights.

**1. Mix risky and risk-free assets:** Let w be the weight in the optimal risky portfolio. The expected complete-portfolio return is \(r_f+w[E(r_R)-r_f]\). **2. Substitute the target:** \(0.10=0.04+w(0.12-0.04)\), so \(0.06=0.08w\). **3. Solve:** \(w=0.06/0.08=0.75\). Invest 75% in the risky portfolio and 25% in the risk-free asset. The risky standard deviation is unnecessary for this target-return calculation. **Answer: C — Selected: 75%.** A: **Incorrect:** 25% risky gives an expected return of 6%. B: **Incorrect:** 50% risky gives an expected return of 8%. C: **Correct:** 75% risky gives 4%+0.75(8%)=10%. D: **Incorrect:** 100% risky gives 12%, exceeding the target.

**1. Explain a short sale:** The investor sells borrowed shares first and later buys shares to return to the lender. A price decline creates a gain. **2. Compute cash flows:** Sale proceeds are \(80\times60=4{,}800\) rupees. Repurchase cost is \(80\times55=4{,}400\) rupees. **3. Subtract:** Profit is \(4{,}800-4{,}400=400\) rupees, or \(80(60-55)\). This ignores financing and transaction costs as requested; no separate dividend payments are specified. **Answer: B — Selected: Rs 400.** A: **Incorrect:** 480 does not equal the five-rupee gain on each of 80 shares. B: **Correct:** 80 shares times a five-rupee price decline gives Rs 400. C: **Incorrect:** 320 understates the total gain. D: **Incorrect:** A lower repurchase price produces a positive gain on a short sale.

**1. State CAPM:** Required expected return is \[E(r)=r_f+\beta[E(r_M)-r_f].\] The bracketed market risk premium is already given as 8%; it should not have 5% subtracted again. **2. Substitute:** \(E(r)=5\%+1.5(8\%)=5\%+12\%=17\%\). **3. Interpret beta:** A beta of 1.5 gives 1.5 times the market risk premium, added to the risk-free return. **Answer: D — Selected: 17%.** A: **Incorrect:** 5% omits all compensation for systematic risk. B: **Incorrect:** 10% does not apply beta to the supplied 8% premium correctly. C: **Incorrect:** 12% is only beta times the market premium and omits the risk-free rate. D: **Correct:** 5%+1.5(8%)=17%.

**1. Identify the random component:** The risk-free asset has zero standard deviation. With risky weight w, the complete portfolio’s random return is scaled by w. **2. Use the risk formula:** \(\sigma_C=|w|\sigma_R\). **3. Substitute:** \(\sigma_C=0.60(15\%)=9\%\). Expected returns affect the portfolio mean, but do not enter this standard-deviation scaling. **Answer: C — Selected: 9%.** A: **Incorrect:** 15% would mean a 100% risky allocation. B: **Incorrect:** 12% would mean an 80% risky allocation. C: **Correct:** A 60% position in a 15%-volatility portfolio gives 9%. D: **Incorrect:** 6% would correspond to a 40% risky weight.

**1. Write the two-asset GMV weight:** Minimizing portfolio variance under \(w_X+w_Y=1\) gives \[w_X=\frac{\sigma_Y^2-\sigma_{XY}}{\sigma_X^2+\sigma_Y^2-2\sigma_{XY}}.\] **2. Substitute the risk values:** \(\sigma_X=0.2\), \(\sigma_Y=0.3\), \(\sigma_{XY}=0\). Thus \(w_X=0.09/(0.04+0.09)=0.09/0.13\). **3. Conclude:** \(w_X\approx0.6923=69.23\%\), with 30.77% in Y. The lower-volatility asset receives the larger weight. Expected returns are irrelevant to pure variance minimization. **Answer: D — Selected: 69.23%.** A: **Incorrect:** 30.77% is the GMV weight in Y, not X. B: **Incorrect:** 40% does not minimize the stated portfolio variance. C: **Incorrect:** 60% is below the calculated 69.23% allocation to X. D: **Correct:** 0.09/0.13 gives 69.23% in X.

**1. Express utility as a function of risky weight:** With a risk-free asset, \[U(w)=r_f+w(\mu-r_f)-\tfrac12Aw^2\sigma^2.\] Here \(A=4\), \(r_f=0.05\), \(\mu=0.13\), \(\sigma^2=0.04\). **2. Optimize:** Setting \(U'(w)=\mu-r_f-Aw\sigma^2=0\) gives \(w^*=(\mu-r_f)/(A\sigma^2)\). **3. Calculate and check:** \(w^*=0.08/(4\times0.04)=0.5=50\%\). Since \(U''=-A\sigma^2=-0.16<0\), this is a maximum. **Answer: B — Selected: 50%.** A: **Incorrect:** 25% is below the utility-maximizing risky allocation. B: **Correct:** The excess return 0.08 divided by risk penalty 0.16 gives 0.50. C: **Incorrect:** 75% exceeds the optimum for A=4. D: **Incorrect:** 100% takes more risk than this investor optimally chooses.

**1. Compute expected payoff:** \[E[X]=0.25(200)+0.40(400)+0.35(600)=50+160+210=420.\] **2. Discount one year:** Using the supplied appropriate annual discount rate, \(PV=E[X]/(1+r)=420/1.10=381.818\). **3. Round:** The present value is 382 units to the nearest integer. The expected payoff and present value refer to different dates. **Answer: A — Selected: 382 units.** A: **Correct:** 420/1.10 rounds to 382 units. B: **Incorrect:** 400 is neither the expected payoff nor its correctly discounted value. C: **Incorrect:** 420 is next year’s expected payoff before discounting. D: **Incorrect:** 462 compounds 420 forward instead of discounting it.

**1. Find the mean return:** \(\mu=0.3(0.20)+0.4(0.08)+0.3(0.02)=0.098\), or 9.8%. **2. Compute probability-weighted squared deviations:** \[\sigma^2=0.3(0.20-0.098)^2+0.4(0.08-0.098)^2+0.3(0.02-0.098)^2.\] The terms are 0.0031212, 0.0001296 and 0.0018252, totaling 0.005076. **3. Take the square root:** \(\sigma=\sqrt{0.005076}=0.071246\approx7.12\%\). **Answer: B — Selected: 7.12%.** A: **Incorrect:** 9.80% is the mean return, not standard deviation. B: **Correct:** The square root of the weighted variance is about 7.12%. C: **Incorrect:** 5.74% is not the square root of 0.005076. D: **Incorrect:** 3.21% understates the dispersion of the three returns.

**1. Use the course leverage convention:** Leverage is total assets divided by equity: \(L=A/E\). **2. Find equity:** With assets of Rs 500 billion and leverage 5, \(E=500/5=100\) billion. **3. Apply the balance-sheet identity:** Assets equal liabilities plus equity. Therefore liabilities are \(500-100=400\) billion rupees. This uses the financial leverage multiple, rather than the separate regulatory capital-ratio convention. **Answer: C — Selected: Rs 400 billion.** A: **Incorrect:** Rs 100 billion is equity under the stated leverage multiple. B: **Incorrect:** Rs 300 billion does not balance the balance sheet with Rs 100 billion equity. C: **Correct:** 500-100=400 billion rupees in liabilities. D: **Incorrect:** Liabilities equal to all assets would leave zero equity and no finite leverage multiple.

**1. Equate utilities:** Indifference means both assets provide equal mean-variance utility: \[0.175-\tfrac12A(0.02)=0.15-\tfrac12A(0.01).\] **2. Rearrange:** The extra expected return from A is \(0.175-0.15=0.025\). Its extra variance penalty is \(\tfrac12A(0.02-0.01)=0.005A\). Therefore \(0.025=0.005A\). **3. Solve and verify:** \(A=5\). Both utilities then equal 0.125. The given risk inputs are already variances, so do not square them again. **Answer: D — Selected: 5.** A: **Incorrect:** A=12.5 would give a larger penalty difference than the return difference. B: **Incorrect:** A=10 doubles the required risk-aversion coefficient. C: **Incorrect:** A=7.5 does not make the two utilities equal. D: **Correct:** A=5 makes both utilities equal to 0.125.

**1. Compute utility without insurance:** With \(u(x)=\sqrt{x}\), \[E[u]=0.5\sqrt{256}+0.5\sqrt{16}=0.5(16)+0.5(4)=10.\] **2. Compute insured wealth:** Insurance adds 240 in the failure state, so wealth before its premium is 256 in either state. After paying premium p, certain wealth is \(256-p\). **3. Find the maximum acceptable premium:** At indifference, \(\sqrt{256-p}=10\). Squaring gives \(256-p=100\), hence \(p=156\) million units. Paying more would reduce certain utility below 10. **Answer: B — Selected: 156 million units.** A: **Incorrect:** 120 is the expected insurance payout, but this risk-averse investor is willing to pay an additional risk premium. B: **Correct:** At p=156, insured wealth is 100 and its utility is 10, matching uninsured expected utility. C: **Incorrect:** At p=180, certain wealth 76 has utility below 10. D: **Incorrect:** At p=240, certain wealth 16 has utility only 4.

**1. Find required expected debt repayment:** The lender advances 400 thousand units and requires 10%, so expected repayment must be \(400(1.10)=440\) thousand. **2. Account for recovery:** In failure, the lender recovers the project’s 100 thousand. If success repayment is D, then \(0.7D+0.3(100)=440\). **3. Solve the promise:** \(D=(440-30)/0.7=585.714\) thousand. Thus \[i=\frac{585.714}{400}-1=0.464286\approx46.43\%.\] This promised repayment is below the successful payoff of 900, so it is feasible. **Answer: C — Selected: 46.43%.** A: **Incorrect:** 28.33% produces insufficient expected repayment for the lender. B: **Incorrect:** 35.27% also falls short of 440 thousand expected repayment. C: **Correct:** A 46.43% promise gives 585.714 in success and expected repayment of 440. D: **Incorrect:** 58.57% exceeds the promise needed under the stated recovery.

**1. Price the debt claim independently:** Required expected repayment is \(400(1.10)=440\) thousand. With 100 recovered on failure, the successful-state debt payment solves \(0.7D+0.3(100)=440\), giving \(D=585.714\). **2. Find residual equity payoffs:** On success, equity receives \(900-585.714=314.286\) thousand. On failure, all 100 recovered goes to debt, leaving equity zero. **3. Average the equity payoff:** \[E[E_1]=0.7(314.286)+0.3(0)=220\text{ thousand}.\] This is next year’s expected payoff, rather than current equity value or profit net of its initial investment. **Answer: D — Selected: 220.00 thousand.** A: **Incorrect:** 110.33 thousand does not equal the probability-weighted residual equity claim. B: **Incorrect:** 147.67 thousand understates the stated successful-state residual after weighting. C: **Incorrect:** 180.19 thousand does not follow from the lender’s required repayment. D: **Correct:** The expected project payoff is 660; subtracting expected debt repayment 440 leaves 220 thousand.

**1. Read the shared data:** \(\sigma_1=0.20\), \(\sigma_2=0.30\), \(\rho=0.3\) and both weights are 0.5. Covariance is \(0.3(0.20)(0.30)=0.018\). **2. Compute variance:** \[\sigma_p^2=0.5^2(0.20)^2+0.5^2(0.30)^2+2(0.5)(0.5)(0.018)=0.01+0.0225+0.009=0.0415.\] **3. Take the square root:** \(\sigma_p=\sqrt{0.0415}=0.203715\approx0.2037\), or 20.37%. **Answer: A — Selected: 0.2037.** A: **Correct:** 0.2037 is the correctly rounded standard deviation. B: **Incorrect:** 0.2243 is not the square root of the given portfolio variance. C: **Incorrect:** 0.2500 averages standard deviations as if correlation were one. D: **Incorrect:** 0.2734 overstates the risk of this equal-weight combination.

**1. Set up the target mean:** Asset 1 has expected return 15%; Asset 2 has 20%. If Asset 1’s weight is w, Asset 2’s is \(1-w\). **2. Solve the weighted average:** \[0.19=0.15w+0.20(1-w)=0.20-0.05w.\] Thus \(0.05w=0.01\) and \(w=0.20\). **3. Check:** \(0.20(15\%)+0.80(20\%)=3\%+16\%=19\%\). Weights are 20% and 80%, respectively. **Answer: A — Selected: 20%, 80%.** A: **Correct:** 20% in Asset 1 and 80% in Asset 2 gives the 19% target. B: **Incorrect:** 40%/60% gives 18%. C: **Incorrect:** 60%/40% gives 17%. D: **Incorrect:** 80%/20% gives 16%.

**1. Compute covariance:** The shared inputs give \(\sigma_{12}=\rho\sigma_1\sigma_2=0.3(0.2)(0.3)=0.018\). **2. Use the minimum-variance weight:** \[w_1=\frac{\sigma_2^2-\sigma_{12}}{\sigma_1^2+\sigma_2^2-2\sigma_{12}}=\frac{0.09-0.018}{0.04+0.09-0.036}=\frac{0.072}{0.094}=0.765957.\] **3. Complete the allocation:** \(w_2=1-w_1=0.234043\). Rounded weights are 76.6% in Asset 1 and 23.4% in Asset 2. These weights minimize variance, rather than target a particular expected return. **Answer: B — Selected: 76.6%, 23.4%.** A: **Incorrect:** 83.2% in Asset 1 is above the variance-minimizing weight. B: **Correct:** 76.6% and 23.4% match the covariance-adjusted GMV calculation. C: **Incorrect:** 62.4% in Asset 1 is below the calculated optimum. D: **Incorrect:** Near-equal weights do not account sufficiently for Asset 1’s lower volatility.