Corporate Finance · End Term · Jan 2026 · FN

**1. Relate nominal, real and inflation rates:** The Fisher relation is \(1+i=(1+r)(1+\pi)\), where i is nominal interest, r is real interest and \(\pi\) is inflation. **2. Solve for real interest:** \[r=\frac{1.155}{1.102}-1=0.048094\approx4.81\%.\] **3. Judge the statement:** The real rate is approximately 4.81%, not 27.3%. Even the simple approximation \(15.5\%-10.2\%=5.3\%\) is nowhere near the claimed number. **Answer: B — Selected: False.** A: **Incorrect:** The claimed real rate does not satisfy the Fisher relation. B: **Correct:** The correct real rate is about 4.81%, so the statement is false.

**1. Define the yield curve:** It plots yields on otherwise comparable debt against time to maturity. **2. Interpret an upward slope:** As maturity increases, the plotted yield increases. Long-term yields are therefore higher than short-term yields. **3. Judge the statement:** It reverses this relationship. Lower long-term yields would describe a downward-sloping or inverted curve. **Answer: B — Selected: False.** A: **Incorrect:** The statement describes the opposite of an upward slope. B: **Correct:** An upward curve has higher, not lower, long-maturity yields.

**1. Read the exchange-rate quote:** Initially one dollar costs Rs 90; afterward it costs Rs 95. More rupees are needed to purchase the same dollar. **2. Express the rupee’s dollar value:** One rupee falls in value from \(1/90\) dollar to \(1/95\) dollar. This is rupee depreciation. **3. Connect to purchasing power:** Holding dollar prices unchanged, the same rupee budget buys fewer dollars and therefore fewer American goods. This is the exchange-rate effect described by the statement. **Answer: A — Selected: True.** A: **Correct:** The higher rupees-per-dollar quote means depreciation and reduced dollar purchasing power. B: **Incorrect:** The rupee does not appreciate when more rupees are required per dollar.

**1. Distinguish call and put:** A call gives its holder the right to buy the underlying asset at the strike price. A put gives the right to sell. **2. Distinguish exercise styles:** An American option can be exercised on or before expiry. A European option is exercisable only at expiry. **3. Judge the statement:** It incorrectly describes both the transaction direction and exercise timing of an American call. **Answer: B — Selected: False.** A: **Incorrect:** The statement describes neither the purchase right nor the exercise timing of an American call correctly. B: **Correct:** An American call permits buying at the strike on or before expiration.

**1. Define the textbook financial leverage multiple:** A bank’s leverage multiple is total assets divided by equity: \(L=A/E\). More assets financed by the same equity mean greater leverage. **2. Identify the stated ratio:** Liabilities divided by assets is a debt-to-assets ratio, not that leverage multiple. **3. Note terminology:** Regulatory leverage requirements can instead quote a capital-to-exposure ratio. That convention also does not make liabilities/assets the definition in the statement. **Answer: B — Selected: False.** A: **Incorrect:** Liabilities/assets is a debt ratio, not the financial leverage multiple used here. B: **Correct:** The given leverage definition is incorrect.

**1. Identify bank-run risk:** Banks can face sudden withdrawals while their assets are less immediately liquid than deposits. **2. Explain the safeguards:** Reserve requirements provide a liquid reserve buffer. Deposit insurance protects eligible depositors within its rules and can reduce the incentive to withdraw solely from fear of a run. **3. Judge the statement:** Both are institutional safeguards aimed at banking stability. Neither eliminates all bank risk, and deposit insurance needs sound supervision because it can also create moral hazard. **Answer: A — Selected: True.** A: **Correct:** Both mechanisms can contribute to bank stability through liquidity and depositor confidence. B: **Incorrect:** The statement correctly identifies two safeguards; it does not claim they guarantee no failures.

**1. Identify a written call:** The writer receives a premium c but must cover the call payoff when exercised. **2. Write expiration profit:** For an uncovered call, \(\Pi=c-\max(S_T-K,0)\). If the stock price \(S_T\) rises without bound, the required payment also rises without bound. **3. Judge the limit:** An uncovered written call has unbounded potential loss. Strike minus premium is the maximum-loss expression for a written put on a nonnegative-price asset, not a written call. A covered-call position has a different combined payoff. **Answer: B — Selected: False.** A: **Incorrect:** The claimed finite bound does not hold for an uncovered written call. B: **Correct:** As the stock rises arbitrarily high, the call writer’s loss is unbounded.

**1. Define in-the-money for a call:** A call has positive intrinsic value when the market price S exceeds the strike K. **2. Calculate intrinsic value:** \(\max(82-80,0)=2\) rupees per share. **3. Conclude:** The call is in-the-money. This does not by itself mean the buyer has made a net profit, since the premium must also be considered for profit. **Answer: A — Selected: True.** A: **Correct:** Rs 82 exceeds the Rs 80 strike, giving positive call intrinsic value. B: **Incorrect:** A call would be out-of-the-money if the stock were below the strike.

**1. State the relevant EMH idea:** Efficient prices incorporate the information covered by the particular efficiency form. Trading skill alone does not imply a reliable opportunity to earn abnormal risk-adjusted returns from that information. **2. Distinguish performance from abnormal performance:** A trader can earn more than the market by taking more risk, or outperform for a period through luck. Those outcomes do not establish persistent informational advantage. **3. Judge the statement:** EMH does not assert that an experienced trader can consistently beat the market. Its central implication is the difficulty of consistently earning abnormal returns using already reflected information. **Answer: B — Selected: False.** A: **Incorrect:** The statement reverses the standard EMH implication and ignores risk adjustment. B: **Correct:** Experience does not establish the persistent abnormal returns claimed here.

**1. Write the SML:** \(E(r_i)=r_f+\beta_i[E(r_M)-r_f]\). The horizontal axis is beta, so its slope is the market risk premium. **2. Compute that slope:** \(8\%-2\%=6\%\), or 0.06 per unit of beta. **3. Identify the misleading number:** \((8\%-2\%)/12\%=0.5\) is the market Sharpe ratio, a return-per-standard-deviation slope for a capital market line. It is not the SML slope. **Answer: B — Selected: False.** A: **Incorrect:** 0.5 uses standard deviation as the risk axis, whereas the SML uses beta. B: **Correct:** The SML slope is the 6% market premium, so the statement is false.

**1. State the Black-Scholes stock model:** Under its standard geometric-Brownian-motion assumption, the logarithm of the future stock price is normally distributed, conditional on the current price. **2. Translate to the price itself:** If the logarithm is normal, the stock price is lognormally distributed. This keeps the modeled price positive. **3. Judge the statement:** It incorrectly assigns the normal distribution to stock-price levels. Normally distributed log returns and lognormally distributed prices are different claims. **Answer: B — Selected: False.** A: **Incorrect:** Normal price levels are not the standard Black-Scholes assumption. B: **Correct:** The price is lognormal; its log change is normal under the model.

**1. Define a long option position:** To be long an option means that the investor has bought/holds the option right. **2. Distinguish maturity:** The option may expire in days, months or years. “Long” in position terminology identifies the side of the contract, not a five-year minimum duration. **3. Contrast a short option:** The short side writes/sells the option and accepts the corresponding obligation. **Answer: B — Selected: False.** A: **Incorrect:** Holding period and contract maturity do not define a long position. B: **Correct:** Long means option buyer/holder, regardless of whether maturity exceeds five years.

**1. Recall a put’s right:** It permits selling the underlying at strike K. Its expiry payoff is \(\max(K-S_T,0)\). **2. Increase the stock price:** A higher stock price makes the right to sell at a fixed strike less valuable. In the standard non-dividend Black-Scholes model, put delta is \(N(d_1)-1<0\). **3. Conclude:** Holding other pricing inputs constant, a put’s value decreases as the stock price rises. At limiting/boundary cases the general monotonic statement is non-increasing. **Answer: A — Selected: True.** A: **Correct:** A fixed-strike selling right becomes less valuable when the underlying price rises. B: **Incorrect:** It is the call, rather than the put, that gains value from a stock-price increase under ordinary assumptions.

**1. Define the combination:** A long straddle buys one call and one put on the same underlying with the same strike K and expiration date. **2. Show its gross expiry payoff:** \[\max(S_T-K,0)+\max(K-S_T,0)=|S_T-K|.\] **3. Interpret:** A large move in either direction can create a gain after premiums. A price close to K can leave the buyer losing the premiums. Matching strike and expiry are defining features. **Answer: A — Selected: True.** A: **Correct:** Buying a matched call and put is exactly a long straddle. B: **Incorrect:** Different strikes would change the strategy, such as to a strangle when expiry matches.

**1. Define the GMV portfolio:** The global minimum-variance portfolio has the smallest attainable variance among the feasible risky portfolios. **2. Compare upper and lower branches:** Above it in expected return, the minimum-variance boundary offers the highest return attainable at the corresponding risk. Points on the lower branch are dominated by higher-return boundary portfolios with the same risk. **3. Conclude:** The upper branch is the efficient frontier in the usual mean-standard-deviation diagram. **Answer: A — Selected: True.** A: **Correct:** The upper minimum-variance branch contains the efficient portfolios. B: **Incorrect:** The statement gives the standard upper-branch definition correctly.

**1. Calculate exercise proceeds:** A call with strike Rs 84 on a stock worth Rs 92 has exercise value \(92-84=8\) rupees. **2. Subtract the original premium:** Net profit is \(8-4.50=3.50\) rupees. **3. Divide by the investment:** \[\mathrm{HPR}=\frac{3.50}{4.50}=0.77778=77.78\%.\] This is the 26-day holding-period return, not an annualized return; transaction costs and financing are not specified. **Answer: D — Selected: 77.78%.** A: **Incorrect:** 4.17% does not divide net option profit by the premium. B: **Incorrect:** 28.57% is below the calculated premium-based return. C: **Incorrect:** 53.17% does not follow from the given cash flows. D: **Correct:** Rs 3.50 profit divided by Rs 4.50 invested gives 77.78%.

**1. Define beta:** \(\beta_i=\mathrm{Cov}(r_i,r_M)/\sigma_M^2=\rho_{iM}\sigma_i/\sigma_M\). **2. Substitute the risk inputs:** \(\rho=0.6\), \(\sigma_i=0.20\), \(\sigma_M=0.12\). \[\beta_i=0.6\frac{0.20}{0.12}=1.0.\] **3. Interpret:** The stock’s systematic exposure is one unit of market beta. Expected returns are not needed to calculate beta from correlation and standard deviations. **Answer: B — Selected: 1.0.** A: **Incorrect:** 0.8 does not equal the correlation-adjusted volatility ratio. B: **Correct:** 0.6(20/12)=1.0. C: **Incorrect:** 1.2 overstates the calculated beta. D: **Incorrect:** 1.5 does not follow from the given risk data.

**1. State the quote convention:** Measure domestic currency value as units of foreign currency per domestic unit. Under the expected interest-parity relation used in the question, the domestic currency’s gross expected appreciation is \((1+i_f)/(1+i_d)\). **2. Substitute:** \[a=\frac{1.09}{1.04}-1=0.0480769\approx4.81\%.\] **3. Explain the sign:** The lower-yield domestic currency is expected to appreciate to offset the foreign interest advantage. The reciprocal exchange-rate quote has the opposite direction and a slightly different percentage. This is a parity-model expectation, not a guarantee of future exchange movements. **Answer: D — Selected: 4.81%.** A: **Incorrect:** The negative sign represents the opposite direction under the stated appreciation convention. B: **Incorrect:** 2.86% does not use the stated 4% and 9% rates. C: **Incorrect:** 2.86% corresponds to different rate inputs. D: **Correct:** (1.09/1.04)-1 gives a 4.81% domestic-currency appreciation.

**1. Write call-buyer profit:** \(\Pi=\max(S_T-48,0)-4\). **2. Find break-even:** Positive profit requires \(S_T>48+4=52\) rupees. **3. Test the options:** At Rs 60, payoff is Rs 12 and net profit is Rs 8. At Rs 50, the call is in-the-money but its Rs 2 payoff is below the Rs 4 premium. The lower prices leave the full premium lost. **Answer: D — Selected: Rs 60.** A: **Incorrect:** At Rs 30 the option expires worthless, giving a Rs 4 loss. B: **Incorrect:** At Rs 40 the option expires worthless, giving a Rs 4 loss. C: **Incorrect:** At Rs 50 the Rs 2 payoff leaves a Rs 2 net loss. D: **Correct:** At Rs 60 the Rs 12 payoff leaves an Rs 8 profit.

**1. Use put delta:** For the standard non-dividend Black-Scholes put, the price sensitivity to the stock is \(\Delta_P=N(d_1)-1\). **2. Substitute the supplied value:** \(N(d_1)=0.5\), so \(\Delta_P=0.5-1=-0.5\). The supplied \(N(d_2)\) is not needed for delta. **3. Estimate the change:** For a Rs 1 stock-price increase, \(\Delta P\approx-0.5(1)=-0.50\) rupees. This is a first-order approximation, not an exact finite-change identity. **Data note:** The printed \(N(d_2)=0.6>N(d_1)=0.5\) is inconsistent with positive-volatility standard Black-Scholes, where \(d_2<d_1\). The intended delta calculation still follows from \(N(d_1)\). **Answer: D — Selected: The put option price decreases by Rs 0.5.** A: **Incorrect:** Put delta is negative here, rather than +1. B: **Incorrect:** A +0.5 change would have the wrong sign for this put. C: **Incorrect:** -0.6 incorrectly uses the supplied d2 value instead of N(d1)-1. D: **Correct:** Delta -0.5 implies an approximate Rs 0.50 decrease.

**1. Use the portfolio-beta rule:** A company’s asset beta is the value-weighted average of its divisional asset betas, using the weights specified in the exercise. **2. Calculate each contribution:** Sales contributes \(0.50(2.7)=1.35\), accounting contributes \(0.20(0.6)=0.12\), and logistics contributes \(0.30(1.5)=0.45\). **3. Add:** \(\beta=1.35+0.12+0.45=1.92\). The weights total one. **Answer: D — Selected: 1.92.** A: **Incorrect:** 1.12 understates the weighted divisional beta. B: **Incorrect:** 1.52 does not equal the sum of the three weighted contributions. C: **Incorrect:** 1.72 is below the calculated 1.92. D: **Correct:** The weighted sum is 1.92.

**1. Apply risk-neutral growth:** In the pricing measure, expected stock price after three months is \(S_0(1+r)=20(1.02)=20.40\). **2. Solve for the up probability q:** \[25q+16(1-q)=20.40.\] Thus \(9q=4.40\) and \(q=4.40/9=0.488889\). **3. Interpret:** The risk-neutral up probability is 48.9%. The given 2% is already the three-month rate, so it is not divided by four. This pricing probability need not equal the real-world probability. **Answer: C — Selected: 48.9%.** A: **Incorrect:** 78.6% would imply too high a risk-neutral expected stock price. B: **Incorrect:** 60% does not give the required expected stock price of Rs 20.40. C: **Correct:** 4.40/9 rounds to 48.9%. D: **Incorrect:** 43.1% gives an expected price below Rs 20.40.

**1. Calculate the CAPM benchmark:** \(r_{\mathrm{CAPM}}=4\%+1.8(12\%-4\%)=4\%+14.4\%=18.4\%\). **2. Define alpha:** Alpha is the stock’s expected return minus its CAPM-required expected return. **3. Subtract:** \(\alpha=16\%-18.4\%=-2.4\%\). The stock offers 2.4 percentage points less than the model benchmark for its beta. **Answer: A — Selected: -2.4%.** A: **Correct:** 16%-18.4%=-2.4%. B: **Incorrect:** -1.5% understates the shortfall. C: **Incorrect:** A positive alpha would require expected return above 18.4%. D: **Incorrect:** +2.4% reverses the subtraction.

**1. Rearrange the Sharpe formula:** \(S=(\mu-r_f)/\sigma\), so \(\sigma=(\mu-r_f)/S\). **2. Substitute:** The excess return is \(16\%-6\%=10\%=0.10\). \[\sigma=0.10/0.75=0.133333.\] **3. Report:** Standard deviation is approximately 13.33%. Check: \(10/13.333\approx0.75\). **Answer: A — Selected: 13.33%.** A: **Correct:** 13.33% gives the required Sharpe ratio of 0.75. B: **Incorrect:** 11.22% would imply a Sharpe ratio higher than 0.75. C: **Incorrect:** 9.35% is too small for a 10% premium and Sharpe ratio 0.75. D: **Incorrect:** 8.28% would overstate reward per unit risk.

**1. State the two-factor APT relation:** Expected return equals the risk-free rate plus each factor sensitivity multiplied by its factor risk premium: \[E(r)=r_f+\beta_W\lambda_W+\beta_C\lambda_C.\] **2. Calculate contributions:** Weather contributes \(1.1(3\%)=3.3\%\); commodities contribute \(1.8(6\%)=10.8\%\). **3. Add the risk-free rate:** \(E(r)=4\%+3.3\%+10.8\%=18.1\%\). The supplied factor values are premiums, so the risk-free rate is not subtracted from them again. **Answer: D — Selected: 18.1%.** A: **Incorrect:** 7.2% omits much of the specified factor compensation. B: **Incorrect:** 9.5% is below the sum of the required contributions. C: **Incorrect:** 12.4% does not apply both sensitivities correctly. D: **Correct:** 4%+1.1(3%)+1.8(6%)=18.1%.

**1. Match each cash flow to its date:** Receive Rs 3,000 at year 1, Rs 2,500 at year 2 and Rs 2,000 at year 3. Discount each at 8% per year. **2. Calculate present values:** \[PV=\frac{3000}{1.08}+\frac{2500}{1.08^2}+\frac{2000}{1.08^3}.\] These terms are approximately Rs 2,777.78, Rs 2,143.35 and Rs 1,587.66. **3. Add and round:** \(PV=6{,}508.7893\), so the maximum fair price under the given discount rate is Rs 6,509. **Answer: 6509.**

**1. Use European put-call parity:** For matching European options on a non-dividend-paying stock, \[C+Ke^{-rT}=P+S_0.\] **2. Discount the strike:** With \(K=450\), \(r=0.10\), \(T=0.5\), its present value is \(450e^{-0.05}=428.0532\). **3. Solve for the call:** \(C=35+430-428.0532=36.9468\) rupees. Rounded to the nearest integer, the call costs Rs 37. Continuous compounding is used because the question specifies it. **Answer: 37.**

**1. Compute terminal call payoffs:** With strike 70, the call pays \(75-70=5\) in the up state and zero in the down state at 50. **2. Calculate hedge delta:** \[\Delta=\frac{C_u-C_d}{S_u-S_d}=\frac{5-0}{75-50}=0.20.\] One call has the same two-state price sensitivity as 0.20 shares. **3. Hedge 50 purchased shares:** Short \(n=50/0.20=250\) calls. The portfolio payoff is \(50(75)-250(5)=2500\) in the up state and \(50(50)=2500\) in the down state. **Price note:** The supplied call price Rs 3 is inconsistent with the model’s zero-rate fair value \(0.4(5)=2\), but the two-state hedge quantity depends on terminal payoffs, not that quote. **Answer: 250.**

**1. State the Gordon Growth formula:** \(P_0=D_1/(k-g)\), where \(D_1\) is next year’s dividend, k is the required return and g is constant dividend growth. **2. Rearrange:** \(k=D_1/P_0+g\). **3. Substitute:** \(k=30/600+0.02=0.05+0.02=0.07\). Therefore the cost of capital is 7% and the requested numeric x is 7. The model condition \(k>g\) is satisfied. **Answer: 7.**

**1. Identify an ordinary annuity:** Assume the stated Rs 100 contributions occur at each year-end, for 35 years, earning an effective 5% annual return. **2. Sum accumulated contributions:** The last contribution earns no further interest by retirement, while the first earns interest for 34 years. The geometric sum is \[FV=100\frac{(1.05)^{35}-1}{0.05}.\] **3. Calculate and round:** \((1.05)^{35}\approx5.516015\), giving \(FV\approx9{,}032.0307\). The retirement balance is Rs 9,032. Beginning-of-year payments would form an annuity due and require multiplying by 1.05; that is not the convention used here. **Answer: 9032.**

**1. Compute option payoffs:** The strike is 90. At a stock price of 100 the call pays 10; at 70 it pays zero. **2. Calculate the risk-neutral probability:** \[q=\frac{80(1.05)-70}{100-70}=\frac{14}{30}.\] The rate 5% already covers the six-month period. **3. Discount expected payoff:** One call is worth \(q(10)/1.05=4.444444\) rupees. One hundred calls cost \(100(4.444444)=444.4444\), rounded to Rs 444. **Answer: 444.**

**1. Calculate initial option cost:** Two calls at Rs 6 and one put at Rs 4 cost \(2(6)+4=16\) rupees. **2. Calculate expiry payoff:** With stock price 96 and strike 80, each call pays \(96-80=16\), while the put pays zero. Total payoff is \(2(16)=32\). **3. Find net profit:** \(32-16=16\) rupees. The initial stock price is not a separate cash outflow because no shares are purchased. This is the usual undiscounted premium-subtracted profit; no financing rate is specified. **Answer: 16.**

**1. Read the utility function:** The source gives \(U=\mu-5\sigma^2\). Let w be the risky weight. Then \(\mu=0.04+0.05w\) and \(\sigma^2=0.15^2w^2=0.0225w^2\). **2. Optimize:** \[U(w)=0.04+0.05w-0.1125w^2.\] The first-order condition is \(0.05-0.225w=0\), giving \(w=2/9\). The negative quadratic coefficient confirms a maximum. **3. Compute expected final wealth:** Expected return is \(0.04+0.05(2/9)=0.051111\). Starting with 100 units gives \(100(1.051111)=105.1111\), rounded to 105 units. **Answer: 105.**

**1. Compute initial premium:** One call costs Rs 7 and two puts cost \(2(5)=10\), totaling Rs 17. **2. Find the smallest terminal payoff:** For common strike 65, \[V_T=\max(S_T-65,0)+2\max(65-S_T,0).\] Both terms are nonnegative and both equal zero at \(S_T=65\). Thus the minimum payoff is zero. **3. Find maximum premium-based loss:** The investor can lose the entire Rs 17 premium, but no more under the usual payoff-minus-premium convention. **Convention note:** The question gives a three-month risk-free rate but its key uses unfinanced premium loss. Including the premium’s foregone 4% return would give a terminal-date economic loss of \(17(1.04)=17.68\), rounded to 18; that is a different profit convention. **Answer: 17.**

**1. Find the CAPM discount rate:** \[k=r_f+\beta[E(r_M)-r_f]=0.04+1.5(0.10-0.04)=0.13.\] **2. Discount the expected one-year payoff:** \(PV=5000/(1+0.13)=5000/1.13=4{,}424.7788\). **3. Round:** The project price is Rs 4,425 to the nearest integer. Beta scales the market risk premium, not the entire market return. **Answer: 4425.**