Module 5: Numeric Questions

For students

Last updated: 17/08/2025 19:04

The questions are based on or inspired by the following references:

• Berk & DeMarzo, Corporate Finance, 5th ed. (2020)
  
• Brealey & Myers, Principles of Corporate Finance, 13th ed. (2020)

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💡 You can also press Ctrl + P (or Cmd + P on Mac) to print or save your responses as a .pdf file.

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⚠️ These exercises are powered by AI-assisted technologies and may contain occasional formatting or logic errors. Please report any issues you encounter so I can improve the experience.

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📘 Part 1 (until Midterm)

Module	Chapter	Slides	T/F	MCQ	Numeric	Long
5	ch13	🎞️	✅	❓	🔢	📝

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Answer the following questions based on the discussions in class.

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Q1.

CAPM Alpha Across Stocks

Stock	Beta	E[R] (%)
A	1.44	16
B	1.19	9
C	1.19	10

You observe three stocks with betas and expected returns (table above).
Using R_f=2%, E[R_m]=8%, compute each stock’s alpha in percent (%).

1a. Stock A alpha (%):

✅ Stock A
CAPM = R_f + β(R_m − R_f) = 10.64%.
α_A = E[R]_A − CAPM = 16% − 10.64% = 5.36%.
Interpretation: α > 0 ⇒ above SML (underpriced → buy); α < 0 ⇒ below SML (overpriced → sell).

1b. Stock B alpha (%):

✅ Stock B
CAPM = 9.14%.
α_B = 9% − 9.14% = -0.14%.

1c. Stock C alpha (%):

✅ Stock C
CAPM = 9.14%.
α_C = 10% − 9.14% = 0.86%.

Q2.

Informed vs. Fad Followers vs. Passive: Who Earns Alpha?

In a market with 71% passive, 5% informed, and 24% fad followers, the informed trade a portfolio with β = 1.74 and E[R] = 15%.
Given R_f = 5%, E[R_m] = 11% (MRP = 6%):

2a. Informed investors’ alpha (bp):

✅ Informed α
CAPM prediction = R_f + β(R_m − R_f) = 15.44%.
Observed E[R] = 15%.
α_informed = E[R] − CAPM = 15% − 15.44% = -44 bp.

2b. Fad followers’ expected return (in %):

✅ Fad followers’ E[R]
Within the trading segment (informed + fad), expected return must match the CAPM prediction for the counterparty exposure:
E[R]_traded (CAPM) = R_f − β(R_m − R_f) = -5.44%.
Let weights inside the trading segment be w_I=17% and w_F=83%. Then:
w_IE[R]_I + w_FE[R]_F = E[R]_traded ⇒ E[R]_F = (E[R]_traded − w_IE[R]_I)/w_F = -9.7%.

2c. Fad followers’ alpha (bp) (vs CAPM with β of informed):

✅ Fad followers’ α
We evaluate α against the same β used by the informed portfolio (as in the textbook exercise). CAPM benchmark = 15.44%.
E[R]_F = -9.7% ⇒ α_F = E[R]_F − CAPM = -2513.8 bp.
Interpretation: informed earn positive α (skill), the counterparty earns negative α; the passive investor (market) earns zero α.

Q3.

Skill, Fees, and Equilibrium AUM

A manager can generate α = 2.2% per year up to $713m of capital; above that, incremental α ≈ 0.
The fund charges 1.1% on total AUM. Assume investors flock in until net α = 0.

3a. Equilibrium AUM (in $ millions):

✅ Equilibrium AUM
Dollar alpha at capacity: α × capacity = 2.2% × $713m = $15.69m per year.
At equilibrium, investors add capital until dollar α = fees:
fee × AUM = 1.1% × AUM = $15.69m ⇒ AUM = $1426m (rounded).

3b. Annual fee revenue at equilibrium (in $ millions):

✅ Fee revenue at equilibrium
Fee revenue = fee × AUM = 1.1% × $1426m = $15.69m per year.
Notice this equals the manager’s dollar α at capacity: $15.69m.
Intuition: competition drives net α to investors ≈ 0; skilled managers capture the rents via fees.

Q4.

Cumulative Abnormal Return (CAR) Around News

A firm-specific news generates an event-day abnormal return = 0.2%, followed by 3 days with an average abnormal return of 0.08% per day.

4a. CAR over the event window (%):

✅ CAR logic (in %): Sum abnormal returns over the window.
CAR = AR₀ + ∑_t>0 AR_t = 0.2% + 3 × 0.08% = 0.44%.

Interpretation: Under semi-strong EMH, prices should adjust quickly to news.
A post-event drift (positive or negative) may indicate delayed reaction or under/overreaction.

Q5.

Momentum Long–Short

A momentum strategy buys past winners and shorts losers (equal weights). Next period: winners return 38%, losers return 13%; the strategy beta is β = 0.11.
Assume R_f = 4%, MRP = 6%.

5a. Long–short return (in %):

✅ Long–short return
Spread = E[R]_winners − E[R]_losers = 38% − 13% = 25%.

5b. CAPM required return for this portfolio (in %):

✅ CAPM required return
E[R]_CAPM = R_f + β × MRP = 4% + (0.11) × 6% = 4.66%.
When β is near zero, CAPM says the expected long–short return should be close to the risk‑free rate.

5c. Alpha (in %):

✅ Alpha
α = Realized long–short return − CAPM required = 25% − 4.66% = 20.34%.
If α > 0 with β ≈ 0, momentum delivers returns not explained by market β → evidence of a factor or behavioral effect.

Q6.

Fama–French–Carhart Expected Return and Factor Contributions

Factor	Beta	Premium (%)
MKT	0.85	5
SMB	-0.06	-1
HML	0.47	2
Mom	-0.04	-1

Given R_f = 3%, the table shows each factor’s beta and premium.

6a. Expected return E[R] (in %):

✅ Expected return under FFC
E[R] = R_f + β_MKT·MKT + β_SMB·SMB + β_HML·HML + β_Mom·Mom.
= 3% + (0.85)×5% + (-0.06)×-1% + (0.47)×2% + (-0.04)×-1%
= 3% + 4.25% + 0.06% + 0.94% + 0.04% = 8.29%.

6b. Contribution from Momentum factor (in %):

✅ Momentum contribution
Contribution = β_Mom × Mom = (-0.04) × -1% = 0.04%.

Q7.

Disposition Effect and Taxes — Brazilian Threshold (Isolated Tax Effect)

Context (read this first):
In Brazil, stock sales are tax-exempt if your total proceeds from stock sales in the month are ≤ BRL 20,000. If your monthly proceeds exceed BRL 20,000, then all capital gains realized that month are taxed at 15%. (Losses do not trigger tax and can typically be used to offset gains, but here we focus only on today’s cash to make the comparison clean.)

This exercise compares two end-of-month choices to raise cash:

• Sell the winner (a position with a gain) vs. Sell the loser (a position with a loss)
  
• We deliberately set the same proceeds (S) for both sales so that any difference in after-tax cash comes only from taxes (not from different sale sizes).
  
• This connects to the disposition effect: investors often prefer selling winners (to “lock in gains”) and holding losers (to avoid realizing losses), but that behaviour can be tax-inefficient when the threshold is binding.

You will: 1) compute after-tax cash if you sell the winner,
2) compute after-tax cash if you sell the loser,
3) compute the advantage (loser − winner).
If S > 20,000, selling the winner may incur tax today, while selling the loser does not.

Leg	Proceeds (BRL)	Basis (BRL)	Gain/Loss (BRL)	Tax (BRL)
Sell winner (gain)	21599	18122	3477	521.55
Sell loser (loss)	21599	33564	-11965	0

Assumptions: One sale in the month, no other offsets, ignore costs; FIFO implicit.
Rule: If monthly proceeds from stock sales > BRL 20,000, all gains that month are taxed at 15%; otherwise, exempt.

6a. After-tax proceeds if selling the winner (BRL):

✅ Step-by-step (winner leg with a gain)
• Check the threshold: Monthly proceeds = 21599. Condition: S > 20,000 ⇒ gains are taxable at 15%.
• Compute tax on the gain (if taxable): Gain = S − basis = 21599 − 18122 = 3477. Tax = 15% × Gain = 0.15 × 3477 = 521.55 BRL. (If the line above shows 0, it means the sale is exempt in this scenario.)
• After-tax cash today: Proceeds − Tax = 21599 − 521.55 = 21077.45 BRL.

Why this matters: If the threshold binds, selling a winner creates immediate tax outflow, reducing today’s cash.

6b. After-tax proceeds if selling the loser (BRL):

✅ Step-by-step (loser leg with a loss)
• Tax on losses: Realizing a loss does not trigger capital gains tax now; the tax due on this sale is 0 BRL. • After-tax cash today: Proceeds − Tax = 21599 − 0 = 21599 BRL.

Why this matters: Selling the loser yields the same gross cash as selling the winner in this setup, but with no tax haircut today. (Separately from this exercise, a realized loss may be used to offset gains in the month/year depending on the rules in force.)

6c. Proceeds advantage (loser − winner) (BRL):

✅ Interpret and connect to investor behavior
Because proceeds are equal (S for both legs), the difference in after-tax cash is purely the tax on the winner (if any):
• Advantage = After-tax(loser) − After-tax(winner) = 21599 − 21077.45 = 521.55 BRL.
• If S > 20,000, then Advantage = the winner’s tax amount.

Key intuition (disposition effect): Many investors feel tempted to sell winners and hold losers. Under the Brazilian threshold rule, when S > 20,000, selling the winner can be tax-inefficient today (you pay 15% on the gain), whereas selling the loser gives you the same cash without a tax haircut. This illustrates how tax rules and behavioral biases interact: sticking to “sell what went up” may reduce your net cash today, even if it feels psychologically rewarding.