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The Finance Challenge: Can You Solve These 10 Brain Teasers?

The Finance Challenge: Can You Solve These?

Are you ready to put your financial acumen to the test? Welcome to "The Finance Challenge," where your skills will be pushed to the limit with ten intriguing brain teasers that blend logic, mathematics, and financial savvy. Whether you're a seasoned finance professional or a curious novice, these puzzles are designed to stimulate your mind and sharpen your problem-solving abilities. Get ready to dive into a world where numbers dance, scenarios twist, and only the sharpest thinkers will emerge victorious. Can you crack the code and prove your prowess in the realm of finance? Let the challenge begin!

Laptop, Rubik's cube, brain, and chart; text reads "The Finance Challenge: Can You Solve These 10 Brain Teasers?" Puzzle piece backdrop.

Question 1: The Stock Market Puzzle

You’re analyzing two stocks, A and B. Stock A has a 60% chance of gaining 20% and a 40% chance of losing 10%. Stock B has a 50% chance of gaining 30% and a 50% chance of losing 5%. Which stock has the higher expected return?

Answer and Explanation:To determine which stock has the higher expected return, we calculate the expected return for each stock using the formula:

Expected Return = (Probability of Gain × Gain) + (Probability of Loss × Loss).

  • Stock A:Expected Return = (0.6 × 20%) + (0.4 × -10%)= (0.6 × 0.20) + (0.4 × -0.10)= 0.12 - 0.04 = 0.08 or 8%.

  • Stock B:Expected Return = (0.5 × 30%) + (0.5 × -5%)= (0.5 × 0.30) + (0.5 × -0.05)= 0.15 - 0.025 = 0.125 or 12.5%.

Conclusion: Stock B has the higher expected return (12.5% vs. 8%).


Explanation: Expected return is a weighted average of possible outcomes, factoring in their probabilities. Stock B’s higher potential gain (30%) outweighs its loss probability, making it a better choice based purely on expected return. However, in practice, consider risk tolerance and volatility (e.g., Stock A’s loss is larger at -10%).


Question 2: The Bond Yield Riddle

A bond with a face value of $1,000 pays a 5% annual coupon and matures in one year. If the bond’s market price is $980, what is its yield to maturity (YTM)?


Answer and Explanation:The yield to maturity (YTM) is the internal rate of return of a bond purchased at its current price and held until maturity. For a one-year bond, we can approximate YTM using the formula:

YTM ≈ (Coupon Payment + (Face Value - Purchase Price)) / Purchase Price.

  • Coupon Payment = 5% of $1,000 = $50.

  • Face Value = $1,000.

  • Purchase Price = $980.

  • YTM ≈ ($50 + ($1,000 - $980)) / $980= ($50 + $20) / $980= $70 / $980 ≈ 0.0714 or 7.14%.


Conclusion: The bond’s YTM is approximately 7.14%.


Explanation: YTM accounts for both the coupon payment and the capital gain (or loss) when the bond matures at face value. Since the bond is purchased below face value ($980 vs. $1,000), the YTM is higher than the coupon rate (5%), reflecting the additional gain at maturity.


Question 3: The Diversification Dilemma

You have $10,000 to invest in two assets: a stock with an expected return of 10% and a standard deviation of 20%, and a bond with an expected return of 5% and a standard deviation of 10%. The correlation between the two is 0. How should you allocate your portfolio to minimize risk?


Answer and Explanation:To minimize risk (measured by portfolio standard deviation), we use the portfolio variance formula for two assets:σ_p = √(w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂),where w₁ and w₂ are the weights of the stock and bond, σ₁ and σ₂ are their standard deviations, and ρ is the correlation (0 in this case). Since ρ = 0, the formula simplifies to:σ_p = √(w₁²σ₁² + w₂²σ₂²).

  • Let w₁ = weight in stock, w₂ = 1 - w₁ (since weights sum to 1).

  • σ₁ = 20% = 0.2, σ₂ = 10% = 0.1.

  • Portfolio variance = w₁²(0.2)² + (1 - w₁)²(0.1)² = 0.04w₁² + 0.01(1 - w₁)².

  • To minimize risk, take the derivative of the variance with respect to w₁, set it to zero, and solve:d/dw₁ [0.04w₁² + 0.01(1 - 2w₁ + w₁²)] = d/dw₁ [0.05w₁² - 0.02w₁ + 0.01] = 0.1w₁ - 0.02 = 0.w₁ = 0.2 (20% in stock).w₂ = 1 - 0.2 = 0.8 (80% in bond).

  • Portfolio standard deviation:σ_p = √(0.2² × 0.2² + 0.8² × 0.1²) = √(0.04 × 0.04 + 0.64 × 0.01) = √(0.0016 + 0.0064) = √0.008 ≈ 0.0894 or 8.94%.


Conclusion: Allocate 20% to the stock and 80% to the bond to minimize risk, achieving a portfolio standard deviation of approximately 8.94%.


Explanation: With zero correlation, diversification reduces risk. The bond’s lower volatility dominates, so a higher allocation to bonds minimizes portfolio risk. The optimal weights balance the squared volatilities.

Question 4: The Option Payoff Puzzle

You buy a call option on a stock with a strike price of $50 for a premium of $5. The stock’s current price is $48. What is your breakeven price, and what is your maximum loss?

Answer and Explanation:

  • Breakeven Price: For a call option, the breakeven price is the strike price plus the premium paid.Breakeven = $50 + $5 = $55.At $55, the option’s payoff ($55 - $50 = $5) equals the premium paid, resulting in zero net profit.

  • Maximum Loss: The maximum loss for a call option buyer is the premium paid, as the option can expire worthless if the stock price is below the strike price.Maximum Loss = $5.


Conclusion: Breakeven price is $55, and maximum loss is $5.


Explanation: A call option gives the right, but not the obligation, to buy the stock at the strike price. The breakeven accounts for the cost of the option (premium), and the loss is limited to the premium since you can choose not to exercise the option.


Question 5: The Dividend Discount Dilemma

A stock pays a dividend of $2 per year, expected to grow at 3% annually forever. If the required rate of return is 8%, what is the stock’s intrinsic value?


Answer and Explanation:We use the Gordon Growth Model to calculate the intrinsic value of a stock with dividends growing at a constant rate:P = D₁ / (r - g),where D₁ is the dividend next year, r is the required return, and g is the growth rate.

  • Current dividend = $2.

  • D₁ = $2 × (1 + 0.03) = $2.06.

  • r = 8% = 0.08, g = 3% = 0.03.

  • P = $2.06 / (0.08 - 0.03) = $2.06 / 0.05 = $41.20.


Conclusion: The stock’s intrinsic value is $41.20.


Explanation: The Gordon Growth Model assumes perpetual dividend growth. The stock’s value is the present value of all future dividends, discounted at the required return minus the growth rate. If r ≤ g, the model would be invalid, as the value would approach infinity.


Question 6: The Leverage Conundrum

A company has a capital structure of 60% debt and 40% equity. The cost of debt is 5%, and the cost of equity is 12%. The tax rate is 30%. What is the company’s weighted average cost of capital (WACC)?


Answer and Explanation:WACC is calculated as:WACC = (w_d × r_d × (1 - T)) + (w_e × r_e),where w_d and w_e are the weights of debt and equity, r_d and r_e are their costs, and T is the tax rate.

  • w_d = 0.6, r_d = 0.05, w_e = 0.4, r_e = 0.12, T = 0.3.

  • WACC = (0.6 × 0.05 × (1 - 0.3)) + (0.4 × 0.12)= (0.6 × 0.05 × 0.7) + (0.4 × 0.12)= (0.6 × 0.035) + 0.048= 0.021 + 0.048 = 0.069 or 6.9%.


Conclusion: The WACC is 6.9%.


Explanation: WACC reflects the average cost of financing the company’s assets, accounting for the tax shield on debt interest (since interest is tax-deductible). The lower cost of debt and tax shield reduce the overall WACC compared to the cost of equity alone.


Question 7: The Currency Conversion Catch

You’re traveling to Europe with $5,000. The exchange rate is $1 = €0.85. Transaction fees are 2% of the converted amount. How many euros will you receive after fees?

Answer and Explanation:

  • Convert dollars to euros: $5,000 × 0.85 = €4,250.

  • Transaction fee = 2% of €4,250 = 0.02 × €4,250 = €85.

  • Euros received = €4,250 - €85 = €4,165.


Conclusion: You will receive €4,165.


Explanation: The exchange rate determines the initial conversion, but transaction fees reduce the final amount. Always account for fees when calculating net proceeds in currency conversions, as they can significantly impact the outcome.


Question 8: The Portfolio Beta Brain Teaser

Your portfolio consists of two stocks: Stock X (weight 70%, beta 1.2) and Stock Y (weight 30%, beta 0.8). What is the portfolio’s beta?


Answer and Explanation:Portfolio beta is the weighted average of the individual betas:β_p = (w_X × β_X) + (w_Y × β_Y).

  • w_X = 0.7, β_X = 1.2, w_Y = 0.3, β_Y = 0.8.

  • β_p = (0.7 × 1.2) + (0.3 × 0.8)= 0.84 + 0.24 = 1.08.


Conclusion: The portfolio’s beta is 1.08.


Explanation: Beta measures a portfolio’s sensitivity to market movements. A beta of 1.08 indicates the portfolio is slightly more volatile than the market (beta = 1). The higher weight of Stock X (with a higher beta) drives the portfolio beta above 1.


Question 9: The Annuity Enigma

You plan to save $10,000 annually for 5 years at an interest rate of 6%, compounded annually. How much will you have at the end of 5 years?


Answer and Explanation:This is a future value of an ordinary annuity problem. The formula is:FV = C × [((1 + r)^n - 1) / r],where C is the annual payment, r is the interest rate, and n is the number of years.

  • C = $10,000, r = 0.06, n = 5.

  • FV = $10,000 × [((1 + 0.06)^5 - 1) / 0.06]= $10,000 × [(1.06^5 - 1) / 0.06].

  • 1.06^5 = 1.338225.

  • FV = $10,000 × [(1.338225 - 1) / 0.06]= $10,000 × (0.338225 / 0.06)= $10,000 × 5.637092 ≈ $56,370.92.


Conclusion: You will have approximately $56,370.92.


Explanation: The future value of an annuity accounts for annual contributions and compound interest. Each $10,000 payment earns interest for the remaining periods, leading to a significantly higher final amount than the sum of contributions ($50,000).


Question 10: The Risk-Return Tradeoff

An investment offers a 10% return with a 15% standard deviation. A risk-free asset yields 3%. If you want a portfolio with a 12% standard deviation, what weight should you assign to the risky investment?


Answer and Explanation:For a portfolio with one risky asset and one risk-free asset, the portfolio standard deviation is the weight of the risky asset times its standard deviation (since the risk-free asset has zero standard deviation):σ_p = w_r × σ_r,where w_r is the weight in the risky asset, and σ_r is its standard deviation.

  • σ_p = 12% = 0.12, σ_r = 15% = 0.15.

  • 0.12 = w_r × 0.15.

  • w_r = 0.12 / 0.15 = 0.8 or 80%.

  • Weight in risk-free asset = 1 - 0.8 = 20%.


Conclusion: Allocate 80% to the risky investment and 20% to the risk-free asset.


Explanation: The portfolio’s risk comes entirely from the risky asset, as the risk-free asset has no volatility. To achieve a 12% standard deviation, you scale down the risky asset’s weight proportionally, as the portfolio’s risk is a linear function of the risky asset’s weight.

Final Thoughts

These 10 brain teasers cover key finance concepts like expected return, YTM, diversification, options, valuation, WACC, currency conversion, beta, annuities, and the risk-return tradeoff. Solving them sharpens your analytical skills and deepens your understanding of financial decision-making. How many did you get right?

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