Mr. Weiss Just Bought A Zero Coupon Bond Issued By Risky Co

Mr Weiss Just Bought A Zero Coupon Bond Issued By Risky Co

Suppose Mr. Weiss holds the bond to maturity. What will be his holding period return if Risky Corp. does not default? What will be his holding period return if the firm defaults?

What is the expected return of the Risky Corp. bond? Is the bond risky or riskfree? Explain.

What is the Yield to Maturity (YTM) of the government bond? Is this YTM the riskfree rate? Explain.

Compare the expected return of the Risky Corp. bond with the riskfree rate. Would a risk-averse investor buy the Risky Corp. bond at $870? Explain.

The standard deviation of the return of the Risky Corp. bond is 34.48%. What is the beta of the bond? What would be the equilibrium expected return of the Risky Corp. bond if the CAPM holds? Does Mr. Weiss overvalue or undervalue the bond relative to the CAPM?

Suppose Mr. Weiss changes his mind and sells his Risky Corp. bond. He invests in a portfolio that allocates 50% of the money on the market portfolio, and the other 50% on the government bond. What are the expected value and the standard deviation of his portfolio return? Is his portfolio efficient? Explain.

Paper For Above instruction

The scenario involving Mr. Weiss's investment in a zero-coupon bond issued by Risky Co. provides a comprehensive context to explore fundamental concepts in fixed income securities, risk assessment, and portfolio theory. This analysis will elucidate what occurs under different default scenarios, estimate expected returns, and evaluate the bond's risk characteristics relative to a risk-free asset, with further insight into market valuation and investment strategies based on CAPM and portfolio diversification principles.

1. Holding Period Return in Default and Non-default Scenarios

Assuming Mr. Weiss holds the bond until maturity, his holding period return (HPR) depends on whether the issuer defaults. If the company does not default, the bond issuer pays the face value of $1000 at maturity, and Mr. Weiss purchased the bond at $870. The HPR in this case is calculated as:

HPR in non-default scenario = (Face value - Purchase price) / Purchase price = ($1000 - $870) / $870 ≈ 0.1494 or 14.94%

If the company defaults, the bondholders recover nothing, resulting in a total loss of the initial investment. The HPR in this scenario is:

HPR in default scenario = (Recovered amount - Purchase price) / Purchase price = (0 - $870) / $870 = -1 or -100%

2. Expected Return and Risk of the Risky Co. Bond

The expected return of the bond incorporates the probabilities of default and non-default. Given a 0.9 probability of expansion (no default) and a 0.1 probability of recession (default), the expected return is:

Expected return = (Probability of no default Return if no default) + (Probability of default Return if default)

= 0.9 14.94% + 0.1 (-100%) = 0.9 0.1494 + 0.1 (-1) = 0.13446 - 0.1 = 0.03446 or 3.45%

This positive expected return reflects the proportion of states where the bond yields profit, yet the significant potential loss renders it risky. Given the potential for total loss, the bond's riskiness is evident, especially since default probability is substantial relative to the expected return. While the expected return is positive, the risk profile suggests that the bond is risky, not riskless, which is typical of corporate bonds compared to government securities.

3. Yield to Maturity (YTM) of the Government Bond and its Riskfree Nature

The government bond with a face value of $1000 and a current price of $952.38 has a YTM calculated as follows:

YTM = [(Face value / Price)^(1 / time)] - 1 = ($1000 / $952.38) - 1 ≈ 0.05 or 5%

This rate is derived assuming annual compounding over one year, consistent with the problem's assumptions. The government bond is considered riskfree, and its YTM reflects the risk-free rate because it is issued by the government, which is presumed to default at negligible probability.

4. Comparison of Expected Return and Investor Behavior

The expected return of the risky bond (3.45%) is notably lower than the risk-free rate implied by the government bond's YTM (5%). This discrepancy arises because the risk premium demanded by investors to hold the riskier bond is not explicitly incorporated into the expected return calculation but reflects the compensation for default risk. A risk-averse investor, valuing safety over higher expected returns, might avoid purchasing the risky bond at $870 unless it offers a sufficiently higher yield or risk premium to compensate for the default probability, aligning with their risk tolerance and investment objectives.

5. Beta and CAPM Valuation of the Risky Bond

The bond's standard deviation of 34.48% and correlation of 0.67 with the market imply a beta, calculated as:

Beta = (Correlation with the market) (Standard deviation of bond / Standard deviation of market) = 0.67 (34.48% / 30%) ≈ 0.67 * 1.149 = 0.77

Using the CAPM, the equilibrium expected return of the bond is:

Expected return = Risk-free rate + Beta * Market risk premium

Assuming the risk-free rate is 5%, and the market risk premium is 15% - 5% = 10%,

Expected return = 5% + 0.77 * 10% = 5% + 7.7% = 12.7%

Compared to the calculated expected return of 3.45%, Mr. Weiss undervalues the bond, implying an overpricing relative to CAPM valuation, or alternatively, the bond's risk is not fully captured by standard CAPM assumptions.

6. Portfolio Construction and Efficiency

If Mr. Weiss reallocates his investment, placing 50% in the market portfolio and 50% in the risk-free government bond, his expected return becomes:

Expected portfolio return = 0.5 Market portfolio expected return + 0.5 Risk-free rate

Assuming the market's expected return is 15%,

Expected return = 0.5 15% + 0.5 5% = 7.5% + 2.5% = 10%

The standard deviation of this portfolio is:

Standard deviation = 0.5 Market standard deviation = 0.5 30% = 15%

This portfolio offers a balanced risk-return profile, characteristic of efficient portfolios on the Capital Market Line. Its efficiency stems from optimal risk diversification, maximizing expected returns for its risk level, consistent with modern portfolio theory.

Conclusion

The analysis of Mr. Weiss's investment choices illustrates key financial principles, including default risk, expected return calculation, beta estimation, and portfolio diversification. Recognizing the risk-return trade-offs and market valuation methods equips investors to make more informed decisions aligned with their risk appetite and investment goals. The interplay of probability, risk premium, and market correlation highlights the complexity and importance of rigorous quantitative assessment in fixed income and portfolio management strategies.

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