How would the following be solved...
Consider a risky portfolio. The end-of-year cash flow derived from the portfolio will be either $50,000 or $150,000, with equal probabilities of 0.5. The alternative riskless investment in T-bills pays 5%.
a. If you require a risk premium of 10%, how much will you be willing to pay for the
portfolio?
b. Suppose the portfolio can be purchased for the amount you found in (a). What will the
expected rate of return on the portfolio be?
c. Now suppose you require a risk premium of 15%. What is the price you will be willing to
pay now?
d. Comparing your answers to (a) and (c), what do you conclude about the relationship
between the required risk premium on a portfolio and the price at which the portfolio will
sell?
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To solve this problem, we will use the concept of expected value and the required risk premium.
a. To determine how much you would be willing to pay for the portfolio, we need to find the present value of the expected cash flows.
The expected cash flow is the weighted average of the possible outcomes, considering the probabilities. In this case, the expected cash flow is calculated as follows:
Expected cash flow = (Cash flow1 * Probability1) + (Cash flow2 * Probability2)
= ($50,000 * 0.5) + ($150,000 * 0.5)
= $25,000 + $75,000
= $100,000
To find the present value of this expected cash flow, we discount it by the risk-free rate of return. The risk-free rate in this case is 5%. Using the formula for present value, we can calculate the price you would be willing to pay for the portfolio.
Price = Expected cash flow / (1 + Risk-free rate)
= $100,000 / (1 + 0.05)
= $100,000 / 1.05
≈ $95,238.10
Therefore, you would be willing to pay approximately $95,238.10 for the portfolio.
b. After purchasing the portfolio for $95,238.10, we can now calculate the expected rate of return. The expected rate of return is also the weighted average of possible returns, considering the probabilities. In this case, we have two possible returns: $50,000 and $150,000, both with equal probabilities of 0.5.
Expected rate of return = (Return1 * Probability1) + (Return2 * Probability2)
= ($50,000 * 0.5) + ($150,000 * 0.5)
= $25,000 + $75,000
= $100,000
To find the expected rate of return as a percentage, we divide the expected rate of return by the purchase price and multiply by 100.
Expected rate of return (%) = (Expected rate of return / Purchase price) * 100
= ($100,000 / $95,238.10) * 100
≈ 105.26%
Therefore, the expected rate of return on the portfolio is approximately 105.26%.
c. Now, let's calculate the price you would be willing to pay for the portfolio if you require a risk premium of 15%. Following the same steps as in part (a), we find:
Price = Expected cash flow / (1 + Risk-free rate + Risk premium)
= $100,000 / (1 + 0.05 + 0.15)
= $100,000 / 1.20
≈ $83,333.33
Therefore, if you require a risk premium of 15%, you would be willing to pay approximately $83,333.33 for the portfolio.
d. Comparing the answers from parts (a) and (c), we can conclude that as the required risk premium on a portfolio increases, the price at which the portfolio will sell decreases. In other words, as you demand a higher risk premium, you are willing to pay less for the portfolio. This relationship is consistent with the concept of risk and return, where higher risk is typically associated with higher expected returns and lower prices.