Economic And Financial Decisions Under Risk Exercise Solution

Economic And Financial Decisions Under Risk Exercise Solution Economic and Financial Decisions Under Risk Exercise Solutions This document provides detailed solutions to exercises related to economic and financial decisions made under risk It explores various concepts and tools used in decisionmaking when faced with uncertain outcomes including expected value risk aversion portfolio optimization and capital budgeting Exercise 1 Expected Value and Risk Aversion A company is considering launching a new product There are three possible scenarios Scenario 1 High demand leading to a profit of 10 million Scenario 2 Medium demand leading to a profit of 5 million Scenario 3 Low demand leading to a loss of 2 million The probabilities of each scenario are Scenario 1 20 Scenario 2 50 Scenario 3 30 Tasks 1 Calculate the expected value of the project 2 Analyze the risk associated with the project 3 Discuss how risk aversion might influence the companys decision Solution 1 Expected Value The expected value EV is calculated by multiplying the payoff of each scenario by its probability and summing the results EV Probability of Scenario 1 Payoff of Scenario 1 Probability of Scenario 2 Payoff of Scenario 2 Probability of Scenario 3 Payoff of Scenario 3 2 EV 020 10 million 050 5 million 030 2 million EV 2 million 25 million 06 million EV 39 million Therefore the expected value of the project is 39 million 2 Risk Analysis Risk is assessed by measuring the variability of possible outcomes A higher variability indicates a higher risk Standard Deviation A commonly used measure of risk is the standard deviation It quantifies the average deviation of outcomes from the expected value Variance The square of the standard deviation representing the overall dispersion of outcomes We can calculate the variance and standard deviation for the project using the following formulas Variance Probability of Scenario i Payoff of Scenario i Expected Value Standard Deviation Variance Using the given probabilities and payoffs Variance 020 10 million 39 million 050 5 million 39 million 030 2 million 39 million Variance 2761 million Standard Deviation 2761 million 525 million The standard deviation of 525 million indicates a significant level of risk associated with the project 3 Risk Aversion Risk aversion describes an individuals or companys preference for certainty over uncertainty Riskaverse individuals or companies demand a higher potential return to compensate for taking on greater risk 3 DecisionMaking A riskaverse company might reject the project despite its positive expected value due to the high standard deviation They might perceive the potential loss of 2 million as too risky even though the expected profit is 39 million Risk Premium Riskaverse companies typically demand a risk premium an additional return on investments with higher risk Exercise 2 Portfolio Optimization An investor has 100000 to allocate between two assets Asset A Expected return of 10 with a standard deviation of 15 Asset B Expected return of 5 with a standard deviation of 8 The correlation coefficient between Asset A and Asset B is 03 Tasks 1 Determine the optimal portfolio allocation to maximize the expected return for a given level of risk 2 Analyze how the correlation coefficient affects the optimal portfolio Solution 1 Optimal Portfolio Allocation Portfolio Expected Return The expected return of a portfolio is the weighted average of the expected returns of individual assets where the weights represent the proportion of the portfolio invested in each asset Portfolio Standard Deviation The standard deviation of a portfolio is not simply the weighted average of individual asset standard deviations It also considers the correlation between assets To determine the optimal portfolio allocation we can use the following steps Calculate the covariance Covariance measures the relationship between the returns of two assets It is calculated by multiplying the correlation coefficient by the standard deviations of the two assets Covariance A B Correlation A B Standard Deviation A Standard Deviation B Covariance A B 03 15 8 0036 Create a table We can construct a table to calculate the expected return and standard deviation for different portfolio weights This involves calculating the weighted average of 4 expected returns and the weighted average of variances and covariances Identify the efficient frontier The efficient frontier is a curve that shows the optimal portfolios with the highest expected return for each level of risk This can be plotted by plotting the expected return and standard deviation of different portfolio weights Select the optimal portfolio The investor chooses the portfolio on the efficient frontier that best aligns with their risk tolerance 2 Correlation Coefficient Positive Correlation A positive correlation 0 Correlation 1 indicates that the returns of the two assets tend to move in the same direction This means that diversifying by combining these assets will reduce portfolio risk less effectively Negative Correlation A negative correlation 1 Correlation 0 indicates that the returns of the two assets tend to move in opposite directions This creates greater diversification benefits as losses in one asset are offset by gains in the other Zero Correlation A correlation of zero indicates that the returns of the two assets are independent Diversification benefits are still realized but not as strong as with negative correlation In our example the correlation coefficient of 03 is positive but relatively low This suggests that there is some diversification benefit but its not as significant as it would be with a negative correlation Exercise 3 Capital Budgeting Under Uncertainty A company is evaluating a project with an initial investment of 100000 The project is expected to generate annual cash flows of 25000 for five years The companys cost of capital is 10 Tasks 1 Calculate the net present value NPV of the project using the traditional approach ignoring uncertainty 2 Analyze the sensitivity of the NPV to changes in the annual cash flows 3 Discuss how decisionmaking under uncertainty might influence the investment decision Solution 1 Traditional NPV Calculation NPV Initial Investment Annual Cash Flow 1 Cost of CapitalYear 5 NPV 100000 25000 1 010 25000 1 010 25000 1 010 NPV 758150 The traditional NPV calculation indicates that the project is profitable with a positive NPV of 758150 2 Sensitivity Analysis Sensitivity analysis assesses how the NPV changes when we vary the projects inputs such as cash flows cost of capital or project life This helps identify the key variables that significantly impact the projects profitability Cash Flow Sensitivity We can analyze how the NPV changes if we assume different cash flows For example if the annual cash flow decreases to 20000 the NPV becomes negative Cost of Capital Sensitivity A higher cost of capital would decrease the NPV Similarly a lower cost of capital would increase the NPV 3 DecisionMaking Under Uncertainty Scenario Analysis Decisionmakers under uncertainty often utilize scenario analysis This involves creating multiple scenarios eg bestcase worstcase and most likely with different assumptions about key variables Decision Tree A decision tree can visually represent the possible outcomes of a project and the associated probabilities It helps to analyze the potential risks and rewards associated with each decision RiskAdjusted Discount Rate A riskadjusted discount rate is used in discounted cash flow analysis to account for the riskiness of a project This involves adding a risk premium to the cost of capital In this case the company might perform a sensitivity analysis or scenario analysis to incorporate uncertainty into the decisionmaking process They might consider adjusting the discount rate to reflect the projects risk Conclusion This document provided solutions to exercises illustrating how to approach economic and financial decisions under risk It emphasized the importance of considering factors like expected value risk aversion portfolio diversification and uncertainty in decisionmaking These tools and concepts are essential for making informed and strategic choices in various 6 economic and financial contexts