Part -1:
1. Within the APT (Arbitrage Pricing Theory) model of equilibrium returns, you are considering an investment in two equities: Company A operating in country 1 (home country) and Company B operating in country 2 (foreign country), with both companies operating in the same sector of the economy in a single country (home country). Both countries share common currency but have differential inflation rates.
Assume that a two-factor APT model holds and the factors are as follows:
• Factor 1: General News Component: Inflation
• Factor 2: Idiosyncratic factor: Company A strategy differential from Company B. Specifically, assume that Company A is operating a high profit margin strategy, while Company B is operating a low margin strategy.
a. How would you model explicitly these two factors' impact on equities A and B? What assumptions will be required to assure general applicability of the standard APT framework to this model?
b. Define the risk premium (or the market price of risk) under APT in the above model setting for both companies.
c. Now, suppose that Company B diversifies its markets to exports into the home country 1. This puts it into direct competition with Company A. How will the APT model change in this case? Explain your answer.
2. Distortionary taxation refers to tax rates that have a direct impact on investment decisions of the agents by altering risk premium paid to investors in the risky assets. Conversely, non-distortionary taxation implies that once the tax is levied on an investor, it has only the effect of reducing her rates of return, without altering her preferences or the degree of risk aversion. Assume both taxes apply to all investors in the market.
a. Graphically, define market portfolio allocations within the CAPM framework in the presence of a non-distortionary tax.
b. Briefly compare and contrast the two market portfolios and investors' portfolio allocations before and after tax for more and less risk-averse investors.
3. Consider the CAPM model of equity returns.
a. In a general CAPM setting, define the market price of risk relationship in the presence of inflation. How will the risk premium evolve over time if inflation is persistent and rising?
b. Now, suppose that in addition to inflation, you are also facing the market with two types of agents. Assume each agent has a distinct information set. Denoting by R, the rate of return on investment in risky asset i, one type has expectations given by E0 (R,) where is the information set available at time t to agent 1.
The other investor has E12),(R,)= Ei2)(R,112.,), where 12., is the information set available at time t to agent 2. Suppose II,, is contained within 12,, so that at any point in time, Agent 1 has less information than Agent 2. What do you expect to happen to the efficiency frontier for the market and for each agent? Explain your answer and all assumptions that form the basis for your answer.
c. What will be the optimal portfolia for Agents 1 and 2? What will be the new market portfolio and how will it relate to Agents 1 and 2 information sets? Please, discuss your answers and state all required assumptions you make.
Part -2:
1.
a. State all of the assumptions required for Capital Asset Pricing Model (CAPM).
b. Define the risk premium decomposition (or the market price of risk) under CAPM in a traditional setting.
c. Assume investors face differential lending and borrowing rates that are (both) risk-free. Define the risk premium decomposition (or the market price of risk) in this setting. Hint: you may assume that the share of net lenders in the market is given by z and the share of borrowers is given by 1-z.
2. The Government levies a tax on the return to all risky assets at some rate t. There is no tax on risk-free returns.
a. Assuming the tax rate is known and certain before the investor makes her decisions, discuss the implications of such a tax on market portfolio within Capital Asset Pricing Mode (CAPM) setting. Do you expect such a tax to change risk (leverage) behaviour of investors? Explain your answer.
b. Suppose there is a general uncertainty about the tax rate (for example, the tax rate might change due to changes in the tax policy, and the investor does not know the exact rate she will be facing in the market prior to making her decisions). How would such a tax impact the market portfolio within CAPM setting? Do you expect uncertainty about the tax rate to change risk / leverage behaviour of investors? Explain your answer.
3.
a. Define Sharpe, Treynor and Jensen performance indices.
b. What does the Roll's Critique imply with respect to Jensen and Treynor indices?
c. What does Capital Asset Pricing Mode imply about the Sharpe ratio? Suppose a new stock is about to be added to the market portfolio. How should it be priced so that investors are willing to hold the stock? Briefly explain your answers.
4. Assume you are considering an investment in two companies, company A and company B. Both companies operate in the same sector in the same country and face the same general uncertain environment defined by one factor: economic growth. The companies also face different idiosyncratic factors. Company A faces idiosyncratic uncertainty concerning its strategy, while Company B faces idiosyncratic uncertainty concerning its plant investment.
a. Assuming that Company A strategy is independent of Company B plant investment, using Arbitrage Pricing Theory, define expected rates of return to Companies A and B shares. Explain and list all assumptions required. Define and briefly interpret all APT 'betas'.
b. Now, assume that Company A strategy factor is negatively correlated with Company B plant investment factor, so that positive outcome for Company B in plant investment is associated with negative payoff on Company A strategy. What do you expect to change in your analysis of companies' returns? Can you adjust your APT identities in (a) to reflect this new assumption? Can you further decompose idiosyncratic factors applying to both firms into correlated and fully independent components? Briefly interpret APT 'betas' under this setting.
5.
a. Define Markowitz's Semi-Variance Model, given a specific exogenously set Minimal Acceptable Return (MAR). [10 points]
b. Assuming individual investor preferences for risk are given by risk-aversion parameter 8, what is the relationship between the MAR, 8, and the Markowitz's semi-variance risk metric? Briefly explain your answer.
c. Can the above be extended to the case where 8 is different for two types of agents, so that one group of agents has higher degree of risk aversion in the market than the other? How would Capital Asset Pricing Mode change to reflect such heterogeneous beliefs? Briefly explain your answer. Note: You can use graphical exposition of CAPM in the setting of heterogeneous beliefs to show these differences.
Part -3:
Question 1: Define the risk premium decomposition (or the market price of risk) under APT and CAPM
Question 2: Non-distortionary taxation implies that once the tax is levied on an investor, it has only the effect of reducing her investable wealth. Graphically, show the implications of non-distortionary tax on optimal portfolio allocations for an investor within CAPM framework. Briefly explain your answer.
Question 3: Show that APT is a general model incorporating as a special case traditional CAPM. Does this fact imply that Higher Moments CAPM is consistent with APT framework? Explain your answers.
Question 4: Define co-skewness and co-kurtosis. Contrast both concepts with their ordinary counterparts, namely simple skewness and simple kurtosis. State and briefly explain the main arguments for including higher moments, such as co-skewness and co-kurtosis in the analysis of the capital markets.
Question 5: Briefly explain how APT model can be extended to cover the case of asymmetric taxation where tax is levied on returns to risk assets, but not on returns to the risk free asset (lending). What assumptions will be required to make sure APT continues to apply in this setting?