Question

1. Using portfolio theory, explain how investors can choose their optimal portfolios in the presence of one risky and one risk free asset and how preferences affect their decisions.

2.Explain intuitively the assumptions underlying the choice of optimal portfolios in the presence of one risky and one risk free asset. Take the example of one risk averse and one risk-tolerant investor and show graphically the impact of risk aversion on their portfolio selection.

Answer 1

Answer:-

1). Investors want to earn the highest return possible for a level of risk that they are willing to take. So how does an investor allocate her capital to maximize her investment utility — the risk-return profile that yields the greatest satisfaction, The simplest way to examine this is to consider a portfolio consisting of 2 assets: a risk-free asset that has a low rate of return but no risk, and a risky asset that has a higher expected return for a higher risk. Investment risk is measured by the standard deviation of investment returns— the greater the standard deviation, the greater the risk. By varying the relative proportions of the 2 assets, an investor can earn a risk-free return by investing all her money in the risk-free asset, or she can potentially earn the maximum return by investing entirely in the risky asset, or she can select a risk-return trade-off that is anywhere between these 2 extremes by selecting varying proportions of the 2 assets.

Asset allocation is the apportionment of funds among different types of assets, such as stocks and bonds, having different ranges of expected returns and risk. Capital allocation, on the other hand, is the apportionment of funds between risk-free investments, such as T-bills, and risky assets, such as stocks. The simplest case of capital allocation is the allocation of funds between a risky asset and a risk-free asset. The risk-return profile of this 2-asset portfolio is determined by the proportion of the risky asset to the risk-free asset. If this portfolio consists of a risky asset with a proportion of y, then the proportion of the risk-free asset must be 1 – y

Portfolio Return = y × Risky Asset Return + (1 – y) × Risk-free Return

One way to adjust the riskiness of a portfolio is by varying the proportion of the risk-free asset and the risky asset. The investment opportunity set is the set of all combinations of the risky and risk-free assets, which graphs as a line when plotted as return against risk, as measured by the standard deviation. The line begins at the intercept with the minimum return and no risk of the risk-free asset, when the entire portfolio is invested in the risk-free asset, to the maximum return and risk when the entire portfolio is invested in the risky asset. Hence, this capital allocation line (CAL) is the graph of all possible combinations of the risk-free asset and the risky asset.

Slope of CAL=Reward-to-Variability Ratio=Portfolio Return – Risk-Free ReturnStandard Deviation of Portfolio / Standard Deviation of Portfolio

2). The efficient frontier rates portfolios (investments) on a scale of return (y-axis) versus risk (x-axis). Compound Annual Growth Rate (CAGR) of an investment is commonly used as the return component while standard deviation (annualized) depicts the risk metric. The efficient frontier theory was introduced by Nobel Laureate Harry Markowitz in 1952 and is a cornerstone of modern portfolio theory (MPT).

The efficient frontier graphically represents portfolios that maximize returns for the risk assumed. Returns are dependent on the investment combinations that make up the portfolio. The standard deviation of a security is synonymous with risk. Ideally, an investor seeks to populate the portfolio with securities offering exceptional returns but whose combined standard deviation is lower than the standard deviations of the individual securities. The less synchronized the securities (lower covariance) then the lower the standard deviation. If this mix of optimizing the return versus risk paradigm is successful then that portfolio should line up along the efficient frontier line.

A key finding of the concept was the benefit of diversification resulting from the curvature of the efficient frontier. The curvature is integral in revealing how diversification improves the portfolio's risk / reward profile. It also reveals that there is a diminishing marginal return to risk. The relationship is not linear. In other words, adding more risk to a portfolio does not gain an equal amount of return. Optimal portfolios that comprise the efficient frontier tend to have a higher degree of diversification than the sub-optimal ones, which are typically less diversified.