Volume 2 & 6 - Portfolio Management Flashcards
Portfolio Risk and Return: Part 1
describe characteristics of the major asset classes that investors consider in forming portfolios
explain risk aversion and its implications for portfolio selection
explain the selection of an optimal portfolio, given an investor’s utility (or risk aversion) and the capital allocation line
calculate and interpret the mean, variance, and covariance (or correlation) of asset returns based on historical data
calculate and interpret portfolio standard deviation
describe the effect on a portfolio’s risk of investing in assets that are less than perfectly correlated
describe and interpret the minimum-variance and efficient frontiers of risky assets and the global minimum-variance portfolio
Historical returns document past performance, while expected returns reflect anticipated future performance. An asset’s expected return is a function of the real risk-free rate, expected inflation, and any risk premiums that investors require as compensation.
Using a mean and variance approach assumes that returns are normally distributed and that markets are informationally and operationally efficient.
However, these assumptions do not necessarily hold.
A normal distributions has three characteristics:
Its mean and median are equal.
It is completely defined by its mean and variance
It is symmetric around its mean.
Most equity return distributions are not normally distributed. They are often asymmetric (or skewed). Stock returns are usually negatively skewed
Distributions also usually have fatter tails than normally distributed variables, which means extreme returns are more likely. This is referred to as kurtosis
The presence of skewness and/or kurtosis is contrary to the assumption that returns are normally distributed. The assumption that markets are operationally efficient is limited by market frictions, such as trading costs. These frictions impact both actual and expected returns.
Risk-seeking investors enjoy the thrill of gambling and will take risks even with a negative expected return.
Risk-neutral investors only care about the expected return. They will prefer an investment that offers a higher return, regardless of its level of risk.
Risk-averse investors will choose the investment that offers the highest return for their desired level of risk (or the least risk for their desired level of return). It is reasonable to assume that most investors are risk-averse.
The key conclusions from utility functions are:
1- Utility has no maximum or minimum
2- A higher return contributes to higher utility
3- Higher variance reduces utility (for risk-averse investors)
4- Utility is only useful in ranking investment options
Utility measures the relative satisfaction gained from a particular portfolio. The utility that investors derive from an asset or portfolio is a function of their degree of risk aversion (A), which is the marginal reward that they require as compensation for taking an additional unit of risk.
The value of A will be positive for risk-averse investors and higher for investors with lower levels of risk tolerance.
A = 0 for risk neutral
A < 0 for risk seeking (ignorant)
Indifference curves plot the risk-return pairs that have the same utility.
The capital allocation line (CAL) represents the investment options for this portfolio of two securities. It is the plot of different risk-return combinations derived by changing the weights of the two securities.
The CAL represents all the investment options. An investor must be somewhere on the line.
The slope represents the additional return required for every increment in risk, which is the market price of risk. The slope is equivalent to the Sharpe ratio.
Indifference curves can be used to determine the optimal investment point on the CAL. The goal is to maximize utility, which is the same as getting on the highest indifference curve. This optimal investment corresponds to the point of tangency between the indifference curve and the capital allocation line.
With respect to risk-averse investors, a risk-free asset will generate a numerical utility that is:
A
the same for all individuals.
B
positive for risk-averse investors.
C
equal to zero for risk seeking investors.
A
When p12 = 1 , the two assets are perfectly positively correlated. An asset is always perfectly positively correlated with itself. If a portfolio is composed of two assets are perfectly positively correlated with each other, its standard deviation is a simple weighted average of the standard deviations of the individual assets.
The lower correlation between two assets in a portfolio, the higher the expected return for a given level of portfolio risk.
If p12 = 0 , the two assets are uncorrelated. The return on the risk-free asset is known in advance with certainty, meaning that it has zero volatility. It follows that the correlation between the risk-free asset and any risky asset is zero. Adding the risk-free asset to a portfolio of risky assets will lower the portfolio’s riskiness as measured by standard deviation.
The power of diversification is its ability to reduce portfolio volatility. There are many ways to diversify a portfolio, notably by making allocations across asset classes (e.g., large-cap stocks, small-cap stocks, corporate bonds, government bonds). Other ways to achieve diversification include:
- Holding international assets, which provides the additional benefit of diversifying currency risk exposure
- Using index funds as a relatively inexpensive and more efficient means of diversification
- Avoiding ownership of your employer’s stock to limit dependence on the source of your employment income to provide investment income as well
- Protecting risky assets by purchasing insurance, which has a negative expected return but is also perfectly negatively correlated with the protected asset
As more assets are added to a portfolio of risky assets, its variance approaches the average covariance of its components.
Each asset that is being considered for inclusion in a portfolio should be evaluated in the portfolio context. Specifically, an asset should only be added to a portfolio if its Sharpe ratios is greater than the Sharpe ratio of the existing portfolio multiplied by the portfolio’s correlation with the new asset
As the number of assets in an equally-weighted portfolio increases, the contribution of each new asset’s variance to the portfolio’s overall variance most likely:
A
increases.
B
approaches zero.
C
remains unchanged.
B
Efficient frontier:
Adding less-correlated asset classes (e.g., international assets) will improve the risk-return trade-off, pushing the curve up and to the left.
The minimum-variance frontier is the left edge of the possibilities in the graph below. It represents the least portfolio risk that can be obtained for a given expected return. The global minimum-variance portfolio, located on the far left of the curve, is the least risky of the minimum variance portfolios.
The section of the minimum-variance frontier that lies above the global minimum-variance portfolio is the Markowitz efficient frontier. Risk-averse investors will not consider portfolios on the lower half of the minimum-variance frontier because, for any portfolio that plots in this section, there is a Markowitz efficient frontier that offers a higher expected return for the same level of risk.
According to the two-fund separation theorem, all investors will use the risky portfolio P to a greater or lesser extent depending on their level of risk aversion. Investors will have different allocations to the risk-free asset, but they will all create portfolios that use the optimal portfolio,
P , and plot on the CAL according to their risk tolerance. Note that CAL portfolios offer better risk-return profiles than efficient frontier portfolios of equivalent risk that have not been combined with the risk-free asset.