Reading 42 LOS's

Question 1

LOS 42a: Calculate and interpret major return measures and describe their appropriate uses

LOS 42b: Describe characteristics of the majoy asset classes that investors consider in forming portfolios

LOS 42c: Calculate and interpret the mean, variance, and covariance (or correlation) of asset returns based on historical data

Answer 1

Financial assets may provide one or both of the following types of returns:

• periodic income (interest or dividends)
• capital gains or losses resulting from changes in market price

Holding Period return is simply the return earned on an investment over a single specified period of time and is calculated as:

• R = (P_t - P_t-1 + D_t) / P_t-1
• where (P_t - P_t-1) / P_t-1 = capital gains and,
• D_t / P_t-1 = dividend yield

Holding period returns may be calculated for more than one period by compounding single period returns:

• R = [(1 + R₁) x (1+ R₂) x …….. (1 + R_n)] - 1

Arithmetic of Mean Return is a simple average of all holding period returns

• R = (R₁ + R₂ + ….. + R_T) / T

This is easy to calculate and has known statistical properties such as standard deviation. However it is upward bias because it assumes that they amount invested each period is the same

Geometric Mean Return accounts for compounding of returns and does not assume that teh amount invested in each period is the same. The geometric is either equal or lower than the arithmetic, and is calculated as:

• R = { [( 1 + R₁) x ( 1+ R₂) ….. X ( 1 + R_n)] ^1/n } - 1

Money Weighted Return of Internal Rate of Return this accounts for the amount of money invested in each period and provides information on the return earned on the actual amount invested. This return eqauls the IRR of an invesment. The drawback is that it does not allow for return comparisons between different individuals or different investment opportunities

Annualized Return

An investment may have a term less than one year long. In such cases, the return on the investment is annualied to enable comparisons across investment instruments with different maturities. They are calculated as:

• r_annual = ( 1 + r_period)ⁿ - 1

Portfolio Return

The return on a portfolio is simply the weighted average of returns on individual assets. For example a return of two-asset portfolio can be calculated as:

• R_p = w₁R₁ + w₂R₂

Gross and New Returns

Gross returns are calculate before deductions for management expenses, custodial fees, taxes and other expenses not directly linked to returns. Net returns is with everything deducted and is what investors care about

Pre-tax and After-tax Nominal Returns

Pre-tax nominal returns do not adjust for taxed or inflation. After tax does account for tax

Real Returns

This is calculated because:

• It is useful in comparing returns across time periods as inflation rates may vary over time
• It is useful in comparing returns among countries when returns are expressed in local currencies in which inflation rates vary between countries
• The after-tax real return is what an investor receives as compensation for postponing consumption and assuming risk after paying taxes on investment returns

Leverage return

This is computed when an investor uses leverage to invest in a security

Variance of a Single ASset

Vairance is the average squared deviation of observed value from their mean. A higher variance indicates higher volatility or dispersion of returns. Calculated as so:

• σ²= Σ (R_t - µ) ² / T

The standard deviation is simply the square root of this forumla

Variance of a Portfolio os Assets

In addition to being a fundtion of individual asset variances and their weights in the portfolio, portfolio vairance also depends on the covariance ( and correlation) between assets in the portfolio. The variance of a two-asset portfolio is calculated as:

σ_p² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂σ₁σ₂þ_1,2 , where þ_1,2 = covariance of asset 1 and 2

Historical Mean Return and Expected Return

Historical returns are calculated from historical data, and where actually earned in the past. Expected return is determined by real risk-free interest rate, expected inflation, and expected risk, and is what an investor expects to earn in the future

Risk Return Trade-off every investment involves a trade-off between the two, with historical evidence showing higher returns associated with higher risks

Other Investment Characteristics

In order to evaluate using mean and variance we need to make the following assumptions:

• Returns follow a normal distribution
• markets are informationally and operationally efficient

Distribution Characteristics

Deviations from normallity may occur because of either:

• Skewness refers to the asymmetry of returns
  
  • When most of the distribution is concentrated on the left, it is referred to as right or potive skewed
  • When it is on the right, it is called left or negative skewed
• Kurtosis refers to fat rails or higher than normal probabilities for extreme returns

market Characteristics

markets are not always operationally efficient. One limitation on operational efficieny in market is liquidity

Question 2

LOS 42d: Explain Risk aversion and its implications for portfolio selections

Answer 2

Risk-averse investors aim to maximize returns for a given level of risk and minimize risk for a given level of return. This is the assumption in the investment area

Risk-seeking investors get extra utility from the uncertainty associated with their investments

Risk Neutral investors seek higher returns irrespective of the level of risk inherent in an investment

Risk tolereance refers to the level of risk that an investor is willing to accept to achieve her investment goals

• The Lower the risk tolerance, the lower the level of risk acceptable to the investor
• The lower the risk tolerance, the higher the risk aversion

Utility Theory and Indifference Curves

In order to quantify the preferences for investment choices using risk and return, utility functions are used. An example is :

• U = E(R) - (1/2) A σ² where A = additional return required by investor to accept an addition unit of risk. It is the measure of risk aversion

This utility function assumes:

• Investors are generally risk averse, but prefer more return to less
• Investors are able to rank different portfolios based on their preferences and these preferences are internally consistent

We can draw the following conclusions from the function:

• utility is unbounded on both sides
• higher return results in higher utility
• higher risk results in lower utility.
• The higher the A, the higher the negative effect of risk on utility

Important Note about risk averison “A”

• A is positive for risk-averse investor
• It is negative for risk-seeking
• It equals zero for risk neutral

Indifference curves

The risk return tradeoff that an investor is willing to bear can be illustrated by an indifference curve. 2 points relating to indifference curves of risk averse investors is worth noting:

1. They are upward sloping, meaning that an investor will be indifferent between two investments with different expected returns only if the investment with the lower expected return entails a lower level of risk
2. They are curved, and their slope becomes steeper as more risk is taken. The increase in return required for every unit of additional risk increases at an increasing rate because of the diminishing marginal utility of wealth

The slope of the indifference curve represents the extra return required by the investor to accept an additional unit of risk.

• So a risk seeking would have a negative slope
• A risk neutral would have no slope

Application of Utility Theory to Portfolio Selection

The Risk-Free Asset

the expected return on a risk-free asset is entirely certain and therefore the standard deviation of its expected returns is 0. The returned earned is the risk free rate (RFR)

Expected return for a Portfolio Containing a Risky Asset and the Risk-Free Asset

Lets assume that we invest a proportion of our investable funds (w_i) in a risk asset (i) that has an expected return, E(R_i) and variance σ_i² and the remainder (1 - w_i) is in the risk-free asset that has an expected return of RFR.

The expected return for this portfolio would be

• E(R_p) =w_iE(R_i) + (1 - w_i) RFR or E(R_p) = RFR + w_i [E (R_i) - RFR]

Standard Deviation of a Portfolio Containing a Risky ASset and the Risk-Free

• σ_port² = w_i²σ_i²

Why is the risk free not accounted for? The variance of the risk free asset is 0 because it has a guranteed return. Also the return on the risk-free asset does not vary with the return on the risky asset. This allows us to leave off the variance measure for the risk free, along with the covariance between the two assets, as both values are 0

The standard deviation will be σ_port = w_iσ_i and it can be arranged to form an expression for w_i:

• w_i = σ_port / σ _i

We can use this weight in our expected portfolio return to get

• E(R_p) = RFR + σ_port { [E(R_i) -RFR] / σ_i }

This eqaution related the return on a portfolio composed of the risk-free asset and a risky asset to the standard deviation of the portfolio know as the capital allocation line (CAL). The CAL has an intercept of RFR and a constant slope that equals [E(Ri) -RFR] / σi
The expression for the slope of the CAL is the extra return required for each additional unit of risk and is also known as the market price of risk

So we will use indifference curves and the CAL to determine which portfolios the investor will choose. The indifference curves represent the investor’s utility function, while the CAL represents the risk-return combinations of the set of portfolios the investor can invest in. Portfolios below the CAL would not be maximizing the potential return given the level of risk, and those that lie about cannot be attained with the given level of assets.

Question 3

LOS 42e: Calculate and interpret portfolio standard deviation

LOS 42f: Describe the effect on a portfolio’s risk of investing in assets that are less than perfectly correlated

Answer 3

We already covered the variance/ standard deviation for a portfolio with 2 risky assets, to understand how more than 2 would be calculated lets examine 3

σ_p² = w₁²σ₁² + w₂²σ₂² + w₃²σ₃²+ 2w₁w₂Cov_1,2 + 2w₂w₃Cov_2,3 + 2 w₃w₁Cov_3,1

As we can see, we just need to make sure we consider the covariance of every asset with every other asset.

Implications:

• When asset returns are negatively correlated, the final term in the standrad deviation formula is negative and serves to reduce the portfolio standard deviation
• If the correlation between two assets equals 0, the portfolio standrad deviation is greater than when correlation is negative
• The risk of a portfolio os risky assets depends on the asset weights and standard deviations and most impotantly on the correlation of asset returns. The higher the correlation between the individual assets, the higher the portfolios standard deviation

Portfolios of Many Risky Assets

As more and more assets are added to a portfolio, the contribution of each individual asset’s risk to portfolio risk diminishes. The covariance among the assets in the portfolio accounts for the bulk of the portfolio risk

Avenues for Diversification

• Investing in a variety of asset classes that are not highly correlated
• Using index funds that minimize the costs of diversification and grant exposure to specific asset classes
• Investing among countries that focus on differnt industries, are undergoining different stages of the business cycle, and have different currencies
• Choosing not to invest a significant portion of their wealth in employee stock plans, as their human capital is already entirely invested in their employing companies
• Only adding a security to the portfolio if its Sharpe ratio is greater than the Sharpe ratio of the portfoilio time the correlation coefficient

Question 4

LOS 42g; Describe and interpret the minimum-variance and efficient frontiers of risky assets and the global minimum-variance portfolio

Answer 4

AS the number of assets available increases, they can be combined into a large number of different portfolios and we can create an opportunity set of investments

The minimum variance frontier (MVF) plots the risk-return characteristics of portfolios that minimize portfolio risk at each given level of return lies further to the left. Note that no risk-averse investor would invest in any portfolio that lies to the right of the MVF, as it would entail a higher level of risk than a portfolio that lies on the MVF for a given level of return

The global minimum-variance portfolio is the portfolio of risky securities that entails the lowest level of risk among all the risk aset portfolios on the MVF. This lies right in the middle of the MVF curve

All points above the global minimum variance portfolio are considered on the Markowitz efficient frontier, and it contains all the possible portfolios that rational, risk-averse investors will consider investing in

A Risk-Free Asset and Many Risky Assets

As investors combine the risk-free asset with portfolios further up the efficient frontier, they keep attaining better portfolio combinations. Each successive portfolio on the efficient frontier has a steeper slope, representing the additional return for unit of extra risk, and the steeper the slope the better the risk-return tradeoff. When the CAL line is tangent to the efficient frontier, we have reached our best risk-return tradeoff possible.

At anypoint on the CAL up to where it touches the efficient frontier, the investor will have a combination of the risk-free asset, with a risky portfolio.

At the point where the CAL touches the frontier, the investor will have all its wealth in the risky portfolio.

Any point past the point that CAL touches the frontier, can be achieved by borrowing additional funds at the RFR and investing these funds into more of the risky portfolio.

Two-Fund Separation Theorem

This states that regardless of risk and return preferences, all investors hold some combination of the risk-free asset and an optimal portfolio of risk assets. Therefore, the invesment problem can be broken down into 2 steps:

1. The investing decision, where an investor identifies her optimal risky portfolio
2. The financing decision, where she determines where exatly on the optimal CAL, she wants her portfolio to lie. Her risk preferences determine whether her desired portfolio requires borrowing or lending at the risk-free rate

Question 5

LOS 42h: Discuss the selection of an optimal portfolio, given an investor’s utility and the CAL

Answer 5

Optimal Investor Portfolio

For a more risk-averse investor, the optimal portfolio would lie close to the y-axes, while a less risk-averse investor’s optimal portfolio will lie further out. Am investor with an even higher tolerance for risk may borrow and extend even further from the y -axis.