Portfolio Risk and Return: Part I

Portfolio Risk and Return: Part I

By Popkarn Arwatchanakarn

​Introduction

• Construction of an optimal portfolio is an important objective for an investor.

• During the process of constructing the optimal portfolio, several factors and investment characteristics are considered.

• The most important of those factors are risk and return of the individual assets under consideration.

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Portfolio Risk and Return

​Introduction

• Correlations among individual assets along with risk and return are important determinants of portfolio risk.

• Creating a portfolio for an investor requires an understanding of the risk profile of the investor.

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Portfolio Risk and Return

2. Investment Characteristics of Assets

1. Return

2. Other Major Return Measures and their Application​

3. Historical Return and Risk​

4. Other Investment Characteristics

Portfolio Risk and Return

2.1 Return

• Financial assets normally generate two types of return for investors.

• First, they may provide periodic income through cash dividends or interest payments.

• Second, the price of a financial asset can increase or decrease, leading to a capital gain or loss.

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Portfolio Risk and Return

2.1.1 Holding Period Return

• Returns can be measured over a single period or over multiple periods.

• A holding period return is the return earned from holding an asset for a single specified period of time.

• The period may be 1 day, 1 week, 1 month, 5 years, or any specified period.

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Portfolio Risk and Return

2.1.1 Holding Period Return

• The holding period return can be computed for a period longer than one year.

• the holding period return is computed by compounding

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Portfolio Risk and Return

2.1.2 Arithmetic or Mean Return

• When assets have returns for multiple holding periods, it is necessary to aggregate those returns into one overall return for ease of comparison and understanding.

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Portfolio Risk and Return

2.1.2 Arithmetic or Mean Return

• The simplest way to compute the return is to take a simple arithmetic average of all holding period returns. Thus, three annual returns of –50 percent, 35 percent, and 27 percent

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Portfolio Risk and Return

2.1.3 Geometric Mean Return

• A geometric mean return provides a more accurate representation of the growth in portfolio value over a given time period than does an arithmetic mean return.

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Portfolio Risk and Return

2.1.4 Money-Weighted Return or Internal Rate of Return

• The money-weighted return accounts for the money invested and provides the investor with information on the return she earns on her actual investment.

• ​The money-weighted return and its calculation are similar to the internal rate of return and the yield to maturity.

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Portfolio Risk and Return

2.1.4 Money-Weighted Return or Internal Rate of Return

• We can compute the internal rate of return by using the preceding equation.

• What is the IRR?

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Portfolio Risk and Return

Fill in the Blank

Compute the holding period return for the five-year period.

2.1.5 Time-Weighted Rate of Return

• An investment measure that is not sensitive to the additions and withdrawals of funds is the time-weighted rate of return.

• The time-weighted rate of return measures the compound rate of growth of $1 initially invested in the portfolio over a stated measurement period.

• If ri is the time-weighted return for year i, we calculate an annualized time-weighted return as the geometric mean of N annual returns, as follows:

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Portfolio Risk and Return

Fill in the Blank

Calculate the time-weighted rate of return for the in-house account.

2.1.6 Annualized Return

• The period during which a return is earned or computed can vary and often we have to annualize a return that was calculated for a period that is shorter (or longer) than one year.

• To annualize any return for a period shorter than one year, the return for the period must be compounded by the number of periods in a year.

  • A monthly return is compounded 12 times,

  • a weekly return is compounded 52 times, and a

  • quarterly return is compounded 4 times.

  • Daily returns are normally compounded 365 times.

  • For an uncommon number of days, we compound by the ratio of 365 to the number of days.

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Portfolio Risk and Return

2.1.6 Annualized Return

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Portfolio Risk and Return

Multiple Select

Which annualized return are correct? (you may choose more than 1)

• Security A = 25.45%

• Security B = 26.39%

• Security C = 21.55%

• Security A = 24.55

• Security B = 29.36%

2.2 Other Major Return Measures

• Gross and Net Return

  • ​A gross return is the return earned by an asset manager prior to deductions for management expenses, custodial fees or taxes.

  • Net return is a measure of what the investment vehicle (mutual fund, etc.) has earned for the investor. Net return accounts for (i.e., deducts) all managerial and administrative expenses that reduce an investor’s return.​

Portfolio Risk and Return

2.2 Other Major Return Measures

• Pre-tax and After-tax Nominal Return

  • all returns are pre-tax nominal returns unless they are otherwise designated.

  • The after-tax nominal return is computed as the total return minus any allowance for taxes on dividends, interest and realized gains.

• ​Real Return

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Portfolio Risk and Return

2.3 Historical Return and Risk

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Portfolio Risk and Return

2.3 Historical Return and Risk

2.3.1 Historical Mean Return and Expected Return

• Historical return is what was actually earned in the past, whereas expected return is what an investor anticipates to earn in the future.

• The relationship between the expected return and the real risk-free interest rate, inflation rate, and risk premium can be expressed by the following equation:

Portfolio Risk and Return

2.3 Historical Return and Risk

2.3.3 Real Returns of Major US Asset Classes

Portfolio Risk and Return

2.3 Historical Return and Risk

2.3.3 Real Returns of Major US Asset Classes

Portfolio Risk and Return

2.3 Historical Return and Risk

2.3.6 Risk-Return Trade-off

Portfolio Risk and Return

• ​The expression “risk–return trade-off” refers to the positive relationship between expected risk and return.

• Exhibit 9 reveals that the risk and return for stocks were the highest of the asset classes, and the risk and return for bonds were lower than stocks

• Another way of looking at the risk–return trade-off is to focus on the risk premium, which is the extra return investors can expect for assuming additional risk, after accounting for the nominal risk-free interest rate

2.4 Other Investment Characteristics

2.4.1 Distributional Characteristics

Portfolio Risk and Return

• ​Normal distribution has three main characteristics: its mean and median are equal; it is completely defined by two parameters, mean and variance

• Skewness refers to asymmetry of the return distribution, that is, returns are not symmetric around the mean.

• Kurtosis refers to fat tails or higher than normal probabilities for extreme returns and has the effect of increasing an asset’s risk that is not captured in a mean– variance framework

2.4 Other Investment Characteristics

2.4.1 Distributional Characteristics

Portfolio Risk and Return

2.4 Other Investment Characteristics

2.4.2 Market Characteristics

Portfolio Risk and Return

• ​The cost of trading has three main components—brokerage commission, bid–ask spread, and price impact. Liquidity affects the latter two.

• ​Liquidity also has implications for the price impact of trade. Price impact refers to how the price moves in response to an order in the market.

• Other market-related characteristics include analyst coverage, availability of information, firm size, etc. These characteristics about companies and financial markets are essential components of investment decision making.

3. Risk Aversion & Portfolio Selection

3.1 Concept of Risk Aversion

• The concept of risk aversion is related to the behavior of individuals under uncertainty

• Assume that an individual is offered two alternatives: one where he will get £50 for sure and the other is a gamble with a 50 percent chance that he gets £100 and 50 percent chance that he gets nothing.

• The expected value in both cases is £50, one with certainty and the other with uncertainty.

Portfolio Risk and Return

3. Risk Aversion & Portfolio Selection

1. Risk Seeking: If an investor chooses the gamble, then the investor is said to be risk loving or risk seeking. The gamble has an uncertain outcome, but with the same expected value as the guaranteed outcome. Thus, an investor choosing the gamble means that the investor gets extra “utility” from the uncertainty associated with the gamble.

2. ​Risk Neutral If an investor is indifferent about the gamble or the guaranteed outcome, then the investor may be risk neutral. Risk neutrality means that the investor cares only about return and not about risk, so higher return investments are more desirable even if they come with higher risk. A risk-neutral investor would maximize return irrespective of risk and a risk-seeking investor would maximize both risk and return.

Portfolio Risk and Return

3. Risk Aversion & Portfolio Selection

3. Risk Averse If an investor chooses the guaranteed outcome, he/she is said to be risk averse because the investor does not want to take the chance of not getting anything at all. In general, investors are likely to shy away from risky investments for a lower, but guaranteed return. That is why they want to minimize their risk for the same amount of return, and maximize their return for the same amount of risk.

4. Risk Tolerance​ refers to the amount of risk an investor can tolerate to achieve an investment goal. The higher the risk tolerance, the greater is the willingness to take risk. Thus, risk tolerance is negatively related to risk aversion.

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3. Risk Aversion & Portfolio Selection

3.2 Utility Theory and Indifference Curves

• A simple implementation of utility theory allows us to quantify the rankings of investment choices using risk and return.

• We assume that investors are risk averse.

• They always prefer more to less (greater return to lesser return).

Portfolio Risk and Return

3. Risk Aversion & Portfolio Selection

3.2 Utility Theory and Indifference Curves

• They are able to rank different portfolios in the order of their preference and that the rankings are internally consistent.

• If an individual prefers X to Y and Y to Z, then he/she must prefer X to Z.

• This property implies that the indifference curves for the same individual can never touch or intersect.

• An example of a utility function is given below

Portfolio Risk and Return

3.2 Utility Theory and Indifference Curves

• An indifference curve plots the combinations of risk–return pairs that an investor would accept to maintain a given level of utility

• the investor is indifferent about the combinations on any one curve because they would provide the same level of overall utility

Portfolio Risk and Return

3.2 Utility Theory and Indifference Curves

Portfolio Risk and Return

• All points on any one of the three curves have the same utility.

• An investor does not care whether he/ she is at Point a or Point b on indifference Curve 1.

• Point a has lower risk and lower return than Point b,

• but the utility of both points is the same because the higher return at Point b is offset by the higher risk

3.2 Utility Theory and Indifference Curves

Portfolio Risk and Return

• ​The most risk-averse investor has an indifference curve with the greatest slope.

• As volatility increases, this investor demands increasingly higher returns to compensate for risk.

• The least risk-averse investor has an indifference curve with the least slope and so the demand for higher return as risk increases is not as acute as for the more risk-averse investor.

3.3 Application of Utility Theory to Portfolio Selection

• The simplest application of utility theory and risk aversion is to a portfolio of two assets, a risk-free asset and a risky asset.

• The risk-free asset has zero risk and a return of Rf .

• The risky asset has a risk of σi (> 0) and an expected return of E(Ri ).

• We can construct a portfolio of these two assets with a portfolio expected return, E(Rp), and portfolio risk, σp,

Portfolio Risk and Return

3.3 Application of Utility Theory to Portfolio Selection

Portfolio Risk and Return

• The capital allocation line represents the portfolios available to an investor.

3.3 Application of Utility Theory to Portfolio Selection

Portfolio Risk and Return

• The capital allocation line consists of the set of feasible portfolios.

• Points under the capital allocation line may be attainable but are not preferred by any investor

• Points above the capital allocation line are desirable but not achievable with available assets.

• Point m and the utility associated with Curve 2 is the best that the investor can do

• The optimal portfolio is the point of tangency between the capital allocation line and the indifference curve for that investor.

4. Portfolio Risk

4.1 Portfolio of Two Risky Assets

4.1.1 ​Portfolio Return

• ​When several individual assets are combined into a portfolio, we can compute the portfolio return as a weighted average of the returns in the portfolio.

• ​If Asset 1 has a return of 20 percent and constitutes 25 percent of the portfolio’s investment, then the contribution to the portfolio return is 5 percent (= 25% of 20%).

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​4.1.1 Portfolio Return

• In general, if Asset i has a return of Ri and has a weight of wi in the portfolio, then the portfolio return, RP, is given as:

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Portfolio Risk and Return

• ​Consider Assets 1 and 2 with weights of 25 percent and 75 percent in a portfolio. If their returns are 20 percent and 5 percent, the weighted average return = (0.25 × 20%) + (0.75 × 5%) = 8.75%.

​4.1.2 Portfolio Risk

• Like a portfolio’s return, we can calculate a portfolio’s variance.

• Variance can be expressed more generally for N securities in a portfolio using the notation from the portfolio return calculation above:

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• ​For a two asset portfolio, the expression for portfolio variance simplifies to the following using covariance and then using correlation:

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Portfolio Risk and Return

Fill in the Blank

What will be the portfolio’s expected return ?

Fill in the Blank

What will be the portfolio’s risk given that the covariance between the S&P 500 and the Emerging Markets index is 0.5 percent or 0.0050?

​4.1.3 Covariance and Correlation

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Portfolio Risk and Return

Fill in the Blank

Calculate the portfolio return and risk if the correlation is 1.0. (Put only number in term of %)

Fill in the Blank

Calculate the portfolio return and risk if the correlation is 0. (Put only number in term of %)

Fill in the Blank

Calculate the portfolio return and risk if the correlation is -1.0. (Put only number in term of %)

​4.1.4 Relationship between Portfolio Risk and Return

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Portfolio Risk and Return

• ​we consider how portfolio risk and return vary with different portfolio weights and different correlations.

• Asset 1 has an annual return of 7 percent and annualized risk of 12 percent, whereas Asset 2 has an annual return of 15 percent and annualized risk of 25 percent.

​4.1.4 Relationship between Portfolio Risk and Return

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Portfolio Risk and Return

• Portfolio risk becomes smaller with each successive decrease in the correlation coefficient, with the smallest risk when ρ12 = –1. The graph in Exhibit  20 shows that the risk–return relationship is a straight line when ρ12 = +1. As the correlation falls, the risk becomes smaller and smaller as in the table.

4.3 The Power of Diversification

• Diversification is one of the most important and powerful concepts in investments.

• Because investors are risk averse, they are interested in reducing risk preferably without reducing return.

• In other cases, investors may accept a lower return if it will reduce the chance of catastrophic losses.

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Portfolio Risk and Return

​4.3.1 Correlation and Risk Diversification

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Portfolio Risk and Return

• Correlation is the key in diversification of risk.

• Correlation among assets remains the primary determinant of portfolio risk.

• Lower correlations are associated with lower risk. Unfortunately, most assets have high positive correlations.

• The challenge in diversifying risk is to find assets that have a correlation that is much lower than +1.0.

​4.3.2 Historical Risk and Correlation

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Portfolio Risk and Return

• ​We were careful to distinguish between historical or past returns and expected or future returns because historical returns may not be a good indicator of future returns.

• Risk for an asset class, however, does not usually change dramatically from one period to the next. Therefore, it is not unreasonable to assume that historical risk can work as a good proxy for future risk.

• A correlation above 0.90 is considered high because the assets do not provide much opportunity for diversification of risk

• ​Low correlations—generally less than 0.50—are desirable for portfolio diversification

​4.3.3 Historical Correlation among Asset Classes

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Portfolio Risk and Return

• ​The low correlations between stocks and bonds are attractive for portfolio diversification

• Including international securities in a portfolio can also control portfolio risk.

• It is not surprising that most diversified portfolios of investors contain domestic stocks, domestic bonds, foreign stocks, foreign bonds, real estate, cash, and other asset classes.

​4.3.4 Avenues for Diversification

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Portfolio Risk and Return

​Diversification thus makes a portfolio more resilient to gyrations in financial markets.

• ​Diversify with asset classes

• ​Diversify with index funds: many investors should consider index mutual funds as an investment vehicle as opposed to individual securities

• ​Diversification among countries: Investment in foreign countries is an essential part of a well-diversified portfolio.

• Diversify by not owning your employer’s stock

• Evaluate each asset before adding to a portfolio.

• Buy insurance for risky portfolio

5. Efficient Frontier and Investor's Optimal Portfolio

• We formalize the effect of diversification and expand the set of investments to include all available risky assets in a mean–variance framework.

• The addition of a risk-free asset generates an optimal risky portfolio and the capital allocation line.

• We can then derive an investor’s optimal portfolio by overlaying the capital allocation line with the indifference curves of investors.

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Portfolio Risk and Return

5.1 Investment Opportunity Set

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Portfolio Risk and Return

• ​As the number of available assets increases, the number of possible combinations increases rapidly.

• When all investable assets are considered, and there are hundreds and thousands of them, we can construct an opportunity set of investments.

• The opportunity set will ordinarily span all points within a frontier because it is also possible to reach every possible point within that curve by judiciously creating a portfolio from the investable assets.

5.1 Investment Opportunity Set

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Portfolio Risk and Return

• ​All points on the curve and points to the right of the curve are attainable by a combination of one or more of the investable assets.

• This set of points is called the investment opportunity set.

5.2 Minimum-Variance Portfolios

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Portfolio Risk and Return

• ​This Exhibit 25 shows the investment opportunity set consisting of all available investable sets

• ​There are a large number of portfolios available for investment, but we must choose a single optimal portfolio.

5.2.1 Minimum-Variance Frontier

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Portfolio Risk and Return

• Risk-averse investors seek to minimize risk for a given return.

• ​Consider Points A, B, and X have the same expected return

• Given a choice, an investor will choose the point with the minimum risk, which is Point X. Point X, however, is unattainable

• Thus, the minimum risk that we can attain for E(R1) is at Point A.

• Minimum-variance portfolio is the one that lies on the solid curve

5.2.3 Efficient Frontier of Risky Assets

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Portfolio Risk and Return

• The minimum-variance frontier gives us portfolios with the minimum variance for a given return.

• Points A and C on the minimum-variance frontier

• Given a choice, an investor will choose Portfolio A because it has a higher return.

• The curve that lies above and to the right of the global minimum-variance portfolio is referred to as the Markowitz efficient frontier

5.3 A Risk-Free Asset and Many Risky Assets

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Portfolio Risk and Return

• ​Most investors, however, have access to a risk-free asset, most notably from securities issued by the government.

• The addition of a risk-free asset makes the investment opportunity set much richer than the investment opportunity set consisting only of risky assets.

5.3.1 Capital Allocation Line and Optimal Risky Portfolio

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Portfolio Risk and Return

• A risk-free asset has zero risk so it must lie on the y-axis in a mean-variance graph.

• The combination of a risk-free asset with a portfolio of risky assets is a straight line.

• ​The portfolios on CAL(P) dominate the portfolios on CAL(A). Therefore, an investor will choose CAL(P) over CAL(A).

• CAL(P) is the optimal capital allocation line and Portfolio P is the optimal risky portfolio.

Multiple Choice

Which of the above points is not achievable?

• Point A

• Point B

• Point P

• Point C

Multiple Select

Which of these portfolios will not be chosen by a rational, risk-averse investor? (Check that all apply)

• Point B

• Point C

• Point D

• Point P

• Point G

Multiple Choice

Which of these portfolios is most suitable for a risk-neutral investor?

• Point B

• Point C

• Point D

• Point A

• Point G

5.4 Optimal Investor Portfolio

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Portfolio Risk and Return

• ​The CAL(P) contains the best possible portfolios available to investors.

• This shows an indifference curve that is tangent to the capital allocation line, CAL(P).

• Thus, the optimal portfolio for the investor with this indifference curve is portfolio C on CAL(P).

Tony Robbins

" The secret of success is learning how to use pain and pleasure instead of having pain and pleasure use you. If you do that, you're in control of your life. If you don't, life controls you."