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Lecture 10

Portfolio Theory and Risk Reduction Strategies II

Dr Jacinta Nwachukwu

Principal Lecturer in Finance

School of Economics, Finance and Accounting

[email protected]

Learning outcomes

Understanding the mean-variance rule

Understanding how the optimal portfolio of risky securities are determined

Understanding the strategies for reducing risk

Understanding how the modern portfolio theory can be used by investors to improve the efficiency of their investment

Lecture Outline

The mean-variance rule

The efficient frontier and the determination of an optimal portfolio of risky securities

Constructing optimal portfolio weights

Constructing optimal portfolio expected return

Constructing optimal portfolio variance and standard deviation

Strategies for managing portfolio risk

Modern portfolio theory and the efficiency of investment

1. The mean-variance

Something to think about

Consider the expected return and standard deviation for securities A, B, C and D shown in the Table below

Q1: Which security (or securities) would you prefer and why?

Q2. Calculate the ratio of excess return to standard deviation, assuming that the expected return on a UK government 3-month security which is typically used as a proxy for risk-free asset is 5 percent per annum.

Securities	Expected Return (%)	Standard deviation (%)	Excess return (% standard deviation)
A	20	10
B	20	40
C	10	10
D	30	40

2. The efficient frontier and optimal portfolio

Figure 1: Portfolio Expected Return-Risk Relationship

Correlation Coefficients (-0.38)

Cor (-0.38)

20 16.890233864573712 14.923806484942105 14.571204480069568 15.939887076137024 18.654758106177631 22.227910383119688 26.311974460310154 30.703745699832787 35.288525047103107 40 15 17 19 21 23 25 27 29 31 33 35

Portfolio standard deviation (% per annum)

Portfolio expected return (% per annum)

Figure 1: Portfolio Expected Return-Risk Relationship

Correlation Coefficients (-0.38)

Cor (-0.38)

Portfolio standard deviation (% per annum)

Portfolio expected return (% per annum)

3. Constructing optimal portfolio weights

Optimal portfolios offer the highest expected excess return for a given level of risk

Risk measurement could be in terms of standard deviation (i.e., Sharpe ratio)

Risk measurement could be in terms of beta coefficient (i.e., Treynor ratio)

Basic Assumptions

(1) All portfolio funds are fully invested.

This means that the sum of the proportions invested in each portfolio asset is equal to 100 percent (or 1.0 in decimal terms)

(2) Short sales are disallowed.

However, unlimited riskless lending and borrowing at the same interest rates are permitted

The portfolio optimisation problem

The construction of optimum portfolio requires the determination of the of the weights to be invested in each asset so as to satisfy the following constrained maximisation problem

Subject to the constraint that the sum of weights is equal to 1:

The techniques available for solving the constrained problems stated in equations 1 and 2 are presented under the following headings:

(1) Two risky asset portfolio

(2) Three risky asset portfolio

Solution for Two Risky Asset Portfolio

The procedure for solving the constrained maximisation problems presented in equations 1 and 2 involves finding the solution to the following system of simultaneous equations for two securities (A and B)

Where:

= the correlation between securities A and A

= the correlation between securities A and B

= the correlation between securities B and A

= the correlation between securities B and B

= the standard deviation of security A

= the standard deviation of security B

Optimising weights for Two Asset portfolio

Solution for Three Risky Asset Portfolio

Where:

= the correlation between securities C and A

= the correlation between securities C and B

= the correlation between securities C and C

= the standard deviation of security C

Optimising weights for Three Asset portfolio

Practice question 1

A two risky asset portfolio

Practice question 1

Consider the two securities (A and B) in practice question 4 in the previous lecture.

Also assume the summary statistics in Table 2 in the previous lecture

Furthermore, assume that the riskless lending and borrowing rate is 5 percent

Q1: Calculate the optimal proportions to be invested in securities A and B so as to satisfy the constrained maximisation problems stated in equations 1 and 2 shown earlier

Q2.Calculate the expected return on the optimum portfolio comprising securities A and B

Q3. Calculate the standard deviation of the return on the optimum portfolio constructed using securities A and B

Q4. Re-do the questions in items 1 to 3 assuming a correlation coefficient of +0.25

Practice question 1 (Solutions: Q2)

Security	Expected return (Percent per annum)	Standard deviation (Percent per annum)	Correlation coefficients
A	35	15	A and B = -0.38
B	40	20	B and A = -0.38
Risk-free asset	5

30 =

35 = -114

Practice question 1 (Solutions: Q2)

Q2.Calculate the expected return on the optimum portfolio comprising securities A and B

Q3. Calculate the standard deviation of the return on the optimum portfolio constructed using securities A and B

= 90.5078

6. Risk Reducing strategies

Diversification

Investment fund management

Asset class allocation decisions

Market timing

Pound cost averaging

6.1: Risk Diversification

Diversification is a strategy for reducing risk by combining a variety of investments, such as stocks issued by different UK companies, shares issued by firms in different countries or a mix of different securities, say stocks, bonds and cash, that are unlikely to move in the same direction at the same time.

The diversification principle is normally illustrated by classifying risk into two categories

Non-systematic risk

Systematic risk

6.2.1: Non-systematic risk

Diversification describes the act of investing in many different assets rather than just one or two with the aim of reducing an asset’s idiosyncratic risk

This component of total risk that can be removed by holding more and more assets in a portfolio is also referred to as non-systematic or firm-specific or diversifiable risk

Non-systematic risk is unique to the nature and activities of the issuing company

Consequently, the degree of its reduction depends on the relative co-movement between the security returns (i.e., the degree correlation among the securities)

The addition of more and more assets with relatively low correlations will significantly reduce the contribution of each security’s own risk to the total portfolio risk

6.2.1: Systematic risk

No matter how many partially correlated securities investors add to the portfolio, they cannot eliminate all of the risk intrinsic in the markets in which they operate

The component of a security’s total risk that seems to be impervious to attempts to remove it through diversification is known as systematic or market or non-diversifiable risk

It is directly related to the overall movements in the general market or the general economy. It arises from interest rate risk, recession, inflation, changes in consumer taste, wars, political instability etc

Figure 2 shows a hypothetical relationship between the level of total risk and the size of the portfolio

It depicts that as more and more partially correlated securities are added, the non-systematic element becomes smaller and smaller, and the total risk for the portfolio approaches its systematic or market risk

Research shows that approximately 50 to 60 randomly selected securities from different industries are needed to ensure adequate diversification

At this point, the total risk curve in Figure 2 levels off and becomes asymptotic to the systematic risk

Figure 2: Systematic and Non-systematic Risk

Measured using the beta coefficient

Concerns about a specific sources of risk such as business, financial, interest rate, inflation, liquidity and currency.

Events relating to environmental, political or economic factors could have affected all businesses in a country or worldwide.

6.3: Investment fund management

Investment fund management

An efficient way to achieve a well-diversified portfolio is to buy investment funds, such as investment funds and unit trusts.

There are many different types of funds, including those specialising in continents or a country, such as Africa, USA, or sector such as technology

Some investment funds will do well or badly according to what assets they hold and how good the fund managers are.

However, it is extremely difficult to identify which managers and funds are good and which are bad

As a result, some investors tend to buy what is known as “tracker” funds which mirror an index, say the FTSE All share index

They are passively managed, compared with actively managed funds where the managers buy and sell to try to get their market timing right.

Passive tracker funds have lower fees in contrast to actively managed funds

6.4: Asset class Allocation Decision

Asset Class Allocation Decision

This refers to the choice on how to divide a savings and investment portfolio across a mix of asset classes, including shares, bonds and cash

An important factor when deciding about the asset split is the investor’s age

The younger investor, say under 30 years, the more likely they are to invest primarily in long-term assets such as equities

Retiring investors focus on fixed income securities such as bonds and cash

Thus, asset allocation decision is more appropriate when the investor is relatively young not when they are close to retirement

Consequently, pension funds regularly revise and review the asset allocation decisions of their clients

Figure 3: The Risk-Return Trade-off

Expected return for taking on extra risk (i.e., risk premium)

Expected return

Risk

Expected return for delaying consumption

T.Bill

CDs

CPs

Bonds

Stocks

Savings deposits

Higher

Lower

Higher

Lower

Eurocurrency

Demand deposits

Properties

6.5:Market Timing

Market Timing

Market timing is a technique of buying securities (say stocks) when markets are low and selling them when markets are high

Such market timing can help investors avoid capital losses

However, there is little evidence that individual or fund managers are particularly good at market timing

They are normally influenced by advertisement of good recent returns which often follow after the market has risen

6.6: pound Cost Averaging

Pound Cost Averaging

This is a technique of allocating a fixed sum for the purchase of particular investments on a regular basis, such as monthly, quarterly or yearly.

The decision to invest equal amount regularly means that when market falls, the investor would buy more units/shares.

When the market rises, the fixed amount of investment would buy fewer units/shares.

In terms of selling, an investor will reduce risk if he/she sells holdings in regular fixed amounts given that it is difficult to tell when the market has reached a peak

Such pound cost averaging strategy helps investors reduce the mistakes of market timing, so enabling them to achieve an average return over time.

7. Modern portfolio theory and the efficiency of investment

Modern Portfolio Theory

Harry Markowitz created Modern Portfolio Theory (MPT) or the mean-variance rule (H. M. Markowitz, “Portfolio Selection,” Journal of Finance 7 (March 1952, pp. 77–91.)

MPT provides a mathematical framework for assembling a portfolio of assets such that the expected excess return is maximized for a given level of risk, typically defined in terms of standard deviation.

That is maximize the function:

MPT informs a rational investor on how to reduce the risk of portfolio expected returns by investing in assets that do not move exactly together in the same direction (i.e., perfectly positively correlated)

In other words, investors can reduce their exposure to individual asset risk by holding a well-diversified portfolio of assets through investment funds.

Such diversification means that portfolios with the same expected returns may have different risk structure depending on the extent of correlation between each asset pairs in the portfolio.

Modern Portfolio Theory (Contd.)

The key insight of MPT is that an asset's risk and return should not be assessed by itself, but by how it contributes to a portfolio's overall risk and return profile.

As portfolio gets larger and larger, its risk approaches the average covariance risk between the securities

In general, the MPT framework provides the guideline for determining and selecting the optimal portfolios which offer the highest ratio of expected excess return to standard return

This optimal portfolio is represented on the efficient frontier

This is concave shaped curve and stretches between the minimum variance (MV) portfolio and the highest return security

End of lecture