2011_Week 7- Diversification - Portfolio Theory
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value of X varies from the mean. A small standard deviation means that the data doesn’t vary a lot from the mean. A large standard deviation means that the data is more spread out.
ρX ,Y =
σ X ,Y σ Xσ Y
8
3.1 Measuring How Assets Move Together
Correlation:
We now consider the following exhaustive list of correlation values between two assets, Asset 1 and Asset 2, ρ12:
11
3.1 Measuring How Assets Move Together
Case 3: Non-perfect correlation (-1< ρ12<+1)
Correlation of 0.46 Between Returns on Assets 1 and 2
0.16 0.14 0.12 Return on Asset 2 0.1 0.08 0.06 0.04 0.02 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Return on Asset 1
2
Βιβλιοθήκη Baidu
2. Review of Statistics
Before we start discussing portfolio theory and the concept of diversification, we will briefly review the statistical terms discussed last lecture. These terms are important to understand as they are central to portfolio theory. More specifically, we will revisit the definitions of:
7
3.1 Measuring How Assets Move Together
Correlation and Covariance (Continued):
However, covariance is sensitive to the scale of measurement of X and Y and therefore the degree of association (as opposed to the sign) is difficult to interpret. Conversely, the correlation coefficient is a standardised measure of association between two variables. It is standardized as correlation measures must lie between negative one and one. This makes it easy to gauge the extent to which two variables are associated. The correlation coefficient, ρxy, is calculated as:
Random variables; Expected values; and, Standard deviation and variance.
3
2. Review of Statistics
Random Variables:
A random variable is one that can take on any number of different values. Each value has an associated probability of occurring. The uncertainty associated with the outcome of a random variable is described by a probability distribution, with the most commonly used distribution being the normal distribution, see below:
0.020 0.018 0.016 0.014 0.012 0.010 0.008 0.006 0.004 0.002 0.000
Probability density
-75
-65
-55
-45
-35
-25
-15
-5
15
25
35
45
55
65
Stock market return
75
5
4
2. Review of Statistics
12
3. Diversification
The optimal investment strategy in terms of constructing a portfolio to reduce risk depends on the properties of the two assets and how they are related to one another. Before we discuss the “ideal” diversification properties, we will go through how to calculate the expected return, standard deviation and variance on a 2-asset portfolio. We will then use these concepts to prove that diversification allows us to reduce risk without sacrificing expected return.
Case 1: Perfect positive correlation (ρ12=+1); Case 2: Perfect negative correlation (ρ12=-1); and. Case 3: Non-perfect correlation (-1< ρ12<+1).
9
3.1 Measuring How Assets Move Together
Expected Values:
The value we expect a random variable (X) to take is known as its expected value or mean.
Standard Deviation: Standard deviation, σ, is a measure of spread. It is based on how far each
Financial Markets and Systems Lecture 7
Diversification: Portfolio Theory
1
1. Lecture Overview
During this lecture, we will discuss the concepts of portfolio theory and diversification. Diversification allows an individual to reduce the risk of their investment without sacrificing any expected return simply by spreading their wealth over a portfolio comprising a number of assets in an appropriate way. During the course of the lecture we will discuss: What diversification is; How including multiple assets in a portfolio can achieve diversification; Which assets to include in order to achieve the greatest level of diversification; and, How to construct a diversified portfolio in practice.
10
3.1 Measuring How Assets Move Together
Case 2: Perfect negative correlation (ρ12=-1)
Perfect Negative Correlation Between Returns on Assets 1 and 2
0 0 -0.02 -0.04 Return on Asset 2 -0.06 -0.08 -0.1 -0.12 -0.14 -0.16 -0.18 Return on Asset 1 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18
Variance:
The variance,
σ2,
is
simply
the
standard
deviation
squared.
5
3. Diversification
Recall from last week the assumption that investors are risk averse and therefore prefer less risk to more. Diversification provides a means of reducing risk faced by investors without sacrificing expected return by combining assets that don’t move perfectly together in a portfolio. Note that the ideas we are about to discuss can be extended to consider more than 2 assets.
Case 1: Perfect positive correlation (ρ12=+1)
Perfect Positive Correlation Between the Returns on Assets 1 and 2
0.18 0.16 0.14 Return on Asset 2 0.12 0.1 0.08 0.06 0.04 0.02 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Return on Asset 1
6
3.1 Measuring How Assets Move Together
Correlation and Covariance:
The covariance between variables X and Y, sXY, is a measure of association between the two variables. For example, we may be interested in whether there is an association between the return on a company’s stock and the return on the stock market in general: If the market always went up at the same time company’s stock went up, the covariance would be positive; If the return on the stock and the return on the market were not associated in any way, the covariance would be near zero; and, If when the market went up, the stock went down, the covariance would be negative.
ρX ,Y =
σ X ,Y σ Xσ Y
8
3.1 Measuring How Assets Move Together
Correlation:
We now consider the following exhaustive list of correlation values between two assets, Asset 1 and Asset 2, ρ12:
11
3.1 Measuring How Assets Move Together
Case 3: Non-perfect correlation (-1< ρ12<+1)
Correlation of 0.46 Between Returns on Assets 1 and 2
0.16 0.14 0.12 Return on Asset 2 0.1 0.08 0.06 0.04 0.02 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Return on Asset 1
2
Βιβλιοθήκη Baidu
2. Review of Statistics
Before we start discussing portfolio theory and the concept of diversification, we will briefly review the statistical terms discussed last lecture. These terms are important to understand as they are central to portfolio theory. More specifically, we will revisit the definitions of:
7
3.1 Measuring How Assets Move Together
Correlation and Covariance (Continued):
However, covariance is sensitive to the scale of measurement of X and Y and therefore the degree of association (as opposed to the sign) is difficult to interpret. Conversely, the correlation coefficient is a standardised measure of association between two variables. It is standardized as correlation measures must lie between negative one and one. This makes it easy to gauge the extent to which two variables are associated. The correlation coefficient, ρxy, is calculated as:
Random variables; Expected values; and, Standard deviation and variance.
3
2. Review of Statistics
Random Variables:
A random variable is one that can take on any number of different values. Each value has an associated probability of occurring. The uncertainty associated with the outcome of a random variable is described by a probability distribution, with the most commonly used distribution being the normal distribution, see below:
0.020 0.018 0.016 0.014 0.012 0.010 0.008 0.006 0.004 0.002 0.000
Probability density
-75
-65
-55
-45
-35
-25
-15
-5
15
25
35
45
55
65
Stock market return
75
5
4
2. Review of Statistics
12
3. Diversification
The optimal investment strategy in terms of constructing a portfolio to reduce risk depends on the properties of the two assets and how they are related to one another. Before we discuss the “ideal” diversification properties, we will go through how to calculate the expected return, standard deviation and variance on a 2-asset portfolio. We will then use these concepts to prove that diversification allows us to reduce risk without sacrificing expected return.
Case 1: Perfect positive correlation (ρ12=+1); Case 2: Perfect negative correlation (ρ12=-1); and. Case 3: Non-perfect correlation (-1< ρ12<+1).
9
3.1 Measuring How Assets Move Together
Expected Values:
The value we expect a random variable (X) to take is known as its expected value or mean.
Standard Deviation: Standard deviation, σ, is a measure of spread. It is based on how far each
Financial Markets and Systems Lecture 7
Diversification: Portfolio Theory
1
1. Lecture Overview
During this lecture, we will discuss the concepts of portfolio theory and diversification. Diversification allows an individual to reduce the risk of their investment without sacrificing any expected return simply by spreading their wealth over a portfolio comprising a number of assets in an appropriate way. During the course of the lecture we will discuss: What diversification is; How including multiple assets in a portfolio can achieve diversification; Which assets to include in order to achieve the greatest level of diversification; and, How to construct a diversified portfolio in practice.
10
3.1 Measuring How Assets Move Together
Case 2: Perfect negative correlation (ρ12=-1)
Perfect Negative Correlation Between Returns on Assets 1 and 2
0 0 -0.02 -0.04 Return on Asset 2 -0.06 -0.08 -0.1 -0.12 -0.14 -0.16 -0.18 Return on Asset 1 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18
Variance:
The variance,
σ2,
is
simply
the
standard
deviation
squared.
5
3. Diversification
Recall from last week the assumption that investors are risk averse and therefore prefer less risk to more. Diversification provides a means of reducing risk faced by investors without sacrificing expected return by combining assets that don’t move perfectly together in a portfolio. Note that the ideas we are about to discuss can be extended to consider more than 2 assets.
Case 1: Perfect positive correlation (ρ12=+1)
Perfect Positive Correlation Between the Returns on Assets 1 and 2
0.18 0.16 0.14 Return on Asset 2 0.12 0.1 0.08 0.06 0.04 0.02 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Return on Asset 1
6
3.1 Measuring How Assets Move Together
Correlation and Covariance:
The covariance between variables X and Y, sXY, is a measure of association between the two variables. For example, we may be interested in whether there is an association between the return on a company’s stock and the return on the stock market in general: If the market always went up at the same time company’s stock went up, the covariance would be positive; If the return on the stock and the return on the market were not associated in any way, the covariance would be near zero; and, If when the market went up, the stock went down, the covariance would be negative.