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FINS5513-fins代写

时间：2023-06-26

FINS5513 Lecture 1

Investment overview:

Risk, return, risk aversion,

diversification

2❑ 1.1 Constructing An Investment Portfolio

➢ Portfolio Basics

➢ Themes in Portfolio Management (Active/passive, traditional/alternative, growth/value)

❑ 1.2 Measuring Return and Risk

➢ Measuring Return (HPR, APR, EAR); Expected Returns

➢ Measuring Risk

➢ Sharpe Ratio

❑ 1.3 Risk Aversion and Investor Preference

➢ Preference, Utility, Risk aversion (mean-variance criterion), Indifference curves

❑ 1.4 Introduction to Modern Portfolio Theory

➢ Portfolio of risky assets

➢ Diversification

Lecture Outline

1.1 Constructing An

Investment Portfolio

FINS5513

4Portfolio Basics

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5❑ A combination of multiple assets and/or securities owned by an investor

➢ The aim of owning multiple assets is to achieve diversification

❑ What asset classes are available?

➢ Real assets vs Financial assets

➢ Financial assets

❖ Equity securities

• Common stock and Preferred stock

• Share market indices

❖ Fixed income securities

• The money market

• The bond market

• Bond market indices

❖ Derivatives securities

What is a Portfolio?

6❑ A combination of multiple assets and/or securities owned by an investor

➢ The aim of owning multiple assets is to achieve diversification

❑ We will be analysing assets/securities both individually as well as in the context of a portfolio

➢ Say an investor owns stocks A and B and wishes to add C

• In an isolated approach, an investor would look at the benefits and costs of C individually

• In a portfolio approach, an investor would compare the benefits and costs of portfolio A+B

to portfolio A+B+C

❑ Investors construct portfolios to achieve diversification and avoid the risks of investing all their

capital into a single asset

➢ As we will see, diversification allows investors to reduce risk without reducing the expected

rate of return on a portfolio

What is a Portfolio?

7❑ The process of portfolio construction can be undertaken top-down or bottom-up

❑ Top-down portfolio construction

➢ Asset allocation – choosing between broad asset classes and determining what

proportions of the portfolio should be invested in each asset class

➢ Top-down starts with asset allocation, then we decide which securities to hold in each asset

class

❑ Bottom-up portfolio construction

➢ Security selection – choosing which individual securities to hold within each asset class

➢ Construct portfolios from securities that are attractively priced with less concern for the

resulting asset allocation

➢ May lead to being overweight or underweight certain sectors or security types

The Investment Process

8Themes in Portfolio Management

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9❑ Active vs Passive management

➢ Active management – attempting to improve performance by identifying mispriced

securities (through security analysis and security selection)

• Active managers attempt to outperform a prescribed market benchmark such as the

S&P500 or the ASX200

➢ Passive management – holding diversified portfolios (little time spent on security selection)

• Passive managers attempt to track a prescribed market benchmark such as the S&P500

or the ASX200

➢ If markets are efficient, why bother with active management?

• If an investor were to believe in efficient markets, the only important decision is asset

allocation, not security selection

❑ There is a significant evidence that the majority of active fund managers underperform their

benchmarks, and overall returns from actively managed funds lag wider stock indices

Active vs Passive Management

10

Other Key Themes

❑ Traditional vs Alternative

➢ Traditional – long-only, unleveraged funds focused on equity, fixed income and/or

balanced (multi-asset) asset classes

➢ Alternative – hedge funds, private equity, venture capital - often leveraged

❑ Growth vs Value

➢ Growth – focuses on early stage emerging companies whose growth is expected to

significantly outperform wider industry trends. Often follows momentum and trends

➢ Value – concentrates on stocks that appear to be trading for less than their intrinsic

value. Focuses on low P/E, low Price/Book, and high free cash flow stocks

➢ Traditionally, value stocks have provided higher returns than growth, however this trend

appears to have reversed post GFC

11

Recent Trends

❑ Increase in passive investing due to lower cost and underperformance of active managers

❑ Increased variety and specialisation in ETFs eg thematic and factor based ETFs

❑ Increased use of high frequency trading and other quant methods using advanced statistical

and programming based techniques

➢ Attempt to take advantage of very short-term anomalies in the market

❑ Wider use of new data sources such as: social media; imagery and sensor data (customer

tracking, carpark monitoring, weather conditions etc); management psychological studies etc

to guide investing

❑ Robo-advisors and other algorithm-driven financial planning digital platforms (with no or little

human involvement)

1.2 Measuring Return and

Risk

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13

Return Measurement

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14

❑ Investor returns from holding an asset come from two basic sources:

➢ Income received periodically such as interest (debt security) and dividends (equity security)

➢ Capital gains (or losses) from the price of the asset increasing (decreasing)

❑ Holding Period Return (HPR) is the return on an asset during the period it is held

➢ The holding period ends when the asset is sold or matures/expires (for finite life assets):

0 1 + = +

Thus, we have:

=

− 0+

0

0 = Price at the beginning of period T

= Price at the end of period T

= Total income received over the holding period T (e.g., interest, coupons, dividends)

Holding Period Return

Income

component

Capital gain

component

15

You buy a share for $75 and sell it 9 months later for $84. It paid a div of $2.25:

o 0 = 75

o = 84

o = 2.25

Thus,

=

(84 − 75) + 2.25

75

= 0.15 = %

Holding Period Return: EXAMPLE

16

❑ The HPR gives the total return over the holding period without regard to the time period

❑ We can annualise the HPR in two ways:

➢ Assuming simple interest – we call this the Annualised Percentage Rate (APR)

➢ Assuming compound interest – we call this the Effective Annual Rate (EAR)

❑ Assume: T = holding period expressed in years (e.g., = 2 for 2-year hold; = 0.25 for 3

months; T=5.5 for 5 years and 6 months)

❑ Annualized percentage rate (APR): =

❑ Effective Annual Rate (EAR): 1 + EAR = 1 + HPR ⇒ = 1 +

1

− 1

APR and EAR

17

❑ The EAR accounts for compounding interest, not just simple interest (i.e. “interest on

interest”)

➢ If compounding is annual: EAR = APR

➢ If compounding is more frequent than annual: EAR > APR

➢ The relationship between EAR and APR is given by:

=

1+ −1

❑ Yields are quoted as APRs for short-term bills/bonds (often called the bond

equivalent yield)

❑ EAR is used to compare returns on investments with different time horizons

APR and EAR

18

You invest $10,000 in a fund. With income reinvestment, your investment is worth $16,000 after

4 years. Calculate the HPR, EAR, and APR

10,000 1 + = 16,000 ⇒ =

16,000 − 10,000

10,000

= 0.60 = 60%

10,000 1 + 4 = 16,000 ⇒ = 1.6

1

4 – 1 = 12.47% . .

=

1 + − 1

=

=

1.1247 4 − 1

4

= 15% . .

APR and EAR: EXAMPLE

19

Measuring Expected Return

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20

❑ On a forward-looking basis under uncertainty, we form return expectations. Ex-ante

analysis is expectations-based analysis before an event

➢ Ex-ante analysis attempts to place probabilities on possible future scenarios

❑ Expected return () is given by:

= σ × ()

where = , () = ()

❑ In practice, expected return is often estimated using the average (or mean) historical

sample rates of return, denoted ത by using the formula:

ҧ =

1

=1

where = , =

Expected Return

21

After extensive simulations, Quant Fund has determined that the distribution of returns for

Walmart (WMT) in different probability weighted economic future scenarios is given by:

The expected return () is the probability weighted return:

() = 0.25 × 0.38 + 0.50 × 0.14 + 0.20 × −0.075 + 0.05 × −0.32 = 13.40%

Expected Return: EXAMPLE 1

Economic Scenario Scenario Probability Scenario Return

Boom 0.25 38.0%

Growth 0.50 14.0%

Flat 0.20 -7.5%

Recession 0.05 -32.0%

22

Suppose Quant Fund analysed 10-year historical returns for WMT as shown:

Estimate the expected return of WMT:

ҧ =

0.233 + 0.302 − 0.034 + 0.465 + 0.16 − 0.266 + 0.119 + 0.182 + 0.17 + 0 .139

10

= 14.70%

Expected Return: EXAMPLE 2

2020 2019 2018 2017 2016 2015 2014 2013 2012 2011

WMT Returns 23.3% 30.2% -3.4% 46.5% 16.0% -26.6% 11.9% 18.2% 17.0% 13.9%

23

Measuring Risk

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24

❑ We seek to maximise return because return maximises wealth. However, we seek return

in a world of uncertainty (i.e., we face risk)

➢ Think of return as the “reward” and risk as the “cost” of that reward. It is important to

incorporate “risk” appropriately into decision making

❑ In finance, risk refers to the likelihood of deviations of the realised outcomes from

expected return. Usually measured by variance (2) or standard deviation ():

2 =

[ − ]2

where = ; () = ()

❑ In practice, we usually estimate the unbiased sample variance 2 from historic data:

2 =

1

−1

σ=1

− ҧ

2

where = ; ҧ = ; =

What is Risk?

25

Determine the standard deviation of WMT based on the information given below:

• = 0.25 × 0.38 + 0.50 × 0.14 + 0.20 × −.075 + 0.05 × −0.32 = 13.40%

• 2 = σ [ − ]

2 = 0.25 × 0.38 − 0.134 2 + 0.50 × 0.14 − 0.134 2 +

0.20 × −0 .075 − 0.134 2 + 0.05 × −0.32 − 0.134 2

=

0.034 ⇒ = 0.034 =

18.49%

Measuring Risk : EXAMPLE 1

Economic Scenario Scenario Probability Scenario Return

Boom 0.25 38.0%

Growth 0.50 14.0%

Flat 0.20 -7.5%

Recession 0.05 -32.0%

26

Suppose Quant Fund analysed 10-year historical returns for WMT as shown:

Estimate the standard deviation of WMT

▪ ҧ =

0.233 +0.302 −0.034 +0.465 +0.16 −0.266 +0.119 +0.182 +0.17 +0 .139

10

= 14.70%

▪ 2 =

1

10−1

ൣ

൧

0.233 − 0.147 2 + 0.302 − 0.147 2 + −0.034 − 0.147 2 + (

)

0.465 −

0.147 2 + 0.16 − 0 .147 2 + −0.266 − 0.147 2 + 0.119 − 0.147 2 + (

)

0.182 −

0.147 2 + 0.17 − 0.147 2 + 0.139 − 0.147 2 = 0.3386 ⇒ ො = 0.3386 = 19.40%

Measuring Risk: EXAMPLE 2

2020 2019 2018 2017 2016 2015 2014 2013 2012 2011

WMT Returns 23.3% 30.2% -3.4% 46.5% 16.0% -26.6% 11.9% 18.2% 17.0% 13.9%

27

Sharpe Ratio

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28

Reward to Volatility (Sharpe) Ratio

❑ Now that we have quantified return and risk individually, how do we relate the

risk/reward relationship in one measure?

❑ We often look at the Risk Premium (i.e., the “excess return” above the risk-free

rate), rather than the (raw) total return

➢ Part of the return can be earned for no risk by investing in a risk-free asset

❑ The Sharpe Ratio for security i is given by:

=

−

= risk-free rate

− = Risk premium for security i

= Standard deviation of excess returns for security i

❑ The Sharpe ratio measures return per unit of risk. The higher the Sharpe ratio - the

higher the incremental return received per unit of risk

1.3 Risk Aversion and

Investor Preference

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30

• Assume you have one million dollars to invest. There are three assets available in

the market with following properties:

= 20%, = 20%

= 25%, = 25%

= 20%, = 25%

, = 0.75, , = 0.5, , = 0.6

• Questions to think about:

➢ What do you care about – your preference?

➢ What combination will suit your preference?

How do you make investment decision?

31

Preference and Utility

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32

❑ Suppose you prefer an apple to a banana. We say that you have a preference for the

apple over the banana

➢ Alternatively, we can say that you enjoy a higher utility from the apple over the

banana

❑ As economists, we are interested in satisfying as many such preferences as possible

❑ In order to work with preferences mathematically, we use utility functions to

reflect/capture preferences

➢ A utility function assigns a value to each outcome so that preferred outcomes get

higher utility values (e.g., in the previous example, apple should be assigned a

higher utility score than banana to reflect your preference)

➢ A perfect utility function should be able to used to assign values to every possible

outcome. However, in practice it is difficult, if not impossible.

Preference and Utility

33

❑ For simplicity, we model utility as depending only on wealth

➢ Essentially assume that the more money an individual has, the better he/she is

able to achieve preferred outcomes.

❑ In finance, we are dealing with uncertain outcomes/wealth from the investment

➢ We are particularly interested in two human natures affecting our

preferences/utilities – greedy and fear:

• More is better (i.e., maximise return)

• More certainty is better (i.e., risk aversion)

Preference and Utility

34

Risk Aversion

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35

❑This is a central concept in finance. It means that we prefer certain outcomes to

stochastic ones. Modern Portfolio Theory rests on the assumption investors are risk

averse.

❑To induce risk averse investors to nevertheless take investments with risk/uncertain

outcomes, we need to give them an incentive – a risk premium

➢ Historical market returns show there is a risk premium in the market (indicates risk

aversion):

• Since 1926, U.S. risk-free assets (1-month T-bills) returned ~3.4% annually while

risky assets (US stocks) returned ~11.7% – resulting in a ~8.3% risk premium with

= 20.4%

• Market takes additional risk only for a commensurate return – indicating risk

aversion

❑How a utility function capture risk aversion:

➢ Let’s take log utility function as an example

Risk aversion

36

❑ A common specification of the wealth utility function is () = ln()

❑ The logarithmic expression results in a concave function

➢ The concavity indicates that the incremental utility we gain from increases in

wealth is less than the utility we lose from equivalent decreases in wealth

➢ The concavity captures risk aversion – risk averse investors would not take a

50/50 bet

Log utility

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

1 11 21 31 41

U

ti

lit

y

Wealth

Utility curve

Wealth level B: 41

Wealth level A: 1

Average utility from A and B

Wealth level: average of A and B

37

❑ As wealth is dependent on risk and return, in this course, we assume a quadratic utility

function: = −

1

2

2

➢ is the Utility score we assign to any investment with a given risk(2) and expected return

(()). is a measure of the risk aversion.

❑ For a risk-free asset, = , as is a known constant and 2 = 0

➢ Therefore, what is the meaning of the utility score U for a risky investment?

• It is the risk-free rate which would result in an investor being indifferent between the risk-

free asset and a risky investment with the same utility score – often called the certainty

equivalent return

Our utility function - Quadratic Utility Function

38

❑ Three investors are analysing the Low-Risk, Medium-Risk and High-Risk funds from earlier.

For an Aggressive investor, the risk aversion coefficient A = 2.0; Moderate investor A = 3.5;

Conservative investor A = 5.0. Which fund would each investor choose?

➢ Replace the risk aversion coefficients and fund return and risk into each utility function.

Then rank each fund based on its utility score:

❑ Aggressive investor chooses the High-Risk fund, the others choose the Medium-Risk fund

➢ Risk aversion doesn’t mean the investor doesn’t take risk – rather it means the investor

puts a higher price (return) on taking risk. For example: even the conservative investor

does not pick the Low-Risk fund

Example: Applying the Utility Function

Investor Risk

Aversion

A

Low Risk Fund

Utility Score

E(r) = 7% σ = 5%

Medium-Risk Fund

Utility Score

E(r) = 9% σ = 10%

High-Risk Fund

Utility Score

E(r) = 13% σ = 20%

Aggressive 2.0 .07 – ½ x 2.0 x .052 = .0675 .09 – ½ x 2.0 x .12 = .0800 .13 – ½ x 2.0 x .22 = .090

Moderate 3.5 .07 – ½ x 3.5 x .052 = .0656 .09 – ½ x 3.5 x .12 = .0725 .13 – ½ x 3.5 x .22 = .060

Conservative 5.0 .07 – ½ x 5.0 x .052 = .0638 .09 – ½ x 5.0 x .12 = .0650 .13 – ½ x 5.0 x .22 = .030

39

❑ The quadratic utility function basically expresses the idea that we like high expected

returns and dislike high risk (return uncertainty):

➢ which is formally known as mean-variance criterion

➢ We can use mean-variance criterion to quickly rank some portfolios

❑ The Mean-Variance Criterion states:

Portfolio A dominates portfolio B if:

≥

and*

≤

* At least one inequality must be strict to rule out indifference

Mean-Variance Criterion

40

Mean-Variance Criterion: EXAMPLE

1

2 3

4

Expected Return

Variance or Standard Deviation

• 2 dominates 1; has a higher return

• 2 dominates 3; has a lower risk

• 4 dominates 3; has a higher return

➢ 2 dominates 1 – same risk but higher return

➢ 2 dominates 3 – some return but lower risk

➢ 4 dominates 3 – some risk but higher return

41

❑ For each individual investor, the unique element in the utility function is the value of

A

❑ So how do we estimate an individual’s risk aversion coefficient?

➢ Often depends on life cycle and personality type

➢ Questionnaires

➢ Discussion with broker/advisor

➢ Observe how much people are willing to pay to avoid risk

➢ Observe individuals’ decisions when confronted with risk

• Would you take $100 for certain or flip of a coin for $200

❑ Note that the higher the risk aversion coefficient A the more risk averse the

investor:

➢ Conservative investors have high risk aversion coefficients

➢ Aggressive investors have low risk aversion coefficients

Estimating Risk Aversion

42

Indifference Curves

FINS5513

43

Indifference Curves

❑ We can illustrate our preferences through indifference curves

➢ Graphical representation of the utility function. Plotted in the risk-return ( − ) space

that connect points giving equal utility

➢ For example, to draw the indifference curve for U = 10%, choose all asset portfolio

combinations of E(r) and σ which yield a utility score of 10%

➢ Note that two indifference curves with different utility levels never intersect

E(r)

σ

U = E(r) – ½* A*σ2 = 10%

U = 15%

U = 20%

U = 25%

❑ Indifference curves for risk-

averse (A > 0) investors are

upward sloping

44

How to Plot an Indifference Curve

❑ Each plot point on an indifference curve represents a risk and return combination which

provides the same utility score

❑ For an Aggressive investor with risk aversion coefficient A=2, and a Conservative investor

with A=5, plot two indifference curves with U=0.03 and U=0.09

Higher Utility

A=5 is

steeper

than A=2

Certainty

equivalent

return.

Plot first.

➢ Given a specific value of A,

indifference curves above and to

the left offer higher utility than

lower curves and don’t intersect

➢ More risk averse investors (A)

have steeper indifference curves

(higher E(r) for each increase in

)

1.4 Forming portfolios

and Diversification

FINS5513

46

❑ What is our objective when we make an investment?

➢ To achieve the optimal investment outcome that maximises our utility

➢ Utility from an investment depends on its expected return and risk

❑ The task is therefore to construct the optimal portfolio from all available assets,

which has the preferred combination of expected return and risk, therefore providing

the highest utility (among all possible portfolios)

❑ Let’s start simply with two assets

Forming Portfolios

47

Portfolios of Two Risky Assets

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48

❑ Portfolio returns are simply a weighted average of individual asset classes in the

portfolio. For example, consider Bonds and Equities asset classes:

= +

where

o =

o = ℎ

o =

o = ℎ

o =

❑ Similarly, portfolio expected return is a weighted average of expected returns of the

two asset classes in the portfolio:

() = () + ()

Portfolios of Two Risky Assets: Return

49

❑ Calculating portfolio risk on the other hand is not simply a weighted average. That

is, it is not simply a case of taking the individual risks of the portfolio’s component

assets

❑ We must also take into account the covariance between asset classes (or individual

assets) within the portfolio:

2 =

2

2 +

2

2 + 2 ,

o

2 =

o

2 =

o

2 =

o ( , ) =

Portfolios of Two Risky Assets: Risk

50

❑ The covariance between two variables refers to their tendency to be higher or lower than their

respective expected values at the same time. It is the sum of deviations from the expected

return of two assets in different states

, = σ()[ − ()][ − ()]

➢ = ;

➢ = ()

➢ = ()

❑ In practice, we usually estimate the unbiased sample covariance from historic data:

, =

1

− 1

=1

− ഥ − ഥ

➢ ( − ഥ ) = ℎ

➢ ( − ഥ ) = ℎ

➢ =

Covariance

51

❑ Correlation scales the covariance to 1:

=

,

❑ Range of values for : − 1.0 ≤ ≤ +1.0

❑ Based on the definition of correlation stated above, covariance of returns between

Bonds and Equities can also be expressed in terms of their correlation:

, =

❑ Note: if you are not familiar with the basic concepts and the related calculation of

variance, covariance, correlation, please do spend extra time and effort to make sure

you are familiar with them since these are the presumed knowledge for this course. Most

of the calculation-based questions could be built upon these basic concepts!!!!!

Correlation

52

Multiple Risky Assets

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53

❑ The return on a portfolio of more than two assets, is also simply the weighted

average of the returns of the assets that make up the portfolio:

= 11 +22 +⋯+ =

=1

= portfolio weight of asset (i.e. the fraction of portfolio value invested in asset )

❑ Similarly, the expected return on a portfolio is a weighted average of the expected

returns on its component assets:

=

=1

()

Portfolios of Multiple Risky Assets: Return

54

❑ For assets (or asset classes), portfolio risk can be calculated with a covariance

matrix, consisting of:

➢ variances

➢ (2 – )/2 covariances

❑ The generalised result for portfolio variance (risk) is given by:

2 = (

=1

,

=1

)

❑ We can set up a variance-covariance matrix from this equation

Portfolios of Multiple Risky Assets: Risk

55

❑ For a 3-asset portfolio:

=

❑ For a 5-asset portfolio

=

Portfolios of Multiple Risky Assets: Risk

11

22

33

11

22

33

11

22

33

44

55

11

22

33

44

55

1

21

2 + 12 1, 2 + 13 1, 3

2

22

2 + 21 2, 1 + 23 2, 3

3

23

2 + 31 3, 1 + 32 3, 2

1

21

2 + 2

22

2 + 3

23

2 + 2[12 1, 2 + 13 1, 3 +23 2, 3 ]

1

21

2 + 12 1, 2 + 13 1, 3 + 14 1, 4 + 15 1, 5

2

22

2 + 21 2, 1 + 23 2, 3 + 24 2, 4 + 25 2, 5

3

23

2 + 31 3, 1 + 32 3, 2 +34 3, 4 + 35 3, 5

4

24

2 + 41 4, 1 + 42 4, 2 + 43 4, 3 + 45 4, 5

5

25

2 + 51 5, 1 + 52 5, 2 + 53 5, 3 + 54 5, 4

1

21

2 + 2

22

2 + 3

23

2 +4

2 4

2 + 5

25

2 + 2[12 1, 2 + 13 1, 3

+ 14 1, 4 + 15 1, 5 + 23 2, 3 + 2 4 2, 4

+ 25 2, 5 + 34 3, 4 + 35 3, 5 + 45 4, 5 ]

56

Diversification Benefit

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57

❑ Recall portfolio variance for a 2-asset portfolio:

2 = 1

21

2 + 2

22

2 + 212(1, 2)

Which can be restated in terms of correlation as:

2 = 1

21

2 +2

22

2 + 2121212

❑ If asset 1 and 2 are perfectly correlated ρ = 1, we have:

2 = 1

21

2 + 2

22

2 + 21212 = (11 + 22)

2

= 11 +22

❑ In other words, the standard deviation of a portfolio with perfectly correlated assets is the

weighted average of the component asset standard deviations

➢ For perfectly correlated assets, both portfolio risk and return are a weighted average

➢ There is no risk reduction benefit in combining perfectly correlated assets

Perfect Correlation

58

❑ For perfectly correlated assets A and B, we can restate the risk and return equations as

follows:

Risk Return

Perfect Correlation

❑ So graphically, tells us where we are along

a line connecting point A (a portfolio consisting

of 100% in A i.e. = 0) and point B (a

portfolio consisting of 100% in B i.e. = 1)

❑ In other words, for perfectly correlated

assets the risk-return relationship is linear

59

❑ For lower values of (i.e., imperfect correlation), the portfolio standard deviation

must be lower:

2 =

2

2 +

2

2 + 2 < ( +)

2

< +

Diversification Benefit

60

❑ Portfolio weights change as we move from

E to B. The lower the correlation, the more

portfolio risk drops as we move from E to B

at each expected return level

❑ The curvature is the diversification benefit.

The lower the correlation the bigger the

bulge (diversification benefit) - same

portfolio expected return for lower portfolio

risk

❑ The diversification benefit depends on the

correlation:

➢ If = +1.0, no risk reduction is possible

➢ If −1 < <1, some level of risk reduction

is possible

➢ If = -1.0, a riskless hedge is possible

Diversification Benefit

0%

2%

4%

6%

8%

10%

12%

0% 5% 10% 15% 20% 25%

P

o

r

tf

o

li

o

E

x

p

e

c

te

d

R

e

tu

r

n

%

Portfolio Standard Deviation %

Diversification Benefit

Correlation = 1 Correlation = .7 Correlation = 0

Correlation = -.71 Correlation = -1

B

E

Diversification benefit

100% invested

in Bonds

100% invested

in Equities

61

❑ Combining two or more imperfectly correlated assets in one portfolio is called diversification

➢ The risk reduction is called a diversification benefit

➢ This concept is at the heart of portfolio theory

➢ Sophisticated version of “not putting all your eggs in one basket”

➢ The lower the correlation the better (same portfolio return for lower portfolio risk)

➢ We typically want to combine many assets with low (ideally negative) correlations to

maximize the diversification benefit

❑ Why is there a diversification benefit?

➢ Low correlated assets are unlikely to return below their respective means at the same time

(cancels out some portfolio risk)

➢ If correlations are negative, they even work as insurance for each other (a riskless hedge

may be possible)

Diversification Benefit

62

❑ In this example, as we add more assets to our portfolio the portfolio standard deviation drops

from ~50% to ~19% - this is the diversification benefit

➢ However, the diversification benefit increases rapidly at first, then diminishes significantly

Portfolio Diversification

63

❑ BKM Chapter 6 and 7

❑ 2.1. Optimal Risky Asset Portfolio Construction

❑ 2.2. Introducing the Risk-Free Asset

❑ 2.3. Deriving the Optimal Portfolio

Next Lecture

Investment overview:

Risk, return, risk aversion,

diversification

2❑ 1.1 Constructing An Investment Portfolio

➢ Portfolio Basics

➢ Themes in Portfolio Management (Active/passive, traditional/alternative, growth/value)

❑ 1.2 Measuring Return and Risk

➢ Measuring Return (HPR, APR, EAR); Expected Returns

➢ Measuring Risk

➢ Sharpe Ratio

❑ 1.3 Risk Aversion and Investor Preference

➢ Preference, Utility, Risk aversion (mean-variance criterion), Indifference curves

❑ 1.4 Introduction to Modern Portfolio Theory

➢ Portfolio of risky assets

➢ Diversification

Lecture Outline

1.1 Constructing An

Investment Portfolio

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4Portfolio Basics

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5❑ A combination of multiple assets and/or securities owned by an investor

➢ The aim of owning multiple assets is to achieve diversification

❑ What asset classes are available?

➢ Real assets vs Financial assets

➢ Financial assets

❖ Equity securities

• Common stock and Preferred stock

• Share market indices

❖ Fixed income securities

• The money market

• The bond market

• Bond market indices

❖ Derivatives securities

What is a Portfolio?

6❑ A combination of multiple assets and/or securities owned by an investor

➢ The aim of owning multiple assets is to achieve diversification

❑ We will be analysing assets/securities both individually as well as in the context of a portfolio

➢ Say an investor owns stocks A and B and wishes to add C

• In an isolated approach, an investor would look at the benefits and costs of C individually

• In a portfolio approach, an investor would compare the benefits and costs of portfolio A+B

to portfolio A+B+C

❑ Investors construct portfolios to achieve diversification and avoid the risks of investing all their

capital into a single asset

➢ As we will see, diversification allows investors to reduce risk without reducing the expected

rate of return on a portfolio

What is a Portfolio?

7❑ The process of portfolio construction can be undertaken top-down or bottom-up

❑ Top-down portfolio construction

➢ Asset allocation – choosing between broad asset classes and determining what

proportions of the portfolio should be invested in each asset class

➢ Top-down starts with asset allocation, then we decide which securities to hold in each asset

class

❑ Bottom-up portfolio construction

➢ Security selection – choosing which individual securities to hold within each asset class

➢ Construct portfolios from securities that are attractively priced with less concern for the

resulting asset allocation

➢ May lead to being overweight or underweight certain sectors or security types

The Investment Process

8Themes in Portfolio Management

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9❑ Active vs Passive management

➢ Active management – attempting to improve performance by identifying mispriced

securities (through security analysis and security selection)

• Active managers attempt to outperform a prescribed market benchmark such as the

S&P500 or the ASX200

➢ Passive management – holding diversified portfolios (little time spent on security selection)

• Passive managers attempt to track a prescribed market benchmark such as the S&P500

or the ASX200

➢ If markets are efficient, why bother with active management?

• If an investor were to believe in efficient markets, the only important decision is asset

allocation, not security selection

❑ There is a significant evidence that the majority of active fund managers underperform their

benchmarks, and overall returns from actively managed funds lag wider stock indices

Active vs Passive Management

10

Other Key Themes

❑ Traditional vs Alternative

➢ Traditional – long-only, unleveraged funds focused on equity, fixed income and/or

balanced (multi-asset) asset classes

➢ Alternative – hedge funds, private equity, venture capital - often leveraged

❑ Growth vs Value

➢ Growth – focuses on early stage emerging companies whose growth is expected to

significantly outperform wider industry trends. Often follows momentum and trends

➢ Value – concentrates on stocks that appear to be trading for less than their intrinsic

value. Focuses on low P/E, low Price/Book, and high free cash flow stocks

➢ Traditionally, value stocks have provided higher returns than growth, however this trend

appears to have reversed post GFC

11

Recent Trends

❑ Increase in passive investing due to lower cost and underperformance of active managers

❑ Increased variety and specialisation in ETFs eg thematic and factor based ETFs

❑ Increased use of high frequency trading and other quant methods using advanced statistical

and programming based techniques

➢ Attempt to take advantage of very short-term anomalies in the market

❑ Wider use of new data sources such as: social media; imagery and sensor data (customer

tracking, carpark monitoring, weather conditions etc); management psychological studies etc

to guide investing

❑ Robo-advisors and other algorithm-driven financial planning digital platforms (with no or little

human involvement)

1.2 Measuring Return and

Risk

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13

Return Measurement

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14

❑ Investor returns from holding an asset come from two basic sources:

➢ Income received periodically such as interest (debt security) and dividends (equity security)

➢ Capital gains (or losses) from the price of the asset increasing (decreasing)

❑ Holding Period Return (HPR) is the return on an asset during the period it is held

➢ The holding period ends when the asset is sold or matures/expires (for finite life assets):

0 1 + = +

Thus, we have:

=

− 0+

0

0 = Price at the beginning of period T

= Price at the end of period T

= Total income received over the holding period T (e.g., interest, coupons, dividends)

Holding Period Return

Income

component

Capital gain

component

15

You buy a share for $75 and sell it 9 months later for $84. It paid a div of $2.25:

o 0 = 75

o = 84

o = 2.25

Thus,

=

(84 − 75) + 2.25

75

= 0.15 = %

Holding Period Return: EXAMPLE

16

❑ The HPR gives the total return over the holding period without regard to the time period

❑ We can annualise the HPR in two ways:

➢ Assuming simple interest – we call this the Annualised Percentage Rate (APR)

➢ Assuming compound interest – we call this the Effective Annual Rate (EAR)

❑ Assume: T = holding period expressed in years (e.g., = 2 for 2-year hold; = 0.25 for 3

months; T=5.5 for 5 years and 6 months)

❑ Annualized percentage rate (APR): =

❑ Effective Annual Rate (EAR): 1 + EAR = 1 + HPR ⇒ = 1 +

1

− 1

APR and EAR

17

❑ The EAR accounts for compounding interest, not just simple interest (i.e. “interest on

interest”)

➢ If compounding is annual: EAR = APR

➢ If compounding is more frequent than annual: EAR > APR

➢ The relationship between EAR and APR is given by:

=

1+ −1

❑ Yields are quoted as APRs for short-term bills/bonds (often called the bond

equivalent yield)

❑ EAR is used to compare returns on investments with different time horizons

APR and EAR

18

You invest $10,000 in a fund. With income reinvestment, your investment is worth $16,000 after

4 years. Calculate the HPR, EAR, and APR

10,000 1 + = 16,000 ⇒ =

16,000 − 10,000

10,000

= 0.60 = 60%

10,000 1 + 4 = 16,000 ⇒ = 1.6

1

4 – 1 = 12.47% . .

=

1 + − 1

=

=

1.1247 4 − 1

4

= 15% . .

APR and EAR: EXAMPLE

19

Measuring Expected Return

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20

❑ On a forward-looking basis under uncertainty, we form return expectations. Ex-ante

analysis is expectations-based analysis before an event

➢ Ex-ante analysis attempts to place probabilities on possible future scenarios

❑ Expected return () is given by:

= σ × ()

where = , () = ()

❑ In practice, expected return is often estimated using the average (or mean) historical

sample rates of return, denoted ത by using the formula:

ҧ =

1

=1

where = , =

Expected Return

21

After extensive simulations, Quant Fund has determined that the distribution of returns for

Walmart (WMT) in different probability weighted economic future scenarios is given by:

The expected return () is the probability weighted return:

() = 0.25 × 0.38 + 0.50 × 0.14 + 0.20 × −0.075 + 0.05 × −0.32 = 13.40%

Expected Return: EXAMPLE 1

Economic Scenario Scenario Probability Scenario Return

Boom 0.25 38.0%

Growth 0.50 14.0%

Flat 0.20 -7.5%

Recession 0.05 -32.0%

22

Suppose Quant Fund analysed 10-year historical returns for WMT as shown:

Estimate the expected return of WMT:

ҧ =

0.233 + 0.302 − 0.034 + 0.465 + 0.16 − 0.266 + 0.119 + 0.182 + 0.17 + 0 .139

10

= 14.70%

Expected Return: EXAMPLE 2

2020 2019 2018 2017 2016 2015 2014 2013 2012 2011

WMT Returns 23.3% 30.2% -3.4% 46.5% 16.0% -26.6% 11.9% 18.2% 17.0% 13.9%

23

Measuring Risk

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24

❑ We seek to maximise return because return maximises wealth. However, we seek return

in a world of uncertainty (i.e., we face risk)

➢ Think of return as the “reward” and risk as the “cost” of that reward. It is important to

incorporate “risk” appropriately into decision making

❑ In finance, risk refers to the likelihood of deviations of the realised outcomes from

expected return. Usually measured by variance (2) or standard deviation ():

2 =

[ − ]2

where = ; () = ()

❑ In practice, we usually estimate the unbiased sample variance 2 from historic data:

2 =

1

−1

σ=1

− ҧ

2

where = ; ҧ = ; =

What is Risk?

25

Determine the standard deviation of WMT based on the information given below:

• = 0.25 × 0.38 + 0.50 × 0.14 + 0.20 × −.075 + 0.05 × −0.32 = 13.40%

• 2 = σ [ − ]

2 = 0.25 × 0.38 − 0.134 2 + 0.50 × 0.14 − 0.134 2 +

0.20 × −0 .075 − 0.134 2 + 0.05 × −0.32 − 0.134 2

=

0.034 ⇒ = 0.034 =

18.49%

Measuring Risk : EXAMPLE 1

Economic Scenario Scenario Probability Scenario Return

Boom 0.25 38.0%

Growth 0.50 14.0%

Flat 0.20 -7.5%

Recession 0.05 -32.0%

26

Suppose Quant Fund analysed 10-year historical returns for WMT as shown:

Estimate the standard deviation of WMT

▪ ҧ =

0.233 +0.302 −0.034 +0.465 +0.16 −0.266 +0.119 +0.182 +0.17 +0 .139

10

= 14.70%

▪ 2 =

1

10−1

ൣ

൧

0.233 − 0.147 2 + 0.302 − 0.147 2 + −0.034 − 0.147 2 + (

)

0.465 −

0.147 2 + 0.16 − 0 .147 2 + −0.266 − 0.147 2 + 0.119 − 0.147 2 + (

)

0.182 −

0.147 2 + 0.17 − 0.147 2 + 0.139 − 0.147 2 = 0.3386 ⇒ ො = 0.3386 = 19.40%

Measuring Risk: EXAMPLE 2

2020 2019 2018 2017 2016 2015 2014 2013 2012 2011

WMT Returns 23.3% 30.2% -3.4% 46.5% 16.0% -26.6% 11.9% 18.2% 17.0% 13.9%

27

Sharpe Ratio

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28

Reward to Volatility (Sharpe) Ratio

❑ Now that we have quantified return and risk individually, how do we relate the

risk/reward relationship in one measure?

❑ We often look at the Risk Premium (i.e., the “excess return” above the risk-free

rate), rather than the (raw) total return

➢ Part of the return can be earned for no risk by investing in a risk-free asset

❑ The Sharpe Ratio for security i is given by:

=

−

= risk-free rate

− = Risk premium for security i

= Standard deviation of excess returns for security i

❑ The Sharpe ratio measures return per unit of risk. The higher the Sharpe ratio - the

higher the incremental return received per unit of risk

1.3 Risk Aversion and

Investor Preference

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30

• Assume you have one million dollars to invest. There are three assets available in

the market with following properties:

= 20%, = 20%

= 25%, = 25%

= 20%, = 25%

, = 0.75, , = 0.5, , = 0.6

• Questions to think about:

➢ What do you care about – your preference?

➢ What combination will suit your preference?

How do you make investment decision?

31

Preference and Utility

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32

❑ Suppose you prefer an apple to a banana. We say that you have a preference for the

apple over the banana

➢ Alternatively, we can say that you enjoy a higher utility from the apple over the

banana

❑ As economists, we are interested in satisfying as many such preferences as possible

❑ In order to work with preferences mathematically, we use utility functions to

reflect/capture preferences

➢ A utility function assigns a value to each outcome so that preferred outcomes get

higher utility values (e.g., in the previous example, apple should be assigned a

higher utility score than banana to reflect your preference)

➢ A perfect utility function should be able to used to assign values to every possible

outcome. However, in practice it is difficult, if not impossible.

Preference and Utility

33

❑ For simplicity, we model utility as depending only on wealth

➢ Essentially assume that the more money an individual has, the better he/she is

able to achieve preferred outcomes.

❑ In finance, we are dealing with uncertain outcomes/wealth from the investment

➢ We are particularly interested in two human natures affecting our

preferences/utilities – greedy and fear:

• More is better (i.e., maximise return)

• More certainty is better (i.e., risk aversion)

Preference and Utility

34

Risk Aversion

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35

❑This is a central concept in finance. It means that we prefer certain outcomes to

stochastic ones. Modern Portfolio Theory rests on the assumption investors are risk

averse.

❑To induce risk averse investors to nevertheless take investments with risk/uncertain

outcomes, we need to give them an incentive – a risk premium

➢ Historical market returns show there is a risk premium in the market (indicates risk

aversion):

• Since 1926, U.S. risk-free assets (1-month T-bills) returned ~3.4% annually while

risky assets (US stocks) returned ~11.7% – resulting in a ~8.3% risk premium with

= 20.4%

• Market takes additional risk only for a commensurate return – indicating risk

aversion

❑How a utility function capture risk aversion:

➢ Let’s take log utility function as an example

Risk aversion

36

❑ A common specification of the wealth utility function is () = ln()

❑ The logarithmic expression results in a concave function

➢ The concavity indicates that the incremental utility we gain from increases in

wealth is less than the utility we lose from equivalent decreases in wealth

➢ The concavity captures risk aversion – risk averse investors would not take a

50/50 bet

Log utility

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

1 11 21 31 41

U

ti

lit

y

Wealth

Utility curve

Wealth level B: 41

Wealth level A: 1

Average utility from A and B

Wealth level: average of A and B

37

❑ As wealth is dependent on risk and return, in this course, we assume a quadratic utility

function: = −

1

2

2

➢ is the Utility score we assign to any investment with a given risk(2) and expected return

(()). is a measure of the risk aversion.

❑ For a risk-free asset, = , as is a known constant and 2 = 0

➢ Therefore, what is the meaning of the utility score U for a risky investment?

• It is the risk-free rate which would result in an investor being indifferent between the risk-

free asset and a risky investment with the same utility score – often called the certainty

equivalent return

Our utility function - Quadratic Utility Function

38

❑ Three investors are analysing the Low-Risk, Medium-Risk and High-Risk funds from earlier.

For an Aggressive investor, the risk aversion coefficient A = 2.0; Moderate investor A = 3.5;

Conservative investor A = 5.0. Which fund would each investor choose?

➢ Replace the risk aversion coefficients and fund return and risk into each utility function.

Then rank each fund based on its utility score:

❑ Aggressive investor chooses the High-Risk fund, the others choose the Medium-Risk fund

➢ Risk aversion doesn’t mean the investor doesn’t take risk – rather it means the investor

puts a higher price (return) on taking risk. For example: even the conservative investor

does not pick the Low-Risk fund

Example: Applying the Utility Function

Investor Risk

Aversion

A

Low Risk Fund

Utility Score

E(r) = 7% σ = 5%

Medium-Risk Fund

Utility Score

E(r) = 9% σ = 10%

High-Risk Fund

Utility Score

E(r) = 13% σ = 20%

Aggressive 2.0 .07 – ½ x 2.0 x .052 = .0675 .09 – ½ x 2.0 x .12 = .0800 .13 – ½ x 2.0 x .22 = .090

Moderate 3.5 .07 – ½ x 3.5 x .052 = .0656 .09 – ½ x 3.5 x .12 = .0725 .13 – ½ x 3.5 x .22 = .060

Conservative 5.0 .07 – ½ x 5.0 x .052 = .0638 .09 – ½ x 5.0 x .12 = .0650 .13 – ½ x 5.0 x .22 = .030

39

❑ The quadratic utility function basically expresses the idea that we like high expected

returns and dislike high risk (return uncertainty):

➢ which is formally known as mean-variance criterion

➢ We can use mean-variance criterion to quickly rank some portfolios

❑ The Mean-Variance Criterion states:

Portfolio A dominates portfolio B if:

≥

and*

≤

* At least one inequality must be strict to rule out indifference

Mean-Variance Criterion

40

Mean-Variance Criterion: EXAMPLE

1

2 3

4

Expected Return

Variance or Standard Deviation

• 2 dominates 1; has a higher return

• 2 dominates 3; has a lower risk

• 4 dominates 3; has a higher return

➢ 2 dominates 1 – same risk but higher return

➢ 2 dominates 3 – some return but lower risk

➢ 4 dominates 3 – some risk but higher return

41

❑ For each individual investor, the unique element in the utility function is the value of

A

❑ So how do we estimate an individual’s risk aversion coefficient?

➢ Often depends on life cycle and personality type

➢ Questionnaires

➢ Discussion with broker/advisor

➢ Observe how much people are willing to pay to avoid risk

➢ Observe individuals’ decisions when confronted with risk

• Would you take $100 for certain or flip of a coin for $200

❑ Note that the higher the risk aversion coefficient A the more risk averse the

investor:

➢ Conservative investors have high risk aversion coefficients

➢ Aggressive investors have low risk aversion coefficients

Estimating Risk Aversion

42

Indifference Curves

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43

Indifference Curves

❑ We can illustrate our preferences through indifference curves

➢ Graphical representation of the utility function. Plotted in the risk-return ( − ) space

that connect points giving equal utility

➢ For example, to draw the indifference curve for U = 10%, choose all asset portfolio

combinations of E(r) and σ which yield a utility score of 10%

➢ Note that two indifference curves with different utility levels never intersect

E(r)

σ

U = E(r) – ½* A*σ2 = 10%

U = 15%

U = 20%

U = 25%

❑ Indifference curves for risk-

averse (A > 0) investors are

upward sloping

44

How to Plot an Indifference Curve

❑ Each plot point on an indifference curve represents a risk and return combination which

provides the same utility score

❑ For an Aggressive investor with risk aversion coefficient A=2, and a Conservative investor

with A=5, plot two indifference curves with U=0.03 and U=0.09

Higher Utility

A=5 is

steeper

than A=2

Certainty

equivalent

return.

Plot first.

➢ Given a specific value of A,

indifference curves above and to

the left offer higher utility than

lower curves and don’t intersect

➢ More risk averse investors (A)

have steeper indifference curves

(higher E(r) for each increase in

)

1.4 Forming portfolios

and Diversification

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46

❑ What is our objective when we make an investment?

➢ To achieve the optimal investment outcome that maximises our utility

➢ Utility from an investment depends on its expected return and risk

❑ The task is therefore to construct the optimal portfolio from all available assets,

which has the preferred combination of expected return and risk, therefore providing

the highest utility (among all possible portfolios)

❑ Let’s start simply with two assets

Forming Portfolios

47

Portfolios of Two Risky Assets

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48

❑ Portfolio returns are simply a weighted average of individual asset classes in the

portfolio. For example, consider Bonds and Equities asset classes:

= +

where

o =

o = ℎ

o =

o = ℎ

o =

❑ Similarly, portfolio expected return is a weighted average of expected returns of the

two asset classes in the portfolio:

() = () + ()

Portfolios of Two Risky Assets: Return

49

❑ Calculating portfolio risk on the other hand is not simply a weighted average. That

is, it is not simply a case of taking the individual risks of the portfolio’s component

assets

❑ We must also take into account the covariance between asset classes (or individual

assets) within the portfolio:

2 =

2

2 +

2

2 + 2 ,

o

2 =

o

2 =

o

2 =

o ( , ) =

Portfolios of Two Risky Assets: Risk

50

❑ The covariance between two variables refers to their tendency to be higher or lower than their

respective expected values at the same time. It is the sum of deviations from the expected

return of two assets in different states

, = σ()[ − ()][ − ()]

➢ = ;

➢ = ()

➢ = ()

❑ In practice, we usually estimate the unbiased sample covariance from historic data:

, =

1

− 1

=1

− ഥ − ഥ

➢ ( − ഥ ) = ℎ

➢ ( − ഥ ) = ℎ

➢ =

Covariance

51

❑ Correlation scales the covariance to 1:

=

,

❑ Range of values for : − 1.0 ≤ ≤ +1.0

❑ Based on the definition of correlation stated above, covariance of returns between

Bonds and Equities can also be expressed in terms of their correlation:

, =

❑ Note: if you are not familiar with the basic concepts and the related calculation of

variance, covariance, correlation, please do spend extra time and effort to make sure

you are familiar with them since these are the presumed knowledge for this course. Most

of the calculation-based questions could be built upon these basic concepts!!!!!

Correlation

52

Multiple Risky Assets

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53

❑ The return on a portfolio of more than two assets, is also simply the weighted

average of the returns of the assets that make up the portfolio:

= 11 +22 +⋯+ =

=1

= portfolio weight of asset (i.e. the fraction of portfolio value invested in asset )

❑ Similarly, the expected return on a portfolio is a weighted average of the expected

returns on its component assets:

=

=1

()

Portfolios of Multiple Risky Assets: Return

54

❑ For assets (or asset classes), portfolio risk can be calculated with a covariance

matrix, consisting of:

➢ variances

➢ (2 – )/2 covariances

❑ The generalised result for portfolio variance (risk) is given by:

2 = (

=1

,

=1

)

❑ We can set up a variance-covariance matrix from this equation

Portfolios of Multiple Risky Assets: Risk

55

❑ For a 3-asset portfolio:

=

❑ For a 5-asset portfolio

=

Portfolios of Multiple Risky Assets: Risk

11

22

33

11

22

33

11

22

33

44

55

11

22

33

44

55

1

21

2 + 12 1, 2 + 13 1, 3

2

22

2 + 21 2, 1 + 23 2, 3

3

23

2 + 31 3, 1 + 32 3, 2

1

21

2 + 2

22

2 + 3

23

2 + 2[12 1, 2 + 13 1, 3 +23 2, 3 ]

1

21

2 + 12 1, 2 + 13 1, 3 + 14 1, 4 + 15 1, 5

2

22

2 + 21 2, 1 + 23 2, 3 + 24 2, 4 + 25 2, 5

3

23

2 + 31 3, 1 + 32 3, 2 +34 3, 4 + 35 3, 5

4

24

2 + 41 4, 1 + 42 4, 2 + 43 4, 3 + 45 4, 5

5

25

2 + 51 5, 1 + 52 5, 2 + 53 5, 3 + 54 5, 4

1

21

2 + 2

22

2 + 3

23

2 +4

2 4

2 + 5

25

2 + 2[12 1, 2 + 13 1, 3

+ 14 1, 4 + 15 1, 5 + 23 2, 3 + 2 4 2, 4

+ 25 2, 5 + 34 3, 4 + 35 3, 5 + 45 4, 5 ]

56

Diversification Benefit

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57

❑ Recall portfolio variance for a 2-asset portfolio:

2 = 1

21

2 + 2

22

2 + 212(1, 2)

Which can be restated in terms of correlation as:

2 = 1

21

2 +2

22

2 + 2121212

❑ If asset 1 and 2 are perfectly correlated ρ = 1, we have:

2 = 1

21

2 + 2

22

2 + 21212 = (11 + 22)

2

= 11 +22

❑ In other words, the standard deviation of a portfolio with perfectly correlated assets is the

weighted average of the component asset standard deviations

➢ For perfectly correlated assets, both portfolio risk and return are a weighted average

➢ There is no risk reduction benefit in combining perfectly correlated assets

Perfect Correlation

58

❑ For perfectly correlated assets A and B, we can restate the risk and return equations as

follows:

Risk Return

Perfect Correlation

❑ So graphically, tells us where we are along

a line connecting point A (a portfolio consisting

of 100% in A i.e. = 0) and point B (a

portfolio consisting of 100% in B i.e. = 1)

❑ In other words, for perfectly correlated

assets the risk-return relationship is linear

59

❑ For lower values of (i.e., imperfect correlation), the portfolio standard deviation

must be lower:

2 =

2

2 +

2

2 + 2 < ( +)

2

< +

Diversification Benefit

60

❑ Portfolio weights change as we move from

E to B. The lower the correlation, the more

portfolio risk drops as we move from E to B

at each expected return level

❑ The curvature is the diversification benefit.

The lower the correlation the bigger the

bulge (diversification benefit) - same

portfolio expected return for lower portfolio

risk

❑ The diversification benefit depends on the

correlation:

➢ If = +1.0, no risk reduction is possible

➢ If −1 < <1, some level of risk reduction

is possible

➢ If = -1.0, a riskless hedge is possible

Diversification Benefit

0%

2%

4%

6%

8%

10%

12%

0% 5% 10% 15% 20% 25%

P

o

r

tf

o

li

o

E

x

p

e

c

te

d

R

e

tu

r

n

%

Portfolio Standard Deviation %

Diversification Benefit

Correlation = 1 Correlation = .7 Correlation = 0

Correlation = -.71 Correlation = -1

B

E

Diversification benefit

100% invested

in Bonds

100% invested

in Equities

61

❑ Combining two or more imperfectly correlated assets in one portfolio is called diversification

➢ The risk reduction is called a diversification benefit

➢ This concept is at the heart of portfolio theory

➢ Sophisticated version of “not putting all your eggs in one basket”

➢ The lower the correlation the better (same portfolio return for lower portfolio risk)

➢ We typically want to combine many assets with low (ideally negative) correlations to

maximize the diversification benefit

❑ Why is there a diversification benefit?

➢ Low correlated assets are unlikely to return below their respective means at the same time

(cancels out some portfolio risk)

➢ If correlations are negative, they even work as insurance for each other (a riskless hedge

may be possible)

Diversification Benefit

62

❑ In this example, as we add more assets to our portfolio the portfolio standard deviation drops

from ~50% to ~19% - this is the diversification benefit

➢ However, the diversification benefit increases rapidly at first, then diminishes significantly

Portfolio Diversification

63

❑ BKM Chapter 6 and 7

❑ 2.1. Optimal Risky Asset Portfolio Construction

❑ 2.2. Introducing the Risk-Free Asset

❑ 2.3. Deriving the Optimal Portfolio

Next Lecture