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Lecture 9

Portfolio Theory and Risk Diversification I

Dr Jacinta Nwachukwu

Principal Lecturer in Finance

School of Economics, Finance and Accounting

[email protected]

Learning Outcomes

1. Understand the meaning and calculation of the expected return and risk measures for an individual security

2. Calculate portfolio expected return and risk measures as formulated by Harry Markowitz

3. Understand the impact of correlation on portfolio risk

4. Understand the impact of portfolio weights on risk

5. Appreciate the significance of the Markowitz Efficient Frontier

The term “expected” means that the return is not known with certainty. So when you make an investment, you are in effect gambling a known value (i.e., present value or purchase price pf the asset) for some unknown expected future outcome that is not known with certainty.

Thus, investors must estimate and manage the expected returns and risk from investments. They will attempt to reduce risk to the extent possible using a range of techniques. This includes building diversified portfolios as part of an effective risk management. So an investor is concerned with the characteristics of total portfolio.

Lecture Outline

1. Calculating expected returns on a security

2. Calculating risk for a security

3. Calculating portfolio expected return

4. Calculating portfolio risk

5. Explaining the components of portfolio risk

Portfolio weights

The correlation coefficient

Covariance

6. The Impact of changes in the portfolio weights on expected return and risk

7. Constructing the Markowitz efficient frontier

8. The Impact of changes in the portfolio weights on the efficient frontier

1. Calculating expected return oN a security

Calculating security expected return

The return expected by investors over some future holding period is calculated as the sum of the weighted average of all possible outcomes

That is:

Practice question 1

Assume that for a £1000, you can purchase a stock whose potential annual returns and subjective probabilities are as given in Table 1

What is the expected return on your investment?

Practice Question 1

Outcomes	Subjective Probability	Potential returns (%)	Expected return (%) (Probability* potential return)
1	0.2	1
2	0.2	7
3	0.3	8
4	0.1	10
5	0.1	15
6	0.1	(10)
Expected return

*Note: A bracket around a figure denotes a negative expected return

2. Calculating risk for a security

Something to think about

What does risk mean to you?

The dictionary definition of risk is “the possibility of loss or injury”

But in finance the definition of risk is one which reflects the fact that the outcomes of financial and economic decisions are almost always unknown at the time the decisions are made.

The definition which we will adopt in this lecture will be: “ Risk is a measure of uncertainty about the future payoff to an investment, assessed over some time horizon and relatively to a benchmark

Risk defined

Risk is measure of uncertainty about the future payoff to an investment, assessed over some time horizon and relative to a benchmark

The important element of the definition include:

(i) Risk arises from uncertainty about which of the many possible outcomes will occur in the future

(ii) Risk relates to future payoff of an investment, which is unknown. But, it is important that investors attempt to identify the possible payoffs and assign the likelihood of each one occurring.

(iii) Risk refers to investment or group of investments, including mutual funds, real estate, shares, the lottery etc.,

(iv) Risk is assessed over the time period for which an investment is held, say short or long term

(v)Risk is assessed relative to a benchmark rather than in isolation.

The common practice is to relate the risk an investment to an asset of the same time horizon with no risk (such as Treasury bonds) or the general market. This risk difference is known as the “risk premium”

(vi) Risk is a measure that can be quantified. The higher the risk, the less desirable the investment and so should command a lower price

Which project is riskier?

Figure 1: Actual Returns on Selected UK Traded Assets

Return on TESCO PLC 38719 38749 38777 38810 38838 38869 38901 38930 38961 38992 39022 39052 39083 39114 39142 39174 39203 39234 39265 39295 39328 39356 39387 39419 39448 39479 39510 39539 39569 39601 39630 39661 39692 39722 39755 39783 39815 39846 39874 39904 39934 39965 39995 40028 40057 40087 40119 40148 40182 40210 40238 40269 40302 40330 40360 40392 40422 40452 40483 40513 40547 40575 40603 40634 40666 40695 40725 40756 40787 40819 -4.0712155222778978 6.2890662563707869 -2.3673814047468182 -3.1815654537591929 2.2524247064829042 4.2121684867394693 7.6347305389221489 4.9374130737134907 -4.5725646123260395 9.3055555555555696 -0.57179161372299958 3.3865814696485597 3.3374536464771332 3.3492822966507179 2.8356481481481421 4.1080472706809168 -0.81081081081081163 -8.7738419618528489 -2.5686977299880542 4.2305334150827791 3.3529411764705848 11.098463289698349 -1.8442622950819672 -0.36534446764091927 -12.624410686223149 -3.9568345323740997 -5.3682896379525555 13.19261213720317 -3.4731934731934677 -10.818642839893762 -2.409964798267004 5.8546059933407388 1.598951507208394 -12.435500515995898 -12.993517972893336 21.909922113105289 -0.50000000000000311 -6.9793411501954292 6.0024009603838129E-2 1.1397720455908853 8.2147093712929991 -3.0967388325568526 3.8885746606334841 2.3274806043282816 6.3048683160415075 2.2247247247247244 3.5521065387157562 1.1820330969267163 -0.80607476635513764 -1.1423860558238201 3.7526804860614731 -6.889424733035053E-2 -5.4119269217511103 -7.6652089407191468 2.7891066964872984 3.8295149110457025 4.5339118858016363 0.67216981132076115 -2.9401429073445029 2.5826695631185075 -5.2235294117647033 0.3475670307845028 -5.7397328055418146 5.9186351706036824 3.8409119068269155 -4.0687268822336327 -4.614427860696507 -1.2648324423001598 -0.15847860538827871 7.2619047619047565 Return on FTSE 100 Index 38719 38749 38777 38810 38838 38869 38901 38930 38961 38992 39022 39052 39083 39114 39142 39174 39203 39234 39265 39295 39328 39356 39387 39419 39448 39479 39510 39539 39569 39601 39630 39661 39692 39722 39755 39783 39815 39846 39874 39904 39934 39965 39995 40028 40057 40087 40119 40148 40182 40210 40238 40269 40302 40330 40360 40392 40422 40452 40483 40513 40547 40575 40603 40634 40666 40695 40725 40756 40787 40819 2.5183313162952952 0.54163845633039776 2.9888629888629952 0.98078664118297865 -4.9692019059952539 1.9148118382892387 1.6268385504165761 -0.37447497596275314 0.92616108768899663 2.82512414441014 -1.3117535730601 2.8435392143896312 -0.28452932098765205 -0.50942270800084333 2.2117799562505072 2.238427393785666 2.6700986168827114 -0.20388437490560907 -3.7500567502534752 -0.89306771906102356 2.5938793965065918 3.9401249458774137 -4.3010592716020053 0.37932374659929524 -8.9377255339249508 7.6533215415490324E-2 -3.0963750998419477 6.7554059030883327 -0.55525438207415734 -7.0636821673412085 -3.8038358307115305 4.1519614183558575 -13.023808678990873 -10.712901580826101 -2.0400703630091637 3.4095149253731267 -6.4182941680573604 -7.6995373048004705 2.5064619722722647 8.0894526374774092 4.104908452529628 -3.8185563276669874 8.4533559258213273 6.5207013280097215 4.5835115810059275 -1.7413662128206526 2.8982059668946327 4.2807328491340284 -4.1456520534279155 3.1993832514214198 6.0715286207862613 -2.223748151278262 -6.5708677723155704 -5.2328270757844431 6.9372978909475584 -0.62381133510841058 6.1892367756258304 2.2816566341058864 -2.5884550324217588 6.72177703814915 -0.62712927337751656 2.2360947653891481 -1.4214214214214178 2.726441917140527 -1.3163314057892164 -0.7395659432387347 -2.1948635148090188 -7.234488925574353 -4.9309481879692347 11.186506775860401 Rate on three month T-Bills 0.36588333333333373 0.36532500000000051 0.36630000000000051 0.36830000000000052 0.37510000000000032 0.37845000000000045 0.3778333333333333 0.39620000000000044 0.4029916666666668 0.41130833333333372 0.41745833333333332 0.42299166666666682 0.442 0.44494166666666662 0.44395000000000001 0.45274166666666665 0.46263333333333329 0.47251666666666725 0.48118333333333335 0.48285833333333372 0.47413333333333324 0.46715000000000001 0.45821666666666688 0.44195000000000001 0.42679166666666668 0.41815000000000002 0.40695833333333331 0.40215000000000001 0.41246666666666737 0.42615000000000008 0.4236916666666668 0.41282500000000039 0.39520833333333338 0.30656666666666738 0.16623333333333357 0.10729166666666677 7.4541666666666659E-2 5.9808333333333463E-2 5.0291666666666672E-2 5.2083333333333461E-2 4.3924999999999999E-2 4.1975000000000005E-2 3.6641666666666725E-2 3.2883333333333389E-2 3.1341666666666691E-2 3.5958333333333335E-2 3.7158333333333335E-2 2.9891666666666691E-2 4.0616666666666704E-2 4.0649999999999985E-2 4.2575000000000002E-2 4.2358333333333442E-2 4.1466666666666693E-2 4.0325E-2 4.1500000000000002E-2 4.1208333333333333E-2 4.1383333333333432E-2 4.2174999999999997E-2 4.1124999999999995E-2 4.0941666666666668E-2 4.2125000000000003E-2 4.4691666666666692E-2 4.6691666666666673E-2 4.7300000000000071E-2 4.3874999999999997E-2 4.3116666666666713E-2 0 0.25832875000000038 0.18722634245958394 2.9891666666666691E-2

Time period in months

Actual return on the selected stock (percent)

Figure 2: The Probabilities of the NPV for Two Projects

Something to think about

1. Which of these two projects is more likely to achieve its expected return of £15,000

2. Which of the two projects would you recommend for an investor who emphasises safety and stability

Calculating variance and standard deviation for a security

Investors must be able to quantify and measure risk as part of a risk management strategy

To calculate the total (stand-alone) risk associated with the expected return, the variance or standard deviation measures is used

To calculate the variance or standard deviation from the probability distribution, the following equation is used:

The measure of variance is normally used by analysts to capture the uncertainty surrounding the performance of an individual asset over a given time period

Variance is defined as the average of the squared deviation of the possible outcomes from their expected outcomes, weighted by their probabilities.

Variance is calculated by:

(i) Computing the expected value of possible outcomes

(ii) Subtracting the expected value from each of the possible payoffs

(iii) Square each of the results

(iv) Multiply each result by its probability and add up the results

The variance and its square root (standard deviation) are measures of the spread or dispersion in he probability distribution.

That is: they measure the dispersion or spread of a random variable around its mean. The larger this dispersion (or spread0 the larger the variance or standard deviation. The tighter the probability distribution of expected returns the smaller the standard deviation and the smaller the risk.

STANDARD DEVIATION

The Standard deviation is a measure of dispersion or spread of each observation around the arithmetic mean of all the observations

The standard deviation is a measure of total risk of an asset. It is calculated using the following equation

Thus, standard deviation is the weighted average of all the deviations from the expected value. So, standard deviation provides some measure of how fare the actual values may be from the expected value, either above or below.

Assuming that the distribution of a security is normally distributed, (i.e., the well known bell-shaped curve), the actual return on the security will be within standard deviation of the expected return, approximately 68 percent of the time. The actual return will be within 2 standard deviation approximately 95 percent of the time.

Standard deviation is a more useful measure of risk than variance because it is measured in the same unit as the payoffs, while variance is the square of the unit of payoff

Standard deviation provides a baseline against which the risk of alternative investments can be measured

The higher the standard deviation, the higher the risk.

So given a choice between two investments with the same expected payoff, a rational investor will choose the one with the lower standard deviation

Practice question 2

Assume that for a £1000, you can purchase a stock whose potential annual returns and subjective probabilities are as given in Table 1

Using the expected return obtained in practice question 1, calculate:

(i) The variance for the investment

(ii) The standard deviation for the investment

Practice question 2

Outcome	Subjective probability	Potential returns (%)	Potential return* probability

1	0.2	1
2	0.2	7
3	0.3	8
4	0.1	10
5	0.1	15
6	0.1	(10)

Note that the probabilities and potential return values are subjective estimates. This is normally based on expert opinion of future economic conditions. Hence, they should be used with caution.

A key point of warning: investors should be careful to remember that past or historical performance cannot always be extrapolated into the future without modifications.

3. Calculating expected return of a portfolio

Using the Markowitz Model

Portfolio Expected Return

An investor’s portfolio is his/her combination assets.

Hence, although individual security returns and risks are important, it is the expected return and risk to the total portfolio that ultimately matters

Mathematically, the expected return on a two asset portfolio (A, B) is represented as follows:

Portfolio expected return is equal to the sum of the weighted average of the expected returns for the constituent assets in the portfolio

The resulting expected portfolio return value must fall between the highest and lowest expected returns for the individual securities making up the portfolio

Practice question 3

Consider the total returns between Southeast Utilities and Precision Instruments for the 2000-2009. The summary statistics for these two stocks are show in Table 2

What is the expected return of your portfolio?

Assume that the investor wishes to invest £1000 in equal amounts in each stock

Table 2: Summary statistics

	Southeast (Stock A)	Precision Instrument (Stock B)
Expected return (% p.a)	35	40
Standard deviation (% p.a)	15	20
Correlation Coefficient	-0.38

4. Calculating portfolio Risk

Using the Markowitz Model

Portfolio Risk

Typically, the risk of a portfolio is stated in terms of standard deviation as per the following equations:

The variance of a two asset (A and B) portfolio is:

Where

The standard deviation of a two asset (A and B) portfolio is:

The equation shows that the risk of a portfolio comprises two components:

(i) The weighted average of individual security risk measures

(ii) The “covariance” or co-movement between the expected returns of each pair of securities in the portfolio

The implication is that portfolio risk is not equal to the sum of the weighted average of the risk of individual securities in the portfolio.

5. Explaining the components of portfolio risk

Portfolio weights

The correlation coefficient

Covariance

Portfolio Weights

This is the percentages of a portfolio’s total value that are invested in each component security.

This is denoted by in our subsequent calculations

The combined portfolio weights are assumed to be 100 percent of total investable funds (or equal to 1)

This indicates that all portfolio funds are invested and that borrowing and/or short selling of assets are forbidden

That is:

For example:

(i) With equal dollar amounts in two securities, the portfolio weights are 0.5 and 0.5

(ii) With equal dollar amounts in three securities, the portfolio weights are 0.333, 0.333 and 0.333

(iii) With equal dollar amounts in five securities, the portfolio weights are 0.20, 0.20, 0.20, 0.20 and 0.20

However, an investor may choose to invest unequal dollar amounts in a portfolio. So for a five security portfolio, this could be: 0.40, 0.10, 0.15, 0.25 and 0.10

The Correlation Coefficient

The correlation coefficient is a statistical measure of the relative co-movement between security returns.

It measures the extent to which the returns on any two securities move together

It is bounded by +1 and -1

(i) +1 represents a perfect positive correlation. This means that when the return on security A goes up (or down), the return on security B goes up (or down) as well by exactly the same amount.

(ii) -1 denotes perfect negative correlation. This implies that the returns of the securities in question have a perfect inverse linear relationship to each other

So when the return on security A goes up (or down), the return on security B goes down (or up) by exactly the same amount

Hence, when the two are combined in a portfolio, the deviations in their returns around the expected return will cancel out. The portfolio will have no risk and so will achieve its expected return with certainty year on year

(iii) 0 signifies zero correlation. This means that the rate of return on individual securities are statistically independent such that any one security’s rate of return is unaffected by another's rate of return

These measures capture how spread out the figures are from the mean.

+1 = perfect positive correlation. This means that when the return on stock A goes up (or down), the return on stock B goes up (or down) as well by exactly the same amount. The same thing happens when the return on stock A goes down, stock B’s return does the same by exactly the same amount. So, knowing what the return on stock A will do allows investors to forecast perfectly what the return on stock B will be.

When stock returns are perfectly correlated, the risk of the portfolio will increase as more securities are added.

(ii) -1 denotes perfect negative correlation. This implies that the returns of the securities in question have a perfect inverse linear relationship to each other

So when the return on security A goes up (or down), the return on security B goes down (or up) by exactly the same amount

(iii) 0 denotes zero correlation. This means that the rate of return on individual securities are statistically independent such that any one security’s rate of return is unaffected by another's rate of return. Under these circumstances, the standard deviation of a portfolio containing independent securities (uncorrelated securites) is given by:

For example, assuming that the risk of each uncorrelated security in a portfolio is 0.20. The risk of the overall portfolio will quickly fall as more and more of these securities are added. Using equation about , we calculate that the total portfolio risk for case of 100 securities will be reduced to 0.02. That is:

=0.02

Evidence in the literature shows that, on average, the correlation coefficient for returns on two randomly selected stocks lies between +0.5 and +0.7. This suggests that the movement in the returns of most stocks are positively correlated with each other. This is what is generally referred to as market risk and cannot be eliminated. For example a rise in the UK interest rates will affect most firms listed on the FTSE exchange adversely because they and their customers will be borrowing funds at a higher rate to finance part of their operations.

Under these circumstances, combining shares into a portfolio reduces risk, but does not eliminate it completely.

Covariance of asset returns

The covariance between the returns of two assets (say A and B) is an absolute measure of the degree of association between the pair of them in a portfolio.

That is; the measure of covariance captures the actual amount of the co-movement among the security returns (i.e., their interaction or co-variability) in absolute terms

Mathematically, the covariance between returns on two securities A and B is summarized by the equation

The covariance equation shows that the covariance between the returns on assets A and B is simply the sum of all weighted products of paired deviations between expected and actual return outcomes.

The weight is the probability of a particular pair of values occurring

The size of the covariance measure depends on the units of the variables involved and the changes when these units are changed. Consequently, the relevance of covariance measures is the sign, whether it is +ve, -Ve or zero. The actual numbers are not useful because there are no logical context to them.

A large and positive covariance suggests that:

A large covariance indicates that the actual returns on the two assets in question deviate substantially from their expected values

A positive figure means that the two assets tend to rise and fall at the same time.

A negative covariance implies that:

The two assets tend to move in opposite directions to one another so that when the return on one asset rises, it falls on the other.

That is: higher than average returns on asset A tend to be paired with lower than average values of asset B

A small or zero covariance means that:

A small covariance indicates that either of the two assets have a small standard deviation and/or

The returns on the two assets move independently of one another with little or no co-variation between their returns

Practice question 4

Consider the total returns between Southeast Utilities and Precision Instruments for the 2000-2009. The summary statistics for these two stocks are show in Table 2

Using the expected return of your portfolio calculated earlier, estimate the risk for the portfolio measured in terms of variance and standard deviation?

Again, assume that the investor wishes to invest £1000 in equal amounts in each stock

Table 2: Summary statistics

	Southeast (Stock A)	Precision Instrument (Stock B)
Expected return (% p.a)	35	40
Standard deviation (% p.a)	15	20
Correlation Coefficient	-0.38

6. The impact of changes in the portfolio weights on expected return and risk

The Impact of Portfolio Weights

As the portfolio risk equation shows, the size of the portfolio weights assigned to each security has an effect on the portfolio risk, holding the values of the all the other parameters including correlation coefficient constant

To illustrate, re-calculate the expected return and risk of the portfolio in practice question 4 assuming that the investor places in each stock the weights shown in Table 3

Southeast (Stock A)	Precision Instrument (Stock B)
0.1	0.9
0.3	0.7
0.5	0.5
0.7	0.3
0.9	0.1

Table 3

Table 3: Portfolio Risk-Return Combinations

Weighting in high risk asset A	Portfolio weighting in low risk asset B	Portfolio Expected return	Portfolio standard deviation	Expected return (% p.a)

0	1	40	20.00	2
0.1	0.9	39.50	17.49	2.26
0.2	0.8	39.00	15.12	2.58
0.3	0.7	38.50	12.98	2.97
0.4	0.6	38.00	11.19	3.40
0.5	0.5	37.50	9.96	3.76
0.6	0.4	37	9.50	3.89
0.7	0.3	36.5	9.92	3.68
0.8	0.2	36.00	11.14	3.24
0.9	0.1	35.5	12.87	2.76
1	0	35	15	2.33

Calculate the efficiency ratio = expected return/ standard deviation

The table shows that the risk of the portfolio varies as the weights for each of the assets changes. Because Southeast (Wa) has substantially lower standard deviation than does Precision, portfolio risk falls as the weight assigned to Southeast increases. But because the correlation is less the perfect negative correlation (-1), the portfolio risk can only decrease so much and starts to risk after a certain minimum point is reached.

Figure 1: Expected return-Risk of Investment in SouthEast and Precision Instruments

20 19.74348753386797 19.487975779952109 19.233504620843284 18.98011591113184 18.727853587637853 18.476763785901468 18.226894963213017 17.978298028456422 17.731026479028223 17.485136545077363 17.240687341286602 16.997741026383476 16.756362970525554 16.516621930649137 16.278590233800958 16.042343968385666 15.807963183155508 15.575532093639692 15.345139295555452 15.116877984557528 14.890846181463298 14.667146961832762 14.445888688481578 14.227185245156541 14.011156269202054 13.797927380588725 13.587630404158039 13.380403581357328 13.176391767096181 12.975746606650423 12.778626686776635 12.585197654387475 12.395632295288529 12.210110564609968 12.028819559707429 11.851953425490668 11.679713181409893 11.512306458742316 11.349947136440768 11.192854863706579 11.041254457714487 10.895375165637942 10.755449781389899 10.621713609394671 10.494403270315088 10.373755347028384 10.260004873293189 10.153383672451268 10.0 54118559078164 9.9624294225856378 9.8785272181636472 9.8026118968364759 9.7348703124386819 9.6754741485882754 9.624577912822982 9.5823170475621406 9.5488062081079015 9.524137756248594 9.508380514051801 9.5015788161757619 9.5037518907008511 9.5148935884748589 9.534972469808185 9.5639322456822118 9.6016925591272706 9.6481500817514245 9.7031798911490874 9.7666370875547539 9.8383586029377881 9.9181651528899231 10.005863281096739 10.101247447716544 10.204102116305972 10.314203798645826 10.431323022512533 10.555226193691919 10.685677329959013 10.822439651021391 10.965277014284682 11.113955191559846 11.268242986375473 11.427913195330106 11.592743419915752 11.762516737501373 11.937022241748567 12.116055463722507 12.299418685450139 12.486921157755422 12.6783792339557 34 12.873616430514 13.072463425077924 13.274758001560706 13.480344951075994 13.689075936673007 13.900809328956354 14.115410018841111 14.332749212903991 14.552704216055515 14.775158205582773 15 40 39.950000000000003 39.900000000000006 39.849999999999994 39.799999999999997 39.75 39.699999999999996 39.65 39.6 39.549999999999997 39.5 39.450000000000003 39.400000000000006 39.349999999999994 39.299999999999997 39.25 39.200000000000003 39.15 39.1 39.050000000000004 39 38.950000000000003 38.900000000000006 38.85 38.799999999999997 38.75 38.700000000000003 38.65 38.599999999999994 38.549999999 999997 38.5 38.449999999999996 38.4 38.349999999999994 38.299999999999997 38.25 38.200000000000003 38.15 38.1 38.049999999999997 38 37.950000000000003 37.900000000000006 37.85 37.800000000000004 37.75 37.700000000000003 37.650000000000006 37.6 37.549999999999997 37.5 37.450000000000003 37.4 37.349999999999994 37.299999999999997 37.25 37.200000000000003 37.150000000000006 37.099999999999994 37.049999999999997 37 36.950000000000003 36.9 36.85 36.799999999999997 36.75 36.700000000000003 36.650000000000006 36.599999999999994 36.549999999999997 36.5 36.450000000000003 36.4 36.35 36.299999999999997 36.25 36.200000000000003 36.15 36.1 36.049999999999997 36 35.950000000000003 35.9 35.85 35.799999999999997 35.75 35.699999999999996 35.65 35.6 35.550000000000004 35.5 35.450000000000003 35.4 35.35 35.299999999999997 35.25 35.200000000000003 35.15 35.099999999999994 35.049999999999997 35

Portfolio standard deviation (% p.a)

Portfolio Expected returns (% p.a)

7. The Markowitz efficient frontier

Figure 3: 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)

Interpreting Figure 3

1. The minimum risk (or minimum variance as it is commonly known) portfolio (MV) has the lowest possible variance (i.e., standard deviation in the case of Figure 3) for any combination of assets A and B for the given correlation coefficient (say, -0.38 in Figure 3)

2. The curve line AB represents the opportunity set or feasible set available to any one contemplating an investment in a portfolio comprising assets A and B

The investor can choose any portfolio on curve AB by placing the appropriate fraction of his or her funds in asset A and B

However, he/she cannot achieve any point above curve AB

This is because points above curve AB can only be achieved through an increase in returns on both securities, a reduction in their standard deviation and/or a decrease in their correlation coefficients

The factors that influence these statistics are beyond the control of the investor

3. The shape of the portfolio possibility curve between the lowest risk-lower return security A and the minimum variance (MV) portfolio is backward bending (convex)

This means that for this portion of the opportunity set, the standard deviation of a portfolio tends to decline as the expected return rises

This result is due to the diversification effect arising from the fact that the correlation between the two assets is not perfectly positive

So the addition of a small proportion of the high return asset B acts as hedge to a portfolio initially composed only of the lower return security A

But as we continue to increase the percentage of the high return asset B in the portfolio, its high standard deviation eventually causes the risk of the entire portfolio to rise

Interpreting Figure 3

4. The shape of the opportunity set (i.e., the portfolio possibility curve) that lies above the minimum risk (MV) portfolio is inward-bending (i.e., concave )

This indicates that as the portfolio becomes more heavily weighted towards the high return asset B; the effect of B's higher risk begins to outweigh the beneficial effect of the less than perfect positive relationship between the two securities

The portion of the curve above the minimum variance portfolio is commonly referred to as the efficient frontier or the efficient set

It extends from the minimum risk portfolio to the maximum return portfolio

The portfolio analysis problem is merely a question of tracing out the efficient frontier starting from the minimum variance portfolio to the maximum return portfolio for a given asset return, standard deviation and correlation coefficient

There is a whole host of software packages that generate an efficient set

The choice of the preferred portfolio within the efficient set is largely dependent on the investor’s attitude towards risk.

A risk-averse investor would select any of the portfolios on the efficient frontier that offers more returns than a specified increase in risk and so might locate towards the lower end of the efficient frontier, say at portfolio P on the efficient frontier

But the question is whether portfolio P is the optimum portfolio which yields the largest expected return for a given level of risk measured by standard deviation or the smallest risk for a given level of standard deviation.

8. The impact of changes in the correlation coefficient on the efficient frontier

The portfolio risk equation shows that this is measurement is affected by the correlation between each pair of securities.

Generally, portfolio risk will decline as the correlation coefficient moves from +1 downwards, everything else held constant

To illustrate, re-calculate the expected return and risk of the portfolio in practice question 4 assuming that:

The investor places equal weights of 0.5 in each stock

The correlation between the two stocks are: (i) -1, (ii) -0.5, (iii) -0.28, (iv) 0, (v) +0.38, (vi) +0.5 and (vii) +1

These graphical representations show the impact that a combination of securities with different correlation will have on portfolio risk.

The portfolio risk decreases as correlation declines from +1 to -1

Diversification as represented by lower correlation coefficient can:

Provide a better expected return for the same level of risk (measured by standard deviation)

Provide lower risk (measured by standard deviation) for the same expected return.

This allows the investment to meet or become closer to the Efficient Frontier.

The following diagram shows the efficient frontier.

The efficient frontier shows the optimum expected return for each level of risk

Figure 4: Portfolio Expected Return-Risk Relationship

for Various Correlation Coefficients

Cor (-0.38) 20 16.890233864573784 14.923806484942105 14.571204480069568 15.939887076137024 18.654758106177631 22.227910383119688 26.311974460310033 30.703745699832787 35.288525047102993 40 15 17 19 21 23 25 27 29 31 33 35 Cor (-0.28) 20 17.31126800670593 15.758172482873691 15.681836627130123 17.102046661145529 19.697715603592208 23.075528162969526 26.94290259047812 31.117840542042831 35.491970923013 40 15 17 19 21 23 25 27 29 31 33 35 Cor (-1) 20 14 8 2 4 10 16 22 28 34 40 15 17 19 21 23 25 27 29 31 33 35 Cor (-0.5) 20 16.370705543744887 13.856406460551026 13.114877048604001 14.422205101855958 17.320508075688775 21.166010488516733 25.534290669607529 30.199337741083003 35.042830935870462 40 15 17 19 21 23 25 27 29 31 33 35 Cor (0) 20 18.439088914585774 17.888543819998169 18.439088914585774 20 22.360679774997777 25.298221281346883 28.635642126552703 32.249030993194211 36.055512754639913 40 15 17 19 21 23 25 27 29 31 33 35 Cor (+0.38) 20 19.867561501100226 20.427432535685735 21.625910385461189 23.364930986416375 25.534290669607529 28.034264748696568 30.784411639659442 33.723582253372811 36.806521161337322 40 15 17 19 21 23 25 27 29 31 33 35 Cor(+0.5) 20 20.297783130184428 21.166010488516733 22.538855339169288 24.331050121193037 26.457513110645902 28.844410203711803 31.43246729100342 34.176014981270129 37.040518354904464 40 15 17 19 21 23 25 27 29 31 33 35 Cor(+1) 20 22 24 26 28 30 32 34 36 38 40 15 17 19 21 23 25 27 29 31 33 35

Portfolio standard deviation (percent per annum)

Portfolio expected return (% per annum)

Interpreting Figure 4

To understand the general relationship between risk, correlation and return, we constructed various combinations of a two asset portfolio A and B for a full range of possible degrees of correlation extending from perfect negative to perfect positive.

These are shown in Figure 4. The important points may be summarized as follows

1. Perfect positive: Combinations of A and B are located on a straight line AB. The straight line connecting the two assets in the expected return-standard deviation space sets the limit for the risk of a portfolio that combines the two securities

An investor who wants to reduce the riskiness of his or her portfolio would have to invest more of his or her fund in the lower risk asset A. The minimum risk portfolio is simply asset A

2. Perfect negative correlation: Assuming the returns of the two assets are moving exactly in opposite direction to each other, it is possible to completely eliminate company-specific variability in the expected portfolio return by combining the lower risk asset A with the higher risk, higher return asset B

Maximum risk reduction is achieved when 67 percent of the portfolio is invested in lower return Asset A and the remaining 33 percent is placed in asset B

The resulting portfolio has a zero standard deviation and expected return of 21.67 percent

This portfolio possibility curve has the greatest bend, indicating that the potential benefits from diversification occurs when assets are perfectly negatively correlated

Interpreting Figure 4

3. Intermediate correlation coefficients: For correlation coefficients between +1 and -1, it is still possible to reduce risk by combining assets into a portfolio

The lower the correlation coefficient, the greater the diversification (or portfolio) effect achievable as represented by the characteristic backward-bending nature of the curves as we progress from the lower correlation of plus 0.5 to -0.5.

4. Although the graph shows different portfolio possibility curves for different correlation coefficients, an investor will only face one correlation between two assets at any one time.

End of lecture