BU5526 Portfolio Analysis Introductory Lecture PDF

BU5526 Portfolio Analysis Introductory Lecture Business School University of Aberdeen Dr Seungho Lee Agenda  Introduction  What is Portfolio Management?  Mathematical Toolkit  Mean-Variance Analysis  Example 2 Lectures Plan ...

BU5526 Portfolio Analysis Introductory Lecture Business School University of Aberdeen Dr Seungho Lee Agenda  Introduction  What is Portfolio Management?  Mathematical Toolkit  Mean-Variance Analysis  Example 2 Lectures Plan  10-Week Teaching Delivery  4 lectures + 3 tutorials + 1quiz with Dr Seungho Lee  4 lectures + 3 tutorials + 1quiz with Dr Pranjal Srivastava  Lecture and tutorial attendance is compulsory.  Main 6 topics: 1) The Fundamentals of Portfolio Theory 2) The Capital Asset Pricing Model 3) Behavioural Finance 4) Debt securities 5) Alternative investments 6) Derivatives 7) Portfolio performance evaluation 3 Course Details  Carefully read the course guide, which contains all the essential information:  Assessments  Two mid-term exams:  Week 31 (24/02/2025) - Dr Lee’s contents (Lectures 1-3):  Week 39 (21/04/2025) - Dr Srivastava (Lectures 4-6):  Final Exam during the Exam Diet;  Tutorials;  Use online tools and regularly check for updates:  MyAberdeen (BlackBoard)  MyTimetable 4 Check Your @abdn.ac.uk Email! 5 Practical Information  Recommended textbooks (Week 26-29):  McMillan, Michael, Jerald E. Pinto, Wendy L. Pirie, Gerhard Van de Venter, and Lawrence E. Kochard, 2011, Investments: Principles of Portfolio and Equity Analysis. 1 edition. (John Wiley & Sons, Hoboken, N.J).  Zvi Bodie, Alex Kane and Alan J. Marcus, Investments, 12th global edition, 2021, or some of earlier editions, published by McGraw‐Hill. 6 Why Do We Study Portfolio Management?  Two main reasons to study portfolio management:  Asset pricing  Managing portfolios of assets  This course will also…  Increase your general knowledge of financial products;  Sharpen your mathematical skills;  Develop your Excel skills. 7 Asset Pricing  What is an asset? Financial Times definition “ Something belonging to an individual or a business that has value or the power to earn money. ” Quite brief definition… 8 Asset Pricing  What is an asset? Business Dictionary definition: 1. Something valuable that an entity owns, benefits from, or has use of, in generating income. 2. Accounting: Something that an entity has acquired or purchased, and that has money value (its cost, book value, market value, or residual value). An asset can be (1) something physical, such as cash, machinery, inventory, land and building, (2) an enforceable claim against others, such as accounts receivable, (3) right, such as copyright, patent, trademark, or (4) an assumption, such as goodwill. Assets shown on their owner's balance sheet are usually classified according to the ease with which they can be converted into cash. 9 Asset Pricing  What is an asset? Reflect: - Could you give some examples of assets? - What are the main financial assets you can think of? Example of financial assets: - Stock - Bonds - Derivatives - … Each type of assets entail different features. 10 Asset Pricing  Why « Pricing »? : Assets are priced by the market. For instance: So, why do we need to price assets? 11 Asset Pricing  Why « Pricing »? Basically, we need to price assets to be able to buy/sell them at a fair price. Participants in a market always do an asset pricing: - Think about local markets How the price is obtained can be fairly different based on markets, buyers/sellers, size, types of assets, etc. ➔ This is called a price discovery process. 12 Asset Pricing  Why « Pricing »? Because an exchange involves a seller and a buyer, any price discovery process stems from demand and supply forces. To illustrate, think about buying a used car… Things are not that different on stock markets: https://markets.cboe.com/us/equities/market_statistics/book_viewer/ Now, the question is how the buyer and the seller obtain a price estimate, that they can offer to their counterpart. 13 Asset Pricing  Why « Pricing »? Pricing an asset is being able to give a price to an asset. You can use your gut feeling. You can use chart analysis – i.e. technical analysis. You can use models. In financial courses, you learn models to price assets. In finance, the price of any asset is simply the current value of its cash-flows. - This is the case for any asset 14 Asset Pricing  Why « Pricing »? For any asset, its value is: You « simply » need two elements: - A prediction of cash-flows; - A discount rate. 15 Asset Pricing  Why « Pricing »? Let us take the example of stocks. To price a stock, you can essentially use: - The Dividend Discount Model; - The FCF model; - The Multipliers and Comparables approach. ➔ This is what you learn to do in a Financial Analysis course. 16 Asset Pricing  Why « Pricing »? For instance, the Dividend Discount Model (DDM) is: As we discussed before, you need two elements: - Future cash-flows (dividend and sale price) - Discount factor: required rate of return of equity. 17 Asset Pricing  Why « Pricing »? Predicting cash-flows is what you learn in a financial analysis course. Estimating the required rate of return is what you do in a Portfolio Analysis course. ? 18 Asset Pricing  Why « Pricing »? Being able to estimate this required rate of return is everything you need to calculate the value of any asset – once you have predicted future cash-flows. This includes estimating the value of a stock; This includes estimating the value of a firm; It is actually the case for any asset. 19 Asset Pricing  The Portfolio Approach gives the formula to estimate what should be the return you should ask.  It allows you to correctly trade this asset…  …and to hope making a profit: this is when pricing assets relates to managing assets.  Managing Portfolios of Assets is a basic skill that any professional in finance should possess. 20 Portfolio Management  Managing assets can be defined as selecting and monitoring assets, on behalf of an investor or a group of investors.  Definition of the Financial Times: The managing of money for investment so that it makes as much profit as possible, for a financial institution or for another person or organization.  Quite pragmatical, but may give a profit-oriented vision only.  Definition of Business Dictionary: Prudent administration of investable (liquid) assets, aimed at achieving an optimum risk-reward ratio.  Preferred. 21 Portfolio Management  Portfolio Management people are:  Investors;  Portfolio Managers - this is an industry title. It can be viewed in a more general way.  Almost all actors in finance « manage » assets, and need skills obtained in a Portfolio Management course: - Traders; - Adviser; - Loan officer; - Analyst; - Etc.  Portfolio Management is an industry: it represents one of the biggest sector in banking and finance. 22 Portfolio Management  To sum it up, there are two key reasons to attend this course:  Understand how to price assets;  Understand how to manage assets. And that is what we will do! 23 Mathematical Toolkit  There are three mathematical concepts to grasp properly:  Mean;  Variance;  Higher moments and distribution.  There are three financial concepts that are related:  Return;  Expected returns;  Portfolio return and variance.  While this part of the course may seem theoretical, it is actually at the core of modern finance. 24 Three Mathematical Concepts  Mean A mean is an average of different observations. There are two types of mean: - Arithmetical mean – that is usually what people refer to when simply saying « mean » - Geometrical mean – that is essential in finance. Both types of mean can be: - Equally weighted: all observations have the same weight; - Unequally weighted. 25 Three Mathematical Concepts  Arithmetical mean: This is the most common mean. It is the sum of the observations, divided by the number of observations (if equal weights). That is exactly what you do when you calculate your GPA across different courses. 26 Three Mathematical Concepts  Arithmetical mean: Mathematically: 𝑁 𝑋1 + 𝑋2 + ⋯ + 𝑋𝑁−1 + 𝑋𝑁 1 𝑋ത = = ෍ 𝑋𝑛 𝑁 𝑁 𝑛=1 This may look complicated… but it is not. Let’s take for instance the mean temperature of 3 days: 10°C, 12°C, and 5°C: 10 + 12 + 5 𝑋ത = = 9°𝐶 3 27 Three Mathematical Concepts  Geometrical mean: Geometrical mean does not make a sum average, but a product average. Mathematically: 𝑋ሜ 𝐺 = 𝑁 𝑋1 × 𝑋2 × ⋯ × 𝑋𝑁−1 × 𝑋𝑁 𝑁 𝑁 = ෑ 𝑋𝑛 𝑛=1 28 Three Mathematical Concepts  Geometrical mean: Again, it looks complicated, but it is not… Taking the same temperatures as before, the geometrical mean temperature of 3 days: 10°C, 12°C, and 5°C is: 3 𝑋ሜ 𝐺 = 10 × 12 × 5 = 8.43°𝐶 A short reminder: taking the N-square root is the same as raising by the power 1/N 29 Three Mathematical Concepts  Arithmetical and geometrical means are two different beasts.  Obviously, choosing when using one and when using the other will depend on the context and objective.  There is a reason to use either geometrical or arithmetical mean  In finance, we frequently use both, so it is important to know why and when. 30 Three Mathematical Concepts  Equally and unequally weighted mean Most of the time, when people refer to mean, that actually « mean » equally weighted mean. For instance, what we have done so far for arithmetical and geometrical mean was equally weighted mean. This makes sense when all observations have the same importance, or weight. 31 Three Mathematical Concepts  Equally and unequally weighted mean However, if you want to give a different importance to different observations, this becomes an unequally weighted mean. It is quite frequent. Let’s for instance consider the GPA of an hypothetical course. - MCQ: 30% of total grade; - Essay: 70% of total grade. What is the grade of someone having 14/22 at the MCQ and 18/22 at the exam? 32 Three Mathematical Concepts  Unequally weighted mean To get the answer, you quite intuitively do: 14 × 30% + 18 × 70% = 16.8 So, the GPA is 16.8/22. Because the importance of each grade is different, you simply put unequal weights. 33 Three Mathematical Concepts  Equally and unequally weighted mean This can be generalised: Equally weighted arithmetic mean: 𝑁 𝑋1 + 𝑋2 + ⋯ + 𝑋𝑁−1 + 𝑋𝑁 1 𝑋ത = = ෍ 𝑋𝑛 𝑁 𝑁 𝑛=1 Unequally weighted arithmetic mean: 𝑁 𝑋ത = 𝑤1 𝑋1 + 𝑤2 𝑋2 + ⋯ + 𝑤𝑁−1 𝑋𝑁−1 + 𝑤𝑁 𝑋𝑁 = ෍ 𝑤𝑛 𝑋𝑛 𝑛=1 34 Three Mathematical Concepts  Equally and unequally weighted mean In the same way for geometric mean: Equally weighted geometric mean: 𝑁 𝑁 𝑋ሜ 𝐺 = 𝑁 𝑋1 × 𝑋2 × ⋯ × 𝑋𝑁−1 × 𝑋𝑁 = ෑ 𝑋𝑛 𝑛=1 Unequally weighted geometric mean: σ𝑁 𝑁 σ𝑁 𝑛=1 𝑤𝑛 𝑛=1 𝑤𝑛 𝑤 𝑤 𝑤 𝑤 𝑤 𝑋ሜ 𝐺 = 𝑋1 1 × 𝑋2 2 × ⋯ × 𝑋𝑁−1 𝑁−1 × 𝑋𝑁 𝑁 = ෑ 𝑋𝑛 𝑛 𝑛=1 35 Three Mathematical Concepts  Equally and unequally weighted mean Using the unequally weighted geometrical mean, your GPA becomes: 1 𝑋ሜ 𝐺 = 140.3 × 180.70 = 16.69 This is not how we calculate GPA!... But it is good to know. Note that the 1-square root is the same as no square root. 36 Three Mathematical Concepts Now that we have a good understanding of what a mean is, let us move to the second key concept: Variance. 37 Three Mathematical Concepts  Variance can basically be seen as an extension of mean.  Mean is useful to describe a population.  It however has huge limitations: two populations with same mean can actually be very different.  Let’s compare Edinburgh and Montréal temperatures.  You would think that these are two very different climates.  But, actually, what does the mean temperature look like? 38  Three Mathematical Concepts  Here is the time-series of max temperature per month: Septembe January February March April May June July August October November December r Edinburgh 7 8 10 12 15 17 19 19 17 13 10 7 Montréal -6 -3 2 11 19 24 26 25 21 13 6 -1  What is the mean temperature:  Edinburgh: 12.83°C  Montréal: 11.42°C  Not that a difference!  Though, you do feel that it does not reflect reality. Something is missing. 39 Three Mathematical Concepts  What is missing is dispersion: how scattered are the temperatures.  We could use any number to measure dispersion:  For instance, how scattered it is from 0°C  However, this would be quickly complicated: what about series with different unities, how do we compare them?  The solution is to measure dispersion from the mean.  This is called variance. 40  Three Mathematical Concepts  Variance is the average of differences from the mean.  The only issue is that we cannot use differences as such: otherwise positive and negative difference will compensate and not reflect dispersion.  We raise the difference by the power 2.  Mathematically: 𝑋1 − 𝜇𝑋 ² + 𝑋2 − 𝜇𝑋 ² + ⋯ + 𝑋𝑁−1 − 𝜇𝑋 ² + 𝑋𝑁 − 𝜇𝑋 ² 𝜎𝑋2 = 𝑁 𝑁 1 = ෍ (𝑋𝑛 − 𝜇𝑋 )² 𝑁 𝑛=1 41  Three Mathematical Concepts  An important statistical concept:  This formula is correct if we possess all the data on the population.  If we only possess a sample, we need to reflect that there is some « impreciseness ».  To do so, we remove « one degree of freedom ».  Mathematically: 𝑋1 − 𝜇𝑋 ² + 𝑋2 − 𝜇𝑋 ² + ⋯ + 𝑋𝑁−1 − 𝜇𝑋 ² + 𝑋𝑁 − 𝜇𝑋 ² 𝜎𝑋2 = 𝑁−1 𝑁 1 = ෍ (𝑋𝑛 − 𝜇𝑋 )² 𝑁−1 𝑛=1 This will always be the case in finance. 42 Three Mathematical Concepts  Can variance help us to better describe differences in termperature between Montréal and Edinburgh?  Variance of temperatures:  Edinburgh: 20.33  Montréal: 135.54 Recalculate it at home. Which formula has been used?  Variance of temperature in Montréal is much higher.  It does reflect well dispersion.  Yet, it does not mean a lot…  We need to obtain a meaningful estimate. 43 Three Mathematical Concepts  To obtain a meaningful estimate of dispersion based on variance, we calculate the standard deviation.  It simply is the square-root of variance: 𝑁 1 𝜎𝑋 = ෍ (𝑋𝑛 − 𝜇𝑋 )² 𝑁−1 𝑛=1 Let’s come back to our temperatures: 44 Three Mathematical Concepts  Standard deviation of temperatures:  Edinburgh: 4.51°C  Montréal: 11.64°C How do you read it?  We now have a meaningful measure of dispersion, that does reflect the difference between the two time-series.  Edinburgh and Montréal weather are different, indeed. 45 Three Mathematical Concepts  Mean and variance are useful to describe a population.  There exist other estimates to describe populations – it answers to different questions.  For instance:  Is it more often cold or hot in Montréal or Edinburgh? * This is what we will call skewness.  Is there more extreme values in Montréal or Edinburgh? This is what we will call kurtosis. 46 Three Mathematical Concepts  You can actually construct as many indicators as you want, by simply raising the power in the variance equation.  That’s what is call in statistics moments.  The third moment, from which we can obtain the Skewness: 𝑋1 − 𝜇𝑋 3 + 𝑋2 − 𝜇𝑋 3 + ⋯ + 𝑋𝑁−1 − 𝜇𝑋 3 + 𝑋𝑁 − 𝜇𝑋 3 𝜎𝑋3 = 𝑁 𝑁 1 3 = ෍ 𝑋𝑛 − 𝜇𝑋 𝑁 𝑛=1 Skewness brings back the sign. A positive skewness means more positive-value, and reversly. 47 Three Mathematical Concepts  The fourth moment, from which we can obtain the Kurtosis: 4 4 4 4 𝑋1 − 𝜇𝑋 + 𝑋2 − 𝜇𝑋 + ⋯ + 𝑋𝑁−1 − 𝜇𝑋 + 𝑋𝑁 − 𝜇𝑋 𝜎𝑋4 = 𝑁 𝑁 1 4 = ෍ 𝑋𝑛 − 𝜇𝑋 𝑁 𝑛=1 Kurtosis outweighs extremum – with no sign. A large kurtosis means a lot of extreme values.  To give a meaningful estimate, we often provide excess kurtosis  This is simply Kurtosis – 3: the reason is that the kurtosis of a normal distribution is 3. 48 Three Mathematical Concepts  A short note:  Third and fourth moment formula are used to calculate Skewness and Kurtosis.  Unbiased equations of these two indicators are slightly more complex, but computer packages provide them automatically.  Unbiased skewness  Unbiased Kurtosis 49 Three Mathematical Concepts  Does it help to better describe our temperatures?  Skewness:  Edinburgh: 0.08  Montréal: -0.16 What does it mean?  Kurtosis:  Edinburgh:1.45  Montréal:1.40 What does it mean? 50 Three Mathematical Concepts  These measures better describe the shape of a distribution: Temperature (°C) 30 25 20 15 10 5 0 1 2 3 4 5 6 7 8 9 10 11 12 -5 -10 Edinburgh Montréal 51 Important Assumptions of Mean-Variance Analysis  In Finance, we mostly use mean and variance (first two moments).  This allows us to describe quite simply different assets.  This lies on the assumption that returns are normally distributed. 52 Important Assumptions of Mean-Variance Analysis  A normal distribution has three main characteristics:  its mean and median are equal;  it is completely defined by two parameters, mean and variance;  it is symmetric around its mean with: 68 percent of the observations within ±1σ of the mean, 95 percent of the observations within ±2σ of the mean, and 99 percent of the observations within ±3σ of the mean.  Using only mean and variance would be appropriate to evaluate instruments if returns were normally distributed.  Returns however are not normally distributed.  They are:  Skewed: they are not symmetric around the mean;  Characterized by a high probability of extreme event. 53 Important Assumptions of Mean-Variance Analysis 54 Important Assumptions of Mean-Variance Analysis 55  Exhibit: Histogram of U.S. Large Company Stock Returns, 1926-2008 2006 Violations of the 2004 2000 2007 1988 2003 1997 normality assumption: 1990 2005 1986 1999 1995 skewness and 1981 1977 1994 1993 1979 1972 1998 1996 1991 1989 kurtosis. 1969 1992 1971 1983 1985 1962 1987 1968 1982 1980 1953 1984 1965 1976 1975 1946 1978 1964 1967 1955 2001 1940 1970 1959 1963 1950 1973 1939 1960 1952 1961 1945 2002 1966 1934 1956 1949 1951 1938 1958 2008 1974 1957 1932 1948 1944 1943 1936 1935 1954 1931 1937 1939 1941 1929 1947 1926 1942 1927 1928 1933 –60 –50 –40 –30 –20 –10 0 10 20 30 40 50 60 70 56 Important Assumptions of Mean-Variance Analysis  Mean-Variance analysis allows us to describe different assets quite simply.  Though, based on the discussion before, remember that it is not a perfect reflection of reality.  Otherwise, it would be as often cold in Montréal as in Edinburgh  And this is not the case…! 57 Mean-Variance Analysis  We now have a good understanding of mean, standard deviation, and distribution shape.  Let’s now move to the three financial concepts that are related to those mathematical concepts:  Returns;  Expected returns;  Portfolio return and variance. 58 Return on Financial Assets Total Return Periodic Capital Gain or Income Loss 59  Holding Period Return A holding period return is the return from holding an asset for a specific period of time. 𝑃𝑡 − 𝑃𝑡−1 + 𝐷𝑡 𝑃𝑡 − 𝑃𝑡−1 𝐷𝑡 𝑅= = + 𝑃𝑡−1 𝑃𝑡−1 𝑃𝑡−1 = Capital gain + Dividend yield For instance, buying a security £100 in year 0, earning a dividend of £2 in year 1 and selling it £105: 105 − 100 2 𝑅= + = 5% + 2% = 7% 100 100 60  Holding Period Returns Holding period returns correspond to compound returns: 𝑅= 1 + 𝑅1 × 1 + 𝑅2 × 1 + 𝑅3 −1 What is the 3-year holding period return if the annual returns are 7%, 9%, and –5%? 𝑅= 1 +.07 (1 +.09)(1 + −.05) − 1 ≈.1080 = 10.80% What is the 3-year holding period return if the annual returns are 7% each year? 𝑅 = (1 + 0.07)3 − 1 ≈ 22.50% 61  Average Returns Average returns Arithmetic Geometric mean return mean return 62  Arithmetic or Mean Return The arithmetic or mean return is the simple average of all holding period returns. 𝑇 𝑅𝑖1 + 𝑅𝑖2 + ⋯ + 𝑅𝑖𝑇−1 + 𝑅𝑖𝑇 1 ሜ 𝑅𝑖 = = ෍ 𝑅𝑖𝑡 𝑇 𝑇 𝑡=1 For instance, the arithmetic or mean return over three years, with a return of -50% on year 1, +35% on year 2 and +27% on year 3 is: −50% + 35% + 27% ሜ 𝑅𝑖 = = 4% 3 This is not what you should use in finance → use geometrical mean 63  Geometric Mean Return The geometric mean return accounts for the compounding of returns. 𝑅ሜ 𝐺𝑖 𝑇 = 1 + 𝑅𝑖1 × 1 + 𝑅𝑖2 × ⋯ × 1 + 𝑅𝑖𝑇−1 × 1 + 𝑅𝑖𝑇 − 1 𝑇 𝑇 = ෑ 1 + 𝑅𝑖𝑡 − 1 𝑡=1 For instance, the geometric mean return over three years, with a return of -50% on year 1, +35% on year 2 and +27% on year 3 is: 3 𝑅𝐺𝑖 = (1 −.50) × (1 +.35) × (1 +.27) − 1 ≈ −5.0% 64  Annualized Return  How to calculate the equivalent annualized return from monthly/weekly/daily returns?  These returns should be capitalized. Drawing from the capitalization of returns: 𝑅= 1 + 𝑅1 × 1 + 𝑅2 × 1 + 𝑅3 −1  Annual returns are such as: 𝑐 𝑟𝑎𝑛𝑛𝑢𝑎𝑙 = 1 + 𝑟𝑝𝑒𝑟𝑖𝑜𝑑 − 1 𝑐: number of periods in a year 65  Annualized Return For instance, weekly return of 0.20% becomes: 𝑟𝑎𝑛𝑛𝑢𝑎𝑙 = (1 + 0.002)52 − 1 =.1095 = 10.95% And 18-month return of 20% becomes: 12ൗ 𝑟𝑎𝑛𝑛𝑢𝑎𝑙 = (1 + 0.20) 18 − 1 = 0.1292 = 12.92% Conversely, an annual return of 10.95% gives a weekly return of: 52 𝑟𝑤𝑒𝑒𝑘𝑙𝑦 = (1 + 10.95%) − 1 = 0.20% 66 Portfolio Return  When several assets are combined into a portfolio, we can compute the portfolio return.  This is the weighted average of the returns of individual assets. 𝑅𝑃 = 𝑤1 𝑅1 + 𝑤2 𝑅2 𝑅𝑃 = 𝑤1 𝑅1 + (1 − 𝑤1 )𝑅2  For example, with 70% in asset 1 (R=0.03) and 30% in asset 2 (R=-0.01): 𝑅𝑃 = 0.7 × 0.03 + 0.3 × (−0.01) = 1.8% 67 Portfolio Return  And for N assets: 𝑁 𝑁 𝑅𝑃 = ෍ 𝑤𝑖 𝑅𝑖 , ෍ 𝑤𝑖 = 1 𝑖=1 𝑖=1 68 Historical and Expected Returns  Historical returns are computed from historical data.  The expected return is what the investor expects to earn.  How to know what you should expect to earn? Ε 𝑅 =? 69 Historical and Expected Returns  In all cases, expected return are deduced from historical returns.  To give you an idea about what that implies, think about life expectancy, is that correct? What is assumed?  But most of the time, historical returns are different from expected returns... 70 Historical and Expected Returns  There are two main “ways” to calculate expected returns:  Gut feeling: “it should earn that”  It is surprisingly frequent…  Modelling  Expected returns are calculated from a formula.  How to select the good formula?  Historical average?  It earned that on average, it should earn the same next year.  Historical average + historical variance?  It earned that on average, but with this standard deviation (volatility), it should be sensibly similar next year. 71 Historical and Expected Returns  One of the objective of this course is to find the best way to calculate expected returns…  …because it will determine the price of an asset. 72 Variance and Standard Deviation of a Single Asset  We have seen how to calculate an average return.  Let’s see how to calculate its variance and standard deviation: Population Sample σ𝑇𝑡=1 𝑅𝑡 − 𝜇 2 σ𝑇𝑡=1 𝑅𝑡 − 𝑅ሜ 2 𝜎2 = 𝑠2= 𝑇 𝑇−1 𝜎 = 𝜎2 𝑠 = 𝑠2 73 Variance and Standard Deviation of a Single Asset  In finance, standard deviation is called volatility.  Volatility of an asset is simply its historical standard deviation.  Volatility is also defined as a risk of equity return. 74 Portfolio Variance  In the same way that we calculated a portfolio return, we might try to calculate a portfolio variance.  However, while you can add two (weighted) returns to obtain a portfolio return, you cannot simply add up two variances.  When one asset moves in one direction, it is unlikely that the other asset will move in the same direction.  To calculate the variance of two assets, you need to calculate the covariance. 75 Portfolio Variance  For two assets, the combined variance will be: 𝜎𝑋21 ,𝑋2 = 𝑤12 𝑉𝑎𝑟 𝑋1 + 𝑤22 𝑉𝑎𝑟 𝑋2 + 2 × 𝑤1 𝑤2 𝐶𝑜𝑣 𝑋1 , 𝑋2  Covariance is calculated as: σ𝑁𝑖=1(𝑅𝑖,𝑋1 − 𝜇𝑋1 ) × (𝑅𝑖,𝑋2 − 𝜇𝑋2 ) 𝐶𝑜𝑣 𝑋1 , 𝑋2 = 𝑁−1  The more the two assets move in the same way, the higher the covariance.  A negative covariance means that assets most of the time move in opposite directions. 76 Portfolio Variance  For the sake of simplicity, let’s just go back to temperatures.  How much do Montréal and Edinburgh temperatures move together?  This is, what is their covariance?  Covariance(Edinburgh, Montréal): 51.71  Most of the time, it moves together. Yeah… that’s seasons.  Try to calculate it home. 77  Portfolio Variance  If someone would live half the time in Edinburgh, and half the time in Montréal, what would be his variance of temperature?  This corresponds to a 50% weighting of each variance, plus the covariance: 2 𝜎𝐸,𝑀 =.52𝐸 𝑉𝑎𝑟 𝐸 +.52𝑀 𝑉𝑎𝑟 𝑀 + 2 ×.5 ×.5 × 𝐶𝑜𝑣 𝐸, 𝑀 = 64.82  Again, this is not very meaningful. 78 Portfolio Variance  To obtain a meaningful estimate, we can calculate the standard-deviation of the portfolio: 2 𝜎𝐸,𝑀 = 𝜎𝐸,𝑀 = 8.05°𝐶 How do you read it? Do it by yourself: - Recalculate these results - What is the average temperature of the two cities? - What if you spend 3 months per year in Montréal? 79  Variance of a Portfolio of Assets Let’s come back to finance Variance can be determined for N securities in a portfolio: 𝑁 𝑁 𝜎𝑃2 = ෍ 𝑤𝑖2 𝑉𝑎𝑟 𝑅𝑖 + ෍ 2 × 𝑤𝑖 𝑤𝑗 𝐶𝑜𝑣 𝑅𝑖 , 𝑅𝑗 𝑖=1 𝑖,𝑗=1, 𝑖≠𝑗 Standard-deviation (volatility of a portfolio) is simply the square root. 80 Variance of a Portfolio of Assets  The issue with covariance is that is it not really meaningful for intuitive understanding.  A bit like variance.  For variance, we use standard-deviation. What about covariance?  For covariance, we use correlation. 81 Correlation and Portfolio Risk  The correlation among assets in the portfolio determines the portfolio’s risk.  It is a measure of the tendency for N investments to act in a similar way.  It can be positive or negative and ranges from –1 to +1.  Covariance is difficult to interpret because it is unbounded on both sides. The correlation coefficient is easier to understand: 𝐶𝑜𝑣 𝑅𝑖 , 𝑅𝑗 𝜌𝑖𝑗 = 𝜎𝑖 𝜎𝑗  Correlation is a measure of the consistency or tendency for two instruments to act in a similar way.  It can be positive or negative and ranges from -1 to +1 82 Correlation and Portfolio Risk  For two assets: ρ1,2 = +1: Returns of the two assets are perfectly positively correlated. Assets 1 and 2 move together 100 percent of the time. ρ1,2 = –1: Returns of the two assets are perfectly negatively correlated. Assets 1 and 2 move in opposite directions 100 percent of the time. ρ1,2 = 0: Returns of the two assets are uncorrelated. Movement of asset 1 provides no prediction regarding the movement of asset 2.  Portfolio risk falls when the two assets are not perfectly correlated, ρ12 < +1 (see the formulas of portfolio risk). 83 Correlation  Coming back to our temperatures, we have calculated the following covariance: 51.90  This was not very telling. We can instead calculate the correlation: 𝐶𝑜𝑣 𝐸, 𝑀 51.90 𝜌𝐸,𝑀 = = ≈ 98.5% 𝜎𝐸 𝜎𝑀 4.51 × 11.64  This is much more meaningful: on average, Montréal and Edinburgh temperatures moves together 98,5% of the time.  We have just demonstrated that north hemisphere countries share the same seasons… 84 Correlation  The variance of a portfolio can also be calculated with the correlation, which is sometimes simpler: 𝑁 𝑁 𝜎𝑃2 = ෍ 𝑤𝑖2 𝑉𝑎𝑟 𝑅𝑖 + ෍ 2 × 𝑤𝑖 𝑤𝑗 𝐶𝑜𝑣 𝑅𝑖 , 𝑅𝑗 𝑖=1 𝑖,𝑗=1, 𝑖≠𝑗 𝑁 𝑁 = ෍ 𝑤𝑖2 𝑉𝑎𝑟 𝑅𝑖 + ෍ 2 × 𝑤𝑖 𝑤𝑗 𝜌𝑖𝑗 𝜎𝑖 𝜎𝑗 𝑖=1 𝑖,𝑗=1, 𝑖≠𝑗 85  Example: Return and Risk of a Two-Asset Portfolio  Assume that as a U.S. investor, you decide to hold a portfolio with 80 percent invested in the S&P 500 U.S. stock index and the remaining 20 percent in the MSCI Emerging Markets index.  The expected return is 9.93 percent for the S&P 500 and 18.20 percent for the Emerging Markets index.  The risk (standard deviation) is 16.21 percent for the S&P 500 and 33.11 percent for the Emerging Markets index.  The covariance between the S&P 500 and the Emerging Markets index is 0.0050  What will be the portfolio’s expected return and risk given that? 86  Example: Return and Risk of a Two-Asset Portfolio (Continued) 𝑅𝑃 = 𝑤1 𝑅1 + 𝑤2 𝑅2 = 0.80 × 0.0993 + 0.20 × 0.1820 = 0.1158 = 11.58% 𝜎𝑃2 = 𝑤12 𝜎12 + 𝑤22 𝜎22 + 2𝑤1 𝑤2 𝐶𝑜𝑣 𝑅1 , 𝑅2 = 0.802 × 0.16212 + 0.202 × 0.33112 + (2 × 0.80 × 0.20 × 0.0050) = 0.02281 𝜎𝑃 = 𝑤12 𝜎12 + 𝑤22 𝜎22 + 2𝑤1 𝑤2 𝐶𝑜𝑣 𝑅1 , 𝑅2 = 0.02281 = 0.1510 = 15.10% 87 References  Chapter 5, McMillan, Michael, Jerald E. Pinto, Wendy L. Pirie, Gerhard Van de Venter, and Lawrence E. Kochard, 2011, Investments: Principles of Portfolio and Equity Analysis. 1 edition. (John Wiley & Sons, Hoboken, N.J). 88