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## Portfolio Management

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**Portfolio Management**Grenoble Ecole de Management MSc Finance 2011**Learning Objectives**Mastering the principles of the portfolio management process: • Investment risk and return analysis • Risk reduction by portfolio diversification • Bet a as a marginal risk**Portfolio Management**Return – Risk analysis Mean – Variance analysis Return and risk are the starting points for any discussion of portfolio management.**Rate of return**Market convention is to report total returns: the sum of capital gains and income received: coupons for bonds or dividend for stocks.**Rate of return**Market’s usual notation for bonds, with P the price of the bond (instead of V value) Market’s usual notation for stocks, with P the price of the stock (instead of V value) Non adjusted for inflation, rate of returns are nominal. Real rate of returns are adjusted for inflation.**Rate of return: examples**• Example 1: rate of return An initial investment is made of EUR 100. One period later the value of the investment has risen to EUR 125. The rate of return of this investment is: • Example 2: total rate of return An initial investment is made of EUR 100. One period later the value of the investment has risen to EUR 125. In the interim the investor has received an income of EUR 9. The total rate of return of this investment is:**Rate of return: examples**• Example 3: real rate of return An initial investment is made of EUR 100. One period later the value of the investment has risen to EUR 125. In the interim the investor has received an income of EUR 9 while prices have grown at a rate of 6%. The total real rate of return of this investment is:**Multi-period rate of return**When the asset is held during several periods of time, the total rate of return is: • But the one-period total rate of return is:**Rate of return: examples**• Example 4: one-year real rate of return An initial investment is made of EUR 100. One period later (the period of investment is 3 years) the value of the investment has risen to EUR 125. In the interim the investor has received an income of EUR 9 while prices have grown at a rate of 6%. The one year total real rate of return of this investment is: This return has been realized over 3 years or 36 months, the one-month total real rate of return of this investment is:**Rate of return: geometric mean**Given several one-period rates of return, how to calculate the one-period mean return ? • Example 5: average rate of return A 3-year investment has registered the following one-year total returns: 5% in year 1, 8% in year 2 and 2% in year 3. The average total rate of return of this investment is:**Historical annual rate of returns: S&P 500**red line:mean 8.05% Do you feel the risk ?**Historical rate of returns**• Average returns are just that – averages. They tell us nothing about the possible range of outcomes, or risk around that central tendency. • Some investors characterize risk as either side of the mean. Others focus on below average outcomes (shortfalls) since the consequences of underperforming a target are more significant than outperforming it.**Risk is not a vague concept**The convention is to define risk as the variance of returns. with**Risk is not a vague concept**The standard deviation which is the square root of the variance is easier to interpret because it has the same dimension as returns Standard-error is also referred as volatility**Risk is not a vague concept**Example 6:A 3-year investment has registered the following one-year total returns: 5% in year 1, 8% in year 2 and 2% in year 3. The variance and standard error are respectively: The mean deviation from the mean is XX%. On average, the outcomes of this investment were distant by XX% of the mean. You can estimate the variability of any stock or bond returns by the procedure just described.**Risk is not a vague concept**27.45% mean 8.05%, 1st-dev 19.4% S&P 500 annual total return, mean return and 1 sd-dev between 1989 and 2009. The average deviation from the mean is 19.4%.**Risk is not a constant (VIX index)**It is conventional to present annualized volatility. When the volatility is estimated on monthly or weekly data you must multiply the standard-error respectively by:**Returns conform closely to a normal distribution**Normal distribution can be completely defined by the mean and the variance. They are the only measure an investor need to consider if the returns are perfectly normally distributed.**Returns conform closely to a normal distribution**• But closely is not perfectly, the normal distribution is just an approximation of the distribution of returns. • Empirical evidence shows that individual securities have more skewed distributions (fat tails), where extreme market moves occur more frequently than normal distribution would suggest. • However often market professional assume normality for routine statistical analyses and portfolio construction.**Risk-return analysis in the process**• You must choose the asset class that suits risk return objectives as reported in the IPS. • One way to compare asset classes is to calculate the mean return per “unit” of risk. This is done by dividing mean returns by standard-deviation. • However this is ex-post, historical analysis. Even though it is reasonable to assume that assets with histories of high variability also have the least predictable future, this analysis says nothing about future returns mean and volatility. Therefore it is not sufficient to determine an asset allocation.**Degrees of risk**Less security-more volatility Absence of residual value risk Default risk Interest rate risk Inflation risk**Portfolio Management**Diversification**Portfolio of assets: Diversification**Returns are additive, so the portfolio’s expected return Rpt will be the weighted sum of the component returns: With w1t and w2t respectively the weight of asset 1 and asset 2 at time t in the portfolio. For n assets**Portfolio of assets: Diversification**We can apply the measures of variability equally well for individual securities and portfolios of securities Variability for selected stocks larger than the variability of the whole portfolio: this the essence of diversification**Diversification reduces risk**• Diversification works because prices of different stocks do not move exactly together. On many occasions a decline in the value of one stock is canceled by a rise in the price of the other. • Statisticians make the same point when they say that stock price changes are less than perfectly correlated. • For example, negative correlation means that when the return of one security is above its mean, the return of the other tends to be below its mean. • The total risk of a two-security portfolio is not simply the weighted average of their individual volatilities. Variances are not additive**Covariance as a measure of diversification**Example 7:A 3-year investment in company AA has registered the following one-year total returns: 5% in year 1, 8% in year 2 and 2% in year 3. A 3-year investment in company BB has registered the following one-year total returns: 3% in year 1, -1% in year 2 and 6% in year 3. What is the risk of a portfolio composed of 50% of AA stocks and 50% of BB stocks ?**Covariance as a measure of diversification**S&P and US Corp covariance: regime dependant**Variance-covariance matrix**We just have to fill in a large number of boxes. Each of those down the diagonal contains the variance weighted by the square of the proportion invested. Each of the other boxes contain the covariance between that pair of securities, weighted by the product of the proportions invested. N 1 Stock AA Stock BB Stock AA Stock BB 2-asset var-covar matrix N n-asset var-covar matrix**Variance-covariance matrix (1989-2009)**Risk decays rather rapidly. Empirical evidences show that a portfolio of 20 to 30 stocks is generally well diversified.**Correlation matrix (1989-2009)**Correlation is more convenient. It lies between -1 and +1.**Diversification reduces risk**• The risk that can be eliminated by diversification is unique risk or idiosyncratic risk. • Unique risk stems from the fact that many of the perils that surround an individual company are peculiar to that company or its immediate competitors. • Economy wide perils which threaten all businesses cannot be avoided by diversification. This is called market risk • Diversification does not insure against markets’ krachs.**Diversification reduces risk**The risk that can be eliminated by diversification is unique risk or idiosyncratic risk.**Beta**• one of the most important idea of this course: the risk of a well-diversified portfolio depends on the sensitivity to market risk (beta) of the securities included in the portfolio not of the unique risk of each security in the portfolio • If we want to know the contribution of an individual security to the risk of a well diversified portfolio, it is no good thinking about how risky that security is if held in isolation. • We need to measure its sensitivity to market risk, market movements. That is because unique risk can be eliminated by diversification. • This sensitivity is called Beta.**Beta**• A stock with a Beta of 1 as a risk in line with the market risk. Beta < 1 indicates lower sensitivity to market risk, beta > 1 indicates higher sensitivity. • The risk of a portfolio is also proportional to the portfolio beta, which equals the weighted average beta of the securities included in the portfolio. • A portfolio with a Beta of 1.5 would amplify on average market moves by 50%**Betas and covariances**A statistician would define the beta of stocki as: The Beta of the IT sector is around 1.5. On average, it amplifies market movements by 50% having 150% of the market risk (average volatility of 30% against 20% for the market)**Beta as a marginal risk**The marginal contribution of an asset to the risk of the portfolio is function of the Beta of this stock.**Beta as a marginal risk**Example 8: A 3-year investment in company AA has registered the following one-year total returns: 5% in year 1, 8% in year 2 and 2% in year 3. A 3-year investment in company BB has registered the following one-year total returns: 3% in year 1, -1% in year 2 and 6% in year 3. What is the risk of a portfolio composed of 50% of AA stocks and 50% of BB stocks ? During the period, the market portfolio has registered the following returns: 4%, 5% and 3%. What is the Beta of the 50% AA / 50% BB stock to the market portfolio. What is the marginal contribution of stock AA to the 50%/50% portfolio ?**Back to the IPS**• We know how to measure risk and return • We know how to classify assets according to their risk-return profile • We know how to add-remove one unity of risk because Beta is a measure of marginal risk • We are able to set a portfolio as to respect the investor’s objectives as mentioned in the IPS • However this is ex-post, historical analysis. this analysis says nothing about future returns mean and volatility. Therefore it is not sufficient to determine an asset allocation.**Summary**• Variance is a measure of total risk. • Market risk accounts for most of the total risk of a well-diversified portfolio • Most individual stocks have higher standard deviations than the market portfolio. This is due to unique risk which can be eliminated by diversification. • The effective risk of any security cannot be judged by an examination of that security alone. • The beta of an individual security measures its sensitivity to market movements • The standard deviation of a well diversified portfolio is proportional to its beta.