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Investment Analysis and Portfolio Management

Learn about the Single-Index Model, Efficient Frontier, and Portfolio Variance in investment analysis and portfolio management. Understand how to calculate portfolio variance, reduce information demand, and model asset returns. Explore diverse assets types, risks, and portfolio returns.

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Investment Analysis and Portfolio Management

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  1. Investment Analysis and Portfolio Management Lecture 6 Gareth Myles

  2. Announcement • There is no lecture next week (Thursday 27th February)

  3. The Single-Index Model • Efficient frontier • Shows achievable risk/return combinations • Permits selection of assets • Can be constructed for any number of assets • Given expected returns, variances and covariances • Calculation is demanding in the information required

  4. The Single-Index Model • More useful if information demand can be reduced • The single-index model is one way to do this • Imposes a statistical model of returns • Simplifies construction of frontier • The model may (or may not) be accurate • The reduced information demand is traded against accuracy

  5. Portfolio Variance • The variance of a portfolio is given by • This requires the knowledge of N variances and N[N – 1] covariances • But symmetry ( ) reduces this to (1/2)N[N – 1] covariances

  6. Portfolio Variance • So N + (1/2)N[N – 1] = (1/2)N[N + 1] pieces of information are required to compute the variance • Example • If a portfolio is composed of all FT 100 shares then (1/2)N[N + 1] = 5050 • This is not even an especially largeportfolio

  7. Portfolio Variance • Where can the information come from? • 1. Data on financial performance (estimation) • 2. From analysts (whose job it is to understand assets) • But brokerages are typically organized into market sectors such as oil, electronics, retailers • This structure can inform about variances but not covariances between sectors • So there is a problem of implementation

  8. Model A possible solution is to relate the returns on assets to some underlying variable Let the return on asset i be modeled by = return on asset i, = return on index, = random error Return is linearly related to return on the index This model is imposed and may not capture the data

  9. Model • Three assumptions are placed on this model • The expected error is zero: • The error and the return on the index are uncorrelated: • The errors are uncorrelated between assets:

  10. Model • The model is estimated using data • Observe the return on the market and the return on the asset • Carry out linear regression to find line of best fit

  11. Example • Monthly data on stock return and FTSE100 return • Observe different scales on vertical axis

  12. Example

  13. Model • The estimated values are • With • The estimation process ensures the average error is zero • The value of is the gradient of the fitted line

  14. Example

  15. Example

  16. Model • If the model is applied to all assets it need not follow that • If the covariance of errors are non-zero this indicates the index is not the only explanatory factor • Some other factor or factors is correlated with (or “explains”) the observed returns

  17. Model • Note: • Note: • And • These observations permits a characterization of assets

  18. Assets Types • If then the asset is more volatile (or risky) than the market • This is termed an “aggressive” asset ri rI

  19. Assets Types • If then the asset is less volatile than the market • This is termed a “defensive” asset

  20. Risk • For an individual asset • If then

  21. Risk • This can be written • So risk is composed of two parts: 1. market (or systematic) risk 2. unique (or unsystematic) risk

  22. Return • Portfolio return

  23. Return • Hence • The portfolio has a value of beta • This also determines its risk

  24. Risk • Portfolio variance is

  25. Risk • The final expression can also be written • Consequence: now need to only know and , i = 1,...,N • For example, for FT 100 need to know 101 variances (reduced from 5050)

  26. Diversified Portfolio • A large portfolio that is evenly held • The non-systematic variance is • This tends to 0 as N tends to infinity, so only market risk is left

  27. Diversified Portfolio • That is • tends to • is undiversifiable market risk • is diversifiable risk

  28. Market Model • A special case of the single-index model • The index is the market • The set of all assets that can be purchased • The market model has two additional properties • Weighted-average beta = 1 • Weighted-average alpha = 0 • Issue: how is the market defined? • This is discussed for CAPM

  29. Adjusting Beta • The value of beta for an asset can be calculated from observed data • This is the historic beta • There are two reasons why this value might be adjusted before being used • Sampling • Fundamentals

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