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Investment Analysis and Portfolio Management

Investment Analysis and Portfolio Management. Lecture 6 Gareth Myles. Announcement. There is no lecture next week (Thursday 27 th February). The Single-Index Model. Efficient frontier Shows achievable risk/return combinations Permits selection of assets

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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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