Document Sample

Investment Analysis and Portfolio Management Lecture 6 Gareth Myles 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 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 Portfolio Variance The variance of a portfolio is given by 12 N N p X i X j ij i 1 j 1 This requires the knowledge of N variances and N[N – 1] covariances But symmetry ( ij ji ) reduces this to (1/2)N[N – 1] covariances 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 large portfolio 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 Model A possible solution is to relate the returns on assets to some underlying variable Let the return on asset i be modelled by ri iI iI rI e iI ri = return on asset i, rI = return on index, eiI = random error Return is linearly related to return on the index This model is imposed and may not capture the data Model Three assumptions are placed on this model The expected error is zero: E e iI 0 The error and the return on the index are uncorrelated: Ee iI rI rI 0 The errors are uncorrelated between assets: Ee iI , e kI 0 Model The model is ri estimated using data Observe the return x on the market and the return on the x x x x asset x x x Carry out linear regression to find line of best fit rI Model The estimated values are T rI , j rI ri , j ri j 1 iI , iI ri iI rI rI , j rI 2 T j 1 With E e iI 0 The estimation process ensures the average error is zero The value of iI is the gradient of the fitted line Model If the model is applied to all assets it need not follow that Ee iI , e kI 0 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 Model Note: iI covarianceof i with index iI 2 I varianceof index Note: II 1 And II 0 These observations permits a characterization of assets Assets Types If iI 1 then the ri asset is more volatile (or risky) than the market This is termed an “aggressive” asset rI Assets Types If iI 1 then the ri asset is less volatile than the market This is termed a “defensive” asset rI Risk For an individual asset If ri iI iI rI e iI then Eri ri 2 i 2 E iI iI rI e iI iI iI rI 2 E iI rI rI e iI 2 E rI rI 2eiI iI rI rI e 2 iI 2 2 iI Risk This can be written i 2 2 iI 2 I 2 ei So risk is composed of two parts: 1. market (or systematic) risk 2 iI 2 I 2. unique (or unsystematic) risk e2i Return Portfolio return N rp X i ri i 1 N X i iI iI rI e iI i 1 N N N X i iI X i iI rI X ie iI i 1 i 1 i 1 pI pI rI e pI Return Hence rp E pI pI rI e pI pI pI rI The portfolio has a value of beta This also determines its risk Risk Portfolio variance is Erp rp 2 p 2 E pI pI rI e pI pI pI rI 2 E 2 pI rI rI 2 2 pI rI rI e pI e 2 pI 2 pI 2 I 2 ep Risk The final expression can also be written 2 N 2 N 2 2 2 X i iI I X i ei p i 1 i 1 Consequence: now need to only know 2 I and ei, i = 1,...,N 2 For example, for FT 100 need to know 101 variances (reduced from 5050) Diversified Portfolio 1 A large portfolio that is evenly held X i N The non-systematic variance is N 2 2 N 1 2 2 ep X i ei ei 2 i 1 i 1 N This tends to 0 as N tends to infinity, so only market risk is left Diversified Portfolio That is 2 p 2 pI 2 I 2 ep tends to 2 p 2 pI 2 I I2 is undiversifiable market risk ep is diversifiable risk 2 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 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

DOCUMENT INFO

Shared By:

Categories:

Tags:

Stats:

views: | 2 |

posted: | 2/10/2012 |

language: | |

pages: | 24 |

OTHER DOCS BY ewghwehws

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.